SOCR - User contributions [en]

SOCR MotionCharts CAOzoneData

2010-03-26T00:58:17Z

Nchristo: /* See also */

== [[SOCR_MotionCharts| SOCR MotionCharts Activities]] - California Ozone Data Activity ==

== Summary==
This activity demonstrates the usage and functionality of [http://socr.ucla.edu/SOCR_MotionCharts/ SOCR MotionCharts] using the [[SOCR_Data_Dinov_121608_OzoneData | SOCR California Ozone dataset]].

==Goals==
The aims of this activity is to:
* use MotionChart to address 2 specific health-related case-studies
* demonstrate data import, MotionChart data manipulations and graphical data interpretation
* explore the interactive graphical visualization of real-life multidimensional datasets
* data navigation from different directions (using data mappings).

==Background ==
[[Image:SOCR_OzoneData_GeoMap_Dinov_121608_Fig1.png|200px|thumbnail|right| [http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_OzoneData_GoogleMap.html Ozone Geo-Map] ]]
Suppose we are asked to analyze a complex dataset that included observational multivariate ozone depletion data. The data included [[SOCR_Data_Dinov_121608_OzoneData |California Ozone measurements from 20 locations between 1980 and 2006]]. The figure on the right illustrates a dynamic interactive map of the geographic locations of the data measurements. This dataset consists of 540 rows and 22 variables. The goals of the study were to identify relationships and associations between the variables and map geographically the significant ozone layer effects. Any such quantitative study requires a preliminary exploratory data analysis. The complexity of the dataset and the intrinsic measurement characteristics of the ozone data demands a new approach to visualization and exploration of these heterogeneous measurements.

==Case Study==
[[Image:SOCR_OzoneData_AQI_Ozone_Chart1.png|300px|thumbnail|right| [http://www.nws.noaa.gov/aq/supplementalpages/aqkey.php Ozone Air Quality Index Map] ]]

This Ozone pollution case study addresses the following specific driving environmental challenges:
* ''Are there temporal changes in California Ozone?''
* ''What is the geographic distribution of the California Ozone pollution and is it changing with time?''

The following chart illustrates the health-related interpretation of the Ozone data in terms of the particulate (particles per million, '''ppm''') recordings, according to the National Oceanic and Atmospheric Administration's (NOAA) Air Quality Index (AQI). [http://www.epa.gov EPA] guidelines on air quality are as follows:
* (2010) The Obama administration’s proposal sets a primary standard for ground-level ozone of no more than 0.060 to 0.070 ppm, to be phased in over two decades.
* (2005) The Bush administration imposed a limit of 0.075 ppm.
* (1997) The Clinton administration introduced a standard of 0.084 ppm.

===Temporal changes in California Ozone===
* Observation: Notice the annualized increase of the ozone pollution with time (increase of the proportion of ''hot-colored bubbles'' with time).
* Motion Chart: Use the following ''variable-mapping'' to demonstrate the significant time effect on the increase of the ozone pollution as measured by ppm recordings:
<center>
{| class="wikitable"
|-
! || colspan=6 | Variables
|-
! [http://socr.ucla.edu/SOCR_MotionCharts/ SOCR MotionChart Property] || Key || X-Axis || Y-Axis || Size || Color || Category
|-
! [[SOCR_Data_Dinov_121608_OzoneData | Data Column Name]] || Year || MTH_1|| MTH_8 || HI_COVER || ANNUAL || Location
|}
</center>
* Note that the variable mapping we used above allows us to track seasonal changes of ozone pollution (Winter, month1, on the X-axis; and Summer, month8, on the Y-axis). If we play this motion chart, the following interpretation of the motion chart is appropriate: For a given bubble (representing one measurement location):
** ''up-down movement'' indicate annual Winter (January) increase or decrease of pollution levels, respectively;
** ''left-right movement'' indicate annual Summer (August) increase or decrease of pollution levels, respectively;
** ''bubble-size increase'' or ''decrease'' indicate corresponding changes in the annual percent coverage during typical periods of high concentration;
** ''color change'' from ''cool-to-hot'' indicates an ''annual increase'' of the ozone measurement for that specific location from one year to the next.

: You should see an image like this one shown below. Play this motion charts by clicking the '''Play''' button and observe the increase of hot-colored bubbles in the chart as time goes from 1980 to 2006.
<center>[[Image:SOCR_OzoneData_AQI_Ozone_Chart2.png|600px]]</center>

===Geographic distribution of California Ozone pollution===
* Observation: The ozone pollution appears to be a more geographically spread out phenomenon in the 2000's, compared to the 1980's -- most of the bubbles cluster together in later years, whereas there were wider geographic-driven fluctuations in the ozone particles in the earlier years. The size of the bubbles reflects the maximum annual pollution and the bubble color indicates the average annual ozone pollution -- ''hot-colors'' represent high and ''cool-colors'' represent low ozone pollution levels, respectively.
* Motion Chart: Use the following variable-mapping to demonstrate the significant geographic temporal re-distribution of the ozone pollution as measured by ppm recordings:
<center>
{| class="wikitable"
|-
! || colspan=6 | Variables
|-
! [http://socr.ucla.edu/SOCR_MotionCharts/ SOCR MotionChart Property] || Key || X-Axis || Y-Axis || Size || Color || Category
|-
! [[SOCR_Data_Dinov_121608_OzoneData | Data Column Name]] || Year || LONGITUDE || LATITUDE || HI_COVER || ANNUAL || Location
|}
</center>
* This second variable mapping allows us to track seasonal changes of ozone pollution for each geographic location. Here the X and Y locations of bubbles are locked at the [http://en.wikipedia.org/wiki/Geographic_coordinate_system GIS longitude and latitude coordinates]. By playing this motion chart, we have the following interpretation: For a given bubble (representing one GIS location):
** ''bubble-size increase'' or ''decrease'' indicate corresponding changes in the annual percent coverage during typical periods of high concentration;
** ''color change'' from ''cool-to-hot'' or ''hot-to-cool'' indicate ''annual increase'' or ''annual decrease'', respectively, of the ozone measurement for that specific location from one year to the next.

: You should see an image like this one shown below. In this mapping, each bubble corresponds spatially to a geographic location, just like in the [http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_OzoneData_GoogleMap.html geographic-map above]. Play this motion charts by clicking the '''Play''' button and observe the increase of hot-colored bubbles in later years at geographic locations which did not show unhealthy ozone pollution levels int he early years.
<center>[[Image:SOCR_OzoneData_AQI_Ozone_Chart3.png|600px]]</center>

==Initial setting==
In addition to this activity, open 2 more browser tabs - one pointing to the [http://socr.ucla.edu/SOCR_MotionCharts/ SOCR MotionCharts applet] and the other displaying the [[SOCR_Data_Dinov_121608_OzoneData | SOCR California Ozone dataset]]. The image below shows this setting.
<center>[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig1.png|250px]] [[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig2.png|250px]] [[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig3.png|250px]]
</center>

==Hands-on activity==
* Using the mouse, copy the [[SOCR_Data_Dinov_121608_OzoneData | SOCR California Ozone dataset]], click on the first cell (top-left) in the DATA tab of the [http://socr.ucla.edu/SOCR_MotionCharts/ SOCR Motion Charts applet], and paste the data in the spreadsheet.
<center>[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig2.png|400px]]</center>

* Next, you need to map the column-variables to different properties it the SOCR MotionChart. For example, you can us the following mapping:
<center>
[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig5.png|150px|thumbnail|right| SOCR MotionChart Data Mapping]]
{| class="wikitable"
|-
! || colspan=6 | Variables
|-
! [http://socr.ucla.edu/SOCR_MotionCharts/ SOCR MotionChart Property] || Key || X-Axis || Y-Axis || Size || Color || Category
|-
! [[SOCR_Data_Dinov_121608_OzoneData | Data Column Name]] || Year || Longitude || Latitude || MTH_1|| MTH_7 || Location
|}
</center>

The figures below represent snapshots of the generated dynamic SOCR motion chart. In the real applet, you can ''play'' (animate) or ''scroll'' (1-year steps) through the years (1980, ..., 2006). Notice the position change between different snapshots of the time slider on the bottom of these figures. Also, mouse-over a blob triggers a dynamic graphical pop-up providing additional information about the data for the specified blob in the chart.

You can also change what variables are mapped to the following SOCR MotionCharts properties:
* ''Key, X-Axis, Y-Axis, Size, Color'' and ''Category''.
<center>
[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig6_10_Animation.png|400px]]</center>
<center>
[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig6.png|150px]]
[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig7.png|150px]]
[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig8.png|150px]]
[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig9.png|150px]]
[[Image:SOCR_Activities_MotionCharts_Ozone_070109_Fig10.png|150px]]
</center>

We can also overlay the Motion-chart above on the [http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_OzoneData_GoogleMap.html Geo-map of of the 20 locations] of the ozone recording stations. This of course, stretches vertically the motion chart, as longitude and latitude coordinates are not isotropic.
<center>[[Image:SOCR_Activities_MotionCharts_Ozone_032510_Fig11.png|400px]]</center>

== Other types of exploratory and statistical analyses of the Ozone data==
Various [ SOCR online tools can also be used to analyze (visually or quantitatively) the Ozone pollution data. The table below contains a summary of the annual pollution rates (ppm) for each of the 20 locations across the span of 27 years:
<center>
{| class="wikitable"
|-
! LOCATION || 1980 || 1981 || 1982 || 1983 || 1984 || 1985 || 1986 || 1987 || 1988 || 1989 || 1990 || 1991 || 1992 || 1993 || 1994 || 1995 || 1996 || 1997 || 1998 || 1999 || 2000 || 2001 || 2002 || 2003 || 2004 || 2005 || 2006
|-
| 2008 || 0.12 || 0.11 || 0.15 || 0.14 || 0.14 || 0.13 || 0.11 || 0.17 || 0.11 || 0.19 || 0.11 || 0.11 || 0.13 || 0.11 || 0.106 || 0.135 || 0.12 || 8.7 || 9.8 || 0.088 || 9.3 || 9.2 || 7.4 || 9.7 || 0.109 || 8.2 || 8.2
|-
| 2040 || 0.18 || 0.2 || 0.23 || 0.28 || 0.28 || 0.22 || 0.15 || 0.17 || 0.22 || 0.23 || 0.2 || 0.18 || 0.16 || 0.146 || 0.102 || 0.12 || 0.12 || 0.115 || 0.125 || 0.104 || 0.118 || 0.135 || 0.112 || 0.107 || 0.105 || 8.3 || 0.108
|-
| 2102 || 0.13 || 0.11 || 0.1 || 0.14 || 0.16 || 0.14 || 0.1 || 0.15 || 0.12 || 0.11 || 0.11 || 7.9 || 0.11 || 0.13 || 0.111 || 0.124 || 0.121 || 8.7 || 0.097 || 9.7 || 0.107 || 0.118 || 0.111 || 9.3 || 8.9 || 9.3 || 0.105
|-
| 2125 || 0.15 || 0.13 || 0.1 || 0.17 || 0.11 || 0.13 || 0.1 || 0.12 || 0.1 || 0.1 || 7.9 || 7.9 || 8.9 || 0.1 || 8.3 || 0.14 || 0.097 || 8.9 || 6.5 || 8.2 || 0.083 || 0.105 || 8.9 || 0.113 || 0.097 || 8.3 || 8.4
|-
| 2199 || 0.21 || 0.19 || 0.19 || 0.19 || 0.2 || 0.24 || 0.18 || 0.17 || 0.2 || 0.19 || 0.17 || 0.18 || 0.15 || 0.17 || 0.165 || 0.16 || 0.16 || 0.155 || 0.173 || 0.126 || 0.124 || 0.137 || 0.136 || 0.141 || 0.125 || 0.139 || 0.126
|-
| 2249 || 0.31 || 0.27 || 0.32 || 0.27 || 0.32 || 0.34 || 0.25 || 0.24 || 0.29 || 0.26 || 0.21 || 0.21 || 0.21 || 0.19 || 0.252 || 0.16 || 0.15 || 0.134 || 0.182 || 0.116 || 0.137 || 0.114 || 0.121 || 0.165 || 9.8 || 9.3 || 0.146
|-
| 2293 || 0.19 || 0.16 || 0.14 || 0.16 || 0.15 || 0.15 || 0.14 || 0.16 || 0.13 || 0.12 || 0.13 || 0.12 || 0.12 || 0.13 || 0.12 || 0.153 || 0.1 || 0.109 || 0.115 || 0.133 || 0.102 || 0.109 || 0.11 || 0.123 || 8.9 || 0.105 || 0.102
|-
| 2410 || 0.14 || 0.12 || 0.1 || 0.13 || 0.14 || 0.12 || 8.9 || 0.11 || 0.12 || 0.12 || 0.11 || 0.11 || 0.1 || 0.11 || 0.1 || 0.133 || 0.112 || 0.103 || 0.119 || 0.113 || 0.079 || 9.1 || 0.109 || 0.101 || 0.104 || 8.7 || 7.9
|-
| 2420 || 0.38 || 0.25 || 0.22 || 0.26 || 0.26 || 0.25 || 0.22 || 0.22 || 0.25 || 0.23 || 0.19 || 0.22 || 0.17 || 0.19 || 0.14 || 0.145 || 0.205 || 0.121 || 0.161 || 0.1 || 0.109 || 0.14 || 0.152 || 0.179 || 0.131 || 0.138 || 0.158
|-
| 2460 || 0.23 || 0.19 || 0.18 || 0.18 || 0.16 || 0.22 || 0.16 || 0.17 || 0.19 || 0.2 || 0.17 || 0.15 || 0.17 || 0.187 || 0.147 || 0.146 || 0.138 || 0.136 || 0.164 || 0.124 || 0.121 || 0.135 || 0.121 || 0.125 || 0.106 || 0.113 || 0.121
|-
| 2484 || 0.41 || 0.35 || 0.36 || 0.39 || 0.31 || 0.36 || 0.31 || 0.3 || 0.3 || 0.33 || 0.23 || 0.28 || 0.27 || 0.24 || 0.251 || 0.212 || 0.196 || 0.162 || 0.195 || 0.137 || 0.174 || 0.189 || 0.136 || 0.15 || 0.134 || 0.145 || 0.165
|-
| 2492 || 0.35 || 0.27 || 0.25 || 0.31 || 0.26 || 0.3 || 0.28 || 0.23 || 0.24 || 0.2 || 0.2 || 0.22 || 0.22 || 0.18 || 0.167 || 0.165 || 0.142 || 0.134 || 0.177 || 0.12 || 0.152 || 0.129 || 0.128 || 0.134 || 0.137 || 0.142 || 0.166
|-
| 2499 || 0.32 || 0.35 || 0.32 || 0.28 || 0.34 || 0.3 || 0.26 || 0.29 || 0.29 || 0.27 || 0.33 || 0.27 || 0.28 || 0.24 || 0.265 || 0.256 || 0.234 || 0.205 || 0.244 || 0.174 || 0.176 || 0.17 || 0.161 || 0.163 || 0.163 || 0.182 || 0.164
|-
| 2525 || 0.29 || 0.24 || 0.28 || 0.26 || 0.22 || 0.29 || 0.22 || 0.2 || 0.23 || 0.21 || 0.19 || 0.2 || 0.21 || 0.2 || 0.184 || 0.202 || 0.18 || 0.136 || 0.149 || 0.112 || 0.164 || 0.152 || 0.147 || 0.155 || 0.128 || 0.126 || 0.169
|-
| 2589 || 0.16 || 0.17 || 0.2 || 0.21 || 0.15 || 0.2 || 0.14 || 0.16 || 0.22 || 0.16 || 0.15 || 0.15 || 0.15 || 0.133 || 9.8 || 0.14 || 9.7 || 0.117 || 9.8 || 0.105 || 9.1 || 0.102 || 0.115 || 7.4 || 0.097 || 9.2 || 8.3
|-
| 2596 || 0.37 || 0.3 || 0.31 || 0.36 || 0.32 || 0.35 || 0.25 || 0.29 || 0.28 || 0.27 || 0.29 || 0.24 || 0.26 || 0.26 || 0.253 || 0.213 || 0.203 || 0.187 || 0.195 || 0.142 || 0.14 || 0.143 || 0.155 || 0.169 || 0.141 || 0.144 || 0.151
|-
| 2655 || 0.12 || 0.11 || 8.9 || 0.11 || 0.11 || 0.11 || 8.9 || 0.11 || 0.1 || 0.1 || 8.9 || 0.11 || 8.9 || 0.12 || 9.2 || 0.13 || 8.9 || 8.3 || 0.125 || 0.115 || 7.7 || 9.8 || 0.116 || 0.105 || 9.2 || 9.1 || 9.6
|-
| 2898 || 0.37 || 0.33 || 0.31 || 0.34 || 0.31 || 0.33 || 0.27 || 0.29 || 0.29 || 0.25 || 0.24 || 0.24 || 0.26 || 0.21 || 0.241 || 0.215 || 0.187 || 0.156 || 0.184 || 0.141 || 0.152 || 0.144 || 0.15 || 0.161 || 0.131 || 0.14 || 0.151
|-
| 2899 || 0.29 || 0.32 || 0.4 || 0.26 || 0.29 || 0.3 || 0.22 || 0.22 || 0.21 || 0.25 || 0.2 || 0.19 || 0.2 || 0.16 || 0.193 || 0.167 || 0.144 || 0.12 || 0.148 || 0.128 || 0.136 || 0.116 || 0.122 || 0.152 || 0.11 || 0.121 || 0.108
|-
| 2991 || 0.13 || 0.16 || 0.15 || 0.15 || 0.14 || 0.15 || 0.18 || 0.18 || 0.16 || 0.19 || 0.12 || 0.12 || 0.141 || 0.138 || 0.115 || 0.124 || 0.121 || 0.102 || 0.106 || 0.103 || 8.3 || 9.3 || 8.6 || 8.0 || 8.3 || 7.5 || 8.8
|}
</center>

=== Spider Chart===
This [[SOCR_EduMaterials_Activities_SpiderWebChart | spider chart]] visually demonstrates the rapid increase of the pollution levels across the years (radial spokes). For each site the 27 annual measurements are connected with lines of the same color as shown on the 20-locations color-mapping below.
<center>[[Image:SOCR_Activities_MotionCharts_Ozone_032510_Fig12.jpg|400px]]
[[Image:SOCR_Activities_MotionCharts_Ozone_032510_Fig13.png|400px]]</center>

=== Box-and-whisker plot ===
This [[ SOCR_EduMaterials_Activities_BoxPlot | Box and Whiskers Plot]] illustrates the same annual (across locations) averages of the ozone pollution for the 27 years on record. Notice the increase in (high-level) outliers, denoted by colored triangles on the top, and the atypically average high pollution levels in the last few years.

<center>[[Image:SOCR_Activities_MotionCharts_Ozone_032510_Fig14.png|400px]]</center>

=== Simple Linear Regression analysis===
We can employ the [[SOCR_EduMaterials_AnalysisActivities_SLR | SOCR simple linear regression analysis]] to determine the relation between the (average) '''Annual''' Ozone pollution (response variable) relative to '''Year''' (time predictor variable). The quantitative results and graphs of this analysis, using only the [[SOCR_Data_121608_OzoneData | Year and Annual variables in the Ozone dataset]], are included below:
*Quantitative analysis:
** Regression Line: <math>OzonePollution = -191.6095127656 + 0.09667\times YEAR</math>
*** Intercept:
****Parameter Estimate: -191.6095127656
****Standard Error: 28.4291851368
****T-Statistics: -6.7398876135
****P-Value: 0.0000000000
*** Slope:
****Parameter Estimate: 0.0966959040
****Standard Error: 0.0142644095
****T-Statistics: 6.7788228014
****P-Value: 0.0000000000
** Correlation(YEAR, OzonePollution) = 0.2805211066
* The prediction interval and the regression line have a slight (but statistically significant) upward trend, which is mostly due to the sporadic extremely high pollution levels in the last few years.
<center>[[Image:SOCR_Activities_MotionCharts_Ozone_032510_Fig15.png|400px]]</center>

== Data type and format ==
SOCR Motion Charts currently accepts three types of data: numbers, dates/time, and strings. With these data types, we feel that the application is able to handle the majority of data out here. We use the natural ordering of these types as defined by Java however. While many types of data can be interpreted as a string, it may not make sense to use lexicological ordering on all the different types. When designing SOCR Motion Charts, we took this into consideration and designed the application so that it can easily be extended to provide a greater variety of interpreted types. Thus, a developer should be able to easily provide better type interpretation for particular types of data.

== Applications ==
The SOCR MotionCharts can be used in a variety of applications to visualize dynamic relationships in multidimensional data in up to four dimensions and a fifth temporal component. Its design and implementation allow for extensions allowing and supporting higher dimensions plug-ins. The overall purpose of SOCR MotionCharts is to provide users with a way to visualize the relationships between multiple variables over a period of time in a simple, intuitive and animated fashion.

==See also==
* [http://www.nytimes.com/2010/01/08/science/earth/08smog.html E.P.A. Seeks Stricter Rules to Curb Smog (New York Times article)]
{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_MotionCharts_CAOzoneData}}

SOCR MotionCharts CAOzoneData

2010-03-24T04:33:43Z

Nchristo: /* Geographic distribution of of California Ozone pollution */

SOCR MotionCharts CAOzoneData

2010-03-24T03:30:49Z

Nchristo: /* Case Study */

SOCR Data LA Neighborhoods Data

2009-10-19T02:31:50Z

Nchristo: /* LA Neighborhoods Data */

== [[SOCR_Data |SOCR Data]] - Los Angeles City Neighborhoods Data==

===Data Description===
[[Image:SOCR_Data_LA_County_Neighborhoods_Dinov_100109_Fig1.png|200px|thumbnail|right| [http://projects.latimes.com/mapping-la/neighborhoods/ LA Times Neighborhoods] ]]

The LA Times map of the neighborhoods of Los Angeles contains data ([http://www.census.gov/ U.S. Census 2000]) on the 110 LA neighborhoods including measures of education, income and population demographics.

