LOS c: Calculate and interpret the price and the value of 1) a forward contract on a fixed income security, 2) a forward rate agreement (FRA), and 3) a forward contract on a currency.

Q1. Consider a 9-month forward contract on a 10-year 7% Treasury note just issued at par. The effective annual risk-free rate is 5% over the near term and the first coupon is to be paid in 182 days. The price of the forward is closest to:

A)   1,037.27.

B)   1,001.84.

C)   965.84.

Q2. The U.S. risk-free rate is 2.96%, the Japanese yen risk-free rate is 1.00%, and the spot exchange rate between the United States and Japan is $0.00757 per yen. Both rates are continuously compounded. The price of a 180-day forward contract on the yen and the value of the forward position 90 days into the contract when the spot rate is $0.00797 are closest to:

          Forward Price        Value After 90 Days

A)    $0.00764                            $0.00212

B)    $0.00764                            $0.00037

Q3. Calculate the price of a 200-day forward contract on an 8% U.S. Treasury bond with a spot price of $1,310. The bond has just paid a coupon and will make another coupon payment in 150 days.  The annual risk-free rate is 5%.

A)   $1,333.50.

B)   $1,305.22.

C)   $1,270.79.

Q4. 30 days ago, J. Klein took a short position in a $10 million 90-day forward rate agreement (FRA) based on the 90-day London Interbank Offered Rate (LIBOR) and priced at 5%. The current LIBOR curve is:

• 30-day = 4.8%
• 60-day = 5.0%
• 90-day = 5.1%
• 120-day = 5.2%
• 150-day = 5.4%

The current value of the FRA, to the short, is closest to:

A)   ?$15,154.

B)   ?$15,495.

C)   ?$15,280.

Q5. What is the value of a 6.00% 1x4 (30 days x 120 days) forward rate agreement (FRA) with a principal amount of $2,000,000, 10 days after initiation if L_10(110) is 6.15% and L₁₀₍₂₀₎ is 6.05%?

A)   $767.40.

B)   $745.76.

C)   $700.00.

Q6. Monica Lewis, CFA, has been hired to review data on a series of forward contracts for a major client. The client has asked for an analysis of a contract with each of the following characteristics:

1.       A forward contract on a U.S. Treasury bond

2.       A forward rate agreement (FRA)

3.       A forward contract on a currency

Information related to a forward contract on a U.S. Treasury bond: The Treasury bond carries a 6% coupon and has a current spot price of $1,071.77 (including accrued interest). A coupon has just been paid and the next coupon is expected in 183 days. The annual risk-free rate is 5%. The forward contract will mature in 195 days.

Information related to a forward rate agreement: The relevant contract is a 3 × 9 FRA. The current annualized 90-day money market rate is 3.5% and the 270-day rate is 4.5%. Based on the best available forecast, the 180-day rate at the expiration of the contract is expected to be 4.2%.

Information related to a forward contract on a currency: The risk-free rate in the U.S. is 5% and 4% in Switzerland. The current spot exchange rate is $0.8611 per Swiss France (SFr). The forward contract will mature in 200 days.

Based on the information given, what initial price should Lewis recommend for a forward contract on the Treasury bond?

A)   $1,073.54.

B)   $1,070.02.

C)   $1,035.12.

Q7. Suppose that the price of the forward contract for the Treasury bond was negotiated off-market and the initial value of the contract was positive as a result. Which party makes a payment and when is the payment made?

A)   The short pays the long at the maturity of the contract.

B)   The long pays the short at the maturity of the contract.

C)   The long pays the short at the initiation of the contract.

Q8. Suppose that instead of a forward contract on the Treasury bond, a similar futures contract was being considered. Which one of the following alternatives correctly gives the preference that an investor would have between a forward and a futures contract on the Treasury bond?

A)   The futures contract will be preferred to the forward contract.

B)   The forward contract will be preferred to the futures contract.

C)   It is impossible to say for certain because it depends on the correlation between the underlying asset and interest rates.

Q9. Based on the information given, what initial price should Lewis recommend for the 3 × 9 FRA?

A)   4.96%.

B)   5.66%.

C)   4.66%.

Q10. Based on the information given and assuming a notional principal of $10 million, what value should Lewis place on the 3 × 9 FRA at time of settlement?

A)   $38,000 paid from short to long.

B)   $37,218 paid from long to short.

C)   $19,000 paid from long to short.

LOS c: Calculate and interpret the price and the value of 1) a forward contract on a fixed income security, 2) a forward rate agreement (FRA), and 3) a forward contract on a currency. fficeffice" />

Q1. Consider a 9-month forward contract on a 10-year 7% Treasury note just issued at par. The effective annual risk-free rate is 5% over the near term and the first coupon is to be paid in 182 days. The price of the forward is closest to:

A)   1,037.27.

B)   1,001.84.

C)   965.84.

