Capital Asset Pricing Model(CAPM, 자본자산가격결정모형)

Capital Market Line (CML)
- If all investors have the same expectations about future security payoffs, they will have the same
risky portfolio regardless of their preferences. As a consequence, the tangent portfolio becomes the
market portfolio because the market portfolio is the aggregation of the portfolios of all the people. In
this case, we call the efficiency set (or capital allocation line) the Capital Market Line.
* Why does the tangent portfolio become the market portfolio?

In the Sharpe-Lintner world, all investors should hold the same portfolio (T) of risky
securities as everybody else. Since everybody holds portfolio T, portfolio T represents the
demand for risky securities.

What is the supply of all risky securities? → Market portfolio (M)

In equilibrium, T should be the M.
E(r)
CML
Market Portfolio : m
rf
σ
0
- The slope of CML :
- Equation :
( )
(
)
(
)
- Interpretation : Expected Return = (Price of Time) + (Market Price of Risk)·(Amount of Risk)
- Notes on CML:
(i) All the portfolios on the CML are efficient because the CML essentially represents the efficient set.
(ii) The correlation coefficient between a portfolio on the CML and the market portfolio is 1 (ρpm=1).

Example 1.
- Suppose E(rm)=20%, σm=15%, rf=10%.
1) What is the slope of CML?
(
)
=
2) What is the expected return and standard deviation if you invest your wealth half in risk-free asset
and half in the market portfolio?
σp=0.5σm+0.5σf=7.5%
E(rp)=rf+(2/3)σp=10%+(2/3)·7.5%=15%
3) What if you borrow half of your wealth and invest the whole amount (including your original
wealth) in the market portfolio?
σp=1.5σm 0.5σf=22.5
E(rp)=rf+(2/3)σp=10%+(2/3)·22.5%=25%

Deficiencies of CML as an Equilibrium Model
- It represents the relationship between return and risk of efficient portfolios only. It does not express
the relationship between return and risk of individual securities.
- It describes the relationship between return and variance (standard deviation) or total risk, not the
market risk (beta risk), which was shown to be more relevant.
- Thus, we need to derive the relationship between return and the market risk (beta risk) for all
individual securities and portfolios. This relationship is represented by the Capital Asset Pricing
Model (CAPM). Its graphical depiction is called the Security Market Line (SML).

Assumptions of the CAPM
- Investors maximize expected utility in terms of mean and variance.
- Investors can borrow and lend unlimited amount at the same rate.
- All assets are perfectly divisible.
- Single period model (No intermediate consumption)
- Perfect competition.
- The supply of assets are given.
- Homogeneous expectations (on expected return, variance and covariance)

The CAPM
(̃)
{ ( ̃)
}
, where
Notes on the CAMP:
(i) Only beta risk matters.
 Y= a + bX
(ii) The expected return is linear in
E(r)
Security Market Line (SML)
slope=E(rm)-rf
E(rm)
m
rf
βm=1
0

β
Derivation of CAPM
- Optimality condition (or condition in the equilibrium where no excess demand occurs):
*Return-risk reward ratios should be the same and constant across the assets in the portfolio
(market PFO’s risk premium per unit risk)= (asset i’s risk premium per unit risk) = asset j’s......
(
)
[ ( )
]
(
)
[ ( )
]
[ ( )

]
............
( )
{ (
)
, where
(
)
∑
(
)
}
* Similar concept in economics textbook
condition of optimal consumption : marginal utility of apple = marginal utility of banana

Example 2.
- E(rm)=20%, σm=15%, rf=10%, Standard deviation of stock A’s return is 25% and its correlation
coefficient with the market is 0.5.
1) What is the beta of stock A?
= (20%)2 = 400%2
(
)·(25%)·(20%)=250%2
= 0.625
2) What is the required return on stock A?
( )
{ (
)
}
10%+5%·0.625=13.125%

The CML and the SML (BD p.362)

Expected Return vs Actual Return
- Expected return: ex-ante (before the fact) or forecasted return
- Actual return: ex-post (after the fact) or realized return

Equilibrium vs Disequilibrium
- Equilibrium may be defined as the situation where no one has incentive to deviate from what it is.
- If the expected return and risk of a stock is represented by a point on SML, the situation is in
equilibrium. Otherwise (if not on SML), it is in disequilibrium.
- In other words, the expected rate of return predicted by the CAPM is the equilibrium expected rate
of return (sometimes, we use the term required rate of return).
- Defining αi as follows
αi = Return anticipated by investors – Equilibrium expected return
if αi = 0, stock i is in equilibrium and if αi ≠0, it is in disequilibrium.

What if a stock is in disequilibrium (not on the SML)?
1) underpriced (undervalued) stock
E(r)
SML
A
A*
m
rf
βA
0
β
- In this case, we could construct A* portfolio by combining the risk-free asset and the market
portfolio, β of which is identical to that of A. If some investors share the same view of stock A’s
prospects, they will buy shares of A and shortsell A* as stock A is underpriced. As a result, the price
of A will rise until A’s expected return falls on the SML (until the equilibrium is restored).
2) overpriced (overvalued) stock
E(r)
SML
B*
rf
m
B
βB
0
β
- In this case, we could construct B* portfolio by combining the risk-free asset and the market
portfolio, β of which is identical to that of B. If some investors share the same view of stock B’s
prospects, they will shortsell shares of B and buy B* as stock B is overpriced. As a result, the price of
B will go down until B’s expected return falls on the SML (until the equilibrium is restored).

Security Characteristic Line (SCL, 증권특성선)
- The extent of disequilibrium is expressed as:
[
{ (
Rearranging yields:
)
}
], where
is the stock i’s anticipated return by investors.
{ (
)
}
Specification of above equation based on the realized values is called the SCL:
̃
(̃
)
̃
- Like the market model (or the single index model), the characteristic line is a return generating
process and essentially a regression model.

The CAPM and capital budgeting
- The SML provides a standard for project acceptance.

If an investment project lies above the SML, it is an attractive investment opportunity.

If an investment project lies below the SML, it is not a good project.

The imprecision in the cost of capital estimate vs. the imprecision in the estimate of future
cash flows (BD. p440,)
- How Firms Calculate the Cost of Capital (Graham and Harvey(2001)_JFE)
 A survey of 392 CFOs