4050 AEM Spring 2012, Exam 2
NAME (Please Print):________________KEY__________
1) The main difference between and forward and a futures contract is (5 points):
a) with a futures contract you always have to make or take delivery of the commodity in the end
b) futures are highly standardized whereas forwards are not
c) forwards are marked-to-market and futures are not
d) forwards are exchange traded and futures are privately negotiated or traded OTC
e) forwards specify a delivery point, and futures do not
Your Answer:______________
Answer: B, futures highly standardized and are exchange traded. This in addition to being
marked-to-market and standardized is what primarily defines a futures. The other options
are all obviously wrong.
2) What is the payoff of a short call option with a strike price of $13.75 if the underlying value is
$14.50 at expiration? (5 points)
a) $0.75
b) $0
c) $1.00
d) -$0.75
e) This is not determinable.
Your Answer:______________
Answer: D, Short call payoff = -Max(0, Spot-K) = -Max(0, 14.5-13.75) =- $0.75
3) What is the payoff of a long call option with a strike price of $4.05 if the underlying value is
$5.15 at expiration? (5 points)
a) -$1.10
b) $0
c) $3.95
d) $5.15
e) $1.10
Your Answer:______________
Answer: b, Long put payoff = Max(0, K-Spot) = Max(04.05 - 5.15 ) = $0
4) A put option effectively gives the option buyer (5 points):
a) The and option to buy a specified underlying at a predetermined price and date in the future
b) The obligation to take delivery of an underlying at a predetermined price and date
c) The obligation to make delivery of an underlying at a predetermined price and date
d) The right but not the obligation to take delivery of an underlying at a predetermined price and date
e) The right but not the obligation to make deliver an underlying at a predetermined price and date
Your Answer:______________
Answer: e, by definition
AEM 4050 Spring 2012, Exam 2
5.) Refer to the Table below. Calculate the Weighted Average Cost of Capital for each capital
structure composition. Explain the significance of WACC. (10 points)
Assets
Tax
Rate
2000
30%
D
E
D/E
I(d)
I(d)(1t)
D/A
I(e)
E/A
400
1600
0.06
0.042
0.2
0.1
0.8
600
1200
1400
800
0.25
0.42857
1
1.5
0.06
0.15
0.042
0.105
0.3
0.6
0.12
0.2
0.7
0.4
WAC
C
0.088
4
0.096
6
0.143
6.) The table below contains asset returns for two assets, X1 and X2, and the associated
probabilities of each outcome. Calculate mean and variance for each asset, as well as their
correlation. Calculate the portfolio standard deviation and expected return at the weight levels
indicated in the Portfolio Risk and Return Table. Graph those points and trace out the
efficient frontier. (15 points)
Extra Credit 1 (3 points): Find the max Sharpe ratio among the weights provided. Draw the
CML.
Asset Returns
Observatio
n
X1
1
-0.1
2
0.15
3
0.1
X2
0.04
-0.1
0.2
prob
0.3
0.2
0.5
Portfolio Risk and Return
w1
w2
Std Port
1
0
0.1
0.6
0.4
0.25
0
0.75
1
E(R)
0.05
0.066
0.080116 8
0.081
0.094652 5
0.118389 0.092
7.) Suppose that today is June 15, 2012. The November 15th 2012 crude oil futures price is $94.00
today (i.e., on June 15, 2012). The spot price today is $72.00. For simplicity assume that you can
AEM 4050 Spring 2012, Exam 2
buy as well as short sell the physical in the spot market freely, and that convergence is perfect. Also,
assume that you can borrow and lend funds at a monthly interest rate of r = 3% per month. The cost
of physical storage for this period is a flat physical carrying charge C = $0.80 per unit of oil per
month, to be paid at the end of the storage period. Assume that there is no basis risk and that you will
cash settle your futures position. At expiration of the futures contract in November, the spot price is
$130.
(5 points) What is the expected future spot price of the November spot price in June using the costof-carry model? Draw a picture/graph like the ones we did in class to depict this. Dont forget to label
your axes. (Hint: Set up your model as: EJUN [SNov] = SJun x (1 + r)M + C x M, where M = 5 is the
number of months between June and November.)
The expected future spot price is equal to the cost of the spot today plus total storage costs for 5
months which equals =72*(1.03)^5+5*0.8= 87.46773335.
[DRAW PICTURE HERE]
(15 points) Is there an arbitrage opportunity? Describe your strategy in words. type What of
transaction/trade is this (i.e., what is it called)? Is there risk involved with this? Why or why not?
