Practice Problems
Financial Derivatives
Spring 2016
Problem Set 5: Arithmetic of Financial Engineering
Problems 1 through 50 are also used in the investments course. We use them here to establish
the foundation of interest rate risk, which plays a significant role in derivatives strategies. Those
who remember them should review them to refresh memory, then move on to problems 51
through 70.
1.
A bond is offered with face value of $1,000, a 10% coupon rate with semi-annual payments,
and twenty years to maturity. If the market interest rate is 12%, what should be the price?
2.
If the market interest rate rises to 15%, what will be the new price for the bond in question
1?
3.
If the market interest rate drops to 9% instead, what will be the new price for the bond in
question 1?
4.
If the market interest rate changes to 8%, what will be the new price for the bond in question
1?
5.
If the market interest rate rises to 10%, what will be the new price for the bond in question
1?
Note that when the market rate equals the coupon rate, the price equals the face value.
6.
If the market interest rate drops to 6% instead, what will be the new price for the bond in
question 1?
7.
If the market interest rate changes to 17%, what will be the new price for the bond in
question 1?
8.
If the market interest rate rises to 18%, what will be the new price for the bond in question
1?
9.
If the market interest rate drops to 16% instead, what will be the new price for the bond in
question 1?
In the first three problems the market rate varied around 12%.1 In the next three problems it
varied around 8% (changing by 2% both up and down). In the final three it varied around 17%
(changing by 1% each way). Note that the capital gain from a decrease of x% in the market
interest rate would be greater that the capital loss from an increase of x%. This relationship is
known as convexity.
10. A bond is offered with face value of $1,000, a 10% coupon rate with semi-annual payments,
and ten years to maturity. If the market interest rate is 12%, what should be the price?
1It
changed by 3%, first going up from 12% to 15% and then going down from 12% to 9%.
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Practice Problems
Financial Derivatives
Spring 2016
11. If the market interest rate rises to 15%, what will be the new price for the bond in question
10?
12. If the market interest rate drops to 9% instead, what will be the new price for the bond in
question 10?
13. If the market interest rate changes to 8%, what will be the new price for the bond in question
10?
14. If the market interest rate rises to 10%, what will be the new price for the bond in question
10?
15. If the market interest rate drops to 6% instead, what will be the new price for the bond in
question 10?
16. If the market interest rate changes to 17%, what will be the new price for the bond in
question 10?
17. If the market interest rate rises to 18%, what will be the new price for the bond in question
10?
18. If the market interest rate drops to 16% instead, what will be the new price for the bond in
question 10?
Note that the capital gains or losses from changes in market interest rates are greater the longer
the time to maturity. That is, bonds with longer maturities are riskier to hold.
19. A bond is offered with face value of $1,000, a 5% coupon rate with semi-annual payments,
and twenty years to maturity. If the market interest rate is 12%, what should be the price?
20. If the market interest rate rises to 15%, what will be the new price for the bond in question
19?
21. If the market interest rate drops to 9% instead, what will be the new price for the bond in
question 19?
22. If the market interest rate changes to 8%, what will be the new price for the bond in question
19?
23. If the market interest rate rises to 10%, what will be the new price for the bond in question
19?
24. If the market interest rate drops to 6% instead, what will be the new price for the bond in
question 19?
25. If the market interest rate changes to 17%, what will be the new price for the bond in
question 19?
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Practice Problems
Financial Derivatives
Spring 2016
26. If the market interest rate rises to 18%, what will be the new price for the bond in question
19?
27. If the market interest rate drops to 16% instead, what will be the new price for the bond in
question 19?
28. A bond is offered with face value of $1,000, a 5% coupon rate with semi-annual payments,
and ten years to maturity. If the market interest rate is 12%, what should be the price?
29. If the market interest rate rises to 15%, what will be the new price for the bond in question
28?
30. If the market interest rate drops to 9% instead, what will be the new price for the bond in
question 28?
31. If the market interest rate changes to 8%, what will be the new price for the bond in question
28?
32. If the market interest rate rises to 10%, what will be the new price for the bond in question
28?
33. If the market interest rate drops to 6% instead, what will be the new price for the bond in
question 28?