===Data Source===
* [http://projects.latimes.com/mapping-la/neighborhoods/ LA Times].
* Variables
** [http://projects.latimes.com/mapping-la/neighborhoods/ LA_Neighborhoods] ('''LA_Nbhd''')
** [http://projects.latimes.com/mapping-la/neighborhoods/income/neighborhood/list/ Highest_Incomes] ('''Income'''): Median household income reports the amount of money earned by the household that falls exactly in the middle of pack.
** [http://projects.latimes.com/mapping-la/neighborhoods/api-test-scores/neighborhood/list/ Highest-Scoring_Public_Schools] ('''Schools'''): The median API score reports the 2008 test results posted by the school that falls exactly in the middle of the pack. California's Academic Performance Index (API) combines several tests into a single number between 200 and 1000 for each school. The tests that make up the API and their weighting are listed on the [http://www.cde.ca.gov/ta/ac/ap/glossary08d.asp California Department of Education website].
** [http://projects.latimes.com/mapping-la/neighborhoods/diversity-index/neighborhood/list/ Most_Diverse_Population] ('''Diversity'''): The diversity index measures the probability that any two residents, chosen at random, would be of different ethnicities. If all residents are of the same ethnic group it's zero. If half are from one group and half from another it's 50%. This figure is used to place areas into 10 groups of even size and then rank those groups. Areas with a diversity rank of 1 are the least ethnically diverse. Those with a diversity rank of 10 are the most ethnically diverse.
** [http://projects.latimes.com/mapping-la/neighborhoods/age/neighborhood/list/ Oldest_Population] ('''Age'''): Median age reports the age of the person who falls exactly in the middle of the pack.
** [http://projects.latimes.com/mapping-la/neighborhoods/owners/neighborhood/list/ Most_Homeowners] ('''Homes'''): The percentage of owners measures the portion of households that are owner occupied. A ranking of the percentage of [http://projects.latimes.com/mapping-la/neighborhoods/renters/neighborhood/list/ households that rent] is also available.
** [http://projects.latimes.com/mapping-la/neighborhoods/veterans/neighborhood/list/ Most_Veterans] ('''Vets'''): The percentage of veterans measures the portion of adult population that once served in the armed forces. A [http://projects.latimes.com/mapping-la/neighborhoods/armed-forces/neighborhood/list/ ranking of active members of the armed forces] is also available.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/asian/neighborhood/list/ Asian] ('''Asian'''): The percentage of population of whose ethnicity is Asian.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/black/neighborhood/list/ Black] ('''Black'''): The percentage of population of whose ethnicity is black.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/latino/neighborhood/list/ Latino] ('''Latino'''): The percentage of population of whose ethnicity is Latino.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/white/neighborhood/list/ White] ('''White'''): The percentage of population of whose ethnicity is white.

===LA Neighborhoods Data===

<center>
{| class="wikitable"
|-
! LA_Nbhd || Income || Schools || Diversity|| Age || Homes || Vets || Asian || Black || Latino || White
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/adams-normandie/ Adams_Normandie] || 29606 || 691 || 0.6 || 26 || 0.26 || 0.05 || 0.05 || 0.25 || 0.62 || 0.06
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/arleta/ Arleta] || 65649 || 719 || 0.4 || 29 || 0.29 || 0.07 || 0.11 || 0.02 || 0.72 || 0.13
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/arlington-heights/ Arlington_Heights] || 31423 || 687 || 0.8 || 31 || 0.31 || 0.05 || 0.13 || 0.25 || 0.57 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/atwater-village/ Atwater_Village] || 53872 || 762 || 0.9 || 34 || 0.34 || 0.06 || 0.20 || 0.01 || 0.51 || 0.22
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/baldwin-hillscrenshaw/ Baldwin_Hills/Crenshaw] || 37948 || 656 || 0.4 || 36 || 0.36 || 0.10 || 0.05 || 0.71 || 0.17 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/bel-air/ Bel-Air] || 208861 || 924 || 0.2 || 46 || 0.46 || 0.13 || 0.08 || 0.01 || 0.05 || 0.83
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverly-crest/ Beverly_Crest] || 168104 || 0 || 0.1 || 45 || 0.45 || 0.10 || 0.04 || 0.02 || 0.03 || 0.88
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverly-grove/ Beverly_Grove] || 63039 || 791 || 0.2 || 38 || 0.38 || 0.05 || 0.05 || 0.02 || 0.06 || 0.82
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverlywood/ Beverlywood] || 105253 || 872 || 0.2 || 39 || 0.39 || 0.08 || 0.07 || 0.04 || 0.06 || 0.80
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/boyle-heights/ Boyle_Heights] || 33235 || 689 || 0.1 || 25 || 0.25 || 0.03 || 0.02 || 0.01 || 0.94 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/brentwood/ Brentwood] || 112927 || 882 || 0.2 || 39 || 0.39 || 0.08 || 0.06 || 0.01 || 0.05 || 0.84
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/broadway-manchester/ Broadway_Manchester] || 29897 || 656 || 0.5 || 23 || 0.23 || 0.04 || 0.00 || 0.39 || 0.59 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/canoga-park/ Canoga_Park] || 51601 || 706 || 0.8 || 30 || 0.3 || 0.06 || 0.11 || 0.04 || 0.51 || 0.31
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/carthay/ Carthay] || 71398 || 762 || 0.8 || 37 || 0.37 || 0.04 || 0.09 || 0.13 || 0.16 || 0.58
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/central-alameda/ Central_Alameda] || 31559 || 669 || 0.1 || 22 || 0.22 || 0.02 || 0.01 || 0.13 || 0.85 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/century-city/ Century_City] || 95135 || 0 || 0.2 || 46 || 0.46 || 0.12 || 0.09 || 0.01 || 0.04 || 0.83
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chatsworth/ Chatsworth] || 84549 || 762 || 0.6 || 40 || 0.4 || 0.11 || 0.14 || 0.02 || 0.14 || 0.65
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chesterfield-square/ Chesterfield_Square] || 37737 || 544 || 0.5 || 31 || 0.31 || 0.07 || 0.01 || 0.59 || 0.37 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/cheviot-hills/ Cheviot_Hills] || 109980 || 0 || 0.3 || 42 || 0.42 || 0.11 || 0.09 || 0.01 || 0.09 || 0.79
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chinatown/ Chinatown] || 22837 || 784 || 1 || 34 || 0.34 || 0.04 || 0.35 || 0.21 || 0.32 || 0.11
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/cypress-park/ Cypress_Park] || 42615 || 697 || 0.2 || 27 || 0.27 || 0.04 || 0.11 || 0.01 || 0.82 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/del-rey/ Del_Rey] || 63317 || 783 || 1 || 35 || 0.35 || 0.08 || 0.14 || 0.04 || 0.42 || 0.36
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/downtown/ Downtown] || 15003 || 724 || 1 || 39 || 0.39 || 0.10 || 0.21 || 0.22 || 0.37 || 0.16
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/eagle-rock/ Eagle_Rock] || 67253 || 830 || 1 || 35 || 0.35 || 0.08 || 0.24 || 0.02 || 0.40 || 0.30
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/east-hollywood/ East_Hollywood] || 29927 || 772 || 0.7 || 31 || 0.31 || 0.03 || 0.16 || 0.02 || 0.60 || 0.18
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/echo-park/ Echo_Park] || 37708 || 755 || 0.6 || 30 || 0.3 || 0.04 || 0.19 || 0.02 || 0.64 || 0.13
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/el-sereno/ El_Sereno] || 45866 || 739 || 0.2 || 28 || 0.28 || 0.05 || 0.11 || 0.02 || 0.81 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/elysian-park/ Elysian_Park] || 28263 || 905 || 0.7 || 31 || 0.31 || 0.04 || 0.43 || 0.02 || 0.48 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/elysian-valley/ Elysian_Valley] || 49013 || 813 || 0.7 || 31 || 0.31 || 0.04 || 0.26 || 0.01 || 0.61 || 0.10
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/encino/ Encino] || 78529 || 899 || 0.2 || 42 || 0.42 || 0.11 || 0.05 || 0.02 || 0.09 || 0.80
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/exposition-park/ Exposition_Park] || 33999 || 733 || 0.6 || 26 || 0.26 || 0.06 || 0.01 || 0.38 || 0.56 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/fairfax/ Fairfax] || 65938 || 734 || 0.2 || 33 || 0.33 || 0.05 || 0.05 || 0.02 || 0.06 || 0.85
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/florence/ Florence] || 29447 || 680 || 0.3 || 23 || 0.23 || 0.04 || 0.00 || 0.28 || 0.70 || 0.00
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/glassell-park/ Glassell_Park] || 50098 || 723 || 0.5 || 30 || 0.3 || 0.05 || 0.17 || 0.01 || 0.66 || 0.14
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/gramercy-park/ Gramercy_Park] || 57983 || 653 || 0.1 || 36 || 0.36 || 0.13 || 0.00 || 0.86 || 0.12 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/granada-hills/ Granada_Hills] || 83911 || 812 || 0.8 || 37 || 0.37 || 0.11 || 0.16 || 0.03 || 0.21 || 0.56
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/green-meadows/ Green_Meadows] || 31347 || 682 || 0.5 || 24 || 0.24 || 0.05 || 0.00 || 0.44 || 0.54 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hancock-park/ Hancock_Park] || 85277 || 861 || 0.4 || 37 || 0.37 || 0.06 || 0.13 || 0.04 || 0.09 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harbor-city/ Harbor_City] || 55447 || 736 || 1 || 30 || 0.3 || 0.08 || 0.13 || 0.11 || 0.48 || 0.25
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harbor-gateway/ Harbor_Gateway] || 47849 || 730 || 0.9 || 27 || 0.27 || 0.08 || 0.16 || 0.16 || 0.53 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harvard-heights/ Harvard_Heights] || 31173 || 712 || 0.5 || 30 || 0.3 || 0.04 || 0.13 || 0.16 || 0.66 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harvard-park/ Harvard_Park] || 37013 || 0 || 0.6 || 28 || 0.28 || 0.07 || 0.01 || 0.48 || 0.48 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/highland-park/ Highland_Park] || 45478 || 744 || 0.4 || 28 || 0.28 || 0.05 || 0.11 || 0.02 || 0.72 || 0.11
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/historic-south-central/ Historic_South_Central] || 30882 || 674 || 0.1 || 23 || 0.23 || 0.03 || 0.01 || 0.10 || 0.87 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood/ Hollywood] || 33694 || 718 || 0.9 || 31 || 0.31 || 0.05 || 0.07 || 0.05 || 0.42 || 0.41
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood-hills/ Hollywood_Hills] || 69277 || 780 || 0.3 || 37 || 0.37 || 0.08 || 0.07 || 0.05 || 0.09 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood-hills-west/ Hollywood_Hills_West] || 108199 || 961 || 0.1 || 41 || 0.41 || 0.08 || 0.04 || 0.03 || 0.06 || 0.85
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hyde-park/ Hyde_Park] || 39460 || 647 || 0.5 || 31 || 0.31 || 0.09 || 0.01 || 0.66 || 0.27 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/jefferson-park/ Jefferson_Park] || 32654 || 680 || 0.7 || 31 || 0.31 || 0.08 || 0.03 || 0.47 || 0.45 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/koreatown/ Koreatown] || 30558 || 740 || 0.8 || 30 || 0.3 || 0.03 || 0.32 || 0.05 || 0.54 || 0.07
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lake-balboa/ Lake_Balboa] || 65336 || 777 || 0.8 || 35 || 0.35 || 0.10 || 0.09 || 0.04 || 0.34 || 0.49
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lake-view-terrace/ Lake_View_Terrace] || 67985 || 746 || 0.9 || 31 || 0.31 || 0.08 || 0.05 || 0.17 || 0.53 || 0.22
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/larchmont/ Larchmont] || 47780 || 800 || 1 || 34 || 0.34 || 0.04 || 0.30 || 0.03 || 0.37 || 0.25
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/leimert-park/ Leimert_Park] || 45865 || 679 || 0.3 || 38 || 0.38 || 0.11 || 0.04 || 0.80 || 0.11 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lincoln-heights/ Lincoln_Heights] || 30579 || 736 || 0.3 || 27 || 0.24 || 0.03 || 0.25 || 0.00 || 0.71 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/los-feliz/ Los_Feliz] || 50793 || 723 || 0.8 || 36 || 0.25 || 0.06 || 0.13 || 0.04 || 0.19 || 0.58
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/manchester-square/ Manchester_Square] || 46093 || 750 || 0.2 || 34 || 0.57 || 0.11 || 0.00 || 0.79 || 0.19 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mar-vista/ Mar_Vista] || 62611 || 709 || 0.9 || 35 || 0.39 || 0.08 || 0.12 || 0.04 || 0.29 || 0.51
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mid-city/ Mid_City] || 43711 || 780 || 0.9 || 31 || 0.31 || 0.06 || 0.04 || 0.38 || 0.45 || 0.10
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mid-wilshire/ Mid_Wilshire] || 58483 || 739 || 1 || 34 || 0.22 || 0.06 || 0.20 || 0.23 || 0.20 || 0.34
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mission-hills/ Mission_Hills] || 75675 || 795 || 0.8 || 34 || 0.78 || 0.10 || 0.11 || 0.04 || 0.54 || 0.29
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/montecito-heights/ Montecito_Heights] || 55901 || 730 || 0.5 || 31 || 0.51 || 0.05 || 0.17 || 0.03 || 0.66 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mount-washington/ Mount_Washington] || 57725 || 825 || 0.7 || 33 || 0.55 || 0.07 || 0.13 || 0.03 || 0.61 || 0.21
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/north-hills/ North_Hills] || 52456 || 760 || 0.8 || 28 || 0.49 || 0.07 || 0.12 || 0.05 || 0.57 || 0.24
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/north-hollywood/ North_Hollywood] || 42791 || 724 || 0.7 || 30 || 0.25 || 0.05 || 0.06 || 0.06 || 0.58 || 0.27
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/northridge/ Northridge] || 67906 || 836 || 0.9 || 32 || 0.54 || 0.09 || 0.14 || 0.05 || 0.26 || 0.50
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pacific-palisades/ Pacific_Palisades] || 168008 || 879 || 0.1 || 43 || 0.83 || 0.12 || 0.05 || 0.00 || 0.03 || 0.89
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pacoima/ Pacoima] || 49066 || 688 || 0.1 || 24 || 0.57 || 0.04 || 0.02 || 0.07 || 0.86 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/palms/ Palms] || 50684 || 793 || 1 || 31 || 0.13 || 0.05 || 0.20 || 0.12 || 0.23 || 0.38
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/panorama-city/ Panorama_City] || 44468 || 721 || 0.4 || 25 || 0.36 || 0.04 || 0.12 || 0.04 || 0.70 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pico-robertson/ Pico_Robertson] || 63356 || 0 || 0.3 || 36 || 0.27 || 0.05 || 0.06 || 0.06 || 0.07 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pico-union/ Pico_Union] || 26424 || 677 || 0.1 || 27 || 0.10 || 0.02 || 0.08 || 0.03 || 0.85 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/playa-del-rey/ Playa_del_Rey] || 91339 || 836 || 0.4 || 38 || 0.47 || 0.11 || 0.08 || 0.04 || 0.10 || 0.73
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/playa-vista/ Playa_Vista] || 68597 || 771 || 1 || 37 || 0.55 || 0.11 || 0.21 || 0.05 || 0.35 || 0.32
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/porter-ranch/ Porter_Ranch] || 121428 || 886 || 0.7 || 41 || 0.92 || 0.09 || 0.27 || 0.02 || 0.08 || 0.61
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/rancho-park/ Rancho_Park] || 78116 || 934 || 0.7 || 37 || 0.58 || 0.09 || 0.16 || 0.03 || 0.14 || 0.63
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/reseda/ Reseda] || 54771 || 779 || 0.9 || 32 || 0.52 || 0.07 || 0.11 || 0.04 || 0.44 || 0.37
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/san-pedro/ San_Pedro] || 57198 || 771 || 0.8 || 34 || 0.44 || 0.11 || 0.05 || 0.06 || 0.41 || 0.45
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sawtelle/ Sawtelle] || 57710 || 733 || 0.9 || 32 || 0.20 || 0.05 || 0.20 || 0.03 || 0.23 || 0.50
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/shadow-hills/ Shadow_Hills] || 82796 || 763 || 0.7 || 39 || 0.78 || 0.11 || 0.07 || 0.01 || 0.29 || 0.59
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sherman-oaks/ Sherman_Oaks] || 69651 || 839 || 0.3 || 37 || 0.41 || 0.08 || 0.06 || 0.04 || 0.12 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/silver-lake/ Silver_Lake] || 54339 || 788 || 1 || 35 || 0.36 || 0.06 || 0.18 || 0.03 || 0.42 || 0.34
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/south-park/ South_Park] || 29518 || 650 || 0.2 || 23 || 0.23 || 0.03 || 0.00 || 0.19 || 0.79 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/studio-city/ Studio_City] || 75657 || 817 || 0.3 || 38 || 0.38 || 0.09 || 0.05 || 0.04 || 0.09 || 0.78
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sun-valley/ Sun_Valley] || 51290 || 685 || 0.4 || 28 || 0.28 || 0.05 || 0.08 || 0.02 || 0.69 || 0.18
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sunland/ Sunland] || 68720 || 816 || 0.6 || 37 || 0.37 || 0.12 || 0.07 || 0.02 || 0.22 || 0.65
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sylmar/ Sylmar] || 65783 || 732 || 0.4 || 28 || 0.28 || 0.08 || 0.03 || 0.04 || 0.70 || 0.21
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/tarzana/ Tarzana] || 73195 || 824 || 0.4 || 38 || 0.38 || 0.09 || 0.05 || 0.04 || 0.15 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/toluca-lake/ Toluca_Lake] || 73111 || 762 || 0.4 || 37 || 0.37 || 0.11 || 0.05 || 0.05 || 0.14 || 0.72
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/tujunga/ Tujunga] || 58001 || 749 || 0.7 || 36 || 0.36 || 0.10 || 0.07 || 0.02 || 0.26 || 0.61
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/university-park/ University_Park] || 18533 || 727 || 1 || 23 || 0.23 || 0.01 || 0.16 || 0.07 || 0.48 || 0.26
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/valley-glen/ Valley_Glen] || 46175 || 726 || 0.8 || 32 || 0.32 || 0.06 || 0.05 || 0.04 || 0.45 || 0.40
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/valley-village/ Valley_Village] || 55470 || 825 || 0.5 || 36 || 0.36 || 0.07 || 0.04 || 0.06 || 0.19 || 0.67
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/van-nuys/ Van_Nuys] || 41134 || 722 || 0.7 || 28 || 0.28 || 0.06 || 0.06 || 0.06 || 0.61 || 0.23
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/venice/ Venice] || 67057 || 754 || 0.6 || 35 || 0.35 || 0.07 || 0.04 || 0.06 || 0.22 || 0.64
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-knolls/ Vermont_Knolls] || 27730 || 682 || 0.5 || 24 || 0.24 || 0.06 || 0.01 || 0.43 || 0.55 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-square/ Vermont_Square] || 29904 || 647 || 0.6 || 26 || 0.26 || 0.06 || 0.01 || 0.39 || 0.57 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-vista/ Vermont_Vista] || 31272 || 685 || 0.6 || 24 || 0.24 || 0.07 || 0.01 || 0.45 || 0.52 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-slauson/ Vermont_Slauson] || 31236 || 679 || 0.5 || 25 || 0.25 || 0.06 || 0.00 || 0.37 || 0.61 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/watts/ Watts] || 24728 || 689 || 0.5 || 21 || 0.21 || 0.04 || 0.00 || 0.38 || 0.61 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-adams/ West_Adams] || 38209 || 746 || 0.6 || 28 || 0.28 || 0.06 || 0.02 || 0.38 || 0.56 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-hills/ West_Hills] || 103012 || 878 || 0.4 || 39 || 0.39 || 0.11 || 0.11 || 0.03 || 0.11 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-los-angeles/ West_Los_Angeles] || 86403 || 905 || 0.3 || 38 || 0.38 || 0.09 || 0.11 || 0.02 || 0.05 || 0.77
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westchester/ Westchester] || 77473 || 800 || 0.9 || 35 || 0.35 || 0.09 || 0.09 || 0.17 || 0.17 || 0.52
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westlake/ Westlake] || 26757 || 718 || 0.3 || 27 || 0.27 || 0.04 || 0.16 || 0.04 || 0.73 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westwood/ Westwood] || 68716 || 952 || 0.6 || 27 || 0.27 || 0.05 || 0.23 || 0.02 || 0.07 || 0.63
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/wilmington/ Wilmington] || 40641 || 703 || 0.1 || 24 || 0.24 || 0.05 || 0.02 || 0.03 || 0.87 || 0.06
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/windsor-square/ Windsor_Square] || 61767 || 0 || 0.9 || 38 || 0.38 || 0.07 || 0.42 || 0.04 || 0.15 || 0.38
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/winnetka/ Winnetka] || 62535 || 734 || 1 || 32 || 0.32 || 0.07 || 0.15 || 0.04 || 0.41 || 0.36
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/woodland-hills/ Woodland_Hills] || 89946 || 826 || 0.3 || 40 || 0.4 || 0.11 || 0.07 || 0.03 || 0.08 || 0.78
|}
</center>

===References===
* [http://www.prb.org/ Pupulation Reference Bureau].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Data_LA_Neighborhoods_Data}}

SOCR EduMaterials Activities Binomial Distributions

2009-10-15T05:26:27Z

Nchristo: /* This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions. */

== This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.==

* '''Description''': You can access the applets for the above distributions at http://www.socr.ucla.edu/htmls/SOCR_Distributions.html .

* '''Exercise 1:''' Use SOCR to graph and print the following distributions and answer the questions below. Also, comment on the shape of each one of these distributions:
**a. <math> X \sim b(10,0.5) </math>, find <math> P(X=3) </math>, <math> E(X) </math>, <math> sd(X) </math>, and verify them with the formulas discussed in class.
**b. <math> X \sim b(10,0.1) </math>, find <math> P(1 \le X \le 3) </math>.
**c. <math> X \sim b(10,0.9) </math>, find <math> P(5 < X < 8), \ P(X < 8), \ P(X \le 7), \ P(X \ge 9) </math>.
**d. <math> X \sim b(30,0.1) </math>, find <math> P(X > 2) </math>.

Below you can see a snapshot of the distribution of <math> X \sim b(20,0.3) </math>

<center>[[Image: SOCR_Activities_Binomial_Christou__binomial.jpg|600px]]</center>

* '''Exercise 2:''' Use SOCR to graph and print the distribution of a geometric random variable with <math> p=0.2, p=0.7 </math>. What is the shape of these distributions? What happens when <math> p </math> is large? What happens when <math> p </math> is small?

Below you can see a snapshot of the distribution of <math> X \sim geometric(0.4) </math>

<center>[[Image: SOCR_Activities_Christou_geometric.jpg|600px]]</center>

* '''Exercise 3:''' Select the geometric probability distribution with <math> p=0.2 </math>. Use SOCR to compute the following:
**a. <math> P(X=5) </math>
**b. <math> P(X > 3) </math>
**c. <math> P(X \le 5) </math>
**d. <math> P(X > 6) </math>
**e. <math> P(X \ge 8) </math>
**f. <math> P(4 \le X \le 9) </math>
**g. <math> P(4 < X < 9) </math>

* '''Exercise 4:''' Verify that your answers in exercise 3 agree with the formulas discussed in class, for example, <math> P(X=x)=(1-p)^{x-1}p </math>, <math> P(X > k)=(1-p)^k </math>, etc. Write all your answers in detail using those formulas.

