Microsoft Word – Practice Exam Questions and Answers.docx
Practice Exam Questions and Answers
Question 1 (Problem 1.29.)
On May 8, 2013, as indicated in Table 1.2, the spot offer price of Google stock is $871.37
and the offer price of a call option with a strike price of $880 and a maturity date of
September is $41.60. A trader is considering two alternatives: buy 100 shares of the stock
and buy 100 September call options. For each alternative, what is (a) the upfront cost, (b)
the total gain if the stock price in September is $950, and (c) the total loss if the stock
price in September is $800. Assume that the option is not exercised before September and
if stock is purchased it is sold in September.
a) The upfront cost for the stock alternative is $87,137. The upfront cost for the option
alternative is $4,160.
b) The gain from the stock alternative is $95,000−$87,137=$7,863. The total gain from
the option alternative is ($950-$880)×100−$4,160=$2,840.
c) The loss from the stock alternative is $87,137−$80,000=$7,137. The loss from the
option alternative is $4,160.
Question 2 (Problem 3.31.)
A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is
concerned about the performance of the market over the next two months and plans to use
three-month futures contracts on the S&P 500 to hedge the risk. The current level of the
index is 1250, one contract is on 250 times the index, the risk-free rate is 6% per annum, and
the dividend yield on the index is 3% per annum. The current 3 month futures price is 1259.
a) What position should the fund manager take to eliminate all exposure to the market
over the next two months?
b) Calculate the effect of your strategy on the fund manager’s returns if the level of the
market in two months is 1,000, 1,100, 1,200, 1,300, and 1,400. Assume that the one-
month futures price is 0.25% higher than the index level at this time.
a) The number of contracts the fund manager should short is
50 000 000
0 87 138 20
1259 250
Rounding to the nearest whole number, 138 contracts should be shorted.
b) The following table shows that the impact of the strategy. To illustrate the
calculations in the table consider the first column. If the index in two months is
1,000, the futures price is 1000×1.0025. The gain on the short futures position is
therefore
(1259 1002 50) 250 138 8 849 250$
The return on the index is 3 2 12 =0.5% in the form of dividend and
250 1250 20% in the form of capital gains. The total return on the index is
therefore 19 5% . The risk-free rate is 1% per two months. The return is therefore
20 5% in excess of the risk-free rate. From the capital asset pricing model we
expect the return on the portfolio to be 0 87 20 5 17 835% % in excess of the
risk-free rate. The portfolio return is therefore 16 835% . The loss on the portfolio
is 0 16835 50 000 000 or $8,417,500. When this is combined with the gain on the
futures the total gain is $431,750.
Index now 1250 1250 1250 1250 1250
Index Level in Two Months 1000 1100 1200 1300 1400
Return on Index in Two Months -0.20 -0.12 -0.04 0.04 0.12
Return on Index incl divs -0.195 -0.115 -0.035 0.045 0.125
Excess Return on Index -0.205 -0.125 -0.045 0.035 0.115
Excess Return on Portfolio -0.178 -0.109 -0.039 0.030 0.100
Return on Portfolio -0.168 -0.099 -0.029 0.040 0.110
Portfolio Gain -8,417,500 -4,937,500 -1,457,500 2,022,500 5,502,500
Futures Now 1259 1259 1259 1259 1259
Futures in Two Months 1002.50 1102.75 1203.00 1303.25 1403.50
Gain on Futures 8,849,250 5,390,625 1,932,000 -1,526,625 -4,985,250
Net Gain on Portfolio 431,750 453,125 474,500 495,875 517,250
Question 3 (Problem 5.28.)
The current USD/euro exchange rate is 1.4000 dollar per euro. The six month forward
exchange rate is 1.3950. The six month USD interest rate is 1% per annum continuously
compounded. Estimate the six month euro interest rate.
If the six-month euro interest rate is rf then
5.0)01.0(
4000.13950.1
fre
so that
00716.0
4000.1
3950.1
ln201.0
fr
and rf = 0.01716. The six-month euro interest rate is 1.716%.
Question 4 (Problem 5.30.)
A stock is expected to pay a dividend of $1 per share in two months and in five months. The
stock price is $50, and the risk-free rate of interest is 8% per annum with continuous
compounding for all maturities. An investor has just taken a short position in a six-month
forward contract on the stock.
a) What are the forward price and the initial value of the forward contract?
b) Three months later, the price of the stock is $48 and the risk-free rate of interest is
still 8% per annum. What are the forward price and the value of the short position in
the forward contract?
a) The present value, I , of the income from the security is given by:
0 08 2 12 0 08 5 121 1 1 9540I e e
From equation (5.2) the forward price, 0F , is given by:
0 08 0 50 (50 1 9540) 50 01F e
or $50.01. The initial value of the forward contract is (by design) zero. The fact that
the forward price is very close to the spot price should come as no surprise. When
the compounding frequency is ignored the dividend yield on the stock equals the
risk-free rate of interest.
b) In three months:
0 08 2 12 0 9868I e
The delivery price, K , is 50.01. From equation (5.6) the value of the short forward
contract, f , is given by
0 08 3 12(48 0 9868 50 01 ) 2 01f e
and the forward price is
0 08 3 12(48 0 9868) 47 96e
Question 5 (Problem 7.23.)
