Finance Theory I

Final Exam

Problem 1.

Suppose that the risk free rate is 5% and that the expected return of a portfolio with a beta of 1 is 10%.

According to the CAPM:

1. What is the expected return on the market portfolio?

2. What is the expected return on a portfolio with a beta of 1.5?

You consider buying a share of stock A currently priced at NOK 50. The stock is expected to pay no

dividends next year and you expect it to then sell at NOK 55.

3. According to the CAPM, what is the stock's beta?

You nd another stock B that you expect to be able to sell for NOK 55 next year. B trades at 45 but

you nd that stocks A and B have the same estimated betas.

4. Is this data consistent with the CAPM? If not, show how you would trade to exploit any deviations

from the CAPM.

Another stock, C, is perfectly correlated with the market.

5. Show that C has zero non-systematic risk.

Problem 2.

There are two periods: period zero (today) and period 1 (next period). In the next period there are

three possible states of the world: recession (with probability 0.3), normal growth (with probability 0.4)

and expansion (with probability 0.3). There are two stocks (stock A and stock B) in the economy. The

stocks will yield payos next period according to the following table:

Stock Recession Normal Growth Expansion

A 6 12 14

B 15 5 10

1. Can you rank the stocks by the criterion of rst order stochastic dominance? If you can rank them,

which stock rst order stochastically dominates the other? You may use a graph to answer this

question if you choose to.

2. Can you rank the stocks by the criterion of second order stochastic dominance? If you can rank

them, which stock second order stochastically dominates the other? You may use a graph to answer

this question if you choose to.

Consider an investor with the following utility function:

u(w) =w.

The investor has a current wealth of $10. This investor is today given one unit of stock A as a present

from his uncle. If the investor sells the stock the proceeds from selling the stock are guaranteed and will

be delivered next period. Also assume that the investor cannot invest in stock B.

3. What is the minimum price the investor would be willing to sell the stock for?

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Problem 3.

A gambler has a utility function based on the logarithm of wealth, i.e. u(W ) = ln(W ). The gambler'scurrent wealth is W0. The gambler is oered a bet on a proposition that has probability p of returningdouble his bet and (1 p) of returning nothing.

1. What fraction of W0 should he bet on this proposition?

2. Discuss how the optimal bet varies with the possible values of p.

Problem 4.

Consider a one-period economy with two dates, 0 and 1. At date 1 the economy has three possible states,

u, n and d. A stock has the following state contingent payos:

State u n dPayo 100 80 50

The current price of the stock is 65. At date 1 a risk free bond with a face value of 1000 matures. This

bond is currently trading at a price of 800.

1. With only this information, is it possible to price a call option on the stock with an exercise price

of 100?

2. With only this information, is it possible to price a call option on the stock with an exercise price

of 80?

Suppose the current price of a call option on the stock with exercise price of 70 is 11.

3. What is the current price of a put option on the stock with an exercise price of 70?

Problem 5.

A company has the following state contingent cash ows.

Probability State price State Cash-

probability Price ow

p q 0.500 0.2 0.16 6500.250 0.3 0.24 2000.125 0.4 0.32 1100.125 0.1 0.08 80

1. What is the risk free interest rate?

2. What is the current value of the rm?

Suppose the rm has debt outstanding with a face value of F . The face value is payable at the sametime the cash ows are realized. The current value of this debt is 117.60.

3. What is the face value F of the debt?

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Problem 6.

A consumer has state independent preferences dened by the utility function u(W ). The consumer caninvest his initial wealth W0 in two assets. The rst is riskless and pays return, per dollar, of Rf . The

second is risky with a return of R per dollar.

1. Show that the fraction of wealth invested in the risky asset is never zero, as long as E(R) > Rf .Why is this behavior referred to as `local risk neutrality'?

Problem 7.

Consider a two-date economy and an agent with utility function over consumption

U(C) =1

1 C1

at each period. Dene the intertemporal utility function as

V (C1, C2) = U(C1) + U(C2)

1. Show that the agent will always prefer a smooth consumption stream to a more variable one with

the same mean, that is

U(C) + U(C) > U(C1) + U(C2)

where C = C1+C22 and C1 6= C2.

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Financial Theory

Solutions

Problem 1.

1. E[rp] = rf + p(E[rm] rf )

0.1 = 0.05 + (E[rm] rf )

E[rm] = 10%

2. E[rp] = 0.05 + 1.5 (0.1 0.05) = 12.5%

3. E[rA] =5550

1 = 10

A = 1

4. E[rB ] =5545

1 = 22%But you are told that

B = A = 1,

which seem to indicate a return of 10%. Stock B is underpriced, buy it and nance this by selling

A.

Cash ow Cash ow Beta

Strategy at date 0 at date 1 risk

buy B 45 55 1sell A 50 55 1net 5 0 0

5. corr(rc, rm) =cov(rc, rm)

cm= 1

cov(rc, rm)2m

=cm

= cm

2c = 2cm

2m +

2c =

2c +

2c

2c = 0

Problem 2.