* '''Exercise 5:''' Let <math> X </math> follow the hypergeometric probability distribution with <math> N=52 </math>, <math> n=10 </math>, and number of "hot" items 13. Use SOCR to graph and print this distribution.

Below you can see a snapshot of the distribution of <math> X \sim hypergeometric(N=100, n=15, r=30) </math>

<center>[[Image: SOCR_Activities_Christou_hypergeometric.jpg|600px]]</center>

* '''Exercise 6:''' Refer to exercise 5. Use SOCR to compute <math> P(X=5) </math> and write down the formula that gives this answer.

* '''Exercise 7:''' Binomial approximation to hypergeometric: Let <math> X </math> follow the hypergeometric probability distribution with <math> N=1000, \ n=10 </math> and number of "hot" items 50. Graph and print this distribution.

* '''Exercise 8:''' Refer to exercise 7. Use SOCR to compute the exact probability: <math> P(X=2) </math>. Approximate <math> P(X=2) </math> using the binomial distribution. Is the approximation good? Why?

* '''Exercise 9:''' Do you think you can approximate well the hypergeometric probability distribution with <math> N=50, \ n=20 </math>, and number of "hot" items 40 using the binomial probability distribution? Graph and print the exact (hypergeometric) and the approximate (binomial) distributions and compare.

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_Binomial_Distributions}}

SOCR Data

2009-10-05T21:14:10Z

Nchristo: /* Observed data */

== [[SOCR_EduMaterials |SOCR Educational Materials]] - SOCR Data==

The links below contain a number of datasets that may be used for demonstration purposes in probability and statistics education. There are two types of data - '''simulated''' (computer-generated using random sampling) and '''observed''' (research, observationally or experimentally acquired).
[[Image:SOCR_Icon_Data.png|150px|thumbnail|right| SOCR Data ]]

==Simulated data==
The [[SOCR]] resources provide a number of mechanisms to simulate data using computer random-number generators. Here are some of the most commonly used SOCR generators of simulated data:
* [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] - each experiment reports random outcomes, sample and population distributions and summary statistics.
* [[SOCR_EduMaterials_Activities_RNG | SOCR random-number generator]] - enables sampling of any size from any of the [http://socr.ucla.edu/htmls/dist SOCR Distributions].
* [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] - all of the [http://socr.ucla.edu/htmls/ana SOCR analyses] allow random sampling from various populations appropriate for the user-specified analysis.

==Observed data==
The following collections include a number of real observed datasets from different disciplines, acquired using different techniques and applicable in different situations.

* [http://gcmd.nasa.gov/ Climate Change Data]
** [[SOCR_Data_Dinov_042108_Antarctic_IceThicknessMawson | Antarctic Ice Thickness at Mawson, Davis and Casey (01/Apr/1954 to 15/Jan/2002). Number of data points is 1636.]]
** [[SOCR_Data_Dinov_071108_OilGasData | Energy Resources, Production and Consumption Dataset]]
** [[SOCR_Data_Dinov_121608_OzoneData | California Ozone Data (1980-2006)]]
** [http://www.stat.ucla.edu/~nchristo/statistics_c173_c273/ca_ozone.txt CA Ozone - 08/08/2005]
* Population Data
** [[SOCR_Data_Dinov_020108_HeightsWeights | 25,000 Records of Human Heights (in) and Weights (lbs)]]
** [[SOCR_Data_Dinov_011709_PRB_Data | Population Data by Country 2000-2006]]
** [[SOCR_Data_LA_Neighborhoods_Data | Los Angeles City Neighborhoods Data (from US Census)]]
* Consumer Price Index (CPI)
** [[SOCR_Data_Dinov_021808_ConsumerPriceIndex | Consumer Price Index (1981-2006) - Fuel and Food Data]]
** [[SOCR_Data_Dinov_021808_ConsumerPriceIndex3Way | Consumer Price Index (1981-2007) - One-, Two- or Three-Way ANOVA Data by items, months and years]]
** [[SOCR_Data_Dinov_010309_HousingPriceIndex | Housing Price Index (2000-2006) (motion charts)]]
** [[SOCR_Data_Dinov_091609_SnP_HomePriceIndex | S&P Home Price Index (1991-2009) (motion charts)]]
* [[SOCR_Data_Dinov_021708_Earthquakes | California Earthquakes Data]] (1969-2007)
* [[SOCR_Data_Dinov_030708_APExamScores | 2007 Advanced Placement (AP) Exam Scores by Discipline]]
* [http://math.whatcom.ctc.edu/content/Links.phtml?cat=18 Online Math Center: A large archive of data from different scientific observations]
* [[NISER_081107_ID_Data | Largemouth Bass Mercury Contamination Dataset]]
* [[SOCR_Data_Dinov_032708_AllometricPlanRels | Allometric relationship between population density, body mass and metabolic activity in Plants]]
* Neuroimaging Data
** [[SOCR_Data_July2009_ID_NI | Neuroimaging study of 27 Alzheimer's disease (AD) subjects, 35 normal controls (NC), and 42 mild cognitive impairment subjects (MCI)]]
** [http://www.stat.ucla.edu/%7Edinov/courses_students.dir/04/Spring/Stat233.dir/HWs.dir/AD_NeuroPsychImagingData1.html Alzheimer's Disease neuroimaging Data]
** [[SOCR_Data_June2008_ID_NI | Neuroimaging study of super-resolution image enhancing]]
** [[SOCR_Data_April2009_ID_NI | Neuroimaging study of Prefrontal Cortex Volume across Species and Tissue Types]]
* [http://wiki.stat.ucla.edu/niser/index.php/NISER_Data NISER Datasets]
* [[SOCR_061708_NC_Data_Aquifer | Texas Wolfcamp aquifer data]]
* [[SOCR_012708_ID_Data_HotDogs | Hot Dog Calorie and Sodium Dataset]]
* Biomedical Data
** [[SOCR_Data_BMI_Regression | Body Density & Body Mass Index (BMI) Data]]
** [[SOCR_Data_KneePainData_041409 | Knee Pain Centroid Locations Data]]
* [http://www.eoddata.com Stock Market Data]
** [[SOCR_Data_Dinov_070108_JAVA | Sun Microsystems (Java) Stock price (2007-2008)]]
** [[SOCR_Data_Dinov_070108_SP500_0608 | S&P 500 (2007-2008)]]
* [[SOCR_Data_Dinov_072108_H_Index_Pubs | Faculty Publications]]
* [[SOCR_US_CensusData | US Census Data]]
** [[SOCR_Data_LA_Neighborhoods_Data | Los Angeles County Neighborhoods Data (from US Census)]]
* [http://www.presidency.ucsb.edu/ US Elections Data]
** [[SOCR_Data_Dinov_11_08_08_PresidentialElections | US Electoral College vs. Popular Vote Presidential Elections Mandate Data (1828-2008)]]

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Data}}

SOCR Data LA Neighborhoods Data

2009-10-05T04:30:02Z

Nchristo: /* LA Neighborhoods Data */

== [[SOCR_Data |SOCR Data]] - Los Angeles City Neighborhoods Data==

===Data Description===
[[Image:SOCR_Data_LA_County_Neighborhoods_Dinov_100109_Fig1.png|200px|thumbnail|right| [http://projects.latimes.com/mapping-la/neighborhoods/ LA Times Neighborhoods] ]]

The LA Times map of the neighborhoods of Los Angeles contains data ([http://www.census.gov/ U.S. Census 2000]) on the 110 LA neighborhoods including measures of education, income and population demographics.

===Data Source===
* [http://projects.latimes.com/mapping-la/neighborhoods/ LA Times].
* Variables
** [http://projects.latimes.com/mapping-la/neighborhoods/ LA_Neighborhoods] ('''LA_Nbhd''')
** [http://projects.latimes.com/mapping-la/neighborhoods/income/neighborhood/list/ Highest_Incomes] ('''Income'''): Median household income reports the amount of money earned by the household that falls exactly in the middle of pack.
** [http://projects.latimes.com/mapping-la/neighborhoods/api-test-scores/neighborhood/list/ Highest-Scoring_Public_Schools] ('''Schools'''): The median API score reports the 2008 test results posted by the school that falls exactly in the middle of the pack. California's Academic Performance Index (API) combines several tests into a single number between 200 and 1000 for each school. The tests that make up the API and their weighting are listed on the [http://www.cde.ca.gov/ta/ac/ap/glossary08d.asp California Department of Education website].
** [http://projects.latimes.com/mapping-la/neighborhoods/diversity-index/neighborhood/list/ Most_Diverse_Population] ('''Diversity'''): The diversity index measures the probability that any two residents, chosen at random, would be of different ethnicities. If all residents are of the same ethnic group it's zero. If half are from one group and half from another it's 50%. This figure is used to place areas into 10 groups of even size and then rank those groups. Areas with a diversity rank of 1 are the least ethnically diverse. Those with a diversity rank of 10 are the most ethnically diverse.
** [http://projects.latimes.com/mapping-la/neighborhoods/age/neighborhood/list/ Oldest_Population] ('''Age'''): Median age reports the age of the person who falls exactly in the middle of the pack.
** [http://projects.latimes.com/mapping-la/neighborhoods/owners/neighborhood/list/ Most_Homeowners] ('''Homes'''): The percentage of owners measures the portion of households that are owner occupied. A ranking of the percentage of [http://projects.latimes.com/mapping-la/neighborhoods/renters/neighborhood/list/ households that rent] is also available.
** [http://projects.latimes.com/mapping-la/neighborhoods/veterans/neighborhood/list/ Most_Veterans] ('''Vets'''): The percentage of veterans measures the portion of adult population that once served in the armed forces. A [http://projects.latimes.com/mapping-la/neighborhoods/armed-forces/neighborhood/list/ ranking of active members of the armed forces] is also available.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/asian/neighborhood/list/ Asian] ('''Asian'''): The percentage of population of whose ethnicity is Asian.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/black/neighborhood/list/ Black] ('''Black'''): The percentage of population of whose ethnicity is black.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/latino/neighborhood/list/ Latino] ('''Latino'''): The percentage of population of whose ethnicity is Latino.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/white/neighborhood/list/ White] ('''White'''): The percentage of population of whose ethnicity is white.

===LA Neighborhoods Data===

<center>
{| class="wikitable"
|-
! LA_Nbhd || Income || Schools || Diversity|| Age || Homes || Vets || Asian || Black || Latino || White
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/adams-normandie/ Adams-Normandie] || 29606 || 691 || 0.6 || 26 || 0.26 || 0.05 || 0.05 || 0.25 || 0.62 || 0.06
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/arleta/ Arleta] || 65649 || 719 || 0.4 || 29 || 0.29 || 0.07 || 0.11 || 0.02 || 0.72 || 0.13
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/arlington-heights/ Arlington_Heights] || 31423 || 687 || 0.8 || 31 || 0.31 || 0.05 || 0.13 || 0.25 || 0.57 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/atwater-village/ Atwater_Village] || 53872 || 762 || 0.9 || 34 || 0.34 || 0.06 || 0.20 || 0.01 || 0.51 || 0.22
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/baldwin-hillscrenshaw/ Baldwin_Hills/Crenshaw] || 37948 || 656 || 0.4 || 36 || 0.36 || 0.10 || 0.05 || 0.71 || 0.17 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/bel-air/ Bel-Air] || 208861 || 924 || 0.2 || 46 || 0.46 || 0.13 || 0.08 || 0.01 || 0.05 || 0.83
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverly-crest/ Beverly_Crest] || 168104 || 791 || 0.1 || 45 || 0.45 || 0.10 || 0.04 || 0.02 || 0.03 || 0.88
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverly-grove/ Beverly_Grove] || 63039 || 0 || 0.2 || 38 || 0.38 || 0.05 || 0.05 || 0.02 || 0.06 || 0.82
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverlywood/ Beverlywood] || 105253 || 872 || 0.2 || 39 || 0.39 || 0.08 || 0.07 || 0.04 || 0.06 || 0.80
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/boyle-heights/ Boyle_Heights] || 33235 || 689 || 0.1 || 25 || 0.25 || 0.03 || 0.02 || 0.01 || 0.94 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/brentwood/ Brentwood] || 112927 || 882 || 0.2 || 39 || 0.39 || 0.08 || 0.06 || 0.01 || 0.05 || 0.84
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/broadway-manchester/ Broadway-Manchester] || 29897 || 656 || 0.5 || 23 || 0.23 || 0.04 || 0.00 || 0.39 || 0.59 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/canoga-park/ Canoga_Park] || 51601 || 706 || 0.8 || 30 || 0.3 || 0.06 || 0.11 || 0.04 || 0.51 || 0.31
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/carthay/ Carthay] || 71398 || 762 || 0.8 || 37 || 0.37 || 0.04 || 0.09 || 0.13 || 0.16 || 0.58
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/central-alameda/ Central-Alameda] || 31559 || 669 || 0.1 || 22 || 0.22 || 0.02 || 0.01 || 0.13 || 0.85 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/century-city/ Century_City] || 95135 || 0 || 0.2 || 46 || 0.46 || 0.12 || 0.09 || 0.01 || 0.04 || 0.83
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chatsworth/ Chatsworth] || 84549 || 762 || 0.6 || 40 || 0.4 || 0.11 || 0.14 || 0.02 || 0.14 || 0.65
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chesterfield-square/ Chesterfield_Square] || 37737 || 544 || 0.5 || 31 || 0.31 || 0.07 || 0.01 || 0.59 || 0.37 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/cheviot-hills/ Cheviot_Hills] || 109980 || 0 || 0.3 || 42 || 0.42 || 0.11 || 0.09 || 0.01 || 0.09 || 0.79
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chinatown/ Chinatown] || 22837 || 784 || 1 || 34 || 0.34 || 0.04 || 0.35 || 0.21 || 0.32 || 0.11
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/cypress-park/ Cypress_Park] || 42615 || 697 || 0.2 || 27 || 0.27 || 0.04 || 0.11 || 0.01 || 0.82 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/del-rey/ Del_Rey] || 63317 || 783 || 1 || 35 || 0.35 || 0.08 || 0.14 || 0.04 || 0.42 || 0.36
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/downtown/ Downtown] || 15003 || 724 || 1 || 39 || 0.39 || 0.10 || 0.21 || 0.22 || 0.37 || 0.16
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/eagle-rock/ Eagle_Rock] || 67253 || 830 || 1 || 35 || 0.35 || 0.08 || 0.24 || 0.02 || 0.40 || 0.30
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/east-hollywood/ East_Hollywood] || 29927 || 772 || 0.7 || 31 || 0.31 || 0.03 || 0.16 || 0.02 || 0.60 || 0.18
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/echo-park/ Echo_Park] || 37708 || 755 || 0.6 || 30 || 0.3 || 0.04 || 0.19 || 0.02 || 0.64 || 0.13
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/el-sereno/ El_Sereno] || 45866 || 739 || 0.2 || 28 || 0.28 || 0.05 || 0.11 || 0.02 || 0.81 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/elysian-park/ Elysian_Park] || 28263 || 905 || 0.7 || 31 || 0.31 || 0.04 || 0.43 || 0.02 || 0.48 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/elysian-valley/ Elysian_Valley] || 49013 || 813 || 0.7 || 31 || 0.31 || 0.04 || 0.26 || 0.01 || 0.61 || 0.10
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/encino/ Encino] || 78529 || 899 || 0.2 || 42 || 0.42 || 0.11 || 0.05 || 0.02 || 0.09 || 0.80
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/exposition-park/ Exposition_Park] || 33999 || 733 || 0.6 || 26 || 0.26 || 0.06 || 0.01 || 0.38 || 0.56 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/fairfax/ Fairfax] || 65938 || 734 || 0.2 || 33 || 0.33 || 0.05 || 0.05 || 0.02 || 0.06 || 0.85
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/florence/ Florence] || 29447 || 680 || 0.3 || 23 || 0.23 || 0.04 || 0.00 || 0.28 || 0.70 || 0.00
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/glassell-park/ Glassell_Park] || 50098 || 723 || 0.5 || 30 || 0.3 || 0.05 || 0.17 || 0.01 || 0.66 || 0.14
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/gramercy-park/ Gramercy_Park] || 57983 || 653 || 0.1 || 36 || 0.36 || 0.13 || 0.00 || 0.86 || 0.12 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/granada-hills/ Granada_Hills] || 83911 || 812 || 0.8 || 37 || 0.37 || 0.11 || 0.16 || 0.03 || 0.21 || 0.56
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/green-meadows/ Green_Meadows] || 31347 || 682 || 0.5 || 24 || 0.24 || 0.05 || 0.00 || 0.44 || 0.54 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hancock-park/ Hancock_Park] || 85277 || 861 || 0.4 || 37 || 0.37 || 0.06 || 0.13 || 0.04 || 0.09 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harbor-city/ Harbor_City] || 55447 || 736 || 1 || 30 || 0.3 || 0.08 || 0.13 || 0.11 || 0.48 || 0.25
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harbor-gateway/ Harbor_Gateway] || 47849 || 730 || 0.9 || 27 || 0.27 || 0.08 || 0.16 || 0.16 || 0.53 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harvard-heights/ Harvard_Heights] || 31173 || 712 || 0.5 || 30 || 0.3 || 0.04 || 0.13 || 0.16 || 0.66 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harvard-park/ Harvard_Park] || 37013 || 0 || 0.6 || 28 || 0.28 || 0.07 || 0.01 || 0.48 || 0.48 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/highland-park/ Highland_Park] || 45478 || 744 || 0.4 || 28 || 0.28 || 0.05 || 0.11 || 0.02 || 0.72 || 0.11
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/historic-south-central/ Historic_South-Central] || 30882 || 674 || 0.1 || 23 || 0.23 || 0.03 || 0.01 || 0.10 || 0.87 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood/ Hollywood] || 33694 || 718 || 0.9 || 31 || 0.31 || 0.05 || 0.07 || 0.05 || 0.42 || 0.41
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood-hills/ Hollywood_Hills] || 69277 || 780 || 0.3 || 37 || 0.37 || 0.08 || 0.07 || 0.05 || 0.09 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood-hills-west/ Hollywood_Hills_West] || 108199 || 961 || 0.1 || 41 || 0.41 || 0.08 || 0.04 || 0.03 || 0.06 || 0.85
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hyde-park/ Hyde_Park] || 39460 || 647 || 0.5 || 31 || 0.31 || 0.09 || 0.01 || 0.66 || 0.27 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/jefferson-park/ Jefferson_Park] || 32654 || 680 || 0.7 || 31 || 0.31 || 0.08 || 0.03 || 0.47 || 0.45 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/koreatown/ Koreatown] || 30558 || 740 || 0.8 || 30 || 0.3 || 0.03 || 0.32 || 0.05 || 0.54 || 0.07
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lake-balboa/ Lake_Balboa] || 65336 || 777 || 0.8 || 35 || 0.35 || 0.10 || 0.09 || 0.04 || 0.34 || 0.49
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lake-view-terrace/ Lake_View_Terrace] || 67985 || 746 || 0.9 || 31 || 0.31 || 0.08 || 0.05 || 0.17 || 0.53 || 0.22
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/larchmont/ Larchmont] || 47780 || 800 || 1 || 34 || 0.34 || 0.04 || 0.30 || 0.03 || 0.37 || 0.25
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/leimert-park/ Leimert_Park] || 45865 || 679 || 0.3 || 38 || 0.38 || 0.11 || 0.04 || 0.80 || 0.11 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lincoln-heights/ Lincoln_Heights] || 30579 || 736 || 0.3 || 27 || 0.27 || 0.03 || 0.25 || 0.00 || 0.71 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/los-feliz/ Los_Feliz] || 50793 || 723 || 0.8 || 36 || 0.36 || 0.06 || 0.13 || 0.04 || 0.19 || 0.58
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/manchester-square/ Manchester_Square] || 46093 || 750 || 0.2 || 34 || 0.34 || 0.11 || 0.00 || 0.79 || 0.19 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mar-vista/ Mar_Vista] || 62611 || 709 || 0.9 || 35 || 0.35 || 0.08 || 0.12 || 0.04 || 0.29 || 0.51
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mid-city/ Mid-City] || 43711 || 780 || 0.9 || 31 || 0.31 || 0.06 || 0.04 || 0.38 || 0.45 || 0.10
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mid-wilshire/ Mid-Wilshire] || 58483 || 739 || 1 || 34 || 0.34 || 0.06 || 0.20 || 0.23 || 0.20 || 0.34
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mission-hills/ Mission_Hills] || 75675 || 795 || 0.8 || 34 || 0.34 || 0.10 || 0.11 || 0.04 || 0.54 || 0.29
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/montecito-heights/ Montecito_Heights] || 55901 || 730 || 0.5 || 31 || 0.31 || 0.05 || 0.17 || 0.03 || 0.66 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mount-washington/ Mount_Washington] || 57725 || 825 || 0.7 || 33 || 0.33 || 0.07 || 0.13 || 0.03 || 0.61 || 0.21
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/north-hills/ North_Hills] || 52456 || 760 || 0.8 || 28 || 0.28 || 0.07 || 0.12 || 0.05 || 0.57 || 0.24
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/north-hollywood/ North_Hollywood] || 42791 || 724 || 0.7 || 30 || 0.3 || 0.05 || 0.06 || 0.06 || 0.58 || 0.27
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/northridge/ Northridge] || 67906 || 836 || 0.9 || 32 || 0.32 || 0.09 || 0.14 || 0.05 || 0.26 || 0.50
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pacific-palisades/ Pacific_Palisades] || 168008 || 879 || 0.1 || 43 || 0.43 || 0.12 || 0.05 || 0.00 || 0.03 || 0.89
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pacoima/ Pacoima] || 49066 || 688 || 0.1 || 24 || 0.24 || 0.04 || 0.02 || 0.07 || 0.86 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/palms/ Palms] || 50684 || 793 || 1 || 31 || 0.31 || 0.05 || 0.20 || 0.12 || 0.23 || 0.38
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/panorama-city/ Panorama_City] || 44468 || 721 || 0.4 || 25 || 0.25 || 0.04 || 0.12 || 0.04 || 0.70 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pico-robertson/ Pico-Robertson] || 63356 || 0 || 0.3 || 36 || 0.36 || 0.05 || 0.06 || 0.06 || 0.07 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pico-union/ Pico-Union] || 26424 || 677 || 0.1 || 27 || 0.27 || 0.02 || 0.08 || 0.03 || 0.85 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/playa-del-rey/ Playa_del_Rey] || 91339 || 836 || 0.4 || 38 || 0.38 || 0.11 || 0.08 || 0.04 || 0.10 || 0.73
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/playa-vista/ Playa_Vista] || 68597 || 771 || 1 || 37 || 0.37 || 0.11 || 0.21 || 0.05 || 0.35 || 0.32
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/porter-ranch/ Porter_Ranch] || 121428 || 886 || 0.7 || 41 || 0.41 || 0.09 || 0.27 || 0.02 || 0.08 || 0.61
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/rancho-park/ Rancho_Park] || 78116 || 934 || 0.7 || 37 || 0.37 || 0.09 || 0.16 || 0.03 || 0.14 || 0.63
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/reseda/ Reseda] || 54771 || 779 || 0.9 || 32 || 0.32 || 0.07 || 0.11 || 0.04 || 0.44 || 0.37
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/san-pedro/ San_Pedro] || 57198 || 771 || 0.8 || 34 || 0.34 || 0.11 || 0.05 || 0.06 || 0.41 || 0.45
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sawtelle/ Sawtelle] || 57710 || 733 || 0.9 || 32 || 0.32 || 0.05 || 0.20 || 0.03 || 0.23 || 0.50
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/shadow-hills/ Shadow_Hills] || 82796 || 763 || 0.7 || 39 || 0.39 || 0.11 || 0.07 || 0.01 || 0.29 || 0.59
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sherman-oaks/ Sherman_Oaks] || 69651 || 839 || 0.3 || 37 || 0.37 || 0.08 || 0.06 || 0.04 || 0.12 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/silver-lake/ Silver_Lake] || 54339 || 788 || 1 || 35 || 0.35 || 0.06 || 0.18 || 0.03 || 0.42 || 0.34
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/south-park/ South_Park] || 29518 || 650 || 0.2 || 23 || 0.23 || 0.03 || 0.00 || 0.19 || 0.79 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/studio-city/ Studio_City] || 75657 || 817 || 0.3 || 38 || 0.38 || 0.09 || 0.05 || 0.04 || 0.09 || 0.78
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sun-valley/ Sun_Valley] || 51290 || 685 || 0.4 || 28 || 0.28 || 0.05 || 0.08 || 0.02 || 0.69 || 0.18
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sunland/ Sunland] || 68720 || 816 || 0.6 || 37 || 0.37 || 0.12 || 0.07 || 0.02 || 0.22 || 0.65
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sylmar/ Sylmar] || 65783 || 732 || 0.4 || 28 || 0.28 || 0.08 || 0.03 || 0.04 || 0.70 || 0.21
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/tarzana/ Tarzana] || 73195 || 824 || 0.4 || 38 || 0.38 || 0.09 || 0.05 || 0.04 || 0.15 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/toluca-lake/ Toluca_Lake] || 73111 || 762 || 0.4 || 37 || 0.37 || 0.11 || 0.05 || 0.05 || 0.14 || 0.72
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/tujunga/ Tujunga] || 58001 || 749 || 0.7 || 36 || 0.36 || 0.10 || 0.07 || 0.02 || 0.26 || 0.61
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/university-park/ University_Park] || 18533 || 727 || 1 || 23 || 0.23 || 0.01 || 0.16 || 0.07 || 0.48 || 0.26
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/valley-glen/ Valley_Glen] || 46175 || 726 || 0.8 || 32 || 0.32 || 0.06 || 0.05 || 0.04 || 0.45 || 0.40
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/valley-village/ Valley_Village] || 55470 || 825 || 0.5 || 36 || 0.36 || 0.07 || 0.04 || 0.06 || 0.19 || 0.67
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/van-nuys/ Van_Nuys] || 41134 || 722 || 0.7 || 28 || 0.28 || 0.06 || 0.06 || 0.06 || 0.61 || 0.23
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/venice/ Venice] || 67057 || 754 || 0.6 || 35 || 0.35 || 0.07 || 0.04 || 0.06 || 0.22 || 0.64
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-knolls/ Vermont_Knolls] || 27730 || 682 || 0.5 || 24 || 0.24 || 0.06 || 0.01 || 0.43 || 0.55 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-square/ Vermont_Square] || 29904 || 647 || 0.6 || 26 || 0.26 || 0.06 || 0.01 || 0.39 || 0.57 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-vista/ Vermont_Vista] || 31272 || 685 || 0.6 || 24 || 0.24 || 0.07 || 0.01 || 0.45 || 0.52 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-slauson/ Vermont-Slauson] || 31236 || 679 || 0.5 || 25 || 0.25 || 0.06 || 0.00 || 0.37 || 0.61 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/watts/ Watts] || 24728 || 689 || 0.5 || 21 || 0.21 || 0.04 || 0.00 || 0.38 || 0.61 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-adams/ West_Adams] || 38209 || 746 || 0.6 || 28 || 0.28 || 0.06 || 0.02 || 0.38 || 0.56 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-hills/ West_Hills] || 103012 || 878 || 0.4 || 39 || 0.39 || 0.11 || 0.11 || 0.03 || 0.11 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-los-angeles/ West_Los_Angeles] || 86403 || 905 || 0.3 || 38 || 0.38 || 0.09 || 0.11 || 0.02 || 0.05 || 0.77
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westchester/ Westchester] || 77473 || 800 || 0.9 || 35 || 0.35 || 0.09 || 0.09 || 0.17 || 0.17 || 0.52
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westlake/ Westlake] || 26757 || 718 || 0.3 || 27 || 0.27 || 0.04 || 0.16 || 0.04 || 0.73 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westwood/ Westwood] || 68716 || 952 || 0.6 || 27 || 0.27 || 0.05 || 0.23 || 0.02 || 0.07 || 0.63
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/wilmington/ Wilmington] || 40641 || 703 || 0.1 || 24 || 0.24 || 0.05 || 0.02 || 0.03 || 0.87 || 0.06
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/windsor-square/ Windsor_Square] || 61767 || 0 || 0.9 || 38 || 0.38 || 0.07 || 0.42 || 0.04 || 0.15 || 0.38
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/winnetka/ Winnetka] || 62535 || 734 || 1 || 32 || 0.32 || 0.07 || 0.15 || 0.04 || 0.41 || 0.36
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/woodland-hills/ Woodland_Hills] || 89946 || 826 || 0.3 || 40 || 0.4 || 0.11 || 0.07 || 0.03 || 0.08 || 0.78
|}
</center>

