investment analysis and portfolio management lecture 7 gareth myles
TRANSCRIPT
The Capital Asset Pricing Model (CAPM)
The CAPM is a model of equilibrium in the market for securities.
Previous lectures have addressed the question of how investors should choose assets given the observed structure of returns.
Now the question is changed to: If investors follow these strategies, how will
returns be determined in equilibrium?
The Capital Asset Pricing Model (CAPM)
The simplest and most fundamental model of equilibrium in the security market Builds on the Markowitz model of portfolio choice Aggregates the choices of individual investors
Trading ensures an equilibrium where returns adjust so that the demand and supply of assets are equal
Many modifications/extensions can be made But basic insights always extend
Assumptions
The CAPM is built on a set of assumptions Individual investors
Investors evaluate portfolios by the mean and variance of returns over a one period horizon
Preferences satisfy non-satiation Investors are risk averse
Trading conditions Assets are infinitely divisible Borrowing and lending can be undertaken at the
risk-free rate of return There are no taxes or transactions costs
Assumptions
The risk-free rate is the same for all Information flows perfectly
The set of investorsAll investors have the same time horizon Investors have identical expectations
Assumptions
The first six assumptions are the Markowitz model
The seventh and eighth assumptions add a perfect capital market and perfect information
The final two assumptions make all investors identical except for their degree of risk aversion
Direct Implications
All investors choose a location on the efficient frontier
The location depends on the degree of risk aversion
The chosen portfolio mixes the risk-free asset and portfolio M of risky assets
pr
pMVP
MVPr
fr
Less riskaverseMore risk
averse
M
Separation Theorem
The optimal combination of risky assets is determined without knowledge of preferences All choose portfolio M This is the Separation Theorem
M must be the market portfolio of risky assets All investors hold it to a greater or lesser extent No other portfolio of risky assets is held There is a question about the interpretation of this
portfolio
Equilibrium
The only assets that need to be marketed are: The risk-free asset A mutual fund representing the market portfolio No other assets are required
In equilibrium there can be no short sales of the risky assets All investors buy the same risky assets No-one can be short since all would be short If all are short the market is not in equilibrium
Equilibrium
Equilibrium occurs when the demand for assets matches the supply This also applies to the risk-free Borrowing must equal lending
This is achieved by the adjustment of asset prices
As prices change so do the returns on the assets
This process generates an equilibrium structure of returns
The Capital Market Line
All efficient portfolios must lie on this line
Slope =
Equation of the line
pr
p
fr
M
fM rr
pM
fMfp
rrrr
Mr
M
Interpretation
rf is the reward for "time" Patience is rewarded Investment delays consumption
is the reward for accepting "risk"
The market price of riskJudged to be equilibrium rewardObtained by matching demand to supply
M
fM rr
Security Market Line
Now consider the implications for individual assets
Graph covariance against returnThe risk on the market portfolio is The covariance of the risk-free asset is zero The covariance of the market with the
market is
M
2M
Security Market Line
Can mix M and the risk-free asset along the line If there was a portfolio
above the line all investors would buy it
No investor would hold one below
The equation of the line is
pr
fr
iM2M
Mr
iMM
fMfi
rrrr
2
M
Security Market Line
Define
The equation of the line becomes
This is the security market line (SML)
2M
iMiM
iMfMfi rrrr
Security Market Line
There is a linear trade-off between risk measured by and return
In equilibrium all assets and portfolios must have risk-return combinations that lie on this line
pr
fr
iM
iMir
Market Model and CAPM
Market model usesCAPM uses is derived from an assumption about
the determination of returns it is derived from a statistical model the index is chosen not specified by any
underlying analysis is derived from an equilibrium theory
iI
iM
iI
iM
Market Model and CAPM
In addition: I is usually assumed to be the market index,
but in principal could be any indexM is always the market portfolio
There is a difference between theseBut they are often used interchangeablyThe market index is taken as an
approximation of the market portfolio
Estimation of CAPM
Use the regression equation
Take the expected value
The security market line implies
It also shows
ifMiMiMfi rrrr
fMiMiMfi rrErrE
fiMiI r 1
0iM
CAPM and Pricing
CAPM also implies the equilibrium asset prices The security market line is
But
where pi(0) is the value of the asset at time 0 and pi(1) is the value at time 1
iMfMfi rrrr
0
01
i
iii p
ppr
CAPM and Pricing
So the security market line gives
This can be rearranged to find
The price today is related to the expected value at the end of the holding period
fMiMfi
ii rrrp
pp
0
01
fMiMf
ii rrr
pp
1
10
CAPM and Project Appraisal
Consider an investment projectIt requires an investment of p(0) todayIt provides a payment of p(1) in a yearShould the project be undertaken?The answer is yes if the present
discounted value (PDV) of the project is positive
CAPM and Project Appraisal
If both p(0) and p(1) are certain then the risk-free interest rate is used to discount
The PDV is
The decision is to accept project if
fr
ppPDV
1
10
fr
pp
1
10
CAPM and Project Appraisal
Now assume p(1) is uncertainCannot simply discount at risk-free rate if
investors are risk averseFor example using
will over-value the projectWith risk aversion the project is worth less
than its expected return
fr
ppPDV
1
10
))1(())1(( pEUpU