34. If the market interest rate changes to 17%, what will be the new price for the bond in
question 28?
35. If the market interest rate rises to 18%, what will be the new price for the bond in question
28?
36. If the market interest rate drops to 16% instead, what will be the new price for the bond in
question 28?
There is something else you should notice from the data you have generated in the above
problems. The relative price changes2 resulting from a change in the market interest rate are
not the same for all bonds of the same maturity. Price volatility is higher when the coupon rate
is lower. This is known as coupon bias.
2That
is,
Pnew –Pold
.
Pold
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Practice Problems
Financial Derivatives
Spring 2016
37. A high tax bracket investor is comparing two bond issues for possible inclusion in her
portfolio. She expects market interest rates to be declining slowly but steadily over the
foreseeable future. Bond A is a 6% ten-year Aaa-rated bond selling at $750.76. Bond B is
an 8% ten-year Aaa-rated bond selling at $875.38. What is the yield to maturity for each
bond, assuming semi-annual payment of coupons? Which would you recommend, based on
the investor's expectations about future interest rates? She is not interested in current
income, but rather in capital appreciation.
38. Calculate the duration for each of the bonds in problems 1, 10, 19, and 28. The duration is
the weighted-average maturity of the package of payments, where the weights are
determined by each payment’s portion of total present value.
See how the duration reveals the effects of convexity and coupon bias. The longer the duration,
the more sensitive the bond price is to interest rate fluctuations.
39. Using the concepts of convexity and duration, explain the process of immunizing a bank’s
exposure to interest rate risk.
Now let’s develop the yield curve.
40. Suppose the interest rate on debt with a 1-year maturity is 6%, and the interest rate on debt
with a 2-year maturity is 6.5%. Theodore Jones expects that the 1-year interest rate will be
7.5% next year. He could invest over a 2-year horizon by purchasing the 2-year bond, or by
purchasing the 1-year bond with the intention of rolling over the money. Which bond will
he choose this year, the 1-year maturity or the 2-year maturity? What risks are involved
with the different strategies?
41. Suppose the interest rate on debt with a 2-year maturity is 7%. Which bond will Theodore
Jones choose this year, the 1-year maturity or the 2-year maturity? What risks are involved
with the different bonds?
42. Assume the yield curve is based upon unbiased expectations. Given Theodore Jones’
expectations, what would be the equilibrium interest rate on debt with a 2-year maturity?
43. Suppose the interest rate on debt with a 1-year maturity is 5%. The 1-year rate is expected
to rise to 6% next year and 7% the following year (in other words, the 1-year forward rate is
6% and the 2-year forward rate is 7%). Assume the yield curve is based upon unbiased
expectations. What would be the equilibrium interest rate on debt with a 3-year maturity?
44. Suppose the interest rate on debt with a 1-year maturity is 6%. The 1-year rate is expected
to rise to 6.5% next year, 7% the following year, and 8% the year after that. Assume the
yield curve is based upon unbiased expectations. What would be the equilibrium interest
rate on debt with a 4-year maturity?
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Practice Problems
Financial Derivatives
Spring 2016
Now let’s do some arbitrage
45. The following prices are observed. Calculate the yield to maturity for each bond, and
formulate a trading strategy to profit from the situation.
•
Treasury bonds with 6% coupon and 10 years remaining to maturity are selling at
$75.08 per $100 of face value (net of accrued interest).
•
Treasury bonds with 8% coupon and 10 years remaining to maturity are selling at
$79.00 per $100 of face value (net of accrued interest).
•
Equivalent risk bonds with 0% coupon and 10 years remaining to maturity are selling at
$36.80 per $100 of face value.
46. The following prices are observed. Calculate the yield to maturity for each bond, and
formulate a trading strategy to profit from the situation.
•
Treasury bonds with 6% coupon and 10 years remaining to maturity are selling at
$67.00 per $100 of face value (net of accrued interest).
•
Treasury bonds with 8% coupon and 10 years remaining to maturity are selling at
$76.00 per $100 of face value (net of accrued interest).
•
Treasury bonds with 10% coupon and 10 years remaining to maturity are selling at
$88.00 per $100 of face value (net of accrued interest).