===References===
* [http://www.prb.org/ Pupulation Reference Bureau].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Data_LA_Neighborhoods_Data}}

SOCR Data LA Neighborhoods Data

2009-10-05T04:22:08Z

Nchristo: /* SOCR Data - Los Angeles County Neighborhoods Data */

== [[SOCR_Data |SOCR Data]] - Los Angeles City Neighborhoods Data==

===Data Description===
[[Image:SOCR_Data_LA_County_Neighborhoods_Dinov_100109_Fig1.png|200px|thumbnail|right| [http://projects.latimes.com/mapping-la/neighborhoods/ LA Times Neighborhoods] ]]

The LA Times map of the neighborhoods of Los Angeles contains data ([http://www.census.gov/ U.S. Census 2000]) on the 110 LA neighborhoods including measures of education, income and population demographics.

===Data Source===
* [http://projects.latimes.com/mapping-la/neighborhoods/ LA Times].
* Variables
** [http://projects.latimes.com/mapping-la/neighborhoods/ LA_Neighborhoods] ('''LA_Nbhd''')
** [http://projects.latimes.com/mapping-la/neighborhoods/income/neighborhood/list/ Highest_Incomes] ('''Income'''): Median household income reports the amount of money earned by the household that falls exactly in the middle of pack.
** [http://projects.latimes.com/mapping-la/neighborhoods/api-test-scores/neighborhood/list/ Highest-Scoring_Public_Schools] ('''Schools'''): The median API score reports the 2008 test results posted by the school that falls exactly in the middle of the pack. California's Academic Performance Index (API) combines several tests into a single number between 200 and 1000 for each school. The tests that make up the API and their weighting are listed on the [http://www.cde.ca.gov/ta/ac/ap/glossary08d.asp California Department of Education website].
** [http://projects.latimes.com/mapping-la/neighborhoods/diversity-index/neighborhood/list/ Most_Diverse_Population] ('''Diversity'''): The diversity index measures the probability that any two residents, chosen at random, would be of different ethnicities. If all residents are of the same ethnic group it's zero. If half are from one group and half from another it's 50%. This figure is used to place areas into 10 groups of even size and then rank those groups. Areas with a diversity rank of 1 are the least ethnically diverse. Those with a diversity rank of 10 are the most ethnically diverse.
** [http://projects.latimes.com/mapping-la/neighborhoods/age/neighborhood/list/ Oldest_Population] ('''Age'''): Median age reports the age of the person who falls exactly in the middle of the pack.
** [http://projects.latimes.com/mapping-la/neighborhoods/owners/neighborhood/list/ Most_Homeowners] ('''Homes'''): The percentage of owners measures the portion of households that are owner occupied. A ranking of the percentage of [http://projects.latimes.com/mapping-la/neighborhoods/renters/neighborhood/list/ households that rent] is also available.
** [http://projects.latimes.com/mapping-la/neighborhoods/veterans/neighborhood/list/ Most_Veterans] ('''Vets'''): The percentage of veterans measures the portion of adult population that once served in the armed forces. A [http://projects.latimes.com/mapping-la/neighborhoods/armed-forces/neighborhood/list/ ranking of active members of the armed forces] is also available.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/asian/neighborhood/list/ Asian] ('''Asian'''): The percentage of population of whose ethnicity is Asian.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/black/neighborhood/list/ Black] ('''Black'''): The percentage of population of whose ethnicity is black.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/latino/neighborhood/list/ Latino] ('''Latino'''): The percentage of population of whose ethnicity is Latino.
** [http://projects.latimes.com/mapping-la/neighborhoods/ethnicity/white/neighborhood/list/ White] ('''White'''): The percentage of population of whose ethnicity is white.

===LA Neighborhoods Data===

<center>
{| class="wikitable"
|-
! LA_Nbhd || Income || Schools || Diversity|| Age || Homes || Vets || Asian || Black || Latino || White
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/adams-normandie/ Adams-Normandie] || 29,606 || 691 || 0.6 || 26 || 0.26 || 0.05 || 0.05 || 0.25 || 0.62 || 0.06
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/arleta/ Arleta] || 65,649 || 719 || 0.4 || 29 || 0.29 || 0.07 || 0.11 || 0.02 || 0.72 || 0.13
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/arlington-heights/ Arlington_Heights] || 31,423 || 687 || 0.8 || 31 || 0.31 || 0.05 || 0.13 || 0.25 || 0.57 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/atwater-village/ Atwater_Village] || 53,872 || 762 || 0.9 || 34 || 0.34 || 0.06 || 0.20 || 0.01 || 0.51 || 0.22
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/baldwin-hillscrenshaw/ Baldwin_Hills/Crenshaw] || 37,948 || 656 || 0.4 || 36 || 0.36 || 0.10 || 0.05 || 0.71 || 0.17 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/bel-air/ Bel-Air] || 208,861 || 924 || 0.2 || 46 || 0.46 || 0.13 || 0.08 || 0.01 || 0.05 || 0.83
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverly-crest/ Beverly_Crest] || 168,104 || 791 || 0.1 || 45 || 0.45 || 0.10 || 0.04 || 0.02 || 0.03 || 0.88
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverly-grove/ Beverly_Grove] || 63,039 || 0 || 0.2 || 38 || 0.38 || 0.05 || 0.05 || 0.02 || 0.06 || 0.82
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/beverlywood/ Beverlywood] || 105,253 || 872 || 0.2 || 39 || 0.39 || 0.08 || 0.07 || 0.04 || 0.06 || 0.80
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/boyle-heights/ Boyle_Heights] || 33,235 || 689 || 0.1 || 25 || 0.25 || 0.03 || 0.02 || 0.01 || 0.94 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/brentwood/ Brentwood] || 112,927 || 882 || 0.2 || 39 || 0.39 || 0.08 || 0.06 || 0.01 || 0.05 || 0.84
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/broadway-manchester/ Broadway-Manchester] || 29,897 || 656 || 0.5 || 23 || 0.23 || 0.04 || 0.00 || 0.39 || 0.59 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/canoga-park/ Canoga_Park] || 51,601 || 706 || 0.8 || 30 || 0.3 || 0.06 || 0.11 || 0.04 || 0.51 || 0.31
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/carthay/ Carthay] || 71,398 || 762 || 0.8 || 37 || 0.37 || 0.04 || 0.09 || 0.13 || 0.16 || 0.58
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/central-alameda/ Central-Alameda] || 31,559 || 669 || 0.1 || 22 || 0.22 || 0.02 || 0.01 || 0.13 || 0.85 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/century-city/ Century_City] || 95,135 || 0 || 0.2 || 46 || 0.46 || 0.12 || 0.09 || 0.01 || 0.04 || 0.83
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chatsworth/ Chatsworth] || 84,549 || 762 || 0.6 || 40 || 0.4 || 0.11 || 0.14 || 0.02 || 0.14 || 0.65
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chesterfield-square/ Chesterfield_Square] || 37,737 || 544 || 0.5 || 31 || 0.31 || 0.07 || 0.01 || 0.59 || 0.37 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/cheviot-hills/ Cheviot_Hills] || 109,980 || 0 || 0.3 || 42 || 0.42 || 0.11 || 0.09 || 0.01 || 0.09 || 0.79
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/chinatown/ Chinatown] || 22,837 || 784 || 1 || 34 || 0.34 || 0.04 || 0.35 || 0.21 || 0.32 || 0.11
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/cypress-park/ Cypress_Park] || 42,615 || 697 || 0.2 || 27 || 0.27 || 0.04 || 0.11 || 0.01 || 0.82 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/del-rey/ Del_Rey] || 63,317 || 783 || 1 || 35 || 0.35 || 0.08 || 0.14 || 0.04 || 0.42 || 0.36
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/downtown/ Downtown] || 15,003 || 724 || 1 || 39 || 0.39 || 0.10 || 0.21 || 0.22 || 0.37 || 0.16
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/eagle-rock/ Eagle_Rock] || 67,253 || 830 || 1 || 35 || 0.35 || 0.08 || 0.24 || 0.02 || 0.40 || 0.30
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/east-hollywood/ East_Hollywood] || 29,927 || 772 || 0.7 || 31 || 0.31 || 0.03 || 0.16 || 0.02 || 0.60 || 0.18
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/echo-park/ Echo_Park] || 37,708 || 755 || 0.6 || 30 || 0.3 || 0.04 || 0.19 || 0.02 || 0.64 || 0.13
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/el-sereno/ El_Sereno] || 45,866 || 739 || 0.2 || 28 || 0.28 || 0.05 || 0.11 || 0.02 || 0.81 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/elysian-park/ Elysian_Park] || 28,263 || 905 || 0.7 || 31 || 0.31 || 0.04 || 0.43 || 0.02 || 0.48 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/elysian-valley/ Elysian_Valley] || 49,013 || 813 || 0.7 || 31 || 0.31 || 0.04 || 0.26 || 0.01 || 0.61 || 0.10
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/encino/ Encino] || 78,529 || 899 || 0.2 || 42 || 0.42 || 0.11 || 0.05 || 0.02 || 0.09 || 0.80
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/exposition-park/ Exposition_Park] || 33,999 || 733 || 0.6 || 26 || 0.26 || 0.06 || 0.01 || 0.38 || 0.56 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/fairfax/ Fairfax] || 65,938 || 734 || 0.2 || 33 || 0.33 || 0.05 || 0.05 || 0.02 || 0.06 || 0.85
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/florence/ Florence] || 29,447 || 680 || 0.3 || 23 || 0.23 || 0.04 || 0.00 || 0.28 || 0.70 || 0.00
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/glassell-park/ Glassell_Park] || 50,098 || 723 || 0.5 || 30 || 0.3 || 0.05 || 0.17 || 0.01 || 0.66 || 0.14
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/gramercy-park/ Gramercy_Park] || 57,983 || 653 || 0.1 || 36 || 0.36 || 0.13 || 0.00 || 0.86 || 0.12 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/granada-hills/ Granada_Hills] || 83,911 || 812 || 0.8 || 37 || 0.37 || 0.11 || 0.16 || 0.03 || 0.21 || 0.56
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/green-meadows/ Green_Meadows] || 31,347 || 682 || 0.5 || 24 || 0.24 || 0.05 || 0.00 || 0.44 || 0.54 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hancock-park/ Hancock_Park] || 85,277 || 861 || 0.4 || 37 || 0.37 || 0.06 || 0.13 || 0.04 || 0.09 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harbor-city/ Harbor_City] || 55,447 || 736 || 1 || 30 || 0.3 || 0.08 || 0.13 || 0.11 || 0.48 || 0.25
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harbor-gateway/ Harbor_Gateway] || 47,849 || 730 || 0.9 || 27 || 0.27 || 0.08 || 0.16 || 0.16 || 0.53 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harvard-heights/ Harvard_Heights] || 31,173 || 712 || 0.5 || 30 || 0.3 || 0.04 || 0.13 || 0.16 || 0.66 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/harvard-park/ Harvard_Park] || 37,013 || 0 || 0.6 || 28 || 0.28 || 0.07 || 0.01 || 0.48 || 0.48 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/highland-park/ Highland_Park] || 45,478 || 744 || 0.4 || 28 || 0.28 || 0.05 || 0.11 || 0.02 || 0.72 || 0.11
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/historic-south-central/ Historic_South-Central] || 30,882 || 674 || 0.1 || 23 || 0.23 || 0.03 || 0.01 || 0.10 || 0.87 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood/ Hollywood] || 33,694 || 718 || 0.9 || 31 || 0.31 || 0.05 || 0.07 || 0.05 || 0.42 || 0.41
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood-hills/ Hollywood_Hills] || 69,277 || 780 || 0.3 || 37 || 0.37 || 0.08 || 0.07 || 0.05 || 0.09 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hollywood-hills-west/ Hollywood_Hills_West] || 108,199 || 961 || 0.1 || 41 || 0.41 || 0.08 || 0.04 || 0.03 || 0.06 || 0.85
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/hyde-park/ Hyde_Park] || 39,460 || 647 || 0.5 || 31 || 0.31 || 0.09 || 0.01 || 0.66 || 0.27 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/jefferson-park/ Jefferson_Park] || 32,654 || 680 || 0.7 || 31 || 0.31 || 0.08 || 0.03 || 0.47 || 0.45 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/koreatown/ Koreatown] || 30,558 || 740 || 0.8 || 30 || 0.3 || 0.03 || 0.32 || 0.05 || 0.54 || 0.07
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lake-balboa/ Lake_Balboa] || 65,336 || 777 || 0.8 || 35 || 0.35 || 0.10 || 0.09 || 0.04 || 0.34 || 0.49
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lake-view-terrace/ Lake_View_Terrace] || 67,985 || 746 || 0.9 || 31 || 0.31 || 0.08 || 0.05 || 0.17 || 0.53 || 0.22
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/larchmont/ Larchmont] || 47,780 || 800 || 1 || 34 || 0.34 || 0.04 || 0.30 || 0.03 || 0.37 || 0.25
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/leimert-park/ Leimert_Park] || 45,865 || 679 || 0.3 || 38 || 0.38 || 0.11 || 0.04 || 0.80 || 0.11 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/lincoln-heights/ Lincoln_Heights] || 30,579 || 736 || 0.3 || 27 || 0.27 || 0.03 || 0.25 || 0.00 || 0.71 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/los-feliz/ Los_Feliz] || 50,793 || 723 || 0.8 || 36 || 0.36 || 0.06 || 0.13 || 0.04 || 0.19 || 0.58
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/manchester-square/ Manchester_Square] || 46,093 || 750 || 0.2 || 34 || 0.34 || 0.11 || 0.00 || 0.79 || 0.19 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mar-vista/ Mar_Vista] || 62,611 || 709 || 0.9 || 35 || 0.35 || 0.08 || 0.12 || 0.04 || 0.29 || 0.51
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mid-city/ Mid-City] || 43,711 || 780 || 0.9 || 31 || 0.31 || 0.06 || 0.04 || 0.38 || 0.45 || 0.10
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mid-wilshire/ Mid-Wilshire] || 58,483 || 739 || 1 || 34 || 0.34 || 0.06 || 0.20 || 0.23 || 0.20 || 0.34
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mission-hills/ Mission_Hills] || 75,675 || 795 || 0.8 || 34 || 0.34 || 0.10 || 0.11 || 0.04 || 0.54 || 0.29
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/montecito-heights/ Montecito_Heights] || 55,901 || 730 || 0.5 || 31 || 0.31 || 0.05 || 0.17 || 0.03 || 0.66 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/mount-washington/ Mount_Washington] || 57,725 || 825 || 0.7 || 33 || 0.33 || 0.07 || 0.13 || 0.03 || 0.61 || 0.21
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/north-hills/ North_Hills] || 52,456 || 760 || 0.8 || 28 || 0.28 || 0.07 || 0.12 || 0.05 || 0.57 || 0.24
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/north-hollywood/ North_Hollywood] || 42,791 || 724 || 0.7 || 30 || 0.3 || 0.05 || 0.06 || 0.06 || 0.58 || 0.27
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/northridge/ Northridge] || 67,906 || 836 || 0.9 || 32 || 0.32 || 0.09 || 0.14 || 0.05 || 0.26 || 0.50
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pacific-palisades/ Pacific_Palisades] || 168,008 || 879 || 0.1 || 43 || 0.43 || 0.12 || 0.05 || 0.00 || 0.03 || 0.89
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pacoima/ Pacoima] || 49,066 || 688 || 0.1 || 24 || 0.24 || 0.04 || 0.02 || 0.07 || 0.86 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/palms/ Palms] || 50,684 || 793 || 1 || 31 || 0.31 || 0.05 || 0.20 || 0.12 || 0.23 || 0.38
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/panorama-city/ Panorama_City] || 44,468 || 721 || 0.4 || 25 || 0.25 || 0.04 || 0.12 || 0.04 || 0.70 || 0.12
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pico-robertson/ Pico-Robertson] || 63,356 || 0 || 0.3 || 36 || 0.36 || 0.05 || 0.06 || 0.06 || 0.07 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/pico-union/ Pico-Union] || 26,424 || 677 || 0.1 || 27 || 0.27 || 0.02 || 0.08 || 0.03 || 0.85 || 0.03
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/playa-del-rey/ Playa_del_Rey] || 91,339 || 836 || 0.4 || 38 || 0.38 || 0.11 || 0.08 || 0.04 || 0.10 || 0.73
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/playa-vista/ Playa_Vista] || 68,597 || 771 || 1 || 37 || 0.37 || 0.11 || 0.21 || 0.05 || 0.35 || 0.32
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/porter-ranch/ Porter_Ranch] || 121,428 || 886 || 0.7 || 41 || 0.41 || 0.09 || 0.27 || 0.02 || 0.08 || 0.61
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/rancho-park/ Rancho_Park] || 78,116 || 934 || 0.7 || 37 || 0.37 || 0.09 || 0.16 || 0.03 || 0.14 || 0.63
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/reseda/ Reseda] || 54,771 || 779 || 0.9 || 32 || 0.32 || 0.07 || 0.11 || 0.04 || 0.44 || 0.37
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/san-pedro/ San_Pedro] || 57,198 || 771 || 0.8 || 34 || 0.34 || 0.11 || 0.05 || 0.06 || 0.41 || 0.45
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sawtelle/ Sawtelle] || 57,710 || 733 || 0.9 || 32 || 0.32 || 0.05 || 0.20 || 0.03 || 0.23 || 0.50
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/shadow-hills/ Shadow_Hills] || 82,796 || 763 || 0.7 || 39 || 0.39 || 0.11 || 0.07 || 0.01 || 0.29 || 0.59
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sherman-oaks/ Sherman_Oaks] || 69,651 || 839 || 0.3 || 37 || 0.37 || 0.08 || 0.06 || 0.04 || 0.12 || 0.74
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/silver-lake/ Silver_Lake] || 54,339 || 788 || 1 || 35 || 0.35 || 0.06 || 0.18 || 0.03 || 0.42 || 0.34
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/south-park/ South_Park] || 29,518 || 650 || 0.2 || 23 || 0.23 || 0.03 || 0.00 || 0.19 || 0.79 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/studio-city/ Studio_City] || 75,657 || 817 || 0.3 || 38 || 0.38 || 0.09 || 0.05 || 0.04 || 0.09 || 0.78
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sun-valley/ Sun_Valley] || 51,290 || 685 || 0.4 || 28 || 0.28 || 0.05 || 0.08 || 0.02 || 0.69 || 0.18
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sunland/ Sunland] || 68,720 || 816 || 0.6 || 37 || 0.37 || 0.12 || 0.07 || 0.02 || 0.22 || 0.65
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/sylmar/ Sylmar] || 65,783 || 732 || 0.4 || 28 || 0.28 || 0.08 || 0.03 || 0.04 || 0.70 || 0.21
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/tarzana/ Tarzana] || 73,195 || 824 || 0.4 || 38 || 0.38 || 0.09 || 0.05 || 0.04 || 0.15 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/toluca-lake/ Toluca_Lake] || 73,111 || 762 || 0.4 || 37 || 0.37 || 0.11 || 0.05 || 0.05 || 0.14 || 0.72
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/tujunga/ Tujunga] || 58,001 || 749 || 0.7 || 36 || 0.36 || 0.10 || 0.07 || 0.02 || 0.26 || 0.61
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/university-park/ University_Park] || 18,533 || 727 || 1 || 23 || 0.23 || 0.01 || 0.16 || 0.07 || 0.48 || 0.26
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/valley-glen/ Valley_Glen] || 46,175 || 726 || 0.8 || 32 || 0.32 || 0.06 || 0.05 || 0.04 || 0.45 || 0.40
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/valley-village/ Valley_Village] || 55,470 || 825 || 0.5 || 36 || 0.36 || 0.07 || 0.04 || 0.06 || 0.19 || 0.67
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/van-nuys/ Van_Nuys] || 41,134 || 722 || 0.7 || 28 || 0.28 || 0.06 || 0.06 || 0.06 || 0.61 || 0.23
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/venice/ Venice] || 67,057 || 754 || 0.6 || 35 || 0.35 || 0.07 || 0.04 || 0.06 || 0.22 || 0.64
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-knolls/ Vermont_Knolls] || 27,730 || 682 || 0.5 || 24 || 0.24 || 0.06 || 0.01 || 0.43 || 0.55 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-square/ Vermont_Square] || 29,904 || 647 || 0.6 || 26 || 0.26 || 0.06 || 0.01 || 0.39 || 0.57 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-vista/ Vermont_Vista] || 31,272 || 685 || 0.6 || 24 || 0.24 || 0.07 || 0.01 || 0.45 || 0.52 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/vermont-slauson/ Vermont-Slauson] || 31,236 || 679 || 0.5 || 25 || 0.25 || 0.06 || 0.00 || 0.37 || 0.61 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/watts/ Watts] || 24,728 || 689 || 0.5 || 21 || 0.21 || 0.04 || 0.00 || 0.38 || 0.61 || 0.01
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-adams/ West_Adams] || 38,209 || 746 || 0.6 || 28 || 0.28 || 0.06 || 0.02 || 0.38 || 0.56 || 0.02
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-hills/ West_Hills] || 103,012 || 878 || 0.4 || 39 || 0.39 || 0.11 || 0.11 || 0.03 || 0.11 || 0.71
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/west-los-angeles/ West_Los_Angeles] || 86,403 || 905 || 0.3 || 38 || 0.38 || 0.09 || 0.11 || 0.02 || 0.05 || 0.77
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westchester/ Westchester] || 77,473 || 800 || 0.9 || 35 || 0.35 || 0.09 || 0.09 || 0.17 || 0.17 || 0.52
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westlake/ Westlake] || 26,757 || 718 || 0.3 || 27 || 0.27 || 0.04 || 0.16 || 0.04 || 0.73 || 0.05
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/westwood/ Westwood] || 68,716 || 952 || 0.6 || 27 || 0.27 || 0.05 || 0.23 || 0.02 || 0.07 || 0.63
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/wilmington/ Wilmington] || 40,641 || 703 || 0.1 || 24 || 0.24 || 0.05 || 0.02 || 0.03 || 0.87 || 0.06
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/windsor-square/ Windsor_Square] || 61,767 || 0 || 0.9 || 38 || 0.38 || 0.07 || 0.42 || 0.04 || 0.15 || 0.38
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/winnetka/ Winnetka] || 62,535 || 734 || 1 || 32 || 0.32 || 0.07 || 0.15 || 0.04 || 0.41 || 0.36
|-
| [http://projects.latimes.com/mapping-la/neighborhoods/neighborhood/woodland-hills/ Woodland_Hills] || 89,946 || 826 || 0.3 || 40 || 0.4 || 0.11 || 0.07 || 0.03 || 0.08 || 0.78
|}
</center>