1. There is no rst order stochastic dominance in this case.

2. Stock A second order stochastically dominates stock B.

3. What is the expected utility from not selling stock A?

E[u(w)] = 0.316 + 0.4

22 + 0.3

24 4.55

If the investor does sell the stock he will get for sure 10 + p. To see the minimum price, solve

4.55 =10 + p

p = 16.67

The the investor sells the stock for 16.67 he will be as well o as he would be by keeping the stock.

If the sale price is larger than 16.67 the investor would be more than happy to sell it.

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Problem 3.

1. Let w be the bet.Final wealth

W ={

W0 + wW0 =W0(1 + w) with probability pW0 wW0 =W0(1 w) with probability 1 p

E[u()] = p ln(W0 + w) + (1 p) ln(W0 w)

First order condition for maximum

p1

W0(1 + w)+ (1 p) 1

W0(1 w) (1) = 0

p1

1 + w+ (1 p) 1

1 w (1) = 0

p1

1 + w= (1 p) 1

1 w

p(1 w) = (1 p)(1 + w)

Solution

w = (2p 1)

Problem 4.

1. This option is never in the money, it has to be priced at zero.

2. With only two traded securities, can not nd state prices for all three states.

3. This can be solved (at least) two ways.

One is to use put call parity

c p = S K(1 + r)

p = c S + K(1 + r)

= 11 65 + 701 + 0.0.25

= 2

Where the risk free rate is found from the bond price:

>> r=1000/800-1

r = 0.25000

Alternatively it is possible to back out the state prices, or alternatively, the state price probabilites.

Three states, three securities gives a cash ow matrix

> X

X =

1000 1000 1000

100 80 50

30 10 0

The three securities have prices

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>> P

P =

800 65 11

Solve for state prices or state price probabilities q. Note that the state price probabilities sumto one.

>> phihat=inv(X)*P'

phihat =

0.20000

0.50000

0.10000

>> qhat=phihat*(1+r)

qhat =

0.25000

0.62500

0.12500

>> qhat'*ones(3,1)

ans = 1.0000

Find the cash ows from the put and price those:

>> S

S =

100 80 50

>> P=max(0,K-S)

P =

0 0 20

>> q=P*qhat/(1+r)

q = 2.0000

>> q=P*phihat

q = 2.0000

Problem 5.

1. Risk free rate can be had from the relationship betwen state price probabilities and state prices

>> r=0.2/0.16-1

r = 0.25000

The risk free rate is 25%.

2. >> X=[650;200;110;80]

X =

650

200

110

80

>> q=[0.2 0.3 0.4 0.1 ]

q =

0.20000 0.30000 0.40000 0.10000

>> phi=q/(1+r)

phi =

0.160000 0.240000 0.320000 0.080000

>> q*X/(1+r)

ans = 193.60

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>> phi*X

ans = 193.60

The rm value is 193.60

3. The face value problem. Since the value of the debt is above 117 clearly the face value can not be

below 117, hence the rm defaults in the states where the cash ow is 110 and 80.

Either the debt has face value below 200, in which case the value is

117.60 =

FF11080

or above 200, where the value is

117.60 =

F20011080

Solving the last one for F, nd that F = 175, which contradicts that F > 200.

>> F=(117.60-phi(2:4)*X(2:4))/phi(1)

F = 175

Therefore, it must be the rst one that is correct. Solve for F

>> F=(117.60-phi(3:4)*X(3:4))/(phi(1)+phi(2))

F = 190

Check by plugging in F=190 and nding a debt value of 117.60.

>> F

F = 190

>> X

X =

650

200

110

80

>> phi*min(F,X)

ans = 117.60

>> q*min(F,X)/(1+r)

ans = 117.60

Problem 6.

1. Let be fraction of initial wealth invested in the risky asset.

End of period wealth will then be

W =W0(1 )Rf +W0(R)

Want to show the fraction invested in the risk free asset is not zero.

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The consumers problem is to maximize expected utility as a function of . The objective functionis

max f() = E[U(W0(1 )Rf +W0R)]= E[U(W0[(1 )Rf + R])]

Calculate the rst and second derivatives.

f () = E[U ()W0(RRf )]

f () = E[U ()W 20 (RRf )2]

If the consumer is risk averse, the second derivative is negative, f () = 0 denes a maximum.

We want to show that 6= 0. Evaluate rst order condition at = 0:

f (0) = E[U (W0Rf )W0(RRf )]= U (W0Rf )W0(E[R]Rf )> 0 if E[R] > Rf

For an optimal , we need f () = 0, Hence 6= 0.This behavior is called local risk neutrality because locally at W0Rf , the decision of whether toinvest in the risk asset is only dependent on whether expected excess return is dierent from Rf .

Problem 7.

1. Note that U() is a concave function and apply Jensens inequality.

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