Under the terms of an interest rate swap, a financial institution has agreed to pay 10% per
annum and receive three-month LIBOR in return on a notional principal of $100 million with
payments being exchanged every three months. The swap has a remaining life of 14 months.
The average of the bid and offer fixed rates currently being swapped for three-month LIBOR
is 12% per annum for all maturities. The three-month LIBOR rate one month ago was 11.8%
per annum. All rates are compounded quarterly. What is the value of the swap?
The swap can be regarded as a long position in a floating-rate bond combined with a short
position in a fixed-rate bond. The correct discount rate is 12% per annum with quarterly
compounding or 11.82% per annum with continuous compounding.
Immediately after the next payment the floating-rate bond will be worth $100 million. The
next floating payment ($ million) is
0 118 100 0 25 2 95
The value of the floating-rate bond is therefore
0 1182 2 12102 95 100 941e
The value of the fixed-rate bond is
0 1182 2 12 0 1182 5 12 0 1182 8 122 5 2 5 2 5e e e
0 1182 11 12 0 1182 14 122 5 102 5 98 678e e
The value of the swap is therefore
100 941 98 678 $2 263million
As an alternative approach we can value the swap as a series of forward rate agreements.
The calculated value is
0 1182 2 12 0 1182 5 12(2 95 2 5) (3 0 2 5)e e
0 1182 8 12 0 1182 11 12(3 0 2 5) (3 0 2 5)e e
0 1182 14 12(3 0 2 5) $2 263 millione
which is in agreement with the answer obtained using the first approach.
Question 6 (Problem 7.1.)
Companies A and B have been offered the following rates per annum on a $20 million five-
year loan:
Fixed Rate Floating Rate
Company A 5.0% LIBOR+0.1%
Company B 6.4% LIBOR+0.6%
Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a
swap that will net a bank, acting as intermediary, 0.1% per annum and that will appear
equally attractive to both companies.
A has an apparent comparative advantage in fixed-rate markets but wants to borrow
floating. B has an apparent comparative advantage in floating-rate markets but wants to
borrow fixed. This provides the basis for the swap. There is a 1.4% per annum differential
between the fixed rates offered to the two companies and a 0.5% per annum differential
between the floating rates offered to the two companies. The total gain to all parties from
the swap is therefore 1 4 0 5 0 9 % per annum. Because the bank gets 0.1% per annum of
this gain, the swap should make each of A and B 0.4% per annum better off. This means that
it should lead to A borrowing at LIBOR 0 3 % and to B borrowing at 6.0%. The appropriate
arrangement is therefore as shown in Figure S7.1.
Figure S7.1: Swap for Problem 7.1
Company
A
Financial
Institution
Company
B
LIBOR LIBOR
LIBOR+0.6%
5.4% 5.3%
5%
Question 7 (Problem 13.9.)
A stock price is currently $50. It is known that at the end of two months it will be either $53
or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is
the value of a two-month European call option with a strikeprice of $49? Use no-arbitrage
arguments.
At the end of two months the value of the option will be either $4 (if the stock price is $53)
or $0 (if the stock price is $48). Consider a portfolio consisting of:
shares
1 option
The value of the portfolio is either 48 or 53 4 in two months. If
48 53 4
i.e.,
0 8
the value of the portfolio is certain to be 38.4. For this value of the portfolio is therefore
riskless. The current value of the portfolio is:
0 8 50 f
where f is the value of the option. Since the portfolio must earn the risk-free rate of
interest
0 10 2 12(0 8 50 ) 38 4f e
i.e.,
2 23f
The value of the option is therefore $2.23.
This can also be calculated directly from equations (13.2) and (13.3). 1 06u , 0 96d so
that
0 10 2 12 0 96
0 5681
1 06 0 96
e
p
and
0 10 2 12 0 5681 4 2 23f e
Question 8 (Problem 15.21.)
Consider an American call option on a stock. The stock price is $50, the time to maturity is 15
months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and the
volatility is 25%. Dividends of $1.50 are expected in 4 months and 10 months. Show that it
can never be optimal to exercise the option on either of the two dividend dates. Calculate the
price of the option.
With the notation in the text
1 2 1 21 50 0 3333 0 8333 1 25 0 08 and 55D D t t T r K
2( ) 0 08 0 41671 55(1 ) 1 80r T tK e e
Hence
2( )2 1
r T tD K e
Also:
2 1( ) 0 08 0 51 55(1 ) 2 16r t tK e e
Hence:
2 1( )1 1
r t tD K e
It follows from the conditions established in Section 15.12 that the option should never be
exercised early.