===References===
* [http://www.prb.org/ Pupulation Reference Bureau].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Data_LA_Neighborhoods_Data}}

SOCR Data

2009-08-10T16:00:41Z

Nchristo: /* Observed data */

== [[SOCR_EduMaterials |SOCR Educational Materials]] - SOCR Data==

The links below contain a number of datasets that may be used for demonstration purposes in probability and statistics education. There are two types of data - '''simulated''' (computer-generated using random sampling) and '''observed''' (research, observationally or experimentally acquired).
[[Image:SOCR_Icon_Data.png|150px|thumbnail|right| SOCR Data ]]

==Simulated data==
The [[SOCR]] resources provide a number of mechanisms to simulate data using computer random-number generators. Here are some of the most commonly used SOCR generators of simulated data:
* [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] - each experiment reports random outcomes, sample and population distributions and summary statistics.
* [[SOCR_EduMaterials_Activities_RNG | SOCR random-number generator]] - enables sampling of any size from any of the [http://socr.ucla.edu/htmls/dist SOCR Distributions].
* [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] - all of the [http://socr.ucla.edu/htmls/ana SOCR analyses] allow random sampling from various populations appropriate for the user-specified analysis.

==Observed data==
The following collections include a number of real observed datasets from different disciplines, acquired using different techniques and applicable in different situations.

* [http://gcmd.nasa.gov/ Climate Change Data]
** [[SOCR_Data_Dinov_042108_Antarctic_IceThicknessMawson | Antarctic Ice Thickness at Mawson, Davis and Casey (01/Apr/1954 to 15/Jan/2002). Number of data points is 1636.]]
** [[SOCR_Data_Dinov_071108_OilGasData | Energy Resources, Production and Consumption Dataset]]
** [[SOCR_Data_Dinov_121608_OzoneData | California Ozone Data (1980-2006)]]
** [http://www.stat.ucla.edu/~nchristo/statistics_c173_c273/ca_ozone.txt CA Ozone - 08/08/2005]
* Population Data
** [[SOCR_Data_Dinov_020108_HeightsWeights | 25,000 Records of Human Heights (in) and Weights (lbs)]]
** [[SOCR_Data_Dinov_011709_PRB_Data | Population Data by Country 2000-2006]]
* Consumer Price Index (CPI)
** [[SOCR_Data_Dinov_021808_ConsumerPriceIndex | Consumer Price Index (1981-2006) - Fuel and Food Data]]
** [[SOCR_Data_Dinov_021808_ConsumerPriceIndex3Way | Consumer Price Index (1981-2007) - One-, Two- or Three-Way ANOVA Data by items, months and years]]
** [[SOCR_Data_Dinov_010309_HousingPriceIndex | Housing Price Index (2000-2006) (motion charts)]]
* [[SOCR_Data_Dinov_021708_Earthquakes | California Earthquakes Data]] (1969-2007)
* [[SOCR_Data_Dinov_030708_APExamScores | 2007 Advanced Placement (AP) Exam Scores by Discipline]]
* [http://math.whatcom.ctc.edu/content/Links.phtml?cat=18 Online Math Center: A large archive of data from different scientific observations]
* [[NISER_081107_ID_Data | Largemouth Bass Mercury Contamination Dataset]]
* [[SOCR_Data_Dinov_032708_AllometricPlanRels | Allometric relationship between population density, body mass and metabolic activity in Plants]]
* Neuroimaging Data
** [[SOCR_Data_July2009_ID_NI | Neuroimaging study of 27 Alzheimer's disease (AD) subjects, 35 normal controls (NC), and 42 mild cognitive impairment subjects (MCI)]]
** [http://www.stat.ucla.edu/%7Edinov/courses_students.dir/04/Spring/Stat233.dir/HWs.dir/AD_NeuroPsychImagingData1.html Alzheimer's Disease neuroimaging Data]
** [[SOCR_Data_June2008_ID_NI | Neuroimaging study of super-resolution image enhancing]]
** [[SOCR_Data_April2009_ID_NI | Neuroimaging study of Prefrontal Cortex Volume across Species and Tissue Types]]
* [http://wiki.stat.ucla.edu/niser/index.php/NISER_Data NISER Datasets]
* [[SOCR_061708_NC_Data_Aquifer | Texas Wolfcamp aquifer data]]
* [[SOCR_012708_ID_Data_HotDogs | Hot Dog Calorie and Sodium Dataset]]
* Biomedical Data
** [[SOCR_Data_BMI_Regression | Body Density & Body Mass Index (BMI) Data]]
** [[SOCR_Data_KneePainData_041409 | Knee Pain Centroid Locations Data]]
* [http://www.eoddata.com Stock Market Data]
** [[SOCR_Data_Dinov_070108_JAVA | Sun Microsystems (Java) Stock price (2007-2008)]]
** [[SOCR_Data_Dinov_070108_SP500_0608 | S&P 500 (2007-2008)]]
* [[SOCR_Data_Dinov_072108_H_Index_Pubs | Faculty Publications]]
* [[SOCR_US_CensusData | US Census Data]]
* [http://www.presidency.ucsb.edu/ US Elections Data]
** [[SOCR_Data_Dinov_11_08_08_PresidentialElections | US Electoral College vs. Popular Vote Presidential Elections Mandate Data (1828-2008)]]

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Data}}

SOCR EduMaterials Activities ConfIntervals

2009-03-02T04:40:35Z

Nchristo: /* This is an activity to explore the confidence intervals for the populatuon mean when the standard deviation is known. */

== This is an activity to explore the confidence intervals for the populatuon mean when the standard deviation is known. ==

* '''Description''': You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the "Confidence Interval Experiment".

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by:
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviaton. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

*'''Exercise 1:'''
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. How many intervals (out of the 200) do you expect to miss the population mean <math>\mu=0</math>? Take a snapshot and describe what you observe.
**'''1.''' What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
**'''2.''' What do the blue lines represent?
**'''3.''' How is the confidence interval represented?
**'''4.''' What does the green dot represent?
**'''5.''' Write down the formula on which the confidence intervals are based and explain why they have different width.

*'''Exercise 2:'''
**'''1.''' Reset and repeat (a) with <math>\alpha=0.01</math>. Take a snapshot and describe what you see.
**'''2.''' Reset and repeat (a) with sample size now <math>n=80</math>. Take a snapshot and describe in detail what you see.
**'''3.''' Reset and repeat (a) with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Take a snapshot and describe in detail what you see.

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ConfIntervals}}

SOCR EduMaterials Activities ConfIntervals

2009-03-02T04:37:55Z

Nchristo: /* This is an activity to explore the confidence intervals for the populatuon mean when the standard deviation is known. */

== This is an activity to explore the confidence intervals for the populatuon mean when the standard deviation is known. ==

* '''Description''': You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the "Confidence Interval Experiment".

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by:
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviaton. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

*'''Exercise 1:'''
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. How many intervals (out of the 200) do you expect to miss the population mean <math>\mu=0</math>? Take a snapshot and describe what you observe.
**'''1.''' What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
**'''2.''' What do the blue lines represent?
**'''3.''' How is the confidence interval represented?
**'''4.''' What does the green dot represent?
**'''5.''' Write down the formula on which the confidence intervals are based.

*'''Exercise 2:'''
**'''1.''' Reset and repeat (a) with <math>\alpha=0.01</math>. Take a snapshot and describe what you see.
**'''2.''' Reset and repeat (a) with sample size now <math>n=80</math>. Take a snapshot and describe in detail what you see.
**'''3.''' Reset and repeat (a) with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Take a snapshot and describe in detail what you see.

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ConfIntervals}}

SOCR EduMaterials Activities ConfIntervals

2009-03-02T04:21:12Z

Nchristo: /* This is an activity to explore the confidence intervals for the populatuon mean when the standard deviation is known. */

== This is an activity to explore the confidence intervals for the populatuon mean when the standard deviation is known. ==

* '''Description''': You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the "Confidence Interval Experiment".

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by:
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviaton. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

*'''Exercise 1:'''
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. How many intervals (out of the 200) do you expect to miss the population mean <math>\mu=0</math>? Take a snapshot and describe what you observe.

**'''1.''' What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
**'''2.''' What do the blue lines represent?
**'''3.''' How is the confidence interval represented?
**'''4.''' What does the green dot represent?
**'''5.''' Write down the formula on which the confidence intervals are based.

*'''Exercise 2:'''
**'''1.''' Reset and repeat (a) with <math>\alpha=0.01</math>. Take a snapshot and describe what you see.
**'''2.''' Reset and repeat (a) with sample size now <math>n=80</math>. Take a snapshot and describe in detail what you see.
**'''3.''' Reset and repeat (a) with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Take a snapshot and describe in detail what you see.

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T19:55:28Z

Nchristo: /* Binomial convergence to Black-Scholes option pricing formula */

== [[SOCR_EduMaterials_ApplicationsActivities | SOCR Applications Activities]] - Black-Scholes Option Pricing Model (with Convergence of Binomial) ==

===Description===
You can access the Black-Scholes Option Pricing Model applet at [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site], select ''Financial Applications'' --> ''BlackScholesOptionPricing''.

===Black-Scholes option pricing formula===
The value <math>C</math> of a European call option at time <math>t=0</math> is:
: <math> C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2) </math>
: <math> d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}
</math>
: <math> d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}=d_1-\sigma \sqrt{t} </math>

Where, 
: <math>S_0</math> Price of the stock at time <math>t=0</math> 
: <math>E</math> Exercise price at expiration 
: <math>r</math> Continuously compounded risk-free interest 
: <math>\sigma</math> Annual standard deviation of the returns of the stock 
: <math>t</math> Time to expiration in years 
: <math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

===Binomial convergence to Black-Scholes option pricing formula===
The binomial formula converges to the Black-Scholes formula when the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
: <math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 

* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the [http://www.socr.ucla.edu/htmls/app/ SOCR Black Scholes Option Pricing model applet] shows the convergence of the call price calculated by the binomial option pricing model to the price of the call calculated using the Black-Scholes model.

 

<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center>

===References===
The materials above was partially taken from:
* ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.
* ''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
* [http://www.socr.ucla.edu/htmls/app/ SOCR Applications Site]

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ApplicationsActivities_BlackScholesOptionPricing}}

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T19:53:15Z

Nchristo: /* Binomial convergence to Black-Scholes option pricing formula */

== [[SOCR_EduMaterials_ApplicationsActivities | SOCR Applications Activities]] - Black-Scholes Option Pricing Model (with Convergence of Binomial) ==

===Description===
You can access the Black-Scholes Option Pricing Model applet at [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site], select ''Financial Applications'' --> ''BlackScholesOptionPricing''.

===Black-Scholes option pricing formula===
The value <math>C</math> of a European call option at time <math>t=0</math> is:
: <math> C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2) </math>
: <math> d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}
</math>
: <math> d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}=d_1-\sigma \sqrt{t} </math>

Where, 
: <math>S_0</math> Price of the stock at time <math>t=0</math> 
: <math>E</math> Exercise price at expiration 
: <math>r</math> Continuously compounded risk-free interest 
: <math>\sigma</math> Annual standard deviation of the returns of the stock 
: <math>t</math> Time to expiration in years 
: <math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

===Binomial convergence to Black-Scholes option pricing formula===
The binomial formula converges to the Black-Scholes formula when the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
: <math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 

* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the [http://www.socr.ucla.edu/htmls/app/ SOCR Black Scholes Option Pricing model applet] shows the convergence of the call price calculated by the binomial option pricing model to the price of the call calculated using the Black-Schole model.

 

<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center>

===References===
The materials above was partially taken from:
* ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.
* ''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
* [http://www.socr.ucla.edu/htmls/app/ SOCR Applications Site]

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ApplicationsActivities_BlackScholesOptionPricing}}

SOCR EduMaterials Activities ApplicationsActivities BinomialOptionPricing

2008-08-03T16:36:03Z

Nchristo:

== Binomial Option Pricing Model =

* '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ .

Define: 

<math>S_0</math> Stock price at <math>t=0</math> 
<math>S_1</math> Stock price at <math>t=1<math> 
<math>E</math> Exercise price of the call option 
<math>u</math> <math>1+ \%</math> change in stock price from <math>t=0</math> to <math>t=1</math> if stock price increases (<math>u>1</math>) 
<math>d</math> <math>1+ \%</math> change in stock price from <math>t=0</math> to <math>t=1</math> if stock price decreases (<math>d<1</math>) 
<math>C</math> The call price 
<math>\alpha</math> The number of shares of stocks purchased per one call (hedge ratio) 
<math>C_u</math> Price of call at <math>t=1</math> if stock price increases: <math>max(S_1-E,0)</math> or <math>max(uS_0-E,0)</math> 
<math>C_d</math> Price of call at <math>t=1</math> if stock price decreases: <math>max(S_1-E,0)</math> or <math>max(dS_0-E,0)</math> 
<math>r</math> Continuous risk-free interest rate 

*The value <math>C</math> of a European call option at time <math>t=0</math> is: 
<math>
C=S_0 \sum_{j=k}^{n} {n \choose j} p'^{j}(1-p')^{n-j} -
\frac{E}{e^{rt}} \sum_{j=k}^{n} {n \choose j} p^{j}(1-p)^{n-j}
</math>
 
Where, 
<math>
u=e^{+\sigma \sqrt{\frac{t}{n}}},
d=e^{-\sigma \sqrt{\frac{t}{n}}}=\frac{1}{u}
</math>
 
<math>
p=\frac{e^{rt}-d}{u-d}, \ \ p'=\frac{up}{e^{rt}}
</math>

 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Risk-free interest rate per period 
<math>n</math> Number of periods 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> &Time to expiration in years 

* The SOCR Binomial Option Pricing applet provides the price of the stock and the price of the call at each node. Note that at expiration the nodes for which the call is in the money (<math>S > E</math> are colored green, while the nodes for which the call is out of the money (<math>S \le E)</math> are colored blue. The example below uses the following data: 
<math>S_0=\$30</math>, <math>E=\$29</math>, <math>r_f=0.05</math>,
<math>\sigma=0.30</math>, <math>\mbox{days to expiration}=73</math>,
<math>\mbox{number of steps}=5</math>.

<center>[[Image: Christou_binomial_applet1.jpg|600px]]</center>

<center>[[Image: Christou_binomial_applet2.jpg|600px]]</center>

 
* The materials above was partially taken from 
''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and 
''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.

SOCR EduMaterials Activities ApplicationsActivities BinomialOptionPricing

2008-08-03T16:35:09Z

Nchristo: New page: == Binomial Option Pricing Model = * '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ . Define: <math>S_0</math> Stock price at <mat...

== Binomial Option Pricing Model =

* '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ .

Define: 

<math>S_0</math> Stock price at <math>t=0</math> 
<math>S_1</math> Stock price at <math>t=1<math> 
<math>E</math> Exercise price of the call option 
<math>u</math> <math>1+ \%</math> change in stock price from <math>t=0</math> to <math>t=1</math> if stock price increases (<math>u>1</math>) 
<math>d</math> <math>1+ \%</math> change in stock price from <math>t=0</math> to <math>t=1</math> if stock price decreases (<math>d<1</math>) 
<math>C</math> The call price 
<math>\alpha</math> The number of shares of stocks purchased per one call (hedge ratio) 
<math>C_u</math> Price of call at <math>t=1</math> if stock price increases: <math>max(S_1-E,0)</math> or <math>max(uS_0-E,0)</math> 
<math>C_d</math> Price of call at <math>t=1</math> if stock price decreases: <math>max(S_1-E,0)</math> or <math>max(dS_0-E,0)</math> 
<math>r</math> Continuous risk-free interest rate 

*The value <math>C</math> of a European call option at time <math>t=0</math> is: 
<math>
C=S_0 \sum_{j=k}^{n} {n \choose j} p'^{j}(1-p')^{n-j} -
\frac{E}{e^{rt}} \sum_{j=k}^{n} {n \choose j} p^{j}(1-p)^{n-j}
</math>
 
Where, 
<math>
u=e^{+\sigma \sqrt{\frac{t}{n}}}, \ \
d=e^{-\sigma \sqrt{\frac{t}{n}}}=\frac{1}{u}
</math>
 
<math>
p=\frac{e^{rt}-d}{u-d}, \ \ p'=\frac{up}{e^{rt}}
</math>

 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Risk-free interest rate per period 
<math>n</math> Number of periods 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> &Time to expiration in years 

* The SOCR Binomial Option Pricing applet provides the price of the stock and the price of the call at each node. Note that at expiration the nodes for which the call is in the money (<math>S > E</math> are colored green, while the nodes for which the call is out of the money (<math>S \le E)</math> are colored blue. The example below uses the following data: 
<math>S_0=\$30</math>, <math>E=\$29</math>, <math>r_f=0.05</math>,
<math>\sigma=0.30</math>, <math>\mbox{days to expiration}=73</math>,
<math>\mbox{number of steps}=5</math>.

