FINM 6900 Finance Theory

University of Queensland

Lecture Note 1The Capital Asset Pricing Model

1. Introduction

Suppose that you, as a risk averse investor, wanted a simple rule for choosing betweenvarious investment alternatives. One rule that you might consider is to select theinvestment that delivers the highest expected return for a given level of variance.That is, you might decide that you wanted to maximize expected return and minimizevariance. Even if you had only a passing familiarity with economic theory, you wouldprobably agree that this approach sounds quite sensible. Later on, we will see thatthe mean-variance rule of portfolio selection is fully consistent with expected utilitymaximization only under special circumstances. For the moment, however, we want toconsider how an investor might behave under circumstances where the mean-varianceapproach is optimal.

1.1. Diversification

We begin with the following two assumptions:

1. Investors are risk averse.

2. Investors seek to maximize expected return and minimize variance.

Under these conditions, all investors will want to hold diversified portfolios. Thereason is that diversification reduces variance.

A Simple ExampleTake the case of a portfolio formed from two assets. The expected return on thisportfolio is given by the formula

Ep = X1E1 +X2E2

1

where X1 = the proportion of wealth invested in asset 1

X2 = 1−X1.

Similarly, the variance of this portfolio is given by

σ2p = X2

1σ21 +X2

2σ22 + 2X1X2σ12.

Recall that the covariance between asset 1 and asset 2 can be written as a functionof the correlation coefficient and the standard deviations

σ12 = r12σ1σ2

Thus, we can write the portfolio variance as

σ2p = X2

1σ21 +X2

2σ22 + 2X1X2r12σ1σ2

There are two possible scenarios depending on the value of the correlation coefficient.Each of these scenarios is considered below.

Scenario 1: r12 = −1

For this situation, it is possible to combine assets 1 and 2 in such a way that theportfolio risk is set to zero. To see this, note that when r12 = −1, the portfoliovariance reduces to:

σ2p = X2

1σ21 +X2

2σ22 − 2X1X2σ1σ2

which implies that

σp =√

(X1σ1 −X2σ2)2

If we setX1 =

σ2σ1 + σ2

the portfolio variance is zero.

Proof:Because

σp =√

(X1σ1 −X2σ2)2,

it is clear that when X1σ1 = X2σ2, the portfolio variance will be equal to zero. Recall,however, that X2 = (1−X1). Thus, to achieve a zero variance we set

X1σ1 = (1−X1)σ2

orX1 =

σ2σ1 + σ2

.

2

Scenario 2: −1 < r12 < 1

Next we take the case where −1 < r12 < 1. Assume without loss of generalitythat asset 2 has the lowest variance. A portfolio of asset 1 and asset 2 will have lowervariance than asset 2 whenever

r12 <σ2σ1

Proof:The minimum variance portfolio is given by:

minx1

σ2p = X2

1σ21 + (1−X1)

2σ22 + 2X1(1−X1)r12σ1σ2

= X21σ

21 + σ2

2 − 2X1σ22 +X2

1σ22 + 2X1r12σ1σ2 − 2X2

1r12σ1σ2.

The first order condition is

2X1σ21 − 2σ2

2 + 2X1σ22 + 2r12σ1σ2 − 4X1r12σ1σ2 = 0.

Rearranging yields

X1(σ21 + σ2

2 − 2r12σ1σ2) = σ22 − r12σ1σ2

or

X1 =σ22 − r12σ1σ2

σ21 + σ2

2 − 2r12σ1σ2

Now, the portfolio will have less risk than asset 2 by itself whenever X1 > 0, whichwill occur whenever

σ22 − r12σ1σ2 > 0

orr12 <

σ2σ1.

1.2. Portfolio Variance

Because diversification reduces risk, investors will want to diversify and hold a largeportfolio of many stocks. For ease of exposition, lets look at an equally weightedportfolio of N stocks. The return on this portfolio is

Rp =1

N(R1 +R2 +R3 + . . .+RN)

3

and the variance is

σ2p =

1

N2

N∑i=1

σ2i +

1

N2

N∑i=1

N∑j=1︸ ︷︷ ︸

i 6=j

σij

The variance of the portfolio reduces to

σ2p =

1

N2N [ Average Variance ] +

1

N2(N2 −N)[ Average Covariance]

so as N →∞,σ2p → Average Covariance.

