IE 5441 1

Chapter 7. The Capital Asset Pricing Model

Shuzhong Zhang

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Market Equilibrium

Equilibrium is a central concept in economics. We will now use the

notion of equilibrium to come up with a method to quantify the fair

price for an asset in the market.

Suppose that everyone in the market are all mean-variance believers.

Then, everyone will hold a portfolio with a very similar structure: one

portion of the fund with the risky assets, and one portion on the

risk-free asset.

The part on the risky assets – the one fund – will become observable:

it is the market portfolio.

Shuzhong Zhang

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Recall what we derived in the mean-variance analysis:

µ− rf = κσ,

for any (µ, σ) pair on the efficient frontier, and the market portfolio is

w =µ− rfκ2

Σ−1(r̄ − rfe),

where

κ =√(r̄ − rfe)TΣ−1(r̄ − rfe).

Substituting this back to the formula before, we have

κ2Σw = (µ− rf )(r̄ − rfe),

and (µ− rfσ2

)Σw = r̄ − rfe.

Shuzhong Zhang

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Now, w is the market portfolio, which is efficient. Let us focus on that

efficient point: (µ, σ) where µ is the mean return of the market

portfolio, and σ2 is the variance of the market portfolio.

Let us read the previous formula componentwise: its i-th component

reads

r̄i − rf =

(µ− rfσ2

)(Σw)i =

(Σw)iσ2

× (µ− rf ) .

The above can be interpreted as:

E[‘return on asset i’]− rf

=‘covariance between asset i and the market portfolio’

‘variance of the market portfolio’

× (‘expected return of the market portfolio’− rf ) .

The above is known as the Capital Asset Pricing Model (CAPM).

Shuzhong Zhang

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The so-called market β is defined as follows

βi =‘covariance between asset i and the market portfolio’

‘variance of the market portfolio’.

Therefore, the CAPM can also be stated as:

The excessive rate of return (compared to the risk-free rate)

should be equal to its β value times the market excessive rate

of return.

Shuzhong Zhang

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Remember that we always have

|σij | ≤ σiσj .

Therefore,

βi =‘covariance between asset i and the market portfolio’

‘variance of the market portfolio’

≤ ‘standard deviation of asset i’

‘standard deviation of the market portfolio’.

It follows that

E[‘return on asset i’]− rf

≤ (‘expected return of the market portfolio’− rf )

‘standard deviation of the market portfolio’

בstandard deviation of asset i’.

The Capital Market Line: (0, r̄f ) with (σM , r̄M ) on the (σ, µ) plot.

Shuzhong Zhang

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Example 7.1. Mr. Smith is young and impatient. He notes that the

risk-free rate is only 6%, and the market portfolio of risky assets has

an expected return of 12% and a standard deviation of 15%. He

figures that it would take 60 years for his $1,000 to grow into $1

million. He cannot wait that long and would like to do it in 10 years.

Mr. Smith determines that he will need 100% rate of return. Even if

he manages the investment in the most efficient way, the standard

deviation still needs to satisfy

1.0 = 0.06 +0.12− 0.06

0.15× σ

or, σ = 235%.

Shuzhong Zhang

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Example 7.2. Consider an oil drilling venture. The price of a share of

this venture is $875. It is expected to yield the equivalent of $1,000

after 1 year, but due to high uncertainty about how much oil is at the

drilling site, the deviation of the return is σ = 40%. Currently the

risk-free rate is 10%. The market expected rate of return is 17%, with

standard deviation 12%. According to CAPM, with this level of risk,

the justifiable rate of return is

r̄ = 0.10 +0.17− 0.10

0.12× 0.40 = 33%.

But the current estimated rate of return is 1000/875− 1 = 14%. It is

below the standard set according to the CAPM.

Shuzhong Zhang

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The security market line

The security market line is the line between r̄i (axis y) and βi (axis x):

they should be on the line y = rf + (r̄M − rf )x if CAPM holds exactly.

Jensen’s α:

αi := r̄i − rf − βi(r̄M − rf ).

If αi > 0 then empirically asset i is an worthy investment.

Let us write

ri = rf + βi(rM − rf ) + ϵi.

The CAPM states that E[ϵi] = 0.

Moreover,

σ2Mβi = cov (ri, rM ) = σ2

Mβi + cov (ϵi, rM ),

and so cov (ϵi, rM ) = 0; i.e. ϵi and the market return is uncorrelated.

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Furthermore,

σ2i = β2

i σ2M + var (ϵi).

• β2i σ

2M is known as the systematic risk.

• var (ϵi) is known as the nonsystematic or idiosyncratic risk.

For a portfolio of assets (w1, ..., wn) where wi is the weight of asset i

with rate of return ri, and βi, i = 1, 2, ..., n, its β is simply

β = cov

(n∑

i=1

wiri, rM

)/σ2

M =n∑

i=1

wiβi.

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Performance evaluation

Step 1. Compute the average rate of return and the average variance

r̄ =1

n

n∑i=1

ri, s2 =1

n− 1

n∑i=1

(ri − r̄)2.

Step 2. Use an index, say S&P 500 index, as the market portfolio,

and compute

cov (r, rM ) =1

n− 1

n∑i=1

(ri − r̄)(rM,i − r̄M ), β =cov (r, rM )

var (rM ).

Step 3. Compute Jensen’s α:

α := r̄ − rf − β(r̄M − rf ).

Shuzhong Zhang

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The Sharpe ratio

Another very important quantity in investment science is known as the

Sharpe ratio of an asset a:

Sa =ra − rf

σa.

By the CAPM theory, any asset, or a portfolio of assets, its Sharpe

ratio cannot exceed κ. Moreover, any efficient portfolio’s Sharpe ratio

attains κ.

It is also a good test to see if an asset or a portfolio is efficient by

computing its Sharpe ratio, and compare it with that of the market

portfolio.

Shuzhong Zhang

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Pricing based on CAPM

Let P be the price where an asset is purchased. Suppose Q is the price

when it is to be sold. Suppose that the asset is efficient. Then

P =Q̄

1 + rf + β(r̄M − rf ).

This formula has a nice interpretation. In the case of deterministic

interest arrangement, the discounting factor is 1/(1 + rf ). Therefore,

1 + rf + β(r̄M − rf ) can be thought as the risk-adjusted discounting

factor.

Shuzhong Zhang

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Example 7.6. Let us consider the investment in an oil venture share

again. The expected payoff is $1,000 and the standard deviation is

40%. The beta of the asset is β = 0.6, which is relatively low because,

although the uncertainty in return due to oil prices is correlated with

the market portfolio, the uncertainty associated with exploration is

not. The risk-free rate is rf = 10%, and the expected rate of return on

the market portfolio is 17%. What is the value of the oil venture?

P =$1, 000

1.10 + 0.6(0.17− 0.10)= $876,

which is quite similar to the market price we saw: $875.

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Linearity of the equivalent pricing formula

By using

β =cov

(QP − 1, rM

)σ2M

and

P =Q̄

1 + rf + β(r̄M − rf )

we have

P =1

1 + rf

[Q̄− cov (Q, rM )(r̄M − rf )

σ2M

].

It is linear because P depends linearly on Q.

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To write it in a different form. Consider an asset. Let its current price

be P and its return be Q. Let r = Q/P − 1. According to CAPM,

E[r]− rf = λcov (r, rM )

where λ = (r̄M − rf )/σ2M . This gives us

1

PE[Q]− rf =

λ

Pcov (Q, rM )

Therefore,

P = (1 + rf )−1(E[Q]− λvar (Q, rM ))

= (1 + rf )−1E[Q(1− λ(rM − E[rM ]))].

Let

fM = 1− λ(rM − E[rM ])

We have a formula for pricing all assets:

P = (1 + rf )−1E[QfM ].

Shuzhong Zhang

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Project Choice

Let us consider a project which requires an outlay of P and generates

a net amount of Q (random) after a year.

If we define

NPV = −P+1

1 + rf

[Q̄− cov (Q, rM )(r̄M − rf )

σ2M

]= −P+(1+rf )

−1E[QfM ]

as the net present value (counting the randomness), and pick a project

with positive NPV, then the selection will be consistent with the cash

flow analysis.

Is this also consistent with the CAPM?

Yes, because this project will alter the market portfolio (hence the

efficient frontier): the current market portfolio violates the equation

P +NPV = (1 + rf )−1E [QfM ] > P.

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However, we will see below that CAPM can also yield strange result in

very specific situations.

Consider the following contingent security:

ν :=

c, if rM > E[rM ] + λ−1,

0, otherwise

where c = (λ(rM − E[rM ])− 1)−1 > 0.

Clearly, the payoff is always nonnegative and with positive probability

it is positive.

Now, if we apply the CAPM pricing formula, then

P (ν) = −(1 + rf )−1Prob {rM > E[rM ] + λ−1}

which is negative!

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In practice, one often compare one asset with a related well priced

asset, instead of using the market portfolio (say S&P 500 index) as a

benchmark.

Recall the CAPM formula

P =1

1 + rf

[Q̄− cov (Q, rM )(r̄M − rf )

σ2M

].

Correlation Pricing Formula

Let M be a marketed asset most correlated with x. If we replace rM

by M/PM − 1, then the projection price of x may be computed as

Px = (1 + rf )−1{x̄− βx,M

[M̄ − PM (1 + rf )

]}.

Shuzhong Zhang

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Example 7.9. The management team of a software firm is considering

the development of a new product, and wants to assign a suitable

value to it (annual worth). The net revenue x from the product (in

one year) has expected value $100,000. The current interest rate is

5%. The team feels that they should discount more heavily. The CEO

suggests that they should use the company’s stock as a most related

marketed security. Historically, the correlation coefficient between

such a product and the stock price is ρ = 0.8. The current stock price

is M = $100 per share, and it is believed to increase by 15% next year.

The ratio between σx/σM is believed to be 900. What is the estimated

value of the product?

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We have

Px =1

1 + rf

[x̄− σxM

σ2M

(M̄ − PM (1 + rf )

)]=

1

1.05

[100, 000− σxM

σxσM

σx

σM(115− 100(1 + 0.05))

]=

1

1.05[100, 000− 0.8× 900× 10]

= $88, 381.

Therefore, the budget for the product, according to the correlation

pricing, should be $88,381.

Shuzhong Zhang