Financial Engineering 3

References: Villalobos, Luenberger, Faerber

Lecture 11

Capital Asset Pricing Model (CAPM)

Lecture Topics • Brief Review of the One-Fund Theorem • Capital Asset Pricing Model (CAPM) • Example

One Fund Theorem • There is a single fund F of risky assets such that any efficient

portfolio can be constructed as a combination of the fund and the risk free asset.

• The mean-variance analysis of Markowitz’ portfolio theory tells us that the portfolio we need to consider for an efficient investment must lie on an “efficient frontier” of the mean- variance graph.

• Because of the one-fund theorem, stating there is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund and the risk free asset, everyone in the market will tend to hold the same portfolio and borrow or lend at the same rate.

• If there were only one portfolio that everyone seems to want to buy what is that portfolio?

The Market Portfolio • The fund everyone buys must be equal to the “market”

portfolio.

• The market portfolio should contain all the assets available.

• The market portfolio should contain each asset in the proportion that they are part of the overall market.

– It should contain shares of each asset in proportion of that stock’s representation in the entire market.

• The weight of an asset in the market portfolio should be equal to the proportion of that asset’s total market capitalization.

• What is the closest thing to this ideal market?

S&P 500

Assumptions of the CAPM • All assets in the world are traded. • All assets are infinitely divisible. • All investors in the world collectively hold all assets. • For every borrower, there is a lender. • There is a risk-free security in the world. • All investors borrow and lend at the risk-free rate. • Everyone agrees on the inputs to the Mean-StDev plot. • Preferences are well-described by simple utility functions. • Security distributions are normal, or at least well described by

two parameters.

The Capital Market Line • If the market portfolio defines the single efficient fund, then the

efficient frontier is a line in the mean-variance plot passing through the mean of the risk-free asset and the market portfolio.

• This line is also called the pricing line because prices adjust so that the efficient portfolios fall on this line.

– This line also identifies the tradeoff between risk and average return.

• The equation of the line is given by:

M f f

M

r r r r σ

σ −

= +

M fr

σ

r

Mσ

Mr

Volatility • The annual standard deviation is considered a measure of a

stock’s risk and it is usually referred to as “Volatility”.

• It is the measure of risk, and usually is computed as the annualized standard deviation of historical returns.

M f f

M

r r r r σ

σ −

= +

• Where is the expected excess rate of return of asset i,

The Pricing Model • The Capital Asset Pricing Model (CAPM) shows how the expected

rate of return of an asset relates to its individual risk. • The Capital Asset Pricing Model states that if the market portfolio

is efficient, the expected return of any of its assets satisfies

( )i f i M fr r r rβ− = − fi rr −

2 M

iM i σ

σ β =

• is known as the “beta” of an asset and is all we need to know about the asset’s risk to be able to use the CAPM formula.

• Beta is really just a normalized covariance of the asset relative to the variance of the market portfolio.

• So, CAPM formula implies that the price of an asset is proportional to the asset’s covariance with the market portfolio.

Beta Examples February 9, 2012

Beta Yahoo Finance

Beta Calculated

Stock Annual StDev

Stock Average Annual Return

S&P500 1.000000 0.163997 0.194619 SPF 3.410000 3.845196 0.940790 0.916651 AA 2.020000 2.063772 0.429527 0.228692 HD 0.890000 0.916191 0.245307 0.284467 KO 0.440000 0.404167 0.157240 0.192754 F 2.530000 2.972545 0.838537 0.780983 APOL 0.890000 0.877320 0.423660 -0.043899

Beta of a Portfolio • The beta of a portfolio is based on the betas of the individual

assets and is calculated by:

– Where the w’s are the weights of the assets in the portfolio. – The summation of the w’s must equal to one.

• For example, a portfolio with the previous six stocks if weighted equally has a beta of 1.85 (using calculated betas).

∑ =

= n

i iip w

1 ββ

Beta of a Portfolio • Suppose that a portfolio with the previous stocks is built

according to the weights shown below.

• If this happened to be efficient, how does this compare to our Market Portfolio?

Beta Calculated Weight

Beta Contribution

SPF 3.845196 40% 1.538079 AA 2.063772 20% 0.412754 HD 0.916191 10% 0.091619 KO 0.404167 10% 0.040417 F 2.972545 10% 0.297254 APOL 0.877320 10% 0.087732 Total 2.467855

Ours is much riskier, but higher return!

The Security Market Line • In the same way that we determine a market line, we can also

determine a line for individual securities by using the formula:

( )fMifi rrrr −+= β

M fr

iβ

ir

1

Risk: Systematic and Non-systematic • Systematic Risk is inherent to the market. • Non-systematic is associated with a particular security. • By including an error term in the security market line we have:

• The expected value and its correlation with the market of the error term is assumed to be zero.