1.3. Risk in a Portfolio Sense

Think of adding a new security to your portfolio. How will this addition affect theriskiness of your holdings? Well, the above result suggests that the answer is by itscontribution to the average covariance of the portfolio. In other words, cov(Ri, Rp),the covariance of the return on a security with that of a well diversified portfolio, isthe appropriate measure of the riskiness of the security when all investors hold welldiversified portfolios.

Now, because each investor holds a large, well diversified portfolio, we can saythat each investor’s portfolio is approximately equal to the market portfolio. Hence,we can assume that

cov(Ri, Rp) ≈ cov(Ri, Rm)

which implies that cov(Ri, Rm) is the basic determinant of the riskiness of an asset.

1.4. The Final Equation

Our intuition tells us that when we invest in a risky asset we should get an expected re-turn of the risk free rate plus a premium for each unit of risk that we undertake. Fromthe above analysis we know that the risk of a security is measured by cov(Ri, Rm).Thus, we have a relation of the form

E[Ri] = Rf + cov(Ri, Rm)[ Risk Premium]

What is the form of the risk premium? Because we are compensated for marketrisk,it is only natural to think that the risk premium will be that of the market. If we

4

define this risk premium to be the expected excess return on the market per unit ofmarket risk, we have

E[Ri] = Rf + cov(Ri, Rm)E[Rm −Rf ]

σ2m

or setting βi = cov(Ri,Rm)σ2m

E[Ri] = Rf + βiE[Rm −Rf ]

This is the Sharpe (1964) – Lintner (1965) CAPM.

5

2. Mathematical Derivation

Assume that there are N risky assets whose unequal expected rates of return aregive by the vector e and whose finite variances are given by the positive definitevariance-covariance matrix V .

2.1. The Sharpe–Lintner CAPM

Step 1Make assumptions sufficient for all investors to want to hold mean-variance efficientportfolios. For example,

1. All investors have quadratic utility functions or

2. Asset returns conform to a multivariate elliptical distribution

Step 2Solve the individual’s portfolio problem. That is, choose portfolio weights (x) to max-imize the expected return on the portfolio (x′e) subject to a given portfolio variance(x′V x)

maxx

x′e+ (1− x′1)rf + λ[c− 1

2x′V x]

This problem yields the following first order necessary and sufficient conditions

e = rf + λV x (1)

c =1

2x′V x (2)

for a given efficient portfolio x.

Step 3Find the expected return on the market portfolio.The market portfolio (xm) is efficient (proof on page 8), which implies from equation(1) that

e = rf + λV xm (3)

Premultiply equation (3) by x′m

x′me = x′mrf + λx′mV xm

orE[Rm] = rf + λvar(Rm) (4)

6

Step 4Find the expected return on an individual security.We know from equation (3) that

e = rf + λV xm

Premultiply equation (3) by xi where xi has a weight of one on asset i and zero onall other assets

x′ie = x′irf + λx′iV xm

orE[Ri] = rf + λcov(Ri, Rm) (5)

Step 5Substitute to obtain the Sharpe–Lintner CAPMRecall equation (4)

E[Rm] = rf + λvar(Rm)

This implies that

λ =E[Rm]− rfvar(Rm)

Substitute this expression for λ into equation (5) to obtain

E[Ri] = rf +E[Rm]− rfvar(Rm)

cov(Ri, Rm)

Rearranging terms we have

E[Ri] = rf +cov(Ri, Rm)

var(Rm)E[Rm − rf ] (6)

orE[Ri] = rf + βiE[Rm − rf ]

which is the Sharpe–Lintner CAPM

7

Proof that market portfolio is efficient

Subscript individuals by v and recall individual v’s first order necessary and sufficientcondition

e = rf + λvV xv (1′)

→ xv = V −1e− rfλv

(7)

Let wv equal the proportion of total wealth held by individual v

→ xm =I∑v=1

wvxv

Substituting for each individual’s portfolios xv (equation (7)) yields

xm =I∑v=1

wvV −1e− rfλv

→ xm = V −1eI∑v=1

wv

λv− V −1rf

I∑v=1

wv

λv

Solving for e we have

eI∑v=1

wv

λv= V xm + rf

I∑v=1

wv

λv

→ e =

[I∑v=1

wv

λv

]−1V xm + rf

[I∑v=1

wv

λv

]−1 I∑v=1

wv

λv

ore = rf + λV xm

which, by equation (1), is a necessary and sufficient condition for efficiency. Hencexm, the market portfolio, is efficient.