<center>[[Image: Christou_binomial_applet1.jpg|600px]]</center>

<center>[[Image: Christou_binomial_applet2.jpg|600px]]</center>

 
* The materials above was partially taken from 
''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and 
''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T16:19:45Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* '''Description''': You can access the Black-Scholes applet at http://www.socr.ucla.edu/htmls/app/ .

* Black-Scholes option pricing formula: 
The value <math>C</math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

 

<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center>

 
* The materials above was partially taken from 
''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and 
''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T16:19:15Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* '''Description''': You can access the Black-Scholes applet at http://www.socr.ucla.edu/htmls/app/ .

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

 

<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center>

 
* The materials above was partially taken from 
''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and 
''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.

SOCR EduMaterials Activities ApplicationsActivities TradingOptions

2008-08-03T16:18:40Z

Nchristo: /* Options */

== Options ==

* '''Description''': You can access the portfolio applets for the options trading strategies at http://www.socr.ucla.edu/htmls/app/ .

An option is a contract between two investors:
 
* Issuer (or seller), holder of a short position. He sells the option.
 
* Holder (buyer), holder of a long position. He buys the option.
 
'''Types of options:'''
 
* Call option: Gives the holder the right to buy an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the option or premium.
 
* Put option: Gives the holder the right to sell an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the put or premium.
The date specified it is called: the expiration date or maturity date. The price specified it is called the exercise price or the strike price.
 
There are European options (can be exercised only on the expiration date) and American options (can be exercised at any time up to the expiration date).
 
'''Stock options mechanics:'''
 
* Options are normally traded in units of 100 shares. The price of the option is on a per share basis. Therefore, if the price of an option is priced at $0.50, the total premium for that option would be <math>\$50</math> (<math>0.50 \times 100 = \$50</math>).
 
* Stock options are on a January, February, or March cycle. Stocks are randomly assigned in one of these three cycles. For example, IBM is on a January cycle (options can be bought on Jan, Apr, Jul, Oct).
 
Stock options expired on the Saturday immediately following the third Friday of the expiration month.
 

The call option will only be exercised if the stock price at expiration is larger than the exercise price. In this case the holder of the call will have a positive payoff. The put option will only be exercised if the stock price at expiration is lower than the exercise price. In this case the holder of the put will have a positive payoff. The two figures below shows when the holder or the seller make a positive payoff.

 

<center>[[Image: Christou_call_faces.jpg|600px]]</center>

 

<center>[[Image: Christou_put_faces.jpg|600px]]</center>

 

There is an infinite number of combinations that one can make using call and put options. Some of the combinations have special names, like straddles, strips, straps, bull spreads, bear spreads, butterfly spreads, covered call, etc. All these are shown in the SOCR Trading Options applet. Here are some snapshots:

 

<center>[[Image: Christou_butterfly_calls.jpg|600px]]</center>

 

<center>[[Image: Christou_bear_spread.jpg|600px]]</center>

 
* The materials above was partially taken from 
''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and 
''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.

SOCR EduMaterials Activities ApplicationsActivities StockSimulation

2008-08-03T16:18:05Z

Nchristo:

== A Model for Stock prices ==

* '''Description''': You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ .

* Process for Stock Prices: Assumed a drift rate equal to <math>\mu S</math> where <math>\mu</math> is the expected return of the stock, and variance <math>\sigma^2 S^2</math> where <math>\sigma^2</math> is the variance of the return of the stock. From Weiner process the model for stock prices is:
<math>
\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}
</math>
or
<math>
\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}.
</math>
Therefore
<math>
\frac{\Delta S}{S} \sim N(\mu \Delta t, \sigma \sqrt{\Delta t}).
</math>
 
<math>S</math> Price of the stock.
<math>\Delta S</math> Change in the stock price.
<math>\Delta t</math> Small interval of time.
<math>\epsilon</math> Follows <math>N(0,1)</math>.

 

*Example: The current price of a stock is <math>S_0=\$100</math>. The expected return is <math>\mu=0.10</math> per year, and the standard deviation of the return is <math>\sigma=0.20</math> (also per year).

* Find an expression for the process of the stock. 
<math>
\frac{\Delta S}{S}=0.14 \Delta t + 0.20 \epsilon \sqrt{\Delta t}
</math>
* Find the distribution of the change in <math>S</math> divided by <math>S</math> at the end of the first year. That is, find the distribution of <math>\frac{\Delta S}{S}</math>. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \Delta t, 0.20 \sqrt{\Delta t}\right).
</math>
* Divide the year in weekly intervals and find the distribution of <math>\frac{\Delta S}{S}</math> at the end of each weekly interval. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \frac{1}{52}, 0.20 \sqrt{\frac{1}{52}}\right).
</math>
 
* Therefore, sampling from this distribution we can simulate the path of the stock. The price of the stock at the end of the first interval will be <math>S_1 = S_0 + \Delta S_1</math>, where <math>\Delta S_1</math> is the change during the first time interval, etc.
 
* Using the SOCR applet we will simulate the stock's path by dividing one year into small intervals each one of length <math>\frac{1}{100}</math> of a year, when <math>S_0=\$20</math>, annual mean and standard deviation: <math>\mu=0.14, \sigma=0.20</math>.
 

* The applet will select a random sample of 100 observations from <math>N(0,1)</math> and will compute 
<math>
\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 \epsilon \sqrt{0.01}.
</math>
Suppose that <math>\epsilon_1=0.58</math>. Then 
<math>
\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 (0.58) \sqrt{0.01}= 0.013 \Rightarrow \Delta S_1= 20(0.013)=0.26.
</math>
 
Therefore
<math>
\Delta S_1 = S_0 + \Delta S_1 = 20 + 0.26=20.26.
</math>
We continue in the same fashion until we reach the end of the year. Here is the SOCR applet.

<center>[[Image: Christou_stock_simulation.jpg|600px]]</center>

 
* The materials above was partially taken from 
''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and 
''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.

SOCR EduMaterials Activities ApplicationsActivities StockSimulation

2008-08-03T16:15:06Z

Nchristo:

SOCR EduMaterials Activities ApplicationsActivities StockSimulation

2008-08-03T16:14:23Z

Nchristo:

== A Model for Stock prices ==

* '''Description''': You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ .

* Process for Stock Prices: Assumed a drift rate equal to <math>\mu S</math> where <math>\mu</math> is the expected return of the stock, and variance <math>\sigma^2 S^2</math> where <math>\sigma^2</math> is the variance of the return of the stock. From Weiner process the model for stock prices is:
<math>
\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}
</math>
or
<math>
\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}.
</math>
Therefore
<math>
\frac{\Delta S}{S} \sim N(\mu \Delta t, \sigma \sqrt{\Delta t}).
</math>
 
<math>S</math> Price of the stock.
<math>\Delta S</math> Change in the stock price.
<math>\Delta t</math> Small interval of time.
<math>\epsilon</math> Follows <math>N(0,1)</math>.

 

*Example: The current price of a stock is <math>S_0=\$100</math>. The expected return is <math>\mu=0.10</math> per year, and the standard deviation of the return is <math>\sigma=0.20</math> (also per year).

* Find an expression for the process of the stock. 
<math>
\frac{\Delta S}{S}=0.14 \Delta t + 0.20 \epsilon \sqrt{\Delta t}
</math>
* Find the distribution of the change in <math>S</math> divided by <math>S</math> at the end of the first year. That is, find the distribution of <math>\frac{\Delta S}{S}</math>. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \Delta t, 0.20 \sqrt{\Delta t}\right).
</math>
* Divide the year in weekly intervals and find the distribution of <math>\frac{\Delta S}{S}</math> at the end of each weekly interval. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \frac{1}{52}, 0.20 \sqrt{\frac{1}{52}}\right).
</math>
 
* Therefore, sampling from this distribution we can simulate the path of the stock. The price of the stock at the end of the first interval will be <math>S_1 = S_0 + \Delta S_1</math>, where <math>\Delta S_1</math> is the change during the first time interval, etc.
 
* Using the SOCR applet we will simulate the stock's path by dividing one year into small intervals each one of length <math>\frac{1}{100}</math> of a year, when <math>S_0=\$20<math>, annual mean and standard deviation: <math>\mu=0.14, \sigma=0.20</math>.
 

* The applet will select a random sample of 100 observations from <math>N(0,1)</math> and will compute 
<math>
\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 \epsilon \sqrt{0.01}.
</math>
Suppose that <math>\epsilon_1=0.58</math>. Then 
<math>
\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 (0.58) \sqrt{0.01}= 0.013 \Rightarrow \Delta S_1= 20(0.013)=0.26.
</math>
 
Therefore
<math>
\Delta S_1 = S_0 + \Delta S_1 = 20 + 0.26=20.26.
</math>
We continue in the same fashion until we reach the end of the year. Here is the SOCR applet.

<center>[[Image: Christou_stock_simulation.jpg|600px]]</center>

SOCR EduMaterials Activities ApplicationsActivities StockSimulation

2008-08-03T16:13:13Z

Nchristo: New page: == A Model for Stock prices == * '''Description''': You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ . * Process for Stock Prices: Assumed a drift rate...

== A Model for Stock prices ==

* '''Description''': You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ .

* Process for Stock Prices: Assumed a drift rate equal to <math>\mu S</math> where <math>\mu</math> is the expected return of the stock, and variance <math>\sigma^2 S^2</math> where <math>\sigma^2</math> is the variance of the return of the stock. From Weiner process the model for stock prices is:
<math>
\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}
</math>
or
<math>
\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}.
</math>
Therefore
<math>
\frac{\Delta S}{S} \sim N(\mu \Delta t, \sigma \sqrt{\Delta t}).
</math>
 
<math>S</math> Price of the stock.
<math>\Delta S</math> Change in the stock price.
<math>\Delta t</math> Small interval of time.
<math>\epsilon</math> Follows <math>N(0,1)</math>.

 

*Example: The current price of a stock is <math>S_0=\$100</math>. The expected return is <math>\mu=0.10</math> per year, and the standard deviation of the return is <math>\sigma=0.20</math> (also per year).

* Find an expression for the process of the stock. 
<math>
\frac{\Delta S}{S}=0.14 \Delta t + 0.20 \epsilon \sqrt{\Delta t}
</math>
* Find the distribution of the change in <math>S<math> divided by <math>S<math> at the end of the first year. That is, find the distribution of <math>\frac{\Delta S}{S}</math>. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \Delta t, 0.20 \sqrt{\Delta t}\right).
</math>
* Divide the year in weekly intervals and find the distribution of <math>\frac{\Delta S}{S}</math> at the end of each weekly interval. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \frac{1}{52}, 0.20 \sqrt{\frac{1}{52}}\right).
</math>
 
* Therefore, sampling from this distribution we can simulate the path of the stock. The price of the stock at the end of the first interval will be <math>S_1 = S_0 + \Delta S_1</math>, where <math>\Delta S_1</math> is the change during the first time interval, etc.
 
* Using the SOCR applet we will simulate the stock's path by dividing one year into small intervals each one of length <math>\frac{1}{100}</math> of a year, when <math>S_0=\$20<math>, annual mean and standard deviation: <math>\mu=0.14, \sigma=0.20</math>.
 

* The applet will select a random sample of 100 observations from <math>N(0,1)</math> and will compute 
<math>
\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 \epsilon \sqrt{0.01}.
</math>
Suppose that <math>\epsilon_1=0.58</math>. Then 
<math>
\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 (0.58) \sqrt{0.01}= 0.013 \Rightarrow \Delta S_1= 20(0.013)=0.26.
</math>
 
Therefore
<math>
\Delta S_1 = S_0 + \Delta S_1 = 20 + 0.26=20.26.
</math>
We continue in the same fashion until we reach the end of the year.

<center>[[Image: Christou_stock_simulation.jpg|600px]]</center>

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T16:02:34Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* '''Description''': You can access the Black-Scholes applet at http://www.socr.ucla.edu/htmls/app/ .

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

 

<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center>

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T16:01:58Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* '''Description''': You can access the Black-Scholes applet at http://www.socr.ucla.edu/htmls/app/ .

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

 

<center>[[Image: Christou_black_scholes.jpg|600px]]</center>

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T15:58:53Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T15:58:26Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>sigma=0.30 </math>,
<math>\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T15:57:07Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30</math>, \ E=\$29,\ R_f=0.05, \sigma=0.30,\
\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T15:56:08Z

Nchristo:

== Black-Scholes option pricing model - Convergence of binomial ==

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n</math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n</math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\
\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

2008-08-03T15:55:09Z

Nchristo: New page: == Black-Scholes option pricing model - Convergence of binomial == * Black-Scholes option pricing formula: The value <math>C<math> of a European call option at time <math>t=0</math> ...

== Black-Scholes option pricing model - Convergence of binomial ==

* Black-Scholes option pricing formula: 
The value <math>C<math> of a European call option at time <math>t=0</math> is:
<math>
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
</math>
 
<math>
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}
</math>
 
<math>
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
</math>
 
Where, 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Continuously compounded risk-free interest 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t<math> Time to expiration in years 
<math>\Phi(d_i)</math> Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math> 

* Binomial convergence to Black-Scholes option pricing formula: 
The binomial formula converges to the Black-Scholes formula when
the number of periods <math>n<math> is large. In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods <math>n<math>. The value of the
call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are:
<math>S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\
\mbox{Days to expiration}=40</math>. 
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).

* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.

File:Christou black scholes binomial.jpg

2008-08-03T15:48:31Z

Nchristo:

File:Christou binomial applet2.jpg

2008-08-03T15:47:24Z

Nchristo:

File:Christou binomial applet1.jpg

2008-08-03T15:46:28Z

Nchristo:

File:Christou stock simulation.jpg

2008-08-03T15:44:02Z

Nchristo:

SOCR EduMaterials Activities ApplicationsActivities Portfolio

2008-08-03T15:41:24Z

Nchristo: /* Portfolio Theory */

== Portfolio Theory ==

* '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ .

* An investor has a certain amount of dollars to invest into two stocks
(<math>IBM</math> and <math>TEXACO</math>). A portion of the available funds will be invested into
IBM (denote this portion of the funds with <math>x_A</math>) and the remaining funds
into TEXACO (denote it with <math>x_B</math>) - so <math>x_A+x_B=1</math>. The resulting portfolio
will be <math>x_A R_A+x_B R_B</math>, where <math>R_A</math> is the monthly return of <math>IBM</math> and <math>R_B</math> is the
monthly return of <math>TEXACO</math>. The goal here is to
find the most efficient portfolios given a certain amount of risk.
Using market data from January 1980 until February 2001 we compute
that <math>E(R_A)=0.010</math>, <math>E(R_B)=0.013</math>, <math>Var(R_A)=0.0061</math>, <math>Var(R_B)=0.0046</math>, and
<math>Cov(R_A,R_B)=0.00062</math>. We first want to minimize the variance of the portfolio.
This will be:
<math>
\mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
Or
 
<math>
\mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
 
* Therefore our goal is to find <math>x_A</math> and <math>x_B</math>, the percentage of the
available funds that will be invested in each stock. Substituting
<math>x_B=1-x_A</math> into the equation of the variance we get
<math>
x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B).
</math>
 
* To minimize the above exression we take the derivative with respect to
<math>x_A</math>, set it equal to zero and solve for <math>x_A</math>. The result is:
<math>
x_A=\frac{Var(R_B) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
and therefore
 
<math>
x_B=\frac{Var(R_A) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
The values of <math>x_A</math> and <math>x_B</math> are:
<math>
x_a=\frac{0.0046-0.0062}{0.0061+0.0046-2(0.00062)} \Rightarrow x_A=0.42.
</math>
and <math>x_B=1-x_A=1-0.42 \Rightarrow x_B=0.58</math>. Therefore if the investor invests
<math>42 \%</math> of the available funds into <math>IBM</math> and the remaining <math>58 \%</math>
into <math>TEXACO</math> the variance of the portfolio will be minimum and equal to:
<math>
Var(0.42R_A+0.58R_B)=0.42^2(0.0061)+0.58^2(0.0046)+2(0.42)(0.58)(0.00062)
=0.002926.
</math>
The corresponding expected return of this porfolio will be:
<math>
E(0.42R_A+0.58R_B)=0.42(0.010)+0.58(0.013)=0.01174.
</math>
* We can try many other combinations of <math>x_A</math> and <math>x_B</math> (but always <math>x_A+x_B=1</math>)
and compute the risk and return for each resulting portfolio. This is
shown in the table and the graph below.

{| class="wikitable" border="1"
|-
! <math>x_A</math>
! <math>x_B</math>
! Risk (<math>\sigma^2</math>)
! Return
! Risk (<math>\sigma</math>)
|-
| 1.00
| 0.00
| 0.006100
| 0.01000
| 0.078102
|-
| 0.95
| 0.05
| 0.005576
| 0.01015
| 0.074670
|-
| 0.90
| 0.10
| 0.005099
| 0.01030
| 0.071404
|-
| 0.85
| 0.15
| 0.004669
| 0.01045
| 0.068329
|-
| 0.80
| 0.20
| 0.004286
| 0.01060
| 0.065471
|-
| 0.75
| 0.25
| 0.003951
| 0.01075
| 0.062859
|-
| 0.70
| 0.30
| 0.003663
| 0.01090
| 0.060526
|-
| 0.65
| 0.35
| 0.003423
| 0.01105
| 0.058505
|-
| 0.60
| 0.40
| 0.003230
| 0.01120
| 0.056830
|-
| 0.55
| 0.45
| 0.003084
| 0.01135
| 0.055531
|-
| 0.50
| 0.50
| 0.002985
| 0.01150
| 0.054635
|-
| 0.42
| 0.58
| 0.002926
| 0.01174
| 0.054088
|-
| 0.40
| 0.60
| 0.002930
| 0.01180
| 0.054126
|-
| 0.35
| 0.65
| 0.002973
| 0.01195
| 0.054524
|-
| 0.30
| 0.70
| 0.003063
| 0.01210
| 0.055348
|-
| 0.25
| 0.75
| 0.003201
| 0.01225
| 0.056580
|-
| 0.20
| 0.80
| 0.003386
| 0.01240
| 0.058193
|-
| 0.15
| 0.85
| 0.003619
| 0.01255
| 0.060157
|-
| 0.10
| 0.90
| 0.003899
| 0.01270
| 0.062439
|-
| 0.05
| 0.95
| 0.004226
| 0.01285
| 0.065005
|-
| 0.00
| 1.00
| 0.004600
| 0.01300
| 0.067823
|}

<center>[[Image: Christou_two_stocks_portfolio.jpg|600px]]</center>

 
* For the above calculations short selling was not allowed (<math>0 \le x_A \le 1</math> and
<math>0 \le x_B \le 1</math>, in addition to <math>x_A+x_B=1</math>). We note here that the efficient portfolios are located on the top part of the graph between the minimum risk portfolio point and the maximum return portfolio point, which is called the efficient frontier (the blue portion of the graph). Efficient portfolios should provide higher expected return for the same level of risk or lower risk for the same level of expected return. 

* If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities. The investor can have very large expected return but this will be associated with very large risk. The constraint here is only <math>x_A+x_B=1</math>, since either <math>x_A<math> or <math>x_B</math> can be negative. The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks. The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed. 

<center>[[Image: Christou_ibm_texaco_short_sales.jpg|600px]]</center>

* When the investor faces the efficient frontier when short sales are allowed and he or she can lend or borrow at the risk-free interest rate the efficient frontier will change in the following way: Let <math>x</math> be the portion of the investor's wealth invested in portfolio <math>A</math> that lies on the efficient frontier, and <math>1-x</math> the the portion invested in a risk-free asset. This combination is a new portfolio and has
<math>
\bar R_p=x\bar R_A + (1-x)R_f
</math>
 
where <math>R_f</math> is the return of the risk-free asset. The variance of this combination is simply
<math>
\sigma_p^2=x^2 \sigma_A^2 \Rightarrow x=\frac{\sigma_p}{\sigma_A}
</math>
 
From the last two equations we get
<math>
\bar R_p = R_f + \left(\frac{\bar R_A-R_f}{\sigma_A}\right)\sigma_p
</math>

 

* This is an equation of a straight line. On this line we find all the possible combinations of portfolio <math>A</math> and the risk-free rate. Another investor can choose to combine the risk-free rate with portfolio <math>B</math> or portfolio <math>C</math>. Clearly, for the same level risk the combinations that lie on the <math>Rf-B</math> line have higher expected return than those on the line <math>Rf-A</math> (see figure below). And <math>Rf-C</math> will produce combinations that have higher return than those on <math>Rf-B</math> for the same level of risk, etc. 