The present value of the dividends is
0 3333 0 08 0 8333 0 081 5 1 5 2 864e e
The option can be valued using the European pricing formula with:
0 50 2 864 47 136 55 0 25 0 08 1 25S K r T
2
1
2 1
ln(47 136 55) (0 08 0 25 2)1 25
0 0545
0 25 1 25
0 25 1 25 0 3340
d
d d
1 2( ) 0 4783 ( ) 0 3692N d N d
and the call price is
0 08 1 2547 136 0 4783 55 0 3692 4 17e
or $4.17.
Question 9 (Problem 22.17.)
Consider a position consisting of a $300,000 investment in gold and a $500,000 investment
in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%
respectively, and that the coefficient of correlation between their returns is 0.6. What is the
10-day 97.5% VaR for the portfolio? By how much does diversification reduce the VaR?
The variance of the portfolio (in thousands of dollars) is
2 2 2 20 018 300 0 012 500 2 300 500 0 6 0 018 0 012 104 04
The standard deviation is 104 04 10 2 . Since ( 1 96) 0 025N , the 1-day 97.5% VaR is
10 2 1 96 19 99 and the 10-day 97.5% VaR is 10 19 99 63 22 . The 10-day 97.5% VaR
is therefore $63,220. The 10-day 97.5% value at risk for the gold investment is
5 400 10 1 96 33 470 . The 10-day 97.5% value at risk for the silver investment is
6 000 10 1 96 37 188 . The diversification benefit is
33 470 37 188 63 220 $7 438
Question 10 (Problem 23.18.)
Suppose that in Problem 23.17 the price of silver at the close of trading yesterday was $16,
its volatility was estimated as 1.5% per day, and its correlation with gold was estimated as
0.8. The price of silver at the close of trading today is unchanged at $16. Update the
volatility of silver and the correlation between silver and gold using the two models in
Problem 23.17. In practice, is the parameter likely to be the same for gold and silver?
The EWMA model with 0 94
The GARCH(1,1) model with 0 000002 , 0 04 , and 0 94
The proportional change in the price of silver is zero. Using the EWMA model the variance is
updated to
20 94 0 015 0 06 0 0 0002115
so that the new daily volatility is 0 0002115 0 01454 or 1.454% per day. Using GARCH
(1,1) the variance is updated to
20 000002 0 94 0 015 0 04 0 0 0002135
so that the new daily volatility is 0 0002135 0 01461 or 1.461% per day. The initial
covariance is 0 8 0 013 0 015 0 000156 Using EWMA the covariance is updated to
0 94 0 000156 0 06 0 0 00014664
so that the new correlation is 0 00014664 (0 01454 0 01271) 0 7934 Using GARCH (1,1)
the covariance is updated to
0 000002 0 94 0 000156 0 04 0 0 00014864
so that the new correlation is 0 00014864 (0 01461 0 01275) 0 7977 .
For a given and , the parameter defines the long run average value of a variance or
a covariance. There is no reason why we should expect the long run average daily variance
for gold and silver should be the same. There is also no reason why we should expect the
long run average covariance between gold and silver to be the same as the long run average
variance of gold or the long run average variance of silver. In practice, therefore, we are
likely to want to allow in a GARCH(1,1) model to vary from market variable to market
variable.
Question 11 (Problem 24.25.)
A company has one- and two-year bonds outstanding, each providing a coupon of 8% per
year payable annually. The yields on the bonds (expressed with continuous compounding are
6.0% and 6.6%, respectively. Risk-free rates are 4.5% for all maturities. The recovery rate is
35%. Defaults can take place half way through each year. Estimate the risk-neutral default
rate each year.
Consider the first bond. Its market price is 0 06 1108 101 71e . Its default-free price is
0 045 1108 103 25e . The present value of the loss from defaults is therefore 1.54. In this
case losses can take place at only one time, halfway through the year. Suppose that the
probability of default at this time is 1Q . The default-free value of the bond is
0 045 0 5108 105 60e . The loss in the event of a default is 105 60 35 70 60 . The present
value of the expected loss is 0 045 0 5 1 170 60 69 03e Q Q
. It follows that
169 03 1 54Q
so that 1 0 0223Q .
Now consider the second bond. It market price is 102.13 and its default-free value is
106.35. The present value of the loss from defaults is therefore 4.22. At time 0.5 the default
free value of the bond is 108.77. The loss in the event of a default is therefore 73.77. The
present value of the loss from defaults at this time is 172 13Q or 1.61. This means that the
present value of the loss from defaults at the 1.5 year point is 4 22 1 61 or 2.61. The
default-free value of the bond at the 1.5 year point is 105.60. The loss in the event of a
default is 70.60. The present value of the expected loss is 265 99Q where 2Q is the
probability of default in the second year. It follows that
265 99 2 61Q
so that 2 0 0396Q .
The probabilities of default in years one and two are therefore 2.23% and 3.96%.