8

2.2. The Black (1972) CAPM

Step 1Make assumptions sufficient for all investors to want to hold mean-variance efficientportfolios.

Step 2Solve the individual’s portfolio problem. That is, choose portfolio weights (x) to max-imize the expected return on the portfolio (x′e) subject to a given portfolio variance(x′V x)

maxx

x′e+ λ[c− 1

2x′V x] + γ[1− x′1]

This problem yields the following first order necessary and sufficient conditions

e = γ1 + λV x (8)

c =1

2x′V x (9)

1 = x′1 (10)

for a given efficient portfolio x.

Step 3Find the expected return on the market portfolioThe market portfolio (xm) is efficient (proof on page 11), which implies from equation(8) that

e = γ1 + λV xm (11)

Premultiply equation (11) by x′m

x′me = γx′m1 + λx′mV xm

orE[Rm] = γ + λvar(Rm) (12)

Step 4Find the expected return on an individual securityWe know from equation(8) that

e = γ1 + λV xm

Premultiply equation (8) by xi where xi has a weight of one on asset i and zero onall other assets

x′ie = γx′i1 + λx′iV xm

9

orE[Ri] = γ + λcov(Ri, Rm) (13)

Step 5Substitute to obtain the Black CAPMRecall equation (12)

E[Rm] = γ + λvar(Rm)

This implies that

λ =E[Rm]− γvar(Rm)

Substitute this expression for λ into equation (13) to get

E[Ri] = γ +E[Rm]− γvar(Rm)

cov(Ri, Rm)

Rearranging terms we have

E[Ri] = γ +cov(Ri, Rm)

var(Rm)E[Rm − γ] (14)

Now, define xz to be a portfolio uncorrelated with the market portfolio and premul-tiply equation (8) by x′z to get

x′ze = γx′z1 + λx′zV xm

orE[Rz] = γ.

Thus, we haveE[Ri] = E[Rz] + βiE[Rm −Rz]

which is the Black CAPM.

10

Proof that market portfolio is efficient

Subscript individuals by v and recall individual v’s first order necessary and sufficientcondition

e = γv1 + λvV xv (8′)

→ xv = V −1e− γv1λv

(15)

Let wv equal the proportion of total wealth held by individual v

→ xm =I∑v=1

wvxv

Substituting for each individual’s portfolios xv (equation (15)) yields

xm =I∑v=1

wvV −1e− γv1λv

→ xm = V −1eI∑v=1

wv

λv− V −1

I∑v=1

wvγv1

λv

Solving for e we have

eI∑v=1

wv

λv= V xm +

I∑v=1

wvγv1

λv

→ e =

[I∑v=1

wv

λv

]−1V xm +

[I∑v=1

wv

λv

]−1 I∑v=1

wvγv1

λv

ore = γ∗1 + λ∗V xm

which, by equation (8), is a necessary and sufficient condition for efficiency. Hencexm, the market portfolio, is efficient.

11

3. Mathematics of the Portfolio Frontier

In the previous section we considered maximizing expected return subject to a givenvariance. Of course, there is another way to approach the same problem — namely,to minimize variance subject to a given expected return. Both approaches yieldequivalent pricing results. However, richer results concerning the characteristics ofthe portfolio frontier can be generated using the minimization approach. This sectionoutlines the solution to the minimization problem and derives a number of interestingresults concerning the portfolio frontier.

Suppose investors choose portfolio weights (x) to minimize variance (x′V x) sub-ject to a given expected return (E[Rp] = x′e). Our problem becomes

minx

1

2x′V x+ λ(E[Rp]− x′e) + γ(1− x′1)

which has the following necessary and sufficient first order conditions:

V x = λe+ γ1 (16)

E[Rp] = x′e (17)

1 = x′1 (18)

Solving equation (16) for x yields

x = λV −1e+ γV −11 (19)

Thus, we see that all frontier portfolios are linear combinations of just two portfolios(V −1e and V −11).