<center>[[Image: Christou_portfolio_risk_free_asset.jpg|600px]]</center>
 

* The solution, therefore, is to find the point of tangency of this line to the efficient frontier. Let's call this point <math>G</math>. To find this point we want to maximize the slope of the line in (1) as follows:
<math>
\mbox{max} \ \ \theta = \frac{\bar R_p - R_f}{\sigma_p}
</math>
Subject to
<math>
\sum_{i=1}^{n} x_i = 1
</math>
Since,
<math>
R_f=\left(\sum_{i=1}^n x_i\right) R_f = \sum_{i=1}^n x_iR_f
</math>
* We can write the maximization problem as
<math>
\mbox{max} \ \ \theta=\frac{\sum_{i=1}^n x_i (\bar R_i - R_f)}
{\left(\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right)^{\frac{1}{2}}}
</math>
 
or
 
<math>
\mbox{max} \ \ \theta=\left[\sum_{i=1}^n x_i (\bar R_i - R_f)\right]
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}}
</math>
 
* Take now the partial derivative with respect to each <math>x_i, i=1, \cdots, n</math>, set them equal to zero and solve. Let's find the partial derivative w.r.t. <math>x_k</math>:
<math>
\frac{\partial \theta}{\partial x_k} =
(\bar R_k - R_f)\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}} +
\left[\sum_{i=1}^n x_i(\bar R_i - R_f)\right]
\left[2x_k\sigma_k^2 + 2 \sum_{j=1, j \ne k}^n x_j \sigma_{kj}\right] \times
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{3}{2}} \times (-\frac{1}{2}) = 0
</math>
 
* Multiply both sides by
<math>
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{\frac{1}{2}} \ \ \ \mbox{to get}
</math>
<math>
(\bar R_k - R_f) -
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
(x_k \sigma_k^2 +\sum_{j=1, j \ne k}^n x_j \sigma_{kj}) =0
</math>
 
* Now, if we let
<math>
\lambda=
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
</math>
 
the previous expression will be
<math>
(\bar R_k - R_f) - \lambda x_k \sigma_k^2 - \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj} = 0
</math>
 
or
 
<math>
\bar R_k - R_f = \lambda x_k \sigma_k^2 + \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj}
</math>
 
* Let's define now a new variable,
<math>
z_k = \lambda x_k
</math>
 
and finally
 
<math>
\bar R_k - R_f = z_k \sigma_k^2 + \sum_{j=1, j \ne k}^n z_j \sigma_{kj}
</math>
* We have one equation like (2) for each <math>i=1, \cdots, n</math>. Here they are:
<math>
\bar R_1 - R_f = z_1 \sigma_1^2 + z_2 \sigma_{12} + z_3 \sigma_{13} + \cdots + z_n \sigma_{1n}
</math>
 
<math>
\bar R_2 - R_f = z_1 \sigma_{21} + z_2 \sigma_2^2 + z_3 \sigma_{23} + \cdots + z_n \sigma_{2n}
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\bar R_n - R_f = z_1 \sigma_{n1} + z_2 \sigma_{n2} + z_3 \sigma_{n3} + \cdots + z_n \sigma_n^2
</math>
 
* The solution involves solving the system of these simultaneous equations, which can be written in matrix form as:
<math>
\bar R = \Sigma Z
</math>
where <math>\Sigma</math> is the variance-covariance matrix of the returns of the <math>n</math> stocks. To solve for <math> Z</math>:
<math>
{Z} = \Sigma^{-1} \bar R
</math>
 
* Once we find the <math>z_i's</math> it is easy to find the <math>x_i's</math> (the fraction of funds to be invested in each security). Earlier we defined
<math>
z_k = \lambda x_k \Rightarrow x_k = \frac{z_k}{\lambda}
</math>
* We need to find <math>\lambda</math> as follows:
<math>
z_1 + z_2 + \cdots + z_n = \sum_{i=1}^n z_i
</math>
 
<math>
\lambda(x_1 + x_2 + \cdots + x_3) = \sum_{i=1}^n z_i
</math>
 
<math>
\Rightarrow \lambda = \sum_{i=1}^n z_i
</math>
 
* Therefore, 
<math>
x_1 = \frac{z_1}{\lambda}
</math>
 
<math>
x_2 = \frac{z_2}{\lambda}
</math>
 
<math>
x_3 = \frac{z_3}{\lambda}
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
x_n = \frac{z_n}{\lambda}
</math>
 

* The snapshot form the SOCR portfolio applet shows an example with 5 stocks. Again, the red points in the applet correspond to the case when short sales are not allowed. The point of tangency can be found with a choice of the risk-free rate that can be entered in the input dialog box.

<center>[[Image: Christou_tangent_point_5_stocks.jpg|600px]]</center>
 

 
* The materials above was partially taken from ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.

File:Christou tangent point 5 stocks.jpg

2008-08-03T15:41:06Z

Nchristo:

File:Christou two stocks portfolio.jpg

2008-08-03T15:40:06Z

Nchristo:

File:Christou ibm texaco short sales.jpg

2008-08-03T15:39:19Z

Nchristo:

File:Christou portfolio risk free asset.jpg

2008-08-03T15:38:25Z

Nchristo:

SOCR EduMaterials Activities ApplicationsActivities Portfolio

2008-08-03T15:35:39Z

Nchristo: /* Portfolio Theory */

== Portfolio Theory ==

* '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ .

* An investor has a certain amount of dollars to invest into two stocks
(<math>IBM</math> and <math>TEXACO</math>). A portion of the available funds will be invested into
IBM (denote this portion of the funds with <math>x_A</math>) and the remaining funds
into TEXACO (denote it with <math>x_B</math>) - so <math>x_A+x_B=1</math>. The resulting portfolio
will be <math>x_A R_A+x_B R_B</math>, where <math>R_A</math> is the monthly return of <math>IBM</math> and <math>R_B</math> is the
monthly return of <math>TEXACO</math>. The goal here is to
find the most efficient portfolios given a certain amount of risk.
Using market data from January 1980 until February 2001 we compute
that <math>E(R_A)=0.010</math>, <math>E(R_B)=0.013</math>, <math>Var(R_A)=0.0061</math>, <math>Var(R_B)=0.0046</math>, and
<math>Cov(R_A,R_B)=0.00062</math>. We first want to minimize the variance of the portfolio.
This will be:
<math>
\mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
Or
 
<math>
\mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
 
* Therefore our goal is to find <math>x_A</math> and <math>x_B</math>, the percentage of the
available funds that will be invested in each stock. Substituting
<math>x_B=1-x_A</math> into the equation of the variance we get
<math>
x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B).
</math>
 
* To minimize the above exression we take the derivative with respect to
<math>x_A</math>, set it equal to zero and solve for <math>x_A</math>. The result is:
<math>
x_A=\frac{Var(R_B) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
and therefore
 
<math>
x_B=\frac{Var(R_A) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
The values of <math>x_A</math> and <math>x_B</math> are:
<math>
x_a=\frac{0.0046-0.0062}{0.0061+0.0046-2(0.00062)} \Rightarrow x_A=0.42.
</math>
and <math>x_B=1-x_A=1-0.42 \Rightarrow x_B=0.58</math>. Therefore if the investor invests
<math>42 \%</math> of the available funds into <math>IBM</math> and the remaining <math>58 \%</math>
into <math>TEXACO</math> the variance of the portfolio will be minimum and equal to:
<math>
Var(0.42R_A+0.58R_B)=0.42^2(0.0061)+0.58^2(0.0046)+2(0.42)(0.58)(0.00062)
=0.002926.
</math>
The corresponding expected return of this porfolio will be:
<math>
E(0.42R_A+0.58R_B)=0.42(0.010)+0.58(0.013)=0.01174.
</math>
* We can try many other combinations of <math>x_A</math> and <math>x_B</math> (but always <math>x_A+x_B=1</math>)
and compute the risk and return for each resulting portfolio. This is
shown in the table and the graph below.

{| class="wikitable" border="1"
|-
! <math>x_A</math>
! <math>x_B</math>
! Risk (<math>\sigma^2</math>)
! Return
! Risk (<math>\sigma</math>)
|-
| 1.00
| 0.00
| 0.006100
| 0.01000
| 0.078102
|-
| 0.95
| 0.05
| 0.005576
| 0.01015
| 0.074670
|-
| 0.90
| 0.10
| 0.005099
| 0.01030
| 0.071404
|-
| 0.85
| 0.15
| 0.004669
| 0.01045
| 0.068329
|-
| 0.80
| 0.20
| 0.004286
| 0.01060
| 0.065471
|-
| 0.75
| 0.25
| 0.003951
| 0.01075
| 0.062859
|-
| 0.70
| 0.30
| 0.003663
| 0.01090
| 0.060526
|-
| 0.65
| 0.35
| 0.003423
| 0.01105
| 0.058505
|-
| 0.60
| 0.40
| 0.003230
| 0.01120
| 0.056830
|-
| 0.55
| 0.45
| 0.003084
| 0.01135
| 0.055531
|-
| 0.50
| 0.50
| 0.002985
| 0.01150
| 0.054635
|-
| 0.42
| 0.58
| 0.002926
| 0.01174
| 0.054088
|-
| 0.40
| 0.60
| 0.002930
| 0.01180
| 0.054126
|-
| 0.35
| 0.65
| 0.002973
| 0.01195
| 0.054524
|-
| 0.30
| 0.70
| 0.003063
| 0.01210
| 0.055348
|-
| 0.25
| 0.75
| 0.003201
| 0.01225
| 0.056580
|-
| 0.20
| 0.80
| 0.003386
| 0.01240
| 0.058193
|-
| 0.15
| 0.85
| 0.003619
| 0.01255
| 0.060157
|-
| 0.10
| 0.90
| 0.003899
| 0.01270
| 0.062439
|-
| 0.05
| 0.95
| 0.004226
| 0.01285
| 0.065005
|-
| 0.00
| 1.00
| 0.004600
| 0.01300
| 0.067823
|}

<center>[[Image: Two_stocks_portfolio.jpg|600px]]</center>

 
* For the above calculations short selling was not allowed (<math>0 \le x_A \le 1</math> and
<math>0 \le x_B \le 1</math>, in addition to <math>x_A+x_B=1</math>). We note here that the efficient portfolios are located on the top part of the graph between the minimum risk portfolio point and the maximum return portfolio point, which is called the efficient frontier (the blue portion of the graph). Efficient portfolios should provide higher expected return for the same level of risk or lower risk for the same level of expected return. 

* If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities. The investor can have very large expected return but this will be associated with very large risk. The constraint here is only <math>x_A+x_B=1</math>, since either <math>x_A<math> or <math>x_B</math> can be negative. The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks. The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed. 

<center>[[Image: Ibm_texaco_short_sales.jpg|600px]]</center>

* When the investor faces the efficient frontier when short sales are allowed and he or she can lend or borrow at the risk-free interest rate the efficient frontier will change in the following way: Let <math>x</math> be the portion of the investor's wealth invested in portfolio <math>A</math> that lies on the efficient frontier, and <math>1-x</math> the the portion invested in a risk-free asset. This combination is a new portfolio and has
<math>
\bar R_p=x\bar R_A + (1-x)R_f
</math>
 
where <math>R_f</math> is the return of the risk-free asset. The variance of this combination is simply
<math>
\sigma_p^2=x^2 \sigma_A^2 \Rightarrow x=\frac{\sigma_p}{\sigma_A}
</math>
 
From the last two equations we get
<math>
\bar R_p = R_f + \left(\frac{\bar R_A-R_f}{\sigma_A}\right)\sigma_p
</math>

 

* This is an equation of a straight line. On this line we find all the possible combinations of portfolio <math>A</math> and the risk-free rate. Another investor can choose to combine the risk-free rate with portfolio <math>B</math> or portfolio <math>C</math>. Clearly, for the same level risk the combinations that lie on the <math>Rf-B</math> line have higher expected return than those on the line <math>Rf-A</math> (see figure below). And <math>Rf-C</math> will produce combinations that have higher return than those on <math>Rf-B</math> for the same level of risk, etc. 

<center>[[Image: Portfolio_risk_free_asset.jpg|600px]]</center>
 

* The solution, therefore, is to find the point of tangency of this line to the efficient frontier. Let's call this point <math>G</math>. To find this point we want to maximize the slope of the line in (1) as follows:
<math>
\mbox{max} \ \ \theta = \frac{\bar R_p - R_f}{\sigma_p}
</math>
Subject to
<math>
\sum_{i=1}^{n} x_i = 1
</math>
Since,
<math>
R_f=\left(\sum_{i=1}^n x_i\right) R_f = \sum_{i=1}^n x_iR_f
</math>
* We can write the maximization problem as
<math>
\mbox{max} \ \ \theta=\frac{\sum_{i=1}^n x_i (\bar R_i - R_f)}
{\left(\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right)^{\frac{1}{2}}}
</math>
 
or
 
<math>
\mbox{max} \ \ \theta=\left[\sum_{i=1}^n x_i (\bar R_i - R_f)\right]
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}}
</math>
 
* Take now the partial derivative with respect to each <math>x_i, i=1, \cdots, n</math>, set them equal to zero and solve. Let's find the partial derivative w.r.t. <math>x_k</math>:
<math>
\frac{\partial \theta}{\partial x_k} =
(\bar R_k - R_f)\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}} +
\left[\sum_{i=1}^n x_i(\bar R_i - R_f)\right]
\left[2x_k\sigma_k^2 + 2 \sum_{j=1, j \ne k}^n x_j \sigma_{kj}\right] \times
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{3}{2}} \times (-\frac{1}{2}) = 0
</math>
 
* Multiply both sides by
<math>
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{\frac{1}{2}} \ \ \ \mbox{to get}
</math>
<math>
(\bar R_k - R_f) -
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
(x_k \sigma_k^2 +\sum_{j=1, j \ne k}^n x_j \sigma_{kj}) =0
</math>
 
* Now, if we let
<math>
\lambda=
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
</math>
 
the previous expression will be
<math>
(\bar R_k - R_f) - \lambda x_k \sigma_k^2 - \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj} = 0
</math>
 
or
 
<math>
\bar R_k - R_f = \lambda x_k \sigma_k^2 + \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj}
</math>
 
* Let's define now a new variable,
<math>
z_k = \lambda x_k
</math>
 
and finally
 
<math>
\bar R_k - R_f = z_k \sigma_k^2 + \sum_{j=1, j \ne k}^n z_j \sigma_{kj}
</math>
* We have one equation like (2) for each <math>i=1, \cdots, n</math>. Here they are:
<math>
\bar R_1 - R_f = z_1 \sigma_1^2 + z_2 \sigma_{12} + z_3 \sigma_{13} + \cdots + z_n \sigma_{1n}
</math>
 
<math>
\bar R_2 - R_f = z_1 \sigma_{21} + z_2 \sigma_2^2 + z_3 \sigma_{23} + \cdots + z_n \sigma_{2n}
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\bar R_n - R_f = z_1 \sigma_{n1} + z_2 \sigma_{n2} + z_3 \sigma_{n3} + \cdots + z_n \sigma_n^2
</math>
 
* The solution involves solving the system of these simultaneous equations, which can be written in matrix form as:
<math>
\bar R = \Sigma Z
</math>
where <math>\Sigma</math> is the variance-covariance matrix of the returns of the <math>n</math> stocks. To solve for <math> Z</math>:
<math>
{Z} = \Sigma^{-1} \bar R
</math>
 
* Once we find the <math>z_i's</math> it is easy to find the <math>x_i's</math> (the fraction of funds to be invested in each security). Earlier we defined
<math>
z_k = \lambda x_k \Rightarrow x_k = \frac{z_k}{\lambda}
</math>
* We need to find <math>\lambda</math> as follows:
<math>
z_1 + z_2 + \cdots + z_n = \sum_{i=1}^n z_i
</math>
 
<math>
\lambda(x_1 + x_2 + \cdots + x_3) = \sum_{i=1}^n z_i
</math>
 
<math>
\Rightarrow \lambda = \sum_{i=1}^n z_i
</math>
 
* Therefore, 
<math>
x_1 = \frac{z_1}{\lambda}
</math>
 
<math>
x_2 = \frac{z_2}{\lambda}
</math>
 
<math>
x_3 = \frac{z_3}{\lambda}
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
x_n = \frac{z_n}{\lambda}
</math>
 

* The snapshot form the SOCR portfolio applet shows an example with 5 stocks. Again, the red points in the applet correspond to the case when short sales are not allowed. The point of tangency can be found with a choice of the risk-free rate that can be entered in the input dialog box.

<center>[[Image: Tangent_point_5_stocks.jpg|600px]]</center>
 

 
* The materials above was partially taken from ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.

SOCR EduMaterials Activities ApplicationsActivities Portfolio

2008-08-03T15:34:04Z

Nchristo: /* Portfolio Theory */

== Portfolio Theory ==

* '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ .

* An investor has a certain amount of dollars to invest into two stocks
(<math>IBM</math> and <math>TEXACO</math>). A portion of the available funds will be invested into
IBM (denote this portion of the funds with <math>x_A</math>) and the remaining funds
into TEXACO (denote it with <math>x_B</math>) - so <math>x_A+x_B=1</math>. The resulting portfolio
will be <math>x_A R_A+x_B R_B</math>, where <math>R_A</math> is the monthly return of <math>IBM</math> and <math>R_B</math> is the
monthly return of <math>TEXACO</math>. The goal here is to
find the most efficient portfolios given a certain amount of risk.
Using market data from January 1980 until February 2001 we compute
that <math>E(R_A)=0.010</math>, <math>E(R_B)=0.013</math>, <math>Var(R_A)=0.0061</math>, <math>Var(R_B)=0.0046</math>, and
<math>Cov(R_A,R_B)=0.00062</math>. We first want to minimize the variance of the portfolio.
This will be:
<math>
\mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
Or
 
<math>
\mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
 
* Therefore our goal is to find <math>x_A</math> and <math>x_B</math>, the percentage of the
available funds that will be invested in each stock. Substituting
<math>x_B=1-x_A</math> into the equation of the variance we get
<math>
x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B).
</math>
 
* To minimize the above exression we take the derivative with respect to
<math>x_A</math>, set it equal to zero and solve for <math>x_A</math>. The result is:
<math>
x_A=\frac{Var(R_B) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
and therefore
 
<math>
x_B=\frac{Var(R_A) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
The values of <math>x_A</math> and <math>x_B</math> are:
<math>
x_a=\frac{0.0046-0.0062}{0.0061+0.0046-2(0.00062)} \Rightarrow x_A=0.42.
</math>
and <math>x_B=1-x_A=1-0.42 \Rightarrow x_B=0.58</math>. Therefore if the investor invests
<math>42 \%</math> of the available funds into <math>IBM</math> and the remaining <math>58 \%</math>
into <math>TEXACO</math> the variance of the portfolio will be minimum and equal to:
<math>
Var(0.42R_A+0.58R_B)=0.42^2(0.0061)+0.58^2(0.0046)+2(0.42)(0.58)(0.00062)
=0.002926.
</math>
The corresponding expected return of this porfolio will be:
<math>
E(0.42R_A+0.58R_B)=0.42(0.010)+0.58(0.013)=0.01174.
</math>
* We can try many other combinations of <math>x_A</math> and <math>x_B</math> (but always <math>x_A+x_B=1</math>)
and compute the risk and return for each resulting portfolio. This is
shown in the table and the graph below.

{| class="wikitable" border="1"
|-
! <math>x_A</math>
! <math>x_B</math>
! Risk (<math>\sigma^2</math>)
! Return
! Risk (<math>\sigma</math>)
|-
| 1.00
| 0.00
| 0.006100
| 0.01000
| 0.078102
|-
| 0.95
| 0.05
| 0.005576
| 0.01015
| 0.074670
|-
| 0.90
| 0.10
| 0.005099
| 0.01030
| 0.071404
|-
| 0.85
| 0.15
| 0.004669
| 0.01045
| 0.068329
|-
| 0.80
| 0.20
| 0.004286
| 0.01060
| 0.065471
|-
| 0.75
| 0.25
| 0.003951
| 0.01075
| 0.062859
|-
| 0.70
| 0.30
| 0.003663
| 0.01090
| 0.060526
|-
| 0.65
| 0.35
| 0.003423
| 0.01105
| 0.058505
|-
| 0.60
| 0.40
| 0.003230
| 0.01120
| 0.056830
|-
| 0.55
| 0.45
| 0.003084
| 0.01135
| 0.055531
|-
| 0.50
| 0.50
| 0.002985
| 0.01150
| 0.054635
|-
| 0.42
| 0.58
| 0.002926
| 0.01174
| 0.054088
|-
| 0.40
| 0.60
| 0.002930
| 0.01180
| 0.054126
|-
| 0.35
| 0.65
| 0.002973
| 0.01195
| 0.054524
|-
| 0.30
| 0.70
| 0.003063
| 0.01210
| 0.055348
|-
| 0.25
| 0.75
| 0.003201
| 0.01225
| 0.056580
|-
| 0.20
| 0.80
| 0.003386
| 0.01240
| 0.058193
|-
| 0.15
| 0.85
| 0.003619
| 0.01255
| 0.060157
|-
| 0.10
| 0.90
| 0.003899
| 0.01270
| 0.062439
|-
| 0.05
| 0.95
| 0.004226
| 0.01285
| 0.065005
|-
| 0.00
| 1.00
| 0.004600
| 0.01300
| 0.067823
|}

<center>[[Image: Two_stocks_portfolio.jpg|600px]]</center>

 
* For the above calculations short selling was not allowed (<math>0 \le x_A \le 1</math> and
<math>0 \le x_B \le 1</math>, in addition to <math>x_A+x_B=1</math>). We note here that the efficient portfolios are located on the top part of the graph between the minimum risk portfolio point and the maximum return portfolio point, which is called the efficient frontier (the blue portion of the graph). Efficient portfolios should provide higher expected return for the same level of risk or lower risk for the same level of expected return. 

* If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities. The investor can have very large expected return but this will be associated with very large risk. The constraint here is only <math>x_A+x_B=1</math>, since either <math>x_A<math> or <math>x_B</math> can be negative. The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks. The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed. 

<center>[[Image: Ibm_texaco_short_sales.jpg|600px]]</center>

* When the investor faces the efficient frontier when short sales are allowed and he or she can lend or borrow at the risk-free interest rate the efficient frontier will change in the following way: Let <math>x</math> be the portion of the investor's wealth invested in portfolio <math>A</math> that lies on the efficient frontier, and <math>1-x</math> the the portion invested in a risk-free asset. This combination is a new portfolio and has
<math>
\bar R_p=x\bar R_A + (1-x)R_f
</math>
 
where <math>R_f</math> is the return of the risk-free asset. The variance of this combination is simply
<math>
\sigma_p^2=x^2 \sigma_A^2 \Rightarrow x=\frac{\sigma_p}{\sigma_A}
</math>
 
From the last two equations we get
<math>
\bar R_p = R_f + \left(\frac{\bar R_A-R_f}{\sigma_A}\right)\sigma_p
</math>

 

* This is an equation of a straight line. On this line we find all the possible combinations of portfolio <math>A</math> and the risk-free rate. Another investor can choose to combine the risk-free rate with portfolio <math>B</math> or portfolio <math>C</math>. Clearly, for the same level risk the combinations that lie on the <math>Rf-B</math> line have higher expected return than those on the line <math>Rf-A</math> (see figure below). And <math>Rf-C</math> will produce combinations that have higher return than those on <math>Rf-B</math> for the same level of risk, etc. 

