FINAL EXAM - Studies2 - HEC Paris

HEC Paris
MBA Program
Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Financial Markets
Prof. Laurent E. Calvet
Fall 2010
FINAL EXAM
180 minutes
Open book
• The exam will be graded out of 200 points.
• Points for each question are shown in parentheses.
• Answer all seven questions.
• You are allowed to use a calculator. All other electronic devices are
strictly prohibited.
• Good luck!
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Total
1

Problem 1. (30 points)
Decide whether the following statements are true or false. No explanations
are necessary.
(a) If the Capital Asset Pricing Model holds, no mutual fund manager can
earn a higher return than the market.
(False)
(b) Over the long run, the stocks of small companies tend to outperform the
stocks of larger companies.
(True)
(c) A credit default swap is a contract that provides insurance against the
risk of default by a company.
(True)
(d) In the Capital Asset Pricing Model, investors are compensated for the
firm-specific volatility of a stock.
(False)
(e) The Black-Scholes formula for a European call assumes that the logarithm
of the price of the underlying stock has a normal distribution.
(True)
(f) It is never optimal to exercise an American call on a non-dividend paying
stock prior to maturity.
(True)
(g) It is never optimal to exercise an American put on a non-dividend paying
stock prior to maturity.
(False)
(h) A put is the obligation to sell an asset at a fixed price on a certain date.
(False)
(i) A put option on the S&P 500 index has a negative β.
(True)
(j) A straddle consists of a long position in a call and a put with the same
strike price.
(True)
2

Problem 2. (20 points)
Consider an economy with two possible states of the world, which occur with
equal probability. Stocks A and B yield the following returns in each state:
Return on stock A Return on stock B
State 1 +40% +25%
State 2 -10% -5%
(a) Compute the expected returns on stocks A and B.
Solution: The expected returns on the stocks are:
E[rA] = (40% − 10%)/2 = 15%
E[rB] = (25% − 5%)/2 = 10%
(b) Compute the standard deviation of returns on A and B.
Solution: The variance of the return on stock A is:
V ar(rA) = [(0.25) 2 + (0.25) 2 ]/2 = (0.25) 2
The standard deviation of stock A is therefore 25%.
The variance of the return on stock B is:
V ar(rB) = [(0.15) 2 + (0.15) 2 ]/2 = (0.15) 2 .
The standard deviation of stock B is therefore σB = 15%.
(c) Compute the covariance of the returns on A and B.
Solution: The covariance is given by:
Cov(rA, rB) = 0.5 × 0.25 × 0.15 + 0.5 × 0.25 × 0.15
= 0.25 × 0.15
= 0.0375.
The above calculation implies that Cov(rA, rB) = σAσB.
(d) Compute the correlation of the returns on A and B.
Solution:
Corr(rA, rB) = Cov(rA, rB)
= 1.
3
σAσB

Problem 3. (30 points)
You analyze the cash flows arising from the purchase of the equity of the
HECMBA Corporation. The company will liquidate after 2 years. It has no
debt.
• In the first year, it will pay either $10 million with 30% probability, or
$1 million with 70% probability.
• In the second year, it will pay either $20 million with 25% probability, or
$3 million with 75% probability.
Suppose the interest rate on both the 1-year and 2-year T-bonds is 3% per
year (with yearly compounding), and you believe that the stock market has
an expected return of 10%. Assume also that HECMBA Corporation has a β
coefficient of 1.7.
(a) What is the expected return on the stock?
Solution: We need to compute the discount rate E[R], and we are told
that the risk-free rate rf = 3%, HECMBA’s beta is β = 1.7, and the
market expected return E[RMkt] = 10%. We use the formula
E[R] = rf + β × (E[RMkt] − rf)
= 0.03 + 1.7 × (0.10 − 0.03)
= 14.90%.
(b) According to the CAPM, what is the value of HECMBA’s equity?
Solution: According to the present value (PV) formula, the price of
HECMBA should be
P = E[C1] E[C2]
+
1 + E[r] (1 + E[r]) 2.
The expected cash flows after after the first quarter are
E[C1] = 30% × $10m + 70% × $1m = $3.7m.
The expected cash flows after after the second quarter are
E[C2] = 25% × $20m + 75% × $3m = $7.25m.
Then the price of Acme is given by the formula:
P = E[C1] E[C2] $3.7m
+ =
1 + E[r] (1 + E[r]) 2 1.149
4
$7.25m
+ = $8.71m.
1.1492

Problem 4. (30 points)
You are hedging the purchase of gold for your jewelry company. You have
been requested to purchase 5 futures contracts to hedge your manufacturing
needs in one year. Each contract is for 100 ounces of gold. The current spot
price of gold is $1500 per ounce. The 1-year risk free rate is 3.5% in annual
units with continuous compounding.
(a) What is the price of one futures contract?
Solution: By the cost of carry relationship, the futures price is:
f0 = 100 × $1, 500 × e 0.035×1 = $155, 342.96
(b) If the spot gold price, at the end of the contract, is $1600, was this hedge
a good idea?
Solution: The futures position is profitable since the spot price at ma-
turity is larger than the initial futures price f0.
(c) What was the profit or loss?
Solution: The profit is:
5 × ($160, 000 − $155, 342.96) = $23, 285.22.
5

Problem 5. (30 points)
Consider a 1-year European call on Apple with strike X = $320. The current
stock price is S0 = $320. The stock has an annualized standard deviation of
20% per year, and is not expected to pay a dividend during the life of the
option. The continuously compounded interest rate is 5% per year.
(a) Is the option in-, at-, or out-of-the-money?
Solution: At the money since S0 = X.
(b) Use the Black-Scholes formula to compute the call price.
Solution: Since the option is at-the-money (S0 = X),
reduces to
We infer that
d1 = ln[S0/(Xe −rT )] + σ 2 /2
σ √ T
d1 = (r + σ2 /2)T
σ √ T
= 0.35.
d2 = d1 − σ √ T = 0.35 − 0.20 = 0.15.
We find in the table that N(d1) = 0.6368 and N(d2) = 0.5596. The call
price C0 is therefore
or C0 = $33.44.
$320 × 0.6368 − $320e −0.05×1 × 0.5596,
(c) What is the price of the American call with identical characteristics?
Solution: An American call on a non-dividend-paying-stock has the
same value as the European call with the same characteristics, that is
$33.44.
(d) What is the price of the European put with the same characteristics?
Solution: The put-call parity P0 = C0 − S0 + Xe −rT implies that the
European put is worth
P0 = 33.44 − 320 + 320e −0.05×1 = $17.84.
(e) What is the price of a 1-year futures contract on the stock?
Solution: The price of the 3-month futures is f = S0e rT = $320e 0.05×1 =
$336.41.
6

Problem 6. (30 points)
A stock price is currently $100. Over each of the next two three-month periods
it will either go up by 6% or down by 3%. The interest rate per period is 2%.
(a) What is today’s value of a six-month European call with a strike of $102?
Solution: The dynamics of the stock price is given by:
S0 = $100
Su = $106
Sd = $97
Suu = $112.36
Sud = $102.82
Sdd = $94.09
The value of the call at the terminal dates is therefore:
C0 =?
Cu =?
Cd =?
At node u, the replicating portfolio contains
unit of the stock, and
B = Cu,u − ∆Su,u
1 + rf
∆ =
= 10.36 − 1 × 112.36
10.36 − 0.82
112.36 − 102.82
1.02
Cuu = $10.36
Cud = $0.82
Cdd = $0
= 1
= − 102
1.02
= −$100.
units of the riskless asset. The value of the call at node u is therefore:
Cu = 1 × 106 − 100 = $6.
At node d, the replicating portfolio contains
∆ =
0.82 − 0
102.82 − 94.09
7
= 0.09393

units of the stock and
B = Cd,u − ∆Sd,u
1 + rf
The call is worth:
Hence
= 0.82 − 0.0939 × 102.82
1.02
Cd = 0.09393 × 97 − 8.6645 = $0.4467.
C0 =?
Cu = $6
Cd = $0.4467
At node 0, the replicating portfolio contains
∆ =
6 − 0.4467
106 − 97
Cuu = $10.36
Cud = $0.82
Cdd = $0.00
= 0.6170.
= −$8.6645.
units of the stock, and B = −$58.24 units of the riskless asset. The value
of the call at node 0 is therefore:
C0 = 0.6170 × 100 − 58.24 = $3.46.
(b) What is the price of the American call with identical characteristics?
Solution: The American call is also worth $3.46.
(c) What is the price of the European put with the same characteristics?
Solution: The put-call parity P0 = C0 − S0 + X/(1 + r) 2 implies that
the European put is worth
P0 = $3.46 − $100 + $102/(1.02 2 ) = $1.50.
8

Problem 7. (30 points)
Three put options on a stock have the same expiration date and strike prices
of $55, $60, and $65. The market prices are $3, $5, and $8, respectively.
(a) Explain how a butterfly spread can be created.
Solution: We can create a butterfly spread by
• Buying 1 put with strike price of $55
• Selling 2 puts with strike price of $60
• Buying 1 put with strike price of $65
(b) How much does it cost today?
Solution: The cost of the spread is:
V0 = 1 × $3 − 2 × $5 + 1 × $8 = $1.
(c) Compute the payoff of the strategy at the terminal date as a function of
the final stock price ST .
Solution: The payoff is:
Thus,
VT = (55 − ST )+ − 2(60 − ST )+ + (65 − ST )+.
• if ST < $55, all puts are exercised at maturity and the payoff of the
portfolio is VT = (55 − ST ) − 2(60 − ST ) + (65 − ST ) = 0;
• if $55 ≤ ST ≤ $60, the payoff is VT = −2(60 − ST ) + (65 − ST ) =
ST − 55;
• if $60 ≤ ST ≤ $65, the payoff is VT = 65 − ST ;
• if ST > $65, the payoff is VT = 0.
(d) For what range of stock prices would the butterfly spread lead to a loss?
Solution: The profit is:
• −$1 if ST < $55;
• ST − 56 if $55 ≤ ST ≤ $60;
• 64 − ST if $60 ≤ ST ≤ $65;
• −$1 if ST > $65.
The butterfly spread therefore leads to a loss if the final stock price is
either lower than $56 or higher than $64.
9