Let’s solve for the Lagrange multipliers λ and γ so that we have an exact expres-sion for the frontier portfolio x. First we define a number of scalars that will reducethe notational burden

B = e′V −1e

A = e′V −11

C = 1′V −11

D = BC − A2.

Multiply equation (19) by e′ to get

e′x = λe′V −1e+ γe′V −11

Using equation (17) and our defined scalars this reduces to

E[Rp] = λB + γA (20)

12

Multiply equation (19) by 1′ to get

1′x = λ1′V −1e+ γ1′V −11

Using equation (18) and our defined scalars this reduces to

1 = λA+ γC (21)

Combining equations (20) and (21) gives us 2 equations in 2 unknowns(B AA C

)(λγ

)=(E[Rp]

1

)We can solve for the constants, λ and γ, as follows(

λγ

)=(B AA C

)−1 (E[Rp]1

)and thus

λ =CE[Rp]− A

Dand

γ =B − AE[Rp]

D.

Recall from equation (19) that the composition of a frontier portfolio is given by

xp = λV −1e+ γV −11.

Plug in the values of λ and γ to obtain

xp =CE[Rp]− A

DV −1e+

B − AE[Rp]

DV −11.

Upon rearranging, we have

xp =BV −11− AV −1e

D+ [

CV −1e− AV −11D

]E[Rp]

orxp = g + hE[Rp] (22)

where the vectors, g and h are defined as

g =BV −11− AV −1e

D

and

h =CV −1e− AV −11

DEquation (22) is a closed form expression for the necessary and sufficient conditionsfor a portfolio, xp, to be a frontier portfolio.

Having characterized a frontier portfolio, we are in a position to establish a numberof additional results:

13

1. The set of frontier portfolios is a hyperbola in standard deviation, expectedreturn space. The hyperbola has its center at (0, A

C) and asymptotes

E[Rp] =A

C+−

√D

Cσp

ProofThe covariance between two arbitrary frontier portfolios, xp and xq, is given by

cov(Rp, Rq) = x′pV xq

= [g + hE[Rp]]′V [g + hE[Rq]]

=C

D[E[Rp]−

A

C][E[Rq]−

A

C] +

1

C

Thus

σpq =C

D[E[Rp]−

A

C][E[Rq]−

A

C] +

1

C(23)

To find the variance of a frontier portfolio, we simply set Rp = Rq in the aboveexpression to obtain

σ2p =

C

D[E[Rp]−

A

C]2 +

1

C(24)

Upon rearranging, we find that

σ2p1C

−[E[Rp]− A

C]2

DC2

= 1

which is the equation of a hyperbola in σ,E[Rp] space with center (0, AC

) andasymptotes

E[Rp] =A

C+−

√D

Cσp

Refer to figure 1 below.

14

2. A frontier portfolio is efficient if E[Rp] >AC

This result is obvious from anexamination of figure 1. An efficient portfolio has the highest expected returnfor a given level of standard deviation.

3. The global minimum variance portfolio has standard deviation of√

1C

and ex-

pected return of AC

ProofFrom equation (24), the variance of a frontier portfolio is

σ2p =

C

D[E[Rp]−

A

C]2 +

1

C

If we choose the value of E[Rp] which minimizes this function we find thatE[Rp] = A

Cwith the minimum variance equal to 1

C

4. The covariance between the global minimum variance portfolio (xmvp) and anyother portfolio (xq) is the variance of the global minimum variance portfolio

cov(Rmvp, Rq) =1

C

ProofForm a portfolio with weight w in portfolio q and weight 1 − w in the globalmvp and consider the following problem:

minw

w2σ2q + 2w(1− w)cov(Rmvp, Rq) + (1− w)2σ2

mvp.

The first order condition is

2wσ2q + (2− 4w)cov(Rmvp, Rq)− 2(1− w)σ2

mvp = 0.

However, the portfolio mvp has the global minimum variance, so w must equalzero. Thus, we have

2cov(Rmvp, Rq)− 2σ2mvp = 0.

orcov(Rmvp, Rq) = σ2

mvp.

