15 CAPM and portfolio management

15

CAPM and portfolio management

15.1

Theoretical foundation for mean-variance analysis

• We assume that investors try to maximize the expected utility,

E[U (W )], where W is their wealth.

• Consider a Taylor expansion of the utility function around

E[W ] (the

expected wealth):

U (W ) = U (E[W ]) + U

(E[W ]) (W − E[W ])

+

1

2

U

(E[W ]) (W − E[W ])

+

X

1

n!

U

_{(E[W ]) (W − E[W ])}

• Next, take the expectation:

E[U (W )] = U (E[W ]) + U

(E[W ])E [(W − E[W ])]

+

1

2

U

′′

(E[W ])E



(W − E[W ])



+

X

nf ty

1

n!

U

(n)

_{(E[W ])E [(W − E[W ])}

]

• With a mean-variance analysis we stop at the second order.

• There are two cases where this can be justified:

– If

W is normally distributed, then the first two moments

characterize all the moments.

15.2

Portfolio with minimum variance

15.2.1

Simple case: 2 assets

• Consider the following two assets

x

and

x

with

–

V ar[x

] = σ

,

V ar[x

] = σ

and

Cov[x

, x

] = σ

–

w

: the weight of the first asset in the portfolio

–

1 − w

: the weight of the second asset in the portfolio

• Denote by

σ

the variance of the portfolio:

σ

= V ar[w

x

+ (1 − w

)x

]

• We want to find the minimum variance portfolio:

min

σ

• The First Order Condition (FOC) is:

2w

σ

−

2(1 − w

)σ

+ 2(1 − w

)σ

+ 2w

(−1)σ

= 0

• Solving for

w

, we get

w

=

σ

2

2

−

σ

• Diversification principle:

– Look at the FOC when we take

w

= 0:

∂σ

∂w

(w

= 0) = 2(σ

−

σ

)

= 2(ρ

σ

σ

−

σ

)

= 2σ

σ



ρ

−

σ

σ



– If

ρ

< 0 or if ρ

> 0 but σ

/σ

> ρ

, then

(w

= 0) < 0 so I

should increase

w

(i.e. buying

x

).

– If

ρ

> 0 but σ

/σ

< ρ

, then

∂w₁

(w

= 0) > 0 so I should

15.2.2

Case with

N assets

• Consider the following elements:

–

w = (w

, w

, . . . , w

)

: portfolio weights

–

x = (x

, x

, . . . , x

)

: asset returns

–

x = (¯

¯

x

, ¯

x

, . . . , ¯

x

)

: expected asset returns

– the variance matrix

Σ =















σ

σ

· · ·

σ

σ

σ

· · ·

σ

..

.

..

.

. ..

..

.

σ

σ

· · ·

σ















–

x

= w

x: portfolio’s return

–

σ

= w

Σw: portfolio’s variance

• Define the following elements:

–

i: N × 1 vector of 1.

–

A = i

Σ

¯

x

–

B = ¯

x

Σ

¯

x

–

C = i

Σ

i

–

D = BC − A

(we can show that

D > 0)

• Show that

w

• We want to characterize the mean-variance frontier (finding the

portfolio with the lowest variance for a given expected return)

• The problem is

min

1

2

w

Σw

subject to

i

w = 1

¯

x

w = µ

• This is a constrained optimization

L =

1

2

w

Σw + γ(1 − i

w) + λ(µ − ¯

x

w)

• The FOCs are

∂L

∂w

=















∂L/∂w

∂L/∂w

..

.

∂L/∂w















= Σw − γi − λ¯

x = 0

(1)

∂L

∂γ

= 1 − i

w = 0

(2)

∂L

∂λ

= µ − ¯

x

w = 0

(3)

• From equation (1), we get

w

= γΣ

i + λΣ

¯

x

• We need to solve for

γ and λ

• From equation (2), we know that

1 − i

w

= 0

1 − i

[γΣ

i + λΣ

¯

x] = 0

1 − γi

Σ

i − λi

Σ

¯

x = 0

1 − γC − λA = 0

(4)

• From equation (3), we know that

µ − ¯

x

w

= 0

µ − ¯

x

[γΣ

i + λΣ

¯

x] = 0

µ − γ ¯

x

Σ

i − λ¯

x

Σ

¯

x = 0

µ − γA − λB = 0

(5)

• We can solve (4) and (5) for

γ and λ. We get

λ =

Cµ − A

D

γ =

B − Aµ

D

• It follows that the optimal portfolio is

w

=

 B − Aµ

D



Σ

i +

 Cµ − A

D



Σ

¯

x

=

 BΣ

i

D

−

AΣ

¯

x

D



+

 CΣ

¯

x

D

−

AΣ

i

D



µ

= g + hµ

where

g =

1

D



BΣ

i − AΣ

¯

x



h =

1

D



CΣ

¯

x − AΣ

i



• Variance of the portfolio when

w = w

σ

= w

Σw

= w

Σ(γΣ

i + λΣ

¯

x)

= w



γΣΣ

i + λΣΣ

¯

x



= w

[γi + λ¯

x]

= γ w

i

|{z}

+λ w

x

¯

|{z}

= γ + λµ

=

B − Aµ

D

+

 Cµ − A

D



µ

=

B − 2Aµ + Cµ

D

⇒

parabola

0 0

Expected return and variance combination for ω=ω*

σ2

• We can find the global minimal variance

∂σ

∂µ

=

2Cµ − 2A

D

= 0

⇒

µ

=

A

C

• What is this variance?

(σ

)

=

B − 2A

+ C

D

=

BC − 2A

_{+ A}

CD

=

BC − A

CD

=

1

C

0 0

Minimum variance portfolio

σ

µ

• What is

γ and λ for this µ

?

λ

=

C

−

A

D

= 0

expected return constraint not binding

γ

=

B − A

D

=

1

C

• What is the portfolio with minimum global variance?

w

= γ

Σ

i + λ

Σ

¯

x

=

1

C

Σ

i + 0Σ

¯

x

=

1

C

Σ

i

=

Σ

i

i

Σ

i

• If we go back to

w

_{(optimal portfolio for a given expected return):}

w

= γΣ

i + λΣ

¯

x

= γC

 Σ

i

C



|

{z

}

+λA

 Σ

¯

x

A



|

{z

}

We see that

w

is a combination of:

– The portfolio with the lowest global variance but lowest expected

return (

w

).

– A second portfolio (

w

) that will increase expected return but will

• But what is

γC + λA?

γC + λA =

 B − Aµ

D



C +

 Cµ − A

D



A

=

BC − ACµ

D

+

ACµ − A

D

=

BC − A

D

= 1

15.3

Covariance properties of minimal variance

portfolios

•

w

has a covariance constant with every asset or portfolio (

= 1/C):

Cov(x

, x

) = E



w

(x − ¯

x)(x − ¯

x)

w



= w

E[(x − ¯

x)(x − ¯

x)

]w

= w

Σw

=

 i

Σ

C



Σw

=

i

w

C

=

1

C

• Covariance of portfolio

w

with any other portfolio:

Cov(x

, x

) = E [w

(x − ¯

x)(x − ¯

x)

w

]

= w

Σw

=

xΣ

¯

A

Σw

=

xw

¯

A

=

x

¯

A

We see that the expected return of any portfolio will be proportional

to its covariance with

w

since

x

¯

= A Cov(x

, x

).

• Consider a portfolio

a on the minimum variance frontier

(

w

= (1 − a)w

+ aw

). What is the covariance between

a and

another portfolio

p?

Cov(x

, x

) = (1 − a)Cov(x

, x

) + aCov(x

, x

)

= (1 − a)

1

C

+ a

¯

x

A

If

x

= x

, then

Cov(x

, x

) = V ar(x

) =

1 − a

C

+

a

A

x

¯

As long as

a is not the minimum variance portfolio, it’s possible to

find a portfolio

z that has a zero covariance with a:

Cov(x

, x

) =

1 − a

C

+ a

¯

x

A

= 0

⇒

x

¯

=

a − 1

a

A

C

This is the expected return of a portfolio with zero covariance with

any portfolio on the minimum variance frontier.

• We saw previously that

V ar(x

) =

1 − a

C

+

a

A

x

¯

= −

a¯

x

A

+

a

A

x

¯

using the result from previous slide

=

a

• Next, define

β

≡

Cov(x

, x

)

V ar(x

)

• Using previous results

β

=

+

x

¯

(¯

x

−

x

¯

)

β

(¯

x

−

x

¯

) =

A

a

 1 − a

C

+

a

A

x

¯



=

A

a

 (1 − a)A + aC ¯x

CA



=

1 − a

a

A

C

|

{z

}

+¯

x

¯

x

= ¯

x

+ β

(¯

x

−

x

¯

)

• The last equality is the CAPM equation without a risk-free asset.

• It is telling us that the expected return on any portfolio

p (i.e. ¯

x

) is

equal to the expected return on a portfolio uncorrelated with portfolio

a (i.e. ¯

x

) plus

β

times the excess return of

a over z.

Portfolio

a is a portfolio on the minimum variance frontier. Portfolio

z is a portfolio uncorrelated with portfolio a.

15.4

Introduction of a riskless asset

• Now assume there is one more asset. This asset is riskless and has a

risk-free rate

r

.

• The problem is now

min

1

2

w

Σw

subject to

w

¯

x + (1 − w

i)r

= µ

• We form the Lagrangian

L =

1

2

w

Σw + λ(µ − w

¯

x − (1 − w

i)r

)

• The FOCs are:

∂L

∂w

= Σw + λ(−¯

x + ir

) = 0

(6)

∂L

∂λ

= µ − w

¯

x − (1 − w

i)r

= 0

(7)

• In equation (6) we can solve for

w:

w

= λΣ

• We can solve for

λ using equation (7):

µ = w

x + (1 − w

¯

i)r

µ = [λΣ

(¯

x − ir

)]

¯

x + (1 − [λΣ

(¯

x − ir

)]

i)r

µ = λ(¯

x

−

r

i

)Σ

x + (1 − λ(¯

¯

x

−

r

i

Σ

i))r

µ = λ(¯

x

Σ

¯

x − r

i

Σ

¯

x) + r

−

λ(¯

x

Σ

i − r

i

Σ

i)r

µ = λ(B − r

A) + r

−

λ(A − r

C)r

µ − r

= λ(B − 2Ar

+ Cr

)

λ =

µ − r

H

(9)

where

H = B − 2Ar

+ Cr

• Equation (9) into equation (8):

w

= Σ

(¯

x − ir

)

 µ − r

H

• The variance of this portfolio is

σ

= w

Σw

= w

Σ



Σ

(¯

x − ir

)

 µ − r

H



= w



(¯

x − ir

)

 µ − r

H



= (w

x

¯

|{z}

−

w

i

|{z}

r

)

(µ − r

)

H

=

(µ − r

)

H

0 0

Expected return and variance combination with a riskless asset

σ2

• Recall the parabola when we have

N risky assets:

σ

=

B − 2Aµ + Cµ

D

Minimum variance portfolio

σ2 µ µ_g = A/C 1/C Efficient portfolios Inefficient portfolios

• In the

µ − σ plane we get

0 0

Expected return and standard deviation combination

σ

µ

A/C

• It can be argued that in equilibrium we should have

r

< A/C

0 0

Expected return and standard deviation combination

σ µ A/C 1/2 r f

• If we create a portfolio by combining the risk-free asset with a

portfolio

b on the frontier, we could get the following combination of

µ and σ

0 0

Expected return and standard deviation combination

σ µ A/C 1/C1/2 r f b

• The portfolio

x

would not be optimal. It is possible to get a higher

µ

for the same

σ by switching from b to m (a portfolio that is tangent)

0 0

Expected return and standard deviation combination

σ µ A/C 1/C1/2 r f m

• It follows that everyone should choose a portfolio which falls on the

r

−

m line.

– If you want higher pair

(µ, σ), you put more weight on m.

– If you want lower pair

(µ, σ), you put more weight on risk-free asset.

– The relative proportion of the risky assets should be the same

regardless of where you are on the

r

−

m line. We refer to m as the

market portfolio.

– Your risk aversion will determine where on the

r

−

m line you are

* Higher risk aversion ⇒ close to r

• What is the CAPM equation when we have a risk-free asset?

• consider the portfolio

m (which is a portfolio on the frontier), then

for a portfolio

q

Cov(x

, x

) = w

Σw

= w

Σ



Σ

(¯

x − ir

)

µ

−

r

H



= w

(¯

x − ir

)

µ

−

r

H

=

(µ

−

r

)(µ

−

r

)

H

• But we also know from slide 28, when we take

µ = µ

, that

σ

=

(µ

−

r

)

H

⇒

µ

−

r

H

=

σ

(µ

−

r

)

• We can next combine the last equation with the

Cov(x

, x

)

equation on the previous slide to get

Cov(x

, x

) = (µ

−

r

)

σ

µ

−

r

µ

−

r

=

Cov(x

, x

)

σ

|

{z

}

(µ

−

r

)

• Conclusion: the expected return of an asset/portfolio:

– Does not depend on its variance.