Using the table below, fill in the transactions, cash flows, and physical inventory figures, and
calculate your final profit.
FJun,Nov = $94
SJun = $72
Snov= $130
Physical Inventory
Date
June
Transaction
Borrow $72 @3%/mo
Buy spot crude and store @ $72
Short sell crude futures @ 94
Nov
Buy Futures to offset (Pay = -130+94=36)
Cash Flow
72
-72
0
-36
130
Cumulativ
Change e
0
0
1
1
0
1
0
1
-1
0
AEM 4050 Spring 2012, Exam 2
Sell stored crude on spot @130
Pay back loan with interest
Pay physical storage
0
0
$6.53
Total
-83.46
-4.00
0
0
Final Profit =130-36-87-4=6 .53
Since the futures price exceeds the expected futures spot price, there will be a cash-and-carry
arbitrage opportunity. A cashand-carry arbitrage means we buy the spot today, sell the
futures today, store, and then deliver on the futures contract at expiration. This is a riskless
position, so is an example of arbitrage, because we had already locked in our price ahead of
time, so we dont care what happens to the spot price. The impact that this will have on the
market is that futures and spot will converge due to supply and demand in the spot and futures
markets until the arbitrage opportunity disappears.
8.) Consider the Cash Flows Below for Investments A and B. Suppose the discount rate is 0.15.
Calculate the NPVs. Which is preferred? Explain why in words using time-value of money
concepts. (15 points)
Cash
Flows
Period
Discounted Cash Flows
Inv A Inv B
Inv A
Inv B
0 -5000
-5000
-5000 -5000
3043.
1
4500
3500 3913.043478
478
2646.
2
2500
3500 1890.359168
503
NPV
803.402646
5
689.9
811
A is preferred because although the sum of the payoffs is the same, with A you get a higher
proportion of them sooner. Because of TVM then, A is preferred.
9.) This question will address optimal hedge ratios. Suppose that we have:
Variance of Spot price: ; Variance of futures price:;
Expected change in spot price: ; Expected change in futures price: ; and correlation between spot and
futures: . Find the optimal hedge ratio, h* assuming risk minimization [i.e., ]. Show work. (20points)
Solution: The risk minimizing hedge ratio can be solved for using the formula discussed in
class. That is,
What is the hedged portfolio variance (i.e., hedged basis risk) at this optimal hedge ratio?
AEM 4050 Spring 2012, Exam 2
[Hint: Just plug in the numbers given using the portfolio variance equation above setting h=h*]
Solution:
Suppose your underlying spot exposure is for 252,000 units of the underlying commodity. Assuming
that the futures contract is for 5,000 units of the underlying commodity, how many futures contracts
do the optimal represent?
Solution:
. That is, 180,000 bushels of short futures, or N= -180,000 / 5000 = -36 Contracts. That is, short
36 futures contracts.
Extra Credit, 3 points: Derive the minimum variance hedge ratio.
Extra Credit:
The Mean-Variance (or Expected Value-Variance or EV model) model is an alternative to the risk
minimization model. It takes into account expected returns from hedging in futures, as well as risk.
The mean variance objective function is written as:
where , and A is a parameter for the agents risk aversion.
Suppose we have an agent with A = 0.2, and all other parameters (variance of spot/futures, etc.) is
above.
Using the numbers in question 9, suppose that this agent is deciding whether to set h = -1 or h= 0.5. Calculate the MVs with these hedge ratios. Which does he/she prefer? Explain. (5 points)
Derive the MV maximizing hedge ratio. You must show your derivation to receive extra credit for
this part. Calculate h* for the MV maximizing hedge ratio and the value of MV function at the
associated h* using the parameters above and your answer for h* (10 points).
[Hint: = 14.44 when h=-1, and = 12.97 when h=-0.5]
Solution:
Case 1: h= -1
MV= [0+(-1 x 3)] 0.5 x 0.2 x 14.44 = -4.444
Case 2: h= -0.5
AEM 4050 Spring 2012, Exam 2
MV= [0+(-0.5 x 3)] 0.5 x 0.2 x 12.97 = -2.797
This hedger prefers h=-0.5 to h=-1 since MV is greater when h=-0.5 (that is, -2.797 > -4.44). The
reason is that this agent is not very risk averse, and since he/she will have expected average
losses associated with short futures positions, he/she will prefer a smaller hedge and will opt to
take on more risk.
Solution: