Pricing and Valuation of Forward Commitments
Literature
Chapter 2: Pricing and Valuation of Forward Commitments, p. 111-176
PIRIE, Wendy L. Derivatives. Hoboken: Wiley, 2017. CFA institute investment series. ISBN 978-1-119-
38181-5.
Chapters 5, 6, p. 107-154
HULL, John. Options, futures, and other derivatives. Ninth edition. Harlow: Pearson, 2018. ISBN
978-1-292-21289-0. Learning Outcomes
Learning Outcomes
• describe and compare how equity, interest rate, fixed-income, and currency forward and futures contracts
are priced and valued;
• calculate and interpret the no-arbitrage value of equity, interest rate, fixed-income, and cur- rency
forward and futures contracts;
• describe and compare how interest rate, currency, and equity swaps are priced and valued;
• calculate and interpret the no-arbitrage value of interest rate, currency, and equity swaps.
Problems
The following information relates to Questions 1-7
Donald Troubadour is a derivatives trader for Southern Shores Investments. The firm seeks arbitrage
opportunities in the forward and futures markets using the carry arbitrage model.
Troubadour identifies an arbitrage opportunity relating to a fixed-income futures contract and its underlying
bond. Current data on the futures contract and underlying bond are presented in Exhibit 1. The current
annual compounded risk-free rate is 0.30%.
Exhibit 1 - Current Data for Futures and Underlying Bond
Futures Contract Underlying Bond
Quoted futures price 125.00 Quoted bond price 112.00
Conversion factor 0.90 Accrued interest since last
coupon payment
0.08
Time remaining to contract
expiration
Three months Accrued interest at futures
contract expiration
0.20
Accrued interest over life of
futures contract
0.00
Troubadour next gathers information on three existing positions.
Position 1 (Nikkei 225 Futures Contract):
Troubadour holds a long position in a Nikkei 225 futures contract that has a remain- ing maturity of three
months.The continuously compounded dividend yield on the Nikkei 225 Stock Index is 1.1%, and the current
stock index level is 16,080. The continuously compounded annual interest rate is 0.2996%.
Position 2 (Euro/JPY Forward Contract):
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One month ago, Troubadour purchased euro/yen forward contracts with three months to expiration at a
quoted price of 100.20 (quoted as a percentage of par). The contract notional amount is 100,000,000. The
current forward price is 100.05.
Position 3 (JPY/USD Currency Forward Contract):
Troubadour holds a short position in a yen/US dollar forward contract with a notional value of $1,000,000.
At contract initiation, the forward rate was 112.10 per $1. The forward contract expires in three months.
The current spot exchange rate is 112.00 per $1, and the annually compounded risk-free rates are -0.20%
for the yen and 0.30% for the US dollar. The current quoted price of the forward contract is equal to the
no-arbitrage price.
Troubadour next considers an equity forward contract for Texas Steel, Inc. (TSI). Information regarding TSI
common shares and a TSI equity forward contract is presented in Exhibit 2.
Exhibit 2 - Selected Information for TSI
Selected Information for TSI
- TSI has historically paid dividends every six months.
- The price per share of TSI’s common shares is $250.
- The forward price per share for a nine-month TSI equity forward contract is $250.562289.
- Assume annual compounding.
Troubadour takes a short position in the TSI equity forward contract. His supervisor asks, “Under which
scenario would our position experience a loss?”
Three months after contract initiation, Troubadour gathers information on TSI and the risk-free rate, which
is presented in Exhibit 3.
Exhibit 3 - Selected Data on TSI and the Risk-Free Rate
Selected Data on TSI and the Risk-Free Rate
- The price per share of TSI’s common shares is $245.
- The risk-free rate is 0.325% (quoted on an annual compounding basis).
- TSI recently announced its regular semiannual dividend of $1.50 per share that will be paid exactly
three months before contract expiration.
- The market price of the TSI equity forward contract is equal to the no-arbitrage forward price.
1. Based on Exhibit 1 and assuming annual compounding, the arbitrage profit on the bond futures contract
is closest to:
• A. 0.4158.
• B. 0.5356.
• C. 0.6195.
2. The current no-arbitrage futures price of the Nikkei 225 futures contract (Position 1) is closest to:
• A. 15,951.81.
• B. 16,047.86.
• C. 16,112.21.
3. The value of Position 2 is closest to:
• A. - 149,925.
• B. - 150,000.
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• C. - 150,075.
4. The value of Position 3 is closest to:
• A. - 40,020.
• B. 139,913.
• C. 239,963.
5. Based on Exhibit 2, Troubadour should find that an arbitrage opportunity relating to TSI shares is
• A. not available.
• B. available based on carry arbitrage.
• C. available based on reverse carry arbitrage.
6. The most appropriate response to Troubadour’s supervisor’s question regarding the TSI forward contract
is:
• A. a decrease in TSI’s share price, all else equal.
• B. an increase in the risk-free rate, all else equal
• C. a decrease in the market price of the forward contract, all else equal.
7. Based on Exhibits 2 and 3, and assuming annual compounding, the per share value of Troubadour’s
short position in the TSI forward contract three months after contract initiation is closest to:
• A. $1.6549.
• B. $5.1561.
• C. $6.6549.
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The following information relates to Questions 8–16
Sonal Johnson is a risk manager for a bank. She manages the bank’s risks using a combination of swaps and
forward rate agreements (FRAs).
Johnson prices a three-year Libor-based interest rate swap with annual resets using the present value factors
presented in Exhibit 1.
Exhibit 1 - Present Value Factors
Maturity (years) Present Value Factors
1 0.990099
2 0.977876
3 0.965136
Johnson also uses the present value factors in Exhibit 1 to value an interest rate swap that the bank entered
into one year ago as the receive-floating party. Selected data for the swap are presented in Exhibit 2. Johnson
notes that the current equilibrium two-year fixed swap rate is 1.00%.
Exhibit 2 - Selected Data on Fixed for Floating Interest Rate Swap
Data
Swap notional amount $50,000,000
Original swap term Three years, with annual resets
Fixed swap rate (since initiation) 3.00%
One of the bank’s investments is exposed to movements in the Japanese yen, and Johnson desires to hedge
the currency exposure. She prices a one-year fixed-for-fixed currency swap involving yen and US dollars, with
a quarterly reset. Johnson uses the interest rate data presented in Exhibit 3 to price the currency swap.
Exhibit 3 - Selected Japanese and US Interest Rate Data
Days to Maturity Yen Spot Interest Rates US Dollar Spot Interest Rates
90 0.05% 0.20%
180 0.10% 0.40%
270 0.15% 0.55%
360 0.25% 0.70%
Johnson next reviews an equity swap with an annual reset that the bank entered into six months ago as the
receive-fixed, pay-equity party. Selected data regarding the equity swap, which is linked to an equity index,
are presented in Exhibit 4. At the time of initiation, the underlying equity index was trading at 100.00.
Exhibit 4 - Selected Data on Equity Swap
Data on Equity Swap
Swap notional amount $20,000,000
Original swap term Five years, with annual resets
Fixed swap rate 2.00%
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The equity index is currently trading at 103.00, and relevant US spot rates, along with their associated
present value factors, are presented in Exhibit 5.
Exhibit 5 - Selected US Spot Rates and Present Value Factors
Maturity (years) Spot Rate Present Value Factors
0.5 0.40% 0.998004
1.5 1.00% 0.985222
2.5 1.20% 0.970874
3.5 2.00% 0.934579
4.5 2.60% 0.895255
Johnson reviews a 6 × 9 FRA that the bank entered into 90 days ago as the pay-fixed/ receive-floating party.
Selected data for the FRA are presented in Exhibit 6, and current Libor data are presented in Exhibit 7.
Based on her interest rate forecast, Johnson also considers whether the bank should enter into new positions
in 1 × 4 and 2 × 5 FRAs.
Exhibit 6 - 6 x 9 FRA Data
Data
FRA term 6x9
FRA rate 0.70%
FRA notional amount US$20,000,000
FRA settlement terms Advanced set, advanced settle
Exhibit 7 - Current Libor
Current Libor
30-day Libor 0.75%
60-day Libor 0.82%
90-day Libor 0.90%
120-day Libor 0.92%
150-day Libor 0.94%
180-day Libor 0.95%
210-day Libor 0.97%
270-day Libor 1.00%
Three months later, the 6 x 9 FRA in Exhibit 6 reaches expiration, at which time the three-month US dollar
Libor is 1.10% and the six-month US dollar Libor is 1.20%. Johnson determines that the appropriate discount
rate for the FRA settlement cash flows is 1.10%.
8. Based on Exhibit 1, Johnson should price the three-year Libor-based interest rate swap at a fixed rate
closest to:
• A. 0.34%.
• B. 1.16%.
• C. 1.19%.
9. From the bank’s perspective, using data from Exhibit 1, the current value of the swap described in
Exhibit 2 is closest to:
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• A. –$2,951,963.
• B. –$1,967,975.
• C. –$1,943,000.
10. Based on Exhibit 3, Johnson should determine that the annualized equilibrium fixed swap rate for
Japanese yen is closest to:
• A. 0.0624%.
• B. 0.1375%.
• C. 0.2496%.
11. From the bank’s perspective, using data from Exhibits 4 and 5, the fair value of the equity swap is
closest to:
• A. –$1,139,425.
• B. –$781,323.
• C. –$181,323.
12. Based on Exhibit 5, the current value of the equity swap described in Exhibit 4 would be zero if the
equity index was currently trading the closest to:
• A. 97.30.
• B. 99.09.
• C. 100.00.
13. From the bank’s perspective, based on Exhibits 6 and 7, the value of the 6 x 9 FRA 90 days after
inception is closest to:
• A. $14,817.
• B. $19,647.
• C. $29,635.
14. Based on Exhibit 7, the no-arbitrage fixed rate on a new 1 × 4 FRA is closest to:
• A. 0.65%.
• B. 0.73%.
• C. 0.98%.
15. Based on Exhibit 7, the fixed rate on a new 2 x 5 FRA is closest to:
• A. 0.61%.
• B. 1.02%.
• C. 1.71%.
16. Based on Exhibit 6 and the three-month US dollar Libor at expiration, the payment amount that the
bank will receive to settle the 6 x 9 FRA is closest to:
• A. $19,945.
• B. $24,925.
• C. $39,781.
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