– Therefore, the variance of the security market line is:

• The first term corresponds to the systematic risk, the second to the risk inherent to the security

( ) ifMifi rrrr εβ +−+=

( )iMii εσβσ var222 +=

Example Yahoo Finance February 9, 2012

Stock Beta (Yahoo

Finance) Calculated

Annual StDev

Std. Dev. (from stock

beta) Price of Stock

S&P500 1.0000 0.1640 $1,351.95 DELL 1.3300 0.2181 $18.06

NVDA 1.5300 0.2509 $16.30 X 2.4800 0.4067 $31.01

ADP 0.7200 0.1181 $54.57 THC 2.7200 0.4461 $5.58

K 0.4300 0.0705 $50.21

( )iMii εσβσ var222 +=

CAPM as a Pricing Formula • Suppose that an asset is purchased at price P and later sold at

a random price Q. • According to the CAPM model, the price that we should be

willing to pay for that asset is given by the equation:

• Equivalently, the price P of an asset with payoff Q is:

( )fMf rrr Q

P −++

= β1

( )( )  

  

 − −

+ = 2

,cov 1

1

M

fMM

f

rrrQ Q

r P

σ

( ) 1i f i M f Q P Q

r r r r P P

β −

= + − = = −

Example • How much should you be willing to pay for a share of Coca

Cola stock (KO) right now if you believe that its price one year from now will be of $50.00?

• The actual current price of KO is $47.70, the current yield of the 3 months T-bill is 5.1%, the current β for KO is 0.6, and the annual return of the S&P 500 is 8.81%.

( ) ( ) 58.46051.0881.06.0051.1 50

1 =

−++ =

−++ =

fMf rrr Q

P β

Jensen Index • Jensen Index measures the deviation of the return of an asset

against its expected return assuming it is efficient.

( )f i M fr r J r rβ− = + −

Mfr

iβ

ir

1

Jensen Index

Old Data Example • Suppose that the current annual return for a stock is 0.1% • The S&P 500 index return is 8.81% • The beta of this is stock is 1.84 • The risk-free rate of return is 5.1%.

• We want to determine the Jensen Index for this stock:

( )f i M fJ r r r rβ= − − −

0.001 0.051 1.84(.0881 .051) 0.11J = − − − = −

( )f i M fr r J r rβ− = + −

Sharpe Index • Sharpe Index measures how efficient the stock is with respect to

the market, and σ is the volatility of the security being analyzed.

• The Sharpe ratio measures excess return per unit of volatility. • Basically, it is the slope of the line.

– We would compare the slope of our asset line to that of the Market line which is efficient.

fr r Sσ− =

M

fr

ir Sharpe Index

fr rS σ −

=

σ

Old Data Example • From previous example using calculated StDev:

• The same ratio for the S&P500 is:

1298.0 3851.0

051.0001.0ˆ −=

− =

− =

σ frrs

ˆ 0.0881 0.051 0.365

0.1014 M fr rs σ − −

= = =

Example • From our original example:

• Assuming a risk free rate of 0.87%, determine the Jensen Index and the Sharpe Ratio for Coca Cola (KO):

Beta Yahoo Finance

Beta Calculated

Stock Annual StDev

Stock Average Annual Return

S&P500 1.000000 0.163997 0.194619 SPF 3.410000 3.845196 0.940790 0.916651 AA 2.020000 2.063772 0.429527 0.228692 HD 0.890000 0.916191 0.245307 0.284467 KO 0.440000 0.404167 0.157240 0.192754 F 2.530000 2.972545 0.838537 0.780983 APOL 0.890000 0.877320 0.423660 -0.043899

( ) 0.1928 0.0087 0.4041(0.1946 .0087) 0.1089KO f i M fJ r r r rβ= − − − = − − − = 0.1928 0.0087

1.1705 0.1572

fr rS σ − −

= = =

Example • The remaining stocks have Jensen and Sharpe indices of:

Beta Yahoo Finance

Beta Calculated

Stock Annual StDev

Stock Average Annual Return Jenson Sharpe

S&P500 1.000000 0.163997 0.194619 1.133672 SPF 3.410000 3.845196 0.940790 0.916651 0.193057 0.965095 AA 2.020000 2.063772 0.429527 0.228692 -0.163702 0.512173 HD 0.890000 0.916191 0.245307 0.284467 0.105430 1.124174 KO 0.440000 0.404167 0.157240 0.192754 0.108912 1.170527 F 2.530000 2.972545 0.838537 0.780983 0.219631 0.920989 APOL 0.890000 0.877320 0.423660 -0.043899 -0.215709 -0.124154

Assignments • Setup your own Excel file and reproduce the examples shown

in this lecture.

• Now down load the weekly (rather than monthly) rates for the stocks, modify your spreadsheet to convert from 12 months per year to 52 weeks per year, and replicate the results with latest 10 year treasury note as the risk free security.

• Choose other stocks to play with and see what you get.

• Luenberger Chapter 7 problems 7.1, 7.3, 7.7

• Read Luenberger Chapters 8 and 9.