5. For any portfolio (xp) on the frontier (except the global minimum varianceportfolio) their exists a unique frontier portfolio (xzp) whose return has zerocovariance with (xp). This portfolio has expected return as follows

E[Rzp] =A

C−

DC2

E[Rp]− AC

(25)

15

ProofRecall from equation (23) that the covariance between the returns on the twofrontier portfolios xp and xzp is given by

cov(Rp, Rzp) =C

D[E[Rp]−

A

C][E[Rzp]−

A

C] +

1

C

Setting this covariance to zero and solving for E[Rzp] we obtain

E[Rzp] =A

C−

DC2

E[Rp]− AC

as desired. This frontier portfolio can be obtained by setting

xzp = g + hE[Rzp].

6. If portfolio xp is efficient, then xzp is inefficient.

ProofFrom equation (25)

E[Rzp] =A

C−

DC2

E[Rp]− AC

.

From result 2 we know that efficient portfolios have E[Rp] >AC

. However,E[Rzp] must be less than A

Cbecause D, C and E[Rp] − A

Care all greater than

zero.

7. Portfolio g is a frontier portfolio whose expected return is zero and portfoliog + h is a frontier portfolio whose expected return is 1.

ProofRecall equation (22), the necessary and sufficient condition for a frontier port-folio:

xp = g + hE[Rp].

Set E[Rp] = 0 to getxp = g.

Set E[Rp] = 1 to getxp = g + h.

8. The portfolio frontier is spanned by the frontier portfolios, g and g + h.

ProofSay we want a frontier portfolio with expected return of E[Rq]. We form ourportfolio (xq) by investing E[Rq] in portfolio

16

g + h and (1− E[Rq]) in portfolio g. Hence

xq = (1− E[Rq])g + E[Rq](g + h)

orxq = g + hE[Rq].

This is the necessary and sufficient condition for a portfolio to be on the frontier.Thus, xq is a frontier portfolio. Because q was chosen arbitrarily, we have shownthat any frontier portfolio can be generated using these two frontier portfolios.

9. The portfolio frontier can be spanned by any two frontier portfolios.

ProofConsider two frontier portfolios xp1 and xp2. Suppose we desire an expectedreturn of

E[Rq] = αE[Rp1] + (1− α)E[Rp2]

We form our portfolio (xq) by investing α in portfolio

xp1 and 1− α in portfolio xp2. Hence

xq = αxp1 + (1− α)xp2.

Because xp1 and xp2 are frontier portfolios

xq = α[g + hE(Rp1)] + (1− α)[g + hE(Rp2)]

which simplifies toxq = g + hE[Rq].

10. A linear combination of frontier portfolios is on the frontier.

ProofLet αi be weights such that

∑mi=1 αi = 1 and let xi be a set of m frontier

portfolios. We form a portfolio

xq =m∑i=1

αixi

Using the fact that xi is a frontier portfolio we have

xq =m∑i=1

αi(g + hE[Ri]).

which simplifies to

xq = g + hm∑i=1

αiE[Ri]

and thus xq is a frontier portfolio.

17

11. If xi, i = 1 . . .m is efficient,∑mi=1 αi = 1, and αi ≥ 0 ∀i, then xq =

∑mi=1 αixi is

efficient.

ProofWe know from the previous result that xq is on the frontier. Now we need toshow that E[Rq] >

AC

. This must be the case because

E[Rq] =m∑i=1

αiE[Ri] >m∑i=1

αiA

C=A

C

Thus, xq is efficient.

12. The covariance between the return on a frontier portfolio (xp) and that on anarbitrary portfolio (xq) which is not necessarily on the frontier is given by

cov(Rp, Rq) = λE[Rq] + γ

ProofBy definition of covariance we have

cov(Rp, Rq) = x′pV xq

Because xp is on the frontier, we can use the first order condition given byequation (19) to substitute for xp

cov(Rp, Rq) = (λV −1e+ γV −11)′V xq

→ cov(Rp, Rq) = λe′V −1V xq + γ1′V −1V xq

→ cov(Rp, Rq) = λe′xq + γ1′xq

orcov(Rp, Rq) = λE[Rq] + γ

13. The expected return of an arbitrary portfolio (xq) which is not necessarily onthe frontier is given by

E[Rq] = E[Rzp] + βqpE[Rp −Rzp] (26)

where xp and xzp are two frontier portfolios such that cov(Rp, Rzp) = 0.

ProofFrom the above result we know

cov(Rp, Rq) = λE[Rq] + γ.

Plug in the values of λ and γ, isolate E[Rq] on the left-hand side and simplifyto obtain the desired result.

18

4. Why Expected Return and Variance

In this section, we look at the conditions under which the mean-variance rule ofportfolio selection is fully consistent with expected utility maximization. We considerassumptions on

1. The Joint Distribution of Asset Returns

2. Utility Functions

either of which are sufficient to imply that investors will want to hold mean-varianceefficient portfolios.

4.1. Assumptions on the Joint Distribution of Returns

First we briefly review a result known as stochastic dominance. We assume no specificform for the utility function. We do assume, however, that individual risk preferencesare such that U , the utility function, is concave.

Second Degree Stochastic DominanceAll risk averse individual prefer A to B

E(UA) ≥ E(UB)

if and only if

RBd= RA + ε

andE[ε|RA] = 0 (27)

Note that one direct consequence of the above conditions is that

E[RA] = E[RB]

andvar(RA) ≤ var(RB)

Proof (Sufficiency Only)

Because RBd= RA + ε, we can write

E[U(RB)] = E[U(RA + ε)]

= E[E[U(RA + ε)|RA]]

≤ E[U(E[RA + ε|RA])] (by Jensens inequality)

= E[U(RA)] (because E[RA|RA] = RA and E[ε|RA] = 0)

19

Thus, A is preferred to B.

ApplicationRecall from result 13 that the expected return on an arbitrary portfolio (xq), whichis not necessarily on the frontier, is given by

E[Rq] = E[Rzp] + βqpE[Rp −Rzp]

where xp and xzp are two frontier portfolios such that cov(Rp, Rzp) = 0. When wedrop the expectations operator, we have

Rq = Rzp + βqp[Rp −Rzp] + ε

which is the first condition shown above. Thus, if the second condition

E[ε|Rzp + βqp(Rp −Rzp)] = 0

is satisfied, we have second degree stochastic dominance, and all risk averse investorswill prefer A (i.e., Rzp + βqp[Rp −Rzp]) to B (i.e., Rq). That is,

E[U(Rzp + βqp[Rp −Rzp])] ≥ E[U(Rq)].

Thus, investors will always choose a portfolio which is a linear combination of twofrontier portfolios because such a combination dominates any other investment alter-native. This result is known as two fund separation. The condition

E[ε|Rzp + βqp(Rp −Rzp)] = 0

is a necessary and sufficient condition for two fund separation. This condition issatisfied when the joint distribution of returns belongs to the class of multivariate el-liptical distributions. The most commonly encountered example from this class is themultivariate normal distribution. Another example is the multivariate t distribution.

4.2. Assumptions on Utility Functions

Quadratic UtilityIf investors have quadratic utility functions, they will choose to maximize expectedreturn and minimize variance. Quadratic utility of end of period wealth implies that

U(W ) = W − aW 2

where a is a constant. Expected utility is given by

E[U(W )] = E[W ]− aE[W 2].

20

Becausevar(W ) = E[W 2]− E[W ]2

we can substituteE[W 2] = var(W ) + E[W ]2

in the above expression for expected utility to get

E[U(W )] = E[W ]− a(E[W ]2 + var(W )).

Quadratic utility, however, has a number of problems

1. It implies that investors become satiated. That is, increasing wealth beyondsome point actually reduces utility.

2. It implies increasing absolute risk aversion. Thus, risky assets are inferior goods

Our intuition tells us that people will always prefer more to less, and that risky assetsare normal goods.

21

5. Implications of the CAPM

The Sharpe Lintner CAPM implies that all investors will hold a linear combinationof the market portfolio and the risk free asset. The Black CAPM, on the other hand,implies that each investor will hold an efficient portfolio; this portfolio need not bethe same for different individuals. Both models, however, imply that the marketportfolio is efficient. This result follows from the fact that the market portfolio is alinear combination of individual portfolios and all the weights are nonnegative. Thus,the testable implication of both the Sharpe–Lintner CAPM and the Black CAPM isthat the market portfolio is efficient.

22