47. The following prices are observed. Calculate the yield to maturity for each bond, and
formulate a trading strategy to profit from the situation.
•
Treasury bonds with 6% coupon and 8 years remaining to maturity are selling at $88.35
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 7 years remaining to maturity are selling at $87.00
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 6 years remaining to maturity are selling at $90.61
per $100 of face value (net of accrued interest).
48. The following prices are observed. Calculate the yield to maturity for each bond, and
formulate a trading strategy to profit from the situation.
•
Treasury bonds with 6% coupon and 8 years remaining to maturity are selling at $86.35
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 7 years remaining to maturity are selling at $89.44
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 6 years remaining to maturity are selling at $88.44
per $100 of face value (net of accrued interest).
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Practice Problems
Financial Derivatives
Spring 2016
49. The following prices are observed. Calculate the yield to maturity for each bond, and
formulate a trading strategy to profit from the situation.
•
Treasury bonds with 6% coupon and 8 years remaining to maturity are selling at $85.70
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 7 years remaining to maturity are selling at $89.44
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 6 years remaining to maturity are selling at $90.61
per $100 of face value (net of accrued interest).
50. The following prices are observed. Calculate the yield to maturity for each bond, and
formulate a trading strategy to profit from the situation.
•
Treasury bonds with 6% coupon and 8 years remaining to maturity are selling at $93.95
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 7 years remaining to maturity are selling at $90.68
per $100 of face value (net of accrued interest).
•
Treasury bonds with 6% coupon and 6 years remaining to maturity are selling at $90.61
per $100 of face value (net of accrued interest).
51. The following prices are observed. Formulate a trading strategy to profit from the situation.
•
US treasury bonds with 0% coupon and 2 years remaining to maturity cost $85.48 per
$100 of face value.
•
UK government bonds with 0% coupon and 2 years remaining to maturity cost £83.86
per £100 of face value.
•
Exchange rates are $1.00 = £0.645 spot, and $1.00 = £0.65 for 2-year forward.
52. The following prices are observed. Formulate a trading strategy to profit from the situation.
•
US treasury bonds with 0% coupon and 2 years remaining to maturity cost $85.48 per
$100 of face value.
•
German government bonds with 0% coupon and 2 years remaining to maturity cost
Euro 88.85 per € 100 of face value.
•
Exchange rates are $1.00=€ 0.74 spot, and $1.00=€ 0.75 for 2-year forward.
53. Suppose an option trader has a call bull spread. The stock price has risen substantially, and
the trader is considering closing the position early. What factors should the trader consider
with regard to closing the transaction before the options expire?
54. Suppose you are following the stock of a firm that has been experiencing severe problems.
Failure is imminent unless the firm is granted government-guaranteed loans. If the firm
fails, its stock will fall substantially. If the loans are granted, it is expected that the stock
will rise substantially. Identify strategies that would be appropriate for this situation.
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Practice Problems
Financial Derivatives
Spring 2016
55. Explain why selecting a strap rather than a straddle implies a somewhat more bullish
outlook.
56. Be prepared to describe a situation in which a strip might be appropriate. First, use the
picture symbols from Donald Smith’s article to explain the payoffs from a strip.
57. Be prepared to describe a situation in which a strangle might be appropriate. First, use the
picture symbols from Donald Smith’s article to explain the payoffs from a strangle.
58. Be prepared to describe a situation in which a calendar spread might be appropriate.
59. Many option traders use a combination of a money spread and a calendar spread called a
diagonal spread. This transaction involves the purchase of a call with a lower exercise price
and a longer time to expiration combined with the sale of a call with a higher exercise price
and a shorter time to expiration. Assume that today is February 11. The underlying stock is
presently at $165.125. Standard deviation is 21%. Option expirations are March 15 and
May 17 (riskfree rates are 5.03% for the March maturity and 5.71% for the May maturity).
Using the option calculation software, evaluate the diagonal spread that involves purchasing
a May 165 call and selling a March 170 call. Estimate the breakeven stock price at the end
of the holding period (assume the position will be unwound at the end of the trading day on
March 14). Explain the nature of the bet.
60. Consider a “plain vanilla” fixed-to-variable interest rate swap. Do the following from the
buyer’s perspective, then from the seller’s perspective: a) Replicate the swap with
transactions in the underlying bonds. b) Explain the exposures to interest rate risk generated
(or reduced) by the swap. c) Explain the bets being made about the direction of future
interest rate movements when one buys (or sells) such a swap.
61. Consider an interest rate cap. Do the following from the buyer’s perspective, then from the
seller’s perspective: a) Replicate the cap with transactions in the underlying bonds. b)
Explain the exposures to interest rate risk generated (or reduced) by the cap. c) Explain the
bets being made about the direction of future interest rate movements when one buys (or
sells) such a cap.
62. Consider an interest rate floor. Do the following from the buyer’s perspective, then from the
seller’s perspective: a) Replicate the floor with transactions in the underlying bonds. b)
Explain the exposures to interest rate risk generated (or reduced) by the floor. c) Explain the
bets being made about the direction of future interest rate movements when one buys (or
sells) such a floor.
63. Consider an interest rate collar. Do the following from the buyer’s perspective, then from
the seller’s perspective: a) Replicate the collar with transactions in the underlying bonds. b)
Explain the exposures to interest rate risk generated (or reduced) by the collar. c) Explain
the bets being made about the direction of future interest rate movements when one buys (or
sells) such a collar.
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Practice Problems
Financial Derivatives
Spring 2016
64. Consider a swap intermediary that enters into multiple fixed-to-variable interest rate swaps,
with various maturities on the fixed side of the swap. Suppose, for example, that the
intermediary swaps to pay LIBOR and receive the fixed rate for five-year maturity treasury
securities. With another counterparty, the same intermediary swaps to receive LIBOR and
pay the fixed rate for twenty-year maturity treasury securities. a) Replicate the package of
swaps with transactions in the underlying bonds. b) Explain the exposures to interest rate
risk generated (or reduced) by the net position created by the two swaps. c) Suppose the
intermediary holds a portfolio of long-term loans with average maturity of twenty years and
principal equal to the notional principal of the swaps. Combine this loan portfolio with the
swap package and explain the impact on interest rate risk exposure.
65. Consider an asset allocation swap in which a pension fund agrees to pay the income from a
bond portfolio in exchange for the returns from an equity market index (say, the S&P 500).
Do the following from the buyer’s perspective, then from the seller’s perspective: a)
Replicate the swap with transactions in the underlying securities. b) Explain the exposures to
interest rate risk, default risk, or equity market risk generated (or reduced) by the swap. c)
Explain the bets being made when one buys (or sells) such a swap.
66. Consider a securitization in which an intermediary purchases a portfolio of bonds, then
divides the cash flow stream so that coupon payments go to one group of investors, while
another group of investors get the principal when bonds in the package mature or are
recalled. (The coupon holders get all of the coupon payments received by the intermediary,
and the zero-coupon holders get all of the principal payments received by the intermediary.
When a bond is recalled prior to maturity, the zero-coupon holders gets all of the principal
portion of the final payment, and coupon-only holders get no further payments.) Do the
following from the buyer’s perspective, then from the seller’s perspective: a) Decompose
the coupon-only tranche into packages of the underlying securities and appropriate
derivatives. b) Decompose the zero-coupon tranche into packages of the underlying
securities and appropriate derivatives. c) Explain the exposures to interest rate risk, default
risk, or other risk generated (or reduced) by the engineered products. c) Explain the bets
being made when one buys (or sells) such products.
67. Consider a securitization in which an intermediary purchases a package of insured home
mortgages, then divides the cash flow stream into an interest-only tranche and a principalonly tranche. (The interest-only holders get all of the interest payments received by the
mortgage service provider, and the principal-only holders get all of the principal payments
received by the mortgage servicer. When a mortgage is repaid early or refinanced, the
principal-only tranche gets all the principal portion of the final payment, and interest-only
holders get no further payments.) Do the following from the buyer’s perspective, then from
the seller’s perspective: a) Decompose the interest-only tranche into packages of the
underlying securities and appropriate derivatives. b) Decompose the principal-only tranche
into packages of the underlying securities and appropriate derivatives. c) Explain the
exposures to interest rate risk, default risk, or other risk generated (or reduced) by the
engineered products. c) Explain the bets being made when one buys (or sells) such products.
page 8
Practice Problems
Financial Derivatives
Spring 2016
68. JunkCo has a low bond rating because it is small and new. JunkCo needs to finance some
new expansion and would like to borrow at a fixed rate for five years, but the lowest rate
available is 11%—which management considers too high. So, JunkCo decides to borrow
for five years at a variable rate 2% over the rate for Treasury Bills (the rate for T-Bills is
now 5%). Meanwhile AAA Corp needs money for only a year, and because of its high
rating can borrow for that maturity at 6%. If it wanted to, AAA Corp could borrow for five
years at a fixed rate of 8%. Suppose you work for CitiCorp. Can you figure out an
alternative borrowing and swap arrangement that would make both JunkCo and AAA Corp
better off?
69. Myron Labs is a British company producing pharmaceuticals and doing research into new
medicines. Myron Labs needs to finance some new expansion and would like to borrow at a
fixed rate for five years, but the lowest rate available to them in England is 11%—which
management considers too high. So, Myron Labs decides to borrow for five years at a
variable rate 2% over the rate for British Treasury Bills (which is now 5%). Meanwhile
Advanced Devices, an American company, needs money for only one year; and can borrow
in the U.S. for that maturity at 6%. If it wanted, Advanced Devices could borrow for five
years at a fixed rate of 8% in the U.S. market. Currency exchange rate is £1= $1.50 spot,
and also £1= $1.50 in the 1-year forward market. Suppose you work for CitiCorp. Can you
figure out an alternative borrowing and swap arrangement that would make both Myron
Labs and Advanced Devices Corp better off?
70. Be prepared to explain the parallel between a box spread and a net present value calculation
(such as the calculations used in corporate capital budgeting).
page 9
Practice Problems
Financial Derivatives
Solutions: Set 5
1.
FV is 1000, PMT is 50, interest per year is 12, P/YR is 2, N is 40, mode is END, calculate PV.
The answer is $849.54. (The negative sign in the display is due to the sign convention.)
2.
The only thing that needs to be changed in the calculator entries would be the interest per year,
which now is 15. (Here are the other entries: FV is 1000, PMT is 50, P/YR is 2, N is 40, mode is
END). When you calculate PV, the answer is $685.14. (The negative sign in the display is due to
the sign convention.)
3.
Once again, the only thing that needs to be changed in the calculator entries would be the interest
per year, which now is 9. The PV is $1,092.01.
4.
Once again, the only thing that needs to be changed in the calculator entries would be the interest
per year, which now is 8. The PV is $1,197.93.
5.
Interest per year is 10 (other entries remain unchanged from the above problems). PV is $1,000.
6.
Interest per year is 6 (other entries remain unchanged from the above problems). PV is $1,462.30
7.
Interest per year is 17 (other entries remain unchanged from the above problems). PV is $603.99.
8.
Interest per year is 18 (other entries remain unchanged from the above problems). PV is $569.71
9.
Interest per year is 16 (other entries remain unchanged from the above problems). PV is $642.26
10. FV is 1000, PMT is 50, interest per year is 12, P/YR is 2, N is 20, mode is END, calculate PV.
The answer is $885.30. (The negative sign in the display is due to the sign convention.)
11. The only thing that needs to be changed in the calculator entries would be the interest per year,
which now is 15. (Here are the other entries: FV is 1000, PMT is 50, P/YR is 2, N is 20, mode is
END). When you calculate PV, the answer is $745.14. (The negative sign in the display is due to
the sign convention.)
12. Interest per year is 9 (other entries remain unchanged from the above problems). PV is $1,065.04
13. Interest per year is 8 (other entries remain unchanged from the above problems). PV is $1,135.90
14. Interest per year is 10 (other entries remain unchanged from the above problems). PV is $1,000.00
15. Interest per year is 6 (other entries remain unchanged from the above problems). PV is $1,297.55
16. Interest per year is 17 (other entries remain unchanged from the above problems). PV is $668.78
17. Interest per year is 18 (other entries remain unchanged from the above problems). PV is $634.86
18. Interest per year is 16 (other entries remain unchanged from the above problems). PV is $705.46
19. FV is 1000, PMT is 25, interest per year is 12, P/YR is 2, N is 40, mode is END, calculate PV.
The answer is $473.38. (The negative sign in the display is due to the sign convention.)
20. The only thing that needs to be changed in the calculator entries would be the interest per year,
which now is 15. (Here are the other entries: FV is 1000, PMT is 25, P/YR is 2, N is 40, mode is
END). When you calculate PV, the answer is $370.28. (The negative sign in the display is due to
the sign convention.)
21. Interest per year is 9 (other entries remain unchanged from the above problems). PV is $631.97
22. Interest per year is 8 (other entries remain unchanged from the above problems). PV is $703.11
Prof. Kensinger
Spring 2013
page 1
Practice Problems
Financial Derivatives
Solutions: Set 5
23. Interest per year is 10 (other entries remain unchanged from the above problems). PV is $571.02
24. Interest per year is 6 (other entries remain unchanged from the above problems). PV is $884.43
25. Interest per year is 17 (other entries remain unchanged from the above problems). PV is $321.13
26. Interest per year is 18 (other entries remain unchanged from the above problems). PV is $300.77
27. Interest per year is 16 (other entries remain unchanged from the above problems). PV is $344.15
28. FV is 1000, PMT is 25, interest per year is 12, P/YR is 2, N is 20, mode is END, calculate PV.
The answer is $598.55. (The negative sign in the display is due to the sign convention.)
29. The only thing that needs to be changed in the calculator entries would be the interest per year,
which now is 15. (Here are the other entries: FV is 1000, PMT is 25, P/YR is 2, N is 20, mode is
END). When you calculate PV, the answer is $490.28. (The negative sign in the display is due to
the sign convention.)
30. Interest per year is 9 (other entries remain unchanged from the above problems). PV is $739.84
31. Interest per year is 8 (other entries remain unchanged from the above problems). PV is $796.15
32. Interest per year is 10 (other entries remain unchanged from the above problems). PV is $688.44
33. Interest per year is 6 (other entries remain unchanged from the above problems). PV is $925.61
34. Interest per year is 17 (other entries remain unchanged from the above problems). PV is $432.20
35. Interest per year is 18 (other entries remain unchanged from the above problems). PV is $406.64
36. Interest per year is 16 (other entries remain unchanged from the above problems). PV is $460.00
37. Both bonds yield 10%; but she would pick Bond A because it gives lower reinvestment risk and a
smaller income tax burden, with the same yield. Given her expectations for declining interest
rates, bond A would also offer a higher expected capital gain.
38. 8.20 years, 6.31 years, 9.42 years, 7.27 years
39. For class discussion.
40. Rollover strategy has expected yield of 6.75%. Since this exceeds the yield on a 2-year obligation,
Theodore will choose the rollover strategy. Risk is that interest rate next year will be less than
expected.
41. Since 7% > 6.75%, Theodore will choose the 2-year maturity. Risk is that short interest rate next
year will be less than expected.
42. If $100 were invested at 6% for the first year followed by 7.5% for the second year, it would grow
to $113.95. Input this as FV, and –100 as PV. This is an average compound annual rate of 6.75%
(in order to find this, additional data inputs are as follows: PMT is zero, P/YR is 1, and N is 2,
compute I/YR).
43. If $100 were invested at 5% for the first year followed by 6% for the second year, and 7% for the
third year, it would grow to $119.09. Input this as FV, and –100 as PV. This is an average
compound annual rate of 6.00% (in order to find this, additional data inputs are as follows: PMT is
zero, P/YR is 1, and N is 3, compute I/YR).
Prof. Kensinger
Spring 2013
page 2
Practice Problems
Financial Derivatives
Solutions: Set 5
44. If $100 were invested at 6% for the first year followed by 6.5% for the second year, 7% for the
third year, and 8% for the fourth year, it would grow to $130.46. Input this as FV, and –100 as
PV. This is an average compound annual rate of 6.87% (in order to find this, additional data inputs
are as follows: PMT is zero, P/YR is 1, and N is 4, compute I/YR).
45. YTM does not follow the normal pattern. It is 11.60% for the 8% coupon, 10.00% for the 6%
coupon, and 10.24% for the 0% coupon. Therefore sell four of the 6%, while buying three of the
8% coupon and one of the 0% coupon bonds. Net cash flow is +$26.52 in the present, and zero in
all future periods. Arbitrage examples like this prove that the yield curve cannot change direction.
46. YTM does not follow the normal pattern. It is 12.10% for the 10% coupon, 12.22% for the 8%
coupon, and 11.68% for the 6% coupon. Therefore sell one each of the 6% and the 10% coupon,
while buying two of the 8% coupon bonds. Net cash flow is +$3 in the present, and zero in all
future periods. Arbitrage examples like this prove that the yield curve cannot change direction.
47. YTM does not follow the normal pattern. It is 8% for both the 6 and 8 year maturities, but 8.50%
for the 7 year. Therefore buy the 7-year bond and hedge. To create two synthetic bonds with 7
year duration, sell one 6-year bond and one 8-year bond. If you buy two 7-year bonds and sell one
each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at any
discount rate. Cash flows are +4.96 in period 0, 0 in periods 1-11, –100 in period 12, +3 in period
13, +203 in period 14, –3 in period 15, and –103 in period 16. Arbitrage examples like this prove
that the yield curve cannot be kinked.
48. YTM does not follow the normal pattern. It is 8.50% for the 6-year maturity, 8.00% for the 7-year,
but 8.38% for the 8-year. Therefore sell the 7-year bond and hedge. To create two synthetic bonds
with 7 year duration, buy one 6-year bond and one 8-year bond. If you sell two 7-year bonds and
buy one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at
any discount rate. Cash flows are +4.09 in period 0, 0 in periods 1-11, +100 in period 12, -3 in
period 13, -203 in period 14, +3 in period 15, and +103 in period 16. Arbitrage examples like this
prove that the yield curve cannot be kinked.
49. YTM does not follow the normal pattern. It is 8.50% for the 8-year maturity, but 8.00% for both
the 7 year and 6-year. Therefore sell the 7-year bond and hedge. To create two synthetic bonds
with 7 year duration, buy one 6-year bond and one 8-year bond. If you sell two 7-year bonds and
buy one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at
any discount rate. Cash flows are +2.57 in period 0, 0 in periods 1-11, +100 in period 12, -3 in
period 13, -203 in period 14, +3 in period 15, and +103 in period 16. Arbitrage examples like this
prove that the yield curve cannot change direction.
50. YTM does not follow the normal pattern. It is 8% for the 6-year maturity, 7.75% for the 7-year
but 7.00% for the 8-year. Therefore buy the 7-year bond and hedge. If you buy two 7-year bonds
and sell one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive
NPV at any discount rate. Cash flows are +3.20 in period 0, 0 in periods 1-11, –100 in period 12,
+3 in period 13, +203 in period 14, –3 in period 15, and –103 in period 16. Arbitrage examples
like this prove that the yield curve cannot be kinked.
51. Given the forward rate, £100=$153.85 two years from now. At the spot exchange rate, that
amount costs $131.51 today in the US and $129.24 in the UK. Money will flow from US to UK in
the spot market and return in the forward market. Dollar will weaken spot and strengthen forward.
Prof. Kensinger
Spring 2013
page 3
Practice Problems
Financial Derivatives
Solutions: Set 5
52. Given the forward rate, $100=€ 75 two years from now. That amount costs $85.48 today in the
US bond market, and € 66.64 in the German bond market (at the spot exchange rate, this is
$90.05). Money will flow from Germany to the US in the spot market and return in the forward
market. Dollar will strengthen spot and weaken forward.
53. The root question is whether or not the trader is still bullish about the underlying. If the trader still
believes the underlying is likely to rise substantially, the holding period should be extended. If
there is just the normal cloud of uncertainty, the profit potential is still greater with the position
alive than dead (but since this is captured in the option prices, immediate action is a matter of
indifference). If the trader believes the underlying has gone too high and is about to reverse,
though, then the position should be unwound immediately.
54. If the information given in the problem were not publicly available, this would be the classic
opportunity for a long straddle (to profit from the prospect of high volatility). The information
sounds like the sort of thing that would be readily predictable for reasonably informed traders,
though, so there may not be any opportunity here.
55-58. For class discussion.
59. For the March 170 call there are 32 days remaining until expiration. For the May 165 call there are
95 days remaining until expiration. The value of the March 170 call is $240.56 for a 100-share
contract. The value of the May 165 call is $835.33 for a 100-share contract (thus the spread is a net
long position costing $594.77 per 100 shares, or $5.9477 per share). Delta for the March 170 call
is 0.3572, and delta for the May 165 call is 0.5790; so the May call is substantially more
responsive to stock price movements (so this is a bullish spread). At the end of the holding period,
there would be 64 days remaining until expiration of the May call. If the T-bill rate and the
volatility don’t change, the spread would break even with the stock price at the end of the holding
period at $163.7743 per share (then the March 170 call would expire worthless, and the May 165
call would be worth $5.9477 per share). Thus this diagonal spread is more expensive than a simple
money spread, but offers more profit over a wider range of terminal stock prices.
60-67. For class discussion.
68. AAA Corp borrows fixed for 5 years at 8% and JunkCo borrows floating. Then AAA Corp swaps
to receive 8% fixed and pay T-bill, while JunkCo swaps to receive T-bill and pay 8% fixed. Thus
during the first year AAA Corp gets to borrow at the T-bill rate. Then instead of paying off its
debt (as originally planned) it establishes a sinking fund invested in in T-Bills and continues to roll
over for the remaining 4 years, while paying interest on the fixed-rate loan. AAA Corp’s annual
cash flow stream for the remaining life of the arrangement is as follows:
Meanwhile, JunkCo in effect borrows at 10% fixed, a full percentage point lower than it could do
on its own. JunkCo’s annual cash flow steam is as follows:
Prof. Kensinger
Spring 2013
page 4
Practice Problems
69-70.
Prof. Kensinger
Financial Derivatives
Solutions: Set 5
For class discussion.
Spring 2013
page 5
Financial Derivatives
Practice Problems
Solutions: Set 5
Table Illustrating Coupon Bias and Convexity
old
rate
new
rate
new price
capital gain
(loss)
relative
change
old price
20-year,
12%
15%
$849.54
$685.14
($164.40)
-19.35%
10% bonds
12%
9%
$849.54
$1,092.01
$242.47
+28.54%
8%
10%
$1,197.93
$1,000.00
($197.93)
-16.52%
8%
6%
$1,197.93
$1,462.30
$264.37
+22.07%
17%
18%
$603.99
$569.71
($34.28)
-5.68%
17%
16%
$603.99
$642.26
$38.27
+6.34%
10-year,
12%
15%
$885.30
$745.14
($140.16)
-15.83%
10% bonds
12%
9%
$885.30
$1,065.04
$179.74
+20.30%
8%
10%
$1,135.90
$1,000.00
($135.90)
-11.96%
8%
6%
$1,135.90
$1,297.55
$161.65
+14.23%
17%
18%
$668.78
$634.86
($33.92)
-5.07%
17%
16%
$668.78
$705.46
$36.68
+5.48%
20-year,
12%
15%
$473.38
$370.28
($103.10)
-21.78%
5% bonds
12%
9%
$473.38
$631.97
$158.59
+33.50%
8%
10%
$703.11
$571.02
($132.09)
-18.79%
8%
6%
$703.11
$884.43
$181.32
+25.79%
17%
18%
$321.13
$300.77
($20.36)
-6.34%
17%
16%
$321.13
$344.15
$23.02
+7.17%
10-year,
12%
15%
$598.55
$490.28
($108.27)
-18.09%
5% bonds
12%
9%
$598.55
$739.84
$141.29
+23.61%
8%
10%
$796.15
$688.44
($107.71)
-13.53%
8%
6%
$796.15
$925.61
$129.46
+16.26%
17%
18%
$432.20
$406.64
($25.56)
-5.91%
17%
16%
$432.20
$460.00
$27.80
+6.43%
Prof. Kensinger
Spring 2009
page 6
Practice Problems
Prof. Kensinger
Financial Derivatives
Spring 2009
Solutions: Set 5
page 7