The Capital Asset Pricing Model

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Lecture 10

• The Capital Asset Pricing Model

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Fundamental or Theoretical Analysis
Preliminaries
Expectation, variance, standard error
(deviation), covariance, and correlation of
returns may be based on (i) fundamental analysis
(ii) historical data
Notation
S possible states ?s probability of
state s 1,2,,S Rs likely return is state s
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Variance
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Covariance measures how two random variables are
related
returns on stock A RAs s 1,,S returns on
stock B RBs s 1,,S
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Example
Suppose we have a theoretical model that predicts
the following returns on stocks A and B in 3
states.
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Covariance
Returns on stocks A and B are perfectly
negatively correlated. Stocks A can be used as a
hedge against the risk in holding stock B
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Historical Data Based Approach
From historical data, calculate the percentage
returns R1, R2, , RT
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Historical Data Based Approach (continued)
Sample correlation of RA and RB
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Expected Return and Variance of Returns on
Portfolios
and the expected return on the portfolio is
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Expected Return and Variance of Returns on
Portfolios (continued)
The variance of the returns on the portfolio is
given by
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Diversification
1. Variances are diversified away 2. Average
covariance converges to covariance from
economy-wide shocks affecting all stocks
- In a diversified portfolio, only systematic
risk affects returns. - Diversifiable or
unsystematic (idiosyncratic) risk is irrelevant
to returns.
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Diversification (continued)
The mean return on the portfolio is 10.
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Diversification (continued)
The standard deviation of the return on the
portfolio is zero. No risk!
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Deriving an appropriate discount rate for risky
cash flows
1. The opportunity set for two assets
2. The opportunity set and efficient set with
many securities
3. The efficient set with a riskless asset
4. The CAPM (capital asset pricing model) equation
5. A risk-return separation theorem
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The opportunity set for two assets
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The opportunity set for two assets (continued)
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Example
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Example (continued)
Opportunity set for assets A and B
Portfolio MV (minimum variance) has the lowest
risk obtainable with assets A and B.
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The opportunity set and efficient set with many
securities
Each pair of securities ((A,B),(A,C),(B,C)) gives
an opportunity set
Except for portfolios close to MV, the efficient
set is very close to a straight line. Also as the
variance of the MV portfolio decreases, the
efficient set gets closer to a straight line.
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The efficient set with a riskless asset
If one asset is riskless, the variance of returns
on that asset, and the covariance with returns on
all other assets will be zero.
In equilibrium, the riskless rate lt return on MV.
Hence, the opportunity set will be the tangent
line from the riskless asset to the efficient set.
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The efficient set with a riskless asset
(continued)
Use of such a broad-based index as a proxy is
justified since most investors hold diversified
portfolios.
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The efficient set with a riskless asset
(continued)
The best measure of the risk of a security in a
large portfolio is the beta of the security,
which measures the responsiveness of the security
to the movements in the market portfolio.
Formula for beta
covariance between the return on asset i and the
return on market
variance of market portfolio
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Example
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Example (continued)
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The CAPM equation
Relationship between risk and expected return
If there is a riskless asset with return r, there
is a straight line trade off between risk and
expected return for a security.
is the contribution of this security to the
portfolio risk.
If the tangency portfolio is the market portfolio
with expected return and standard
deviation , then
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The CAPM equation (continued)
Equilibrium expected return on asset j
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The CAPM equation (continued)
(Expected return on a security) (current risk
free interest rate) (beta coefficient of the
security)(historical market risk premium)
Finally, we established a way of determining
appropriate discount rate for risky cash flows.
We first measure its risk by its beta
coefficient, and then obtain the required return
from the CAPM equation.
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The CAPM equation (continued)
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The CAPM equation (continued)
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The Security Market Line (SML)
The Security Market Line (SML) below graphs
expected return against beta, using the CAPM
equation.
Slope of the SML is the risk premium. For the
SP500 and US treasury bills, the risk premium is
about 8.5. (The book uses 9.2, which is based
on Ibbotson et. al study). This estimate is often
used as a forecast for the risk premium on stocks
in the future.
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SML (continued)
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A Risk-Return Separation Theorem
An investment will be worth taking only if it is
at least as desirable as what is already
available in the financial markets.
A new investment will be worthwhile if and only
if it is outside (above) the efficient set (or
the risk-return budget constraint).
No matter where individual would choose to be on
the efficient set, an investment can only make
them better off if it is above the efficient set.
If the two financial separation theorems did not
hold, then the firms would need to know the
inter-temporal and risk-return preferences of
each owner to decide desirable investments.
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Problem 10.13 from the text
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Problem 10.13 from the text (continued)
What are covariances and correlations between the
returns? For j A,B,C and k A,B,C
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Problem 10.13 from the text (continued)
What are expected returns and standard deviations
of the portfolios?
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Problem 10.39 from the text
Suppose you have invested 30,000 in the
following 4 stocks
The risk free rate is 4 and the expected return
on the market portfolio is 15. Based on the
CAPM, what is the expected return on the above
portfolio?
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Problem 10.39 from the text (continued)
There are two ways to answer the question.