<center>[[Image: Portfolio_risk_free_asset.jpg|600px]]</center>>
 

* The solution, therefore, is to find the point of tangency of this line to the efficient frontier. Let's call this point <math>G</math>. To find this point we want to maximize the slope of the line in (1) as follows:
<math>
\mbox{max} \ \ \theta = \frac{\bar R_p - R_f}{\sigma_p}
</math>
Subject to
<math>
\sum_{i=1}^{n} x_i = 1
</math>
Since,
<math>
R_f=\left(\sum_{i=1}^n x_i\right) R_f = \sum_{i=1}^n x_iR_f
</math>
* We can write the maximization problem as
<math>
\mbox{max} \ \ \theta=\frac{\sum_{i=1}^n x_i (\bar R_i - R_f)}
{\left(\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right)^{\frac{1}{2}}}
</math>
 
or
 
<math>
\mbox{max} \ \ \theta=\left[\sum_{i=1}^n x_i (\bar R_i - R_f)\right]
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}}
</math>
 
* Take now the partial derivative with respect to each <math>x_i, i=1, \cdots, n</math>, set them equal to zero and solve. Let's find the partial derivative w.r.t. <math>x_k</math>:
<math>
\frac{\partial \theta}{\partial x_k} =
(\bar R_k - R_f)\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}} +
\left[\sum_{i=1}^n x_i(\bar R_i - R_f)\right]
\left[2x_k\sigma_k^2 + 2 \sum_{j=1, j \ne k}^n x_j \sigma_{kj}\right] \times
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{3}{2}} \times (-\frac{1}{2}) = 0
</math>
 
* Multiply both sides by
<math>
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{\frac{1}{2}} \ \ \ \mbox{to get}
</math>
<math>
(\bar R_k - R_f) -
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
(x_k \sigma_k^2 +\sum_{j=1, j \ne k}^n x_j \sigma_{kj}) =0
</math>
 
* Now, if we let
<math>
\lambda=
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
</math>
 
the previous expression will be
<math>
(\bar R_k - R_f) - \lambda x_k \sigma_k^2 - \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj} = 0
</math>
 
or
 
<math>
\bar R_k - R_f = \lambda x_k \sigma_k^2 + \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj}
</math>
 
* Let's define now a new variable,
<math>
z_k = \lambda x_k
</math>
 
and finally
 
<math>
\bar R_k - R_f = z_k \sigma_k^2 + \sum_{j=1, j \ne k}^n z_j \sigma_{kj}
</math>
* We have one equation like (2) for each <math>i=1, \cdots, n</math>. Here they are:
<math>
\bar R_1 - R_f = z_1 \sigma_1^2 + z_2 \sigma_{12} + z_3 \sigma_{13} + \cdots + z_n \sigma_{1n}
</math>
 
<math>
\bar R_2 - R_f = z_1 \sigma_{21} + z_2 \sigma_2^2 + z_3 \sigma_{23} + \cdots + z_n \sigma_{2n}
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\bar R_n - R_f = z_1 \sigma_{n1} + z_2 \sigma_{n2} + z_3 \sigma_{n3} + \cdots + z_n \sigma_n^2
</math>
 
* The solution involves solving the system of these simultaneous equations, which can be written in matrix form as:
<math>
\bar R = \Sigma Z
</math>
where <math>\Sigma</math> is the variance-covariance matrix of the returns of the <math>n</math> stocks. To solve for <math> Z</math>:
<math>
{Z} = \Sigma^{-1} \bar R
</math>
 
* Once we find the <math>z_i's</math> it is easy to find the <math>x_i's</math> (the fraction of funds to be invested in each security). Earlier we defined
<math>
z_k = \lambda x_k \Rightarrow x_k = \frac{z_k}{\lambda}
</math>
* We need to find <math>\lambda</math> as follows:
<math>
z_1 + z_2 + \cdots + z_n = \sum_{i=1}^n z_i
</math>
 
<math>
\lambda(x_1 + x_2 + \cdots + x_3) = \sum_{i=1}^n z_i
</math>
 
<math>
\Rightarrow \lambda = \sum_{i=1}^n z_i
</math>
 
* Therefore, 
<math>
x_1 = \frac{z_1}{\lambda}
</math>
 
<math>
x_2 = \frac{z_2}{\lambda}
</math>
 
<math>
x_3 = \frac{z_3}{\lambda}
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
x_n = \frac{z_n}{\lambda}
</math>
 

* The snapshot form the SOCR portfolio applet shows an example with 5 stocks. Again, the red points in the applet correspond to the case when short sales are not allowed. The point of tangency can be found with a choice of the risk-free rate that can be entered in the input dialog box.

<center>[[Image: Tangent_point_5_stocks.jpg|600px]]</center>
 

 
* The materials above was partially taken from ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.

SOCR EduMaterials Activities ApplicationsActivities TradingOptions

2008-08-03T15:29:53Z

Nchristo: /* Options */

SOCR EduMaterials Activities ApplicationsActivities Portfolio

2008-08-03T15:28:11Z

Nchristo: /* Portfolio Theory */

== Portfolio Theory ==

* '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ .

An investor has a certain amount of dollars to invest into two stocks
(<math>IBM</math> and <math>TEXACO</math>). A portion of the available funds will be invested into
IBM (denote this portion of the funds with <math>x_A</math>) and the remaining funds
into TEXACO (denote it with <math>x_B</math>) - so <math>x_A+x_B=1</math>. The resulting portfolio
will be <math>x_A R_A+x_B R_B</math>, where <math>R_A</math> is the monthly return of <math>IBM</math> and <math>R_B</math> is the
monthly return of <math>TEXACO</math>. The goal here is to
find the most efficient portfolios given a certain amount of risk.
Using market data from January 1980 until February 2001 we compute
that <math>E(R_A)=0.010</math>, <math>E(R_B)=0.013</math>, <math>Var(R_A)=0.0061</math>, <math>Var(R_B)=0.0046</math>, and
<math>Cov(R_A,R_B)=0.00062</math>. We first want to minimize the variance of the portfolio.
This will be:
<math>
\mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
Or
 
<math>
\mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B)
</math>
 
<math>
\mbox{subject to} \ \ x_A+x_B=1
</math>
 
Therefore our goal is to find <math>x_A</math> and <math>x_B</math>, the percentage of the
available funds that will be invested in each stock. Substituting
<math>x_B=1-x_A</math> into the equation of the variance we get
<math>
x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B).
</math>
 
To minimize the above exression we take the derivative with respect to
<math>x_A</math>, set it equal to zero and solve for <math>x_A</math>. The result is:
<math>
x_A=\frac{Var(R_B) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
and therefore
 
<math>
x_B=\frac{Var(R_A) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
</math>
 
The values of <math>x_A</math> and <math>x_B</math> are:
<math>
x_a=\frac{0.0046-0.0062}{0.0061+0.0046-2(0.00062)} \Rightarrow x_A=0.42.
</math>
and <math>x_B=1-x_A=1-0.42 \Rightarrow x_B=0.58</math>. Therefore if the investor invests
<math>42 \%</math> of the available funds into <math>IBM</math> and the remaining <math>58 \%</math>
into <math>TEXACO</math> the variance of the portfolio will be minimum and equal to:
<math>
Var(0.42R_A+0.58R_B)=0.42^2(0.0061)+0.58^2(0.0046)+2(0.42)(0.58)(0.00062)
=0.002926.
</math>
The corresponding expected return of this porfolio will be:
<math>
E(0.42R_A+0.58R_B)=0.42(0.010)+0.58(0.013)=0.01174.
</math>
We can try many other combinations of <math>x_A</math> and <math>x_B</math> (but always <math>x_A+x_B=1</math>)
and compute the risk and return for each resulting portfolio. This is
shown in the table and the graph below.

{| class="wikitable" border="1"
|-
! <math>x_A</math>
! <math>x_B</math>
! Risk (<math>\sigma^2</math>)
! Return
! Risk (<math>\sigma</math>)
|-
| 1.00
| 0.00
| 0.006100
| 0.01000
| 0.078102
|-
| 0.95
| 0.05
| 0.005576
| 0.01015
| 0.074670
|-
| 0.90
| 0.10
| 0.005099
| 0.01030
| 0.071404
|-
| 0.85
| 0.15
| 0.004669
| 0.01045
| 0.068329
|-
| 0.80
| 0.20
| 0.004286
| 0.01060
| 0.065471
|-
| 0.75
| 0.25
| 0.003951
| 0.01075
| 0.062859
|-
| 0.70
| 0.30
| 0.003663
| 0.01090
| 0.060526
|-
| 0.65
| 0.35
| 0.003423
| 0.01105
| 0.058505
|-
| 0.60
| 0.40
| 0.003230
| 0.01120
| 0.056830
|-
| 0.55
| 0.45
| 0.003084
| 0.01135
| 0.055531
|-
| 0.50
| 0.50
| 0.002985
| 0.01150
| 0.054635
|-
| 0.42
| 0.58
| 0.002926
| 0.01174
| 0.054088
|-
| 0.40
| 0.60
| 0.002930
| 0.01180
| 0.054126
|-
| 0.35
| 0.65
| 0.002973
| 0.01195
| 0.054524
|-
| 0.30
| 0.70
| 0.003063
| 0.01210
| 0.055348
|-
| 0.25
| 0.75
| 0.003201
| 0.01225
| 0.056580
|-
| 0.20
| 0.80
| 0.003386
| 0.01240
| 0.058193
|-
| 0.15
| 0.85
| 0.003619
| 0.01255
| 0.060157
|-
| 0.10
| 0.90
| 0.003899
| 0.01270
| 0.062439
|-
| 0.05
| 0.95
| 0.004226
| 0.01285
| 0.065005
|-
| 0.00
| 1.00
| 0.004600
| 0.01300
| 0.067823
|}

<center>[[Image: Two_stocks_portfolio.jpg|600px]]</center>

 
For the above calculations short selling was not allowed (<math>0 \le x_A \le 1</math> and
<math>0 \le x_B \le 1</math>, in addition to <math>x_A+x_B=1</math>). We note here that the efficient portfolios are located on the top part of the graph between the minimum risk portfolio point and the maximum return portfolio point, which is called the efficient frontier (the blue portion of the graph). Efficient portfolios should provide higher expected return for the same level of risk or lower risk for the same level of expected return. 

If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities. The investor can have very large expected return but this will be associated with very large risk. The constraint here is only <math>x_A+x_B=1</math>, since either <math>x_A<math> or <math>x_B</math> can be negative. The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks. The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed. 

<center>[[Image: Ibm_texaco_short_sales.jpg|600px]]</center>

When the investor faces the efficient frontier when short sales are allowed and he or she can lend or borrow at the risk-free interest rate the efficient frontier will change in the following way: Let <math>x</math> be the portion of the investor's wealth invested in portfolio <math>A</math> that lies on the efficient frontier, and <math>1-x</math> the the portion invested in a risk-free asset. This combination is a new portfolio and has
<math>
\bar R_p=x\bar R_A + (1-x)R_f
</math>
 
where <math>R_f</math> is the return of the risk-free asset. The variance of this combination is simply
<math>
\sigma_p^2=x^2 \sigma_A^2 \Rightarrow x=\frac{\sigma_p}{\sigma_A}
</math>
 
From the last two equations we get
<math>
\bar R_p = R_f + \left(\frac{\bar R_A-R_f}{\sigma_A}\right)\sigma_p
</math>

 

This is an equation of a straight line. On this line we find all the possible combinations of portfolio <math>A</math> and the risk-free rate. Another investor can choose to combine the risk-free rate with portfolio <math>B</math> or portfolio <math>C</math>. Clearly, for the same level risk the combinations that lie on the <math>Rf-B</math> line have higher expected return than those on the line <math>Rf-A</math> (see figure below). And <math>Rf-C</math> will produce combinations that have higher return than those on <math>Rf-B</math> for the same level of risk, etc. 

<center>[[Image: Portfolio_risk_free_asset.jpg|600px]]</center>>
 

The solution, therefore, is to find the point of tangency of this line to the efficient frontier. Let's call this point <math>G</math>. To find this point we want to maximize the slope of the line in (1) as follows:
<math>
\mbox{max} \ \ \theta = \frac{\bar R_p - R_f}{\sigma_p}
</math>
Subject to
<math>
\sum_{i=1}^{n} x_i = 1
</math>
Since,
<math>
R_f=\left(\sum_{i=1}^n x_i\right) R_f = \sum_{i=1}^n x_iR_f
</math>
we can write the maximization problem as
<math>
\mbox{max} \ \ \theta=\frac{\sum_{i=1}^n x_i (\bar R_i - R_f)}
{\left(\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right)^{\frac{1}{2}}}
</math>
 
or
 
<math>
\mbox{max} \ \ \theta=\left[\sum_{i=1}^n x_i (\bar R_i - R_f)\right]
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}}
</math>
 
Take now the partial derivative with respect to each <math>x_i, i=1, \cdots, n</math>, set them equal to zero and solve. Let's find the partial derivative w.r.t. <math>x_k</math>:
<math>
\frac{\partial \theta}{\partial x_k} =
(\bar R_k - R_f)\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{1}{2}} +
\left[\sum_{i=1}^n x_i(\bar R_i - R_f)\right]
\left[2x_k\sigma_k^2 + 2 \sum_{j=1, j \ne k}^n x_j \sigma_{kj}\right] \times
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{-\frac{3}{2}} \times (-\frac{1}{2}) = 0
</math>
 
Multiply both sides by
<math>
\left[\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}\right]^{\frac{1}{2}} \ \ \ \mbox{to get}
</math>
<math>
(\bar R_k - R_f) -
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
(x_k \sigma_k^2 +\sum_{j=1, j \ne k}^n x_j \sigma_{kj}) =0
</math>
 
Now, if we let
<math>
\lambda=
\frac{\sum_{i=1}^n x_i(\bar R_i - R_f)}
{\sum_{i=1}^n x_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j=1, j \ne i}^n x_i x_j \sigma_{ij}}
</math>
 
the previous expression will be
<math>
(\bar R_k - R_f) - \lambda x_k \sigma_k^2 - \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj} = 0
</math>
 
or
 
<math>
\bar R_k - R_f = \lambda x_k \sigma_k^2 + \sum_{j=1, j \ne k}^n \lambda x_j \sigma_{kj}
</math>
 
Let's define now a new variable,
<math>
z_k = \lambda x_k
</math>
 
and finally
 
<math>
\bar R_k - R_f = z_k \sigma_k^2 + \sum_{j=1, j \ne k}^n z_j \sigma_{kj}
</math>
We have one equation like (2) for each <math>i=1, \cdots, n</math>. Here they are:
<math>
\bar R_1 - R_f = z_1 \sigma_1^2 + z_2 \sigma_{12} + z_3 \sigma_{13} + \cdots + z_n \sigma_{1n}
</math>
 
<math>
\bar R_2 - R_f = z_1 \sigma_{21} + z_2 \sigma_2^2 + z_3 \sigma_{23} + \cdots + z_n \sigma_{2n}
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\cdots \cdots
</math>
 
<math>
\bar R_n - R_f = z_1 \sigma_{n1} + z_2 \sigma_{n2} + z_3 \sigma_{n3} + \cdots + z_n \sigma_n^2
</math>
 
The solution involves solving the system of these simultaneous equations, which can be written in matrix form as:
<math>
\bar R = \Sigma Z
</math>
where <math>\Sigma</math> is the variance-covariance matrix of the returns of the <math>n</math> stocks. To solve for <math> Z</math>:
<math>
{Z} = \Sigma^{-1} \bar R
</math>
 
Once we find the <math>z_i's</math> it is easy to find the <math>x_i's</math> (the fraction of funds to be invested in each security). Earlier we defined
<math>
z_k = \lambda x_k \Rightarrow x_k = \frac{z_k}{\lambda}
</math>
We need to find <math>\lambda</math> as follows:
<math>
z_1 + z_2 + \cdots + z_n = \sum_{i=1}^n z_i
</math>
 
<math>
\lambda(x_1 + x_2 + \cdots + x_3) = \sum_{i=1}^n z_i
</math>
 
<math>
\Rightarrow \lambda = \sum_{i=1}^n z_i
</math>
 
Therefore, 
<math>
x_1 = \frac{z_1}{\lambda}
</math>
 
<math>
x_2 = \frac{z_2}{\lambda}
</math>
 
<math>
x_3 = \frac{z_3}{\lambda}
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
\cdots
</math>
 
<math>
x_n = \frac{z_n}{\lambda}
</math>
 

The snapshot form the SOCR portfolio applet shows an example with 5 stocks. Again, the red points in the applet correspond to the case when short sales are not allowed. The point of tangency can be found with a choice of the risk-free rate that can be entered in the input dialog box.

<center>[[Image: Tangent_point_5_stocks.jpg|600px]]</center>
 

 
The materials above was partially taken from ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.

SOCR EduMaterials Activities ApplicationsActivities TradingOptions

2008-08-03T15:21:51Z

Nchristo: /* Options */

== Options ==

An option is a contract between two investors:
 
* Issuer (or seller), holder of a short position. He sells the option.
 
* Holder (buyer), holder of a long position. He buys the option.
 
'''Types of options:'''
 
* Call option: Gives the holder the right to buy an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the option or premium.
 
* Put option: Gives the holder the right to sell an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the put or premium.
The date specified it is called: the expiration date or maturity date. The price specified it is called the exercise price or the strike price.
 
There are European options (can be exercised only on the expiration date) and American options (can be exercised at any time up to the expiration date).
 
'''Stock options mechanics:'''
 
* Options are normally traded in units of 100 shares. The price of the option is on a per share basis. Therefore, if the price of an option is priced at $0.50, the total premium for that option would be <math>\$50</math> (<math>0.50 \times 100 = \$50</math>).
 
* Stock options are on a January, February, or March cycle. Stocks are randomly assigned in one of these three cycles. For example, IBM is on a January cycle (options can be bought on Jan, Apr, Jul, Oct).
 
Stock options expired on the Saturday immediately following the third Friday of the expiration month.
 

The call option will only be exercised if the stock price at expiration is larger than the exercise price. In this case the holder of the call will have a positive payoff. The put option will only be exercised if the stock price at expiration is lower than the exercise price. In this case the holder of the put will have a positive payoff. The two figures below shows when the holder or the seller make a positive payoff.

 

<center>[[Image: Christou_call_faces.jpg|600px]]</center>

 

<center>[[Image: Christou_put_faces.jpg|600px]]</center>

 

There is an infinite number of combinations that one can make using call and put options. Some of the combinations have special names, like straddles, strips, straps, bull spreads, bear spreads, butterfly spreads, covered call, etc. All these are shown in the SOCR Trading Options applet. Here are some snapshots:

 

<center>[[Image: Christou_butterfly_calls.jpg|600px]]</center>

 

<center>[[Image: Christou_bear_spread.jpg|600px]]</center>

SOCR EduMaterials Activities ApplicationsActivities TradingOptions

2008-08-03T15:19:59Z

Nchristo: /* Options */

== Options ==

An option is a contract between two investors:
 
* Issuer (or seller), holder of a short position. He sells the option.
 
* Holder (buyer), holder of a long position. He buys the option.
 
'''Types of options:'''
 
* Call option: Gives the holder the right to buy an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the option or premium.
 
* Put option: Gives the holder the right to sell an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the put or premium.
The date specified it is called: the expiration date or maturity date. The price specified it is called the exercise price or the strike price.
 
There are European options (can be exercised only on the expiration date) and American options (can be exercised at any time up to the expiration date).
 
'''Stock options mechanics:'''
 
* Options are normally traded in units of 100 shares. The price of the option is on a per share basis. Therefore, if the price of an option is priced at $\$0.50$, the total premium for that option would be <math>\$50</math> (<math>0.50 \times 100 = \$50</math>.)
 
* Stock options are on a January, February, or March cycle. Stocks are randomly assigned in one of these three cycles. For example, IBM is on a January cycle (options can be bought on Jan, Apr, Jul, Oct).
 
Stock options expired on the Saturday immediately following the third Friday of the expiration month.
 

The call option will only be exercised if the stock price at expiration is larger than the exercise price. In this case the holder of the call will have a positive payoff. The put option will only be exercised if the stock price at expiration is lower than the exercise price. In this case the holder of the put will have a positive payoff. The two figures below shows when the holder or the seller make a positive payoff.

 

<center>[[Image: Christou_call_faces.jpg|600px]]</center>

 

<center>[[Image: Christou_put_faces.jpg|600px]]</center>

 

There is an infinite number of combinations that one can make using call and put options. Some of the combinations have special names, like straddles, strips, straps, bull spreads, bear spreads, butterfly spreads, covered call, etc. All these are shown in the SOCR Trading Options applet. Here are some snapshots:

 

<center>[[Image: Christou_butterfly_calls.jpg|600px]]</center>

 

<center>[[Image: Christou_bear_spread.jpg|600px]]</center>

SOCR EduMaterials Activities ApplicationsActivities TradingOptions

2008-08-03T15:18:47Z

Nchristo: /* Options */

== Options ==

An option is a contract between two investors:
 
* Issuer (or seller), holder of a short position. He sells the option.
 
* Holder (buyer), holder of a long position. He buys the option.
 
'''Types of options:'''
 
* Call option: Gives the holder the right to buy an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the option or premium.
 
* Put option: Gives the holder the right to sell an asset by a certain date for a certain price called exercise price with a fee. This fee it is the price of the put or premium.
The date specified it is called: the expiration date or maturity date. The price specified it is called the exercise price or the strike price.
 
There are European options (can be exercised only on the expiration date) and American options (can be exercised at any time up to the expiration date).
 
'''Stock options mechanics:'''
 
* Options are normally traded in units of 100 shares. The price of the option is on a per share basis. Therefore, if the price of an option is priced at $\$0.50$, the total premium for that option would be <math>\$50</math> (<math>0.50 \times 100 = \$50</math>.)
 
* Stock options are on a January, February, or March cycle. Stocks are randomly assigned in one of these three cycles. For example, IBM is on a January cycle (options can be bought on Jan, Apr, Jul, Oct).
 
Stock options expired on the Saturday immediately following the third Friday of the expiration month.
 

The call option will only be exercised if the stock price at expiration is larger than the exercise price. In this case the holder of the call will have a positive payoff. The put option will only be exercised if the stock price at expiration is lower than the exercise price. In this case the holder of the put will have a positive payoff. The two figures below shows when the holder or the seller make a positive payoff.

 

<center>[[Image: Christou_call_faces.jpg|600px]]</center>

 

<center>[[Image: Christou_put_faces.jpg|600px]]</center>

 

There is an infinite number of combinations that one can make using call and put options. Some of the combinations have special names, like straddles, strips, straps, bull spreads, bear spreads, butterfly spreads, covered call, etc. All these are shown in the SOCR Trading Options applet. Here are some snapshots:

 

<center>[[Image: Christou_butterflyl_calls.jpg|600px]]</center>

 

<center>[[Image: Christou_bear_spread.jpg|600px]]</center>

File:Christou bear spread.jpg

2008-08-03T15:14:53Z

Nchristo:

File:Christou butterfly calls.jpg

2008-08-03T15:13:48Z

Nchristo:

File:Christou put faces.jpg

2008-08-03T15:12:31Z

Nchristo:

File:Christou call faces.jpg

2008-08-03T15:11:08Z

Nchristo: