Advanced Financial EconomicsHomework 2

Due on April 14th before class

March 30, 2015

1. (20 points) An agent has Y0 = 1 to invest. On the market two financial

assets exist. The first one is riskless. The price is one and its return rf = 2.

Short selling on this asset is allowed. The second asset is risky. Its price is 1

and its return r, where r is a random variable with probability distribution

r =

1 with probability π1,

2 with probability π2,

3 with probability π3.

No short selling is allowed on this asset.

a) If the agent invests a in the risky asset, what is the probability distribution

of the agent’s portfolio return rP ?

b) A risk averse agent maximizes a von Neumann utility,

E{u[Y0rf + a(r − rf )]}.

Show the optimal choice of a is positive if and only if the expectation

of r is greater than 2.

c) Find the optimal choice of a when u(Y ) = 1− exp(−bY ), b > 0 and when

u(Y ) = 11−γY

1−γ, 0 < γ < 1.

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d) Find the absolute risk aversion coefficient in either case of question c). If

Y increases, how will the agent react?

Answer:

a)

rP =

2(1− a) + a = 2− a with probability π1

2(1− a) + 2a = 2 with probability π2

2(1− a) + 3a = 2 + a with probability π3

b) Define

W (a) = E{u[Y0rf + a(r − rf )]} = π1u(2− a) + π2u(2) + π3u(2 + a)

The optimal choice of a, denoted by a, satisfies the first order condition,

W ′(a) = −π1u′(2− a) + π3u′(2 + a) = 0

By risk aversion (u′′ < 0), W ′′(a) = π1u′′(2− a) + π3u

′′(2 + a) < 0, i.e.

W ′(a) is everywhere decreasing. We have

a > 0⇔ W ′(0) > W ′(a)⇔ u′(2)(π3 − π1) > 0⇔ π3 > π1 ⇔ E(r) > 2,

where E(r) = π1 + 2π2 + 3π3 = 2− π1 + π3.

c) • In the case that u(Y ) = 1− exp(−bY ), b > 0,

W (a) = π1[1− exp(−b(2− a))] + π2[1− exp(−ba)] + π3[1− exp(−b(2 + a))]

a satisfies the first order condition,

W ′(a) = −π1b exp(−b(2− a)) + π3b exp(−b(2 + a)) = 0

We have,

ln(π3b)[−b(2 + a)] = ln(π1b)[−b(2− a)]

a =ln(π3)− ln(π1)

2b

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• In the case that u(Y ) = 11−γY

1−γ, 0 < γ < 1,

W (a) = π1

[1

1− γ(2− a)1−γ

]+ π2

[1

1− γ(2)1−γ

]+ π3

[1

1− γ(2 + a)1−γ

]a satisfies the first order condition,

W ′(a) = −π1(2− a)−γ + π3(2 + a)−γ = 0

We have,

a = 2π1/γ3 − (−π1)1/γ

π1/γ3 + (−π1)1/γ

d) • In the case that u(Y ) = 1− exp(−bY ),

RA = −u′′(Y )

u′(Y )= b.

Since R′A(Y ) = 0, the amount invested in the risky asset is not

affected by the increase in the wealth.

• In the case that u(Y ) = 11−γY

1−γ,

RA = −u′′(Y )

u′(Y )=γ

Y.

Since R′A(Y ) = − γY 2 < 0, as the wealth increases, da/dY > 0.

2. (10 points) An individual with a well-behaved utility function and an

initial wealth of $1 can invest in two assets. Each asset has a price of $1.

The first is a riskless asset that pays rf = 1. The second pays amount r1

and r2 where r1 < r2 with probabilities of π and 1− π, respectively. Denote

the units demanded to the riskless asset and the risky asset by x1 and x2,

respectively, with x1, x2 ∈ [0, 1].

a) Given a simple necessary condition (involving r1 and r2 only) for the

demand for the riskless asset to be strictly positive. Given a simple

necessary condition (involving r1, r2 and π only) so that the demand

for the risky asset is strictly positive.

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b) Assume now that the conditions in item a) are satisfied. Formulate the

optimization problem and write down the first order condition. Can you

intuitively guess the sign of dx1/da? Verify your guess by assuming that

x1 is a function of r1 written as x1(r1), and taking the total differential

of the first order condition with respect to r1. Can you conjecture a

sign for dx1/dπ? Provide an economic interpretation without verifying

it as done previously.

Answer: Since the price of both assets is 1, the number of units purchased

equals the value of the purchase. Let x1 be units (value) of risk free asset

demanded and x2 be units (value) of risky asset demanded. Since Y0 = 1,

x1 + x2 = 1 and x2 = 1− x1.a) Necessary condition for x1 > 0 is rf > r1. Since rf = 1, or in this case,

r1 < 1. Necessary condition for the demand of the risky asset to be positive:

πr1 + (1− π)r2 > 1.

b) We would expect dx1/dr1 < 0.

The optimization problem is,

maxx1

πu(x1 + (1− x1)r1) + (1− π)u(x1 + (1− x1)r2)

s. t. x1 + x2 ≤ 1 or 0 ≤ x1 ≤ 1

The first order condition for an interior solution is,

πu′(x1 + (1− x1)r1)(1− r1) + (1− π)u′(x1 + (1− x1)r2)(1− r2) = 0

This equation implicitly defines x1 = x1(r1). Taking the total differential

yields,

− πu′(x1 + (1− x1)r1) + πu′′(x1 + (1− x1)r1)(1− r1)[(1− r1)

dx1dr1

+ 1− x1]

+ (1− π)u′′(x1 + (1− x1)r2)(1− r2)2dx1dr1

= 0

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We have,

dx1dr1

=πu′(x1 + (1− x1)r1)− πu′′(x1 + (1− x1)r1)(1− r1)(1− x1)

(1− π)u′′(x1 + (1− x1)r2)(1− r2)2 + πu′′(x1 + (1− x1)r1)(1− r1)2< 0

as the numerator is positive and the denominator negative.

We would expect that an increase in the probability of the unfavorable

risky asset state r1, would tend to decrease the demand for the risky asset.

The world is becoming riskier. Thus dx1/dπ > 0.

3.(20 points) You are a portfolio manager considering whether or not to

allocate some of the money to SP500 index. Denote by ryour and rSP the

returns of your portfolio and returns of SP500 Index, respectively. Let σ(r)

be the standard deviation of r and corr(ryour, rSP ) be the correlation between

ryour and rSP . Your assistant provides you with the following historical return

information:

Year ryour (%) rSP (%)1992 54 501993 24 101994 -6 -101995 24 601996 -6 -201997 54 80

a) Show that the addition of the SP500 Index to your portfolio will reduce

risk (at no loss in return) provided

corr(ryour, rSP ) <σyourσSP

assuming as in the case, σyour > σSP .

b) Based on this historical data, could you receive higher returns for the

same level of risk (standard deviation) by allocating some of your wealth

to the SP500?

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c) Based on this historical experience, would it be possible to reduce your

portfolio’s risk below its current level by investing something in the

SP500?

d) What fraction of the variation in SP500 can be explained by variation in

your portfolio’s returns?

Answer: a) Let wyour = 1 − wSP . Let P denote the new portfolio that

is composed of the fraction wyour of your portfolio and the fraction wSP of

SP500.

σ2P = (1− wSP )2σ2

your + w2SPσ

2SP + 2wSP (1− wSP )cov(ryour, rSP ).

Then∂σ2

P

∂wSP

∣∣∣∣wSP=0

= 2[cov(ryour, rSP )− σ2your].

If cov(ryour, rSP ) − σ2your < 0,

∂σ2P

∂wSP

∣∣∣∣wSP=0

< 0. In this case, you add SP500

to your portfolio, the new portfolio’s risk will decline below your portfolio.

Equivalently,

cov(ryour, rSP ) < σ2your,

or

corr(ryour, rSP ) <σyourσSP

.

b) and c). We compute these data from the sample statistics that we are

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given.

µyour =1

6(0.54 + 0.24− 0.06 + 0.24− 0.06 + 0.54) = 0.24

µSP =1

6(0.50 + 0.10− 0.10 + 0.60− 0.20 + 0.80) = 0.2833

σ2your = 0.072

σyour = 0.2683

σ2SP = 0.1657

σSP = 0.4070

cov(ryour, rSP ) = 0.08

now check the above inequality :

corr(ryour, rSP ) =cov(ryour, rSP )

σyourσSP=

0.08

0.2683× 0.4070= 0.7325

Clearly corr(ryour, rSP ) = 0.7325 > σyourσSP

= 0.26830.4070

= 0.6592 . So the answers

to b) and c) are no. These are, in fact, two ways of asking the same questions.

d) By the regression,

rSP = αSP + βSP ryour + ε

⇒ σ2SP = β2

SPσ2your + σ2

ε

The fraction of SP500’s variation explained by variations in your portfolios

return is,

β2SPσ

2your

σ2SP

=

(corr(ryour,rSP )σyourσSP

σ2your

)2σ2your

σ2SP

= (corr(ryour, rSP ))2 = (0.7325)2 = 0.5366

4.(25 points) Given two random variables X and Y

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probability state of nature X Y0.2 I 18 00.2 II 5 -30.2 III 12 150.2 IV 4 120.2 V 6 1

a) Calculate the mean and variance of each variable, and the covariance

between X and Y

b) Suppose X and Y represent the returns from two assets. Calculate the

mean and variance for the following portfolio:

portfolio 1 2 3 4 5 6 7% invested in X 125 100 75 50 25 0 -25% invested in Y -25 0 25 50 75 100 125

c) Find the portfolio that has the minimum variance.

d) Let portfolio A have 75% in X and portfolio B have 25% in X. Calculate

the covariance between the two portfolios.

e) Calculate the covariance between the minimum variance portfolio and

portfolio A, and the covariance between the minimum variance portfolio

and portfolio B.

f) What is the covariance between the minimum variance portfolio and any

other portfolio along the efficient set?

g) What is the relationship between the covariance of the minimum variance

portfolio with other efficient portfolio, and the variance of the minimum

variance portfolio?

Answer: a)

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Table 1:pi Xi piXi pi(Xi − EX)2 Yi piYi pi(Yi − EY )2 pi(Xi − EX)(Yi − EY )0.2 18 3.6 16.2 0 0 5 -90.2 5 1 3.2 -3 -0.6 12.8 6.40.2 12 2.4 1.8 15 3 20 60.2 4 0.8 5 12 2.4 9.8 -70.2 6 1.2 1.8 1 0.2 3.2 2.4

EX = 9 var(X) = 28 EY = 5 var(Y ) = 50.8 cov(X, Y ) = −1.2

b)

Table 2:% in X % in Y E(rP ) var(rP ) σP125 -25 10 47.675 6.905100 0 9 28 5.29275 25 8 18.475 4.29850 50 7 19.1 4.3725 75 6 29.875 5.4660 100 5 50.8 7.127-25 125 4 81.875 9.048

mean = aE(X) + (1− a)E(Y )

variance = a2σ2x + (1− a)2σ2

y + 2a(1− a)cov(x, y)

The minimum variance frontier is shown in Figure 1.

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Figure 1: The minimum variance frontier

c) The minimum variance portfolio is given by

dvar(rP )

da= 0

and solve for a∗

a∗ =σ2y − σxσyρxy

σ2x + σ2

y − 2σxσyρxy

with cov(x, y) = ρxyσxσy

a∗ =σ2y − cov(x, y)

σ2x + σ2

y − 2cov(x, y)= 0.64

Thus, the minimum variance portfolio has 64 percent invested in asset X and

36 percent in asset Y .

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The portfolio mean and variance at this point are:

E(rP ) = 0.64 ∗ 9 + 0.36 ∗ 5 = 7.56

var(rP ) = 0.642 ∗ 28 + 0.362 ∗ 50.8 + 2 ∗ 0.64 ∗ 0.36 ∗ (−1.2) = 17.5

σ(rP ) = 4.183

d)Portfolio A is 75 percent in X and 25 percent in Y; portfolio B is 25

percent in X and 75 percent in Y. The covariance between the two is

cov(A,B) =[0.75 0.25

] 28 −1.2

−1.2 50.8

0.75

0.25

= 14.025

e)The covariance between the minimum variance portfolio Vmin and port-

folio A is

cov(Vmin, A) =[0.64 0.36

] 28 −1.2

−1.2 50.8

0.75

0.25

= 17.5

The covariance between the minimum variance portfolio Vmin and portfolio

B is

cov(Vmin, B) =[0.64 0.36

] 28 −1.2

−1.2 50.8

0.25

0.75

= 17.5

f) It is no coincidence that the covariance between the minimum variance

portfolio and portfolios A and B is the same (except for rounding error).

It can be proven that the covariance between the global minimum variance

portfolio and any portfolio along the efficient set is a constant. (See Merton

(1972).)

g) The variance of the global minimum variance portfolio is

var(Vmin) =[0.64 0.36

] 28 −1.2

−1.2 50.8

0.64

0.36

= 17.5

Again, this is not a coincidence. The variance of the minimum variance

portfolio equals the covariance between the minimum variance portfolio and

any other portfolio along the efficient set.

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5.(15 points) Two securities have the following joint distribution of re-

turns, r1 and r2:

P (r1 = −1 and r2 = 0.15) = 0.1

P (r1 = 0.5 and r2 = 0.15) = 0.8

P (r1 = 0.5 and r2 = 1.65) = 0.1

a) Compute the means, variances and covariance of returns for the two se-

curities

b) Plot the feasible mean-standard deviation [ER, σ] combinations, assum-

ing that the two securities are the only investments available.

c) Which portfolios belong to the mean-variance efficient set?

d) Show that security 2 is mean-variance dominated by security 1, yet enters

all efficient portfolios but one. How do you explain this?

e) Suppose the possibility of lending, but not borrowing, at 5% (without

risk) is added to the previous opportunities. Draw the new set of

[ER, σ] combinations. Which portfolios are now efficient?

Answer: a)

E(r1) = 0.1 ∗ (−1) + 0.8 ∗ 0.5 + 0.1 ∗ 0.5 = 0.35

V ar(r1) = 0.1(−1.0− 0.35)2 + 0.8(0.5− 0.35)2 + 0.1(0.5− 0.35)2 = 0.2025

E(r2) = 0.1 ∗ (0.15) + 0.8 ∗ 0.15 + 0.1 ∗ 1.65 = 0.3

V ar(r2) = 0.1(0.15− 0.3)2 + 0.8(0.15− 0.3)2 + 0.1(1.65− 0.3)2 = 0.2025

cov(r1, r2) = 0.1(−1.0− 0.35)(0.15− 0.3) + 0.8(0.5− 0.35)(0.15− 0.3)

+ 0.1(0.5− 0.35)(1.65− 0.3) = 0.0225

b) See the table below

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Table 3:% in 1 % in 2 E(rP ) var(rP ) σP100 0 0.35 0.2025 0.4575 25 0.3375 0.1350 0.36750 50 0.325 0.1125 0.33525 75 0.3125 0.135 0.3670 100 0.3 0.2025 0.45

Because the efficient set is the upper half of the minimum variance fron-

tier, we also need to know the mean and variance of the minimum variance

portfolio.

a∗ =σ2y − cov(x, y)

σ2x + σ2

y − 2cov(x, y)= 0.5

The minimum variance frontier is as Figure 2.

Figure 2: minimum variance frontier

c) The efficient set is the solid line in the above figure. It starts with the

minimum variance portfolio (50 percent in 1, 50 percent in 2) and ends with

100 percent in asset 1 (assuming no short sales).

d) If mean and variance are the only relevant decision criteria (even

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though the returns are not jointly normally distributed), then asset 1 dom-

inates asset 2 because it has a higher mean and the same variance. Asset 2

enters into efficient portfolios because of the diversification benefits it pro-

vides since it is not perfectly correlated with asset 1.

e)Figure 3 shows the new efficient set as the solid line.

Figure 3: Efficient set with no riskless borrowing

6.(10 points) Given that assets X and Y are perfectly correlated such

that Y = 6 + 0.2X and the probability distribution for X is as shown in the

table below.

probability X(%)0.1 300.2 200.4 150.2 100.1 -50

What is the percentage of your wealth to put into asset X to achieve zero

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variance? Graph the minimum variance frontier and the zero variance point.

Answer: First compute the expected returns and standard deviations for

asset X.

Table 4:pi Xi piXi pi(Xi − EX)2

0.1 30 3 400.2 20 4 200.4 15 6 100.2 10 2 00.1 -50 -5 360

EX = 10 var(X) = 430

Using the probability properties, we can immediately write the expected

value and variance of asset Y .

E(Y ) = 6 + 0.2EX = 8

V ar(Y ) = 0.22V ar(X) = 17.2

Also, the standard deviations of X and Y are

σX = 20.74 σY = 4.15

The simplest way to solve the problem is to make use of the fact that the

opportunity set for a portfolio of two perfectly correlated assets is a straight

line. The opportunity set is graphed in Figure 4. If we find the expected

return represented by its intercept then we can determine the correct weights

for the zero variance portfolio. The linear equation for the minimum variance

frontier is

E(RP ) = a+ bσ(RP )

We know the coordinates of the points X and Y , therefore the slope is

b =EX − EYσX − σY

= 0.1206

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Figure 4: Minimum variance frontier for two perfectly correlated assets

Using the coordinates for X and the slope we have

E(X) = a+ 0.1206σX

a = 7.5

The zero variance portfolio has an expected return of 7.5%. If we let α be

the percent invested in X then the proportions of X and Y in this portfolio

are

E(RP ) = αEX + (1− α)EY

α = −0.25

Therefore, we should sell short 25% of our wealth in asset X and go long

125% in asset Y . To confirm this result we can plug the appropriate weights

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into the definition of the variance of a portfolio of two assets as follows:

V ar(RP ) = α2V ar(X) + 2α(1− α)cov(x, y) + (1− α)2V ar(Y )

= (−0.25)2 ∗ 430 + 2(−0.25)(1.25)(1.0)(20.74)(4.15) + (1.25)2 ∗ (17.2)

≈ 0

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