Correct answer is B)

The forward price is calculated as the bond price minus the present value of the coupon, times one plus the risk-free rate for the term of the forward.
        (1,000 – 35/1.05^182/365) 1.05^9/12 = $1,001.84

Q2. The ffice:smarttags" />U.S. risk-free rate is 2.96%, the Japanese yen risk-free rate is 1.00%, and the spot exchange rate between the United States and Japan is $0.00757 per yen. Both rates are continuously compounded. The price of a 180-day forward contract on the yen and the value of the forward position 90 days into the contract when the spot rate is $0.00797 are closest to:

          Forward Price        Value After 90 Days

A)    $0.00764                            $0.00212

B)    $0.00764                            $0.00037

C)    $0.00750                            $0.00212

Correct answer is B)

The no-arbitrage price of the 180-day forward contract is:

F_T = $0.00757 × e^{(0.0296 ? 0.0100) × (180 / 365)} = $0.00764

The value of the contract in 90 days with 180 – 90 = 90 days remaining is:

Q3. Calculate the price of a 200-day forward contract on an 8% U.S. Treasury bond with a spot price of $1,310. The bond has just paid a coupon and will make another coupon payment in 150 days.  The annual risk-free rate is 5%.

A)   $1,333.50.

B)   $1,305.22.

C)   $1,270.79.

Correct answer is B)

Coupon = (1,000 × 0.08) / 2 = $40.00

Present value of coupon payment = $40.00 / 1.05^150/365 = $39.21

Forward price on the fixed income security = ($1,310 - $39.21) × (1.05)^200/365 = $1,305.22

Q4. 30 days ago, J. Klein took a short position in a $10 million 90-day forward rate agreement (FRA) based on the 90-day London Interbank Offered Rate (LIBOR) and priced at 5%. The current LIBOR curve is:

• 30-day = 4.8%
• 60-day = 5.0%
• 90-day = 5.1%
• 120-day = 5.2%
• 150-day = 5.4%

The current value of the FRA, to the short, is closest to:

A)   ?$15,154.

B)   ?$15,495.

C)   ?$15,280.

Correct answer is A)

FRAs are entered in to hedge against interest rate risk. A person would buy a FRA anticipating an increase in interest rates. If interest rates increase more than the rate agreed upon in the FRA (5% in this case) then the long position is owed a payment from the short position.

Step 1: Find the forward 90-day LIBOR 60-days from now.

[(1 + 0.054(150 / 360)) / (1 + 0.05(60 / 360)) ? 1](360 / 90) = 0.056198. Since projected interest rates at the end of the FRA have increased to approximately 5.6%, which is above the contracted rate of 5%, the short position currently owes the long position.

Step 2: Find the interest differential between a loan at the projected forward rate and a loan at the forward contract rate.

(0.056198 ? 0.05) × (90 / 360) = 0.0015495 × 10,000,000 = $15,495

Step 3: Find the present value of this amount ‘payable’ 90 days after contract expiration (or 60 + 90 = 150 days from now) and note once again that the short (who must ‘deliver’ the loan at the forward contract rate) loses because the forward 90-day LIBOR of 5.6198% is greater than the contract rate of 5%.

[15,495 / (1 + 0.054(150 / 360))] = $15,154.03

This is the negative value to the short.

Q5. What is the value of a 6.00% 1x4 (30 days x 120 days) forward rate agreement (FRA) with a principal amount of $2,000,000, 10 days after initiation if L_10(110) is 6.15% and L₁₀₍₂₀₎ is 6.05%?

A)   $767.40.

B)   $745.76.

C)   $700.00.

Correct answer is B)

The current 90-day forward rate at the settlement date, 20 days from now is:
([1+ (0.0615 x 110/360)]/[1+ (0.0605 x 20/360)] – 1) x 360/90 = 0.061517

The interest difference on a $2 million, 90-day loan made 20 days from now at the above rate compared to the FRA rate of 6.0% is:
[(0.061517 x 90/360) – (0.060 x 90/360)] x 2,000,000 = $758.50

Discount this amount at the current 110-day rate:
758.50/[1+ (0.0615 x 110/360)] = $745.76

Q6. Monica Lewis, CFA, has been hired to review data on a series of forward contracts for a major client. The client has asked for an analysis of a contract with each of the following characteristics:

1.       A forward contract on a U.S. Treasury bond

2.       A forward rate agreement (FRA)

3.       A forward contract on a currency

Information related to a forward contract on a U.S. Treasury bond: The Treasury bond carries a 6% coupon and has a current spot price of $1,071.77 (including accrued interest). A coupon has just been paid and the next coupon is expected in 183 days. The annual risk-free rate is 5%. The forward contract will mature in 195 days.

Information related to a forward rate agreement: The relevant contract is a 3 × 9 FRA. The current annualized 90-day money market rate is 3.5% and the 270-day rate is 4.5%. Based on the best available forecast, the 180-day rate at the expiration of the contract is expected to be 4.2%.

Information related to a forward contract on a currency: The risk-free rate in the U.S. is 5% and 4% in Switzerland. The current spot exchange rate is $0.8611 per Swiss France (SFr). The forward contract will mature in 200 days.

Based on the information given, what initial price should Lewis recommend for a forward contract on the Treasury bond?

A)   $1,073.54.

B)   $1,070.02.

C)   $1,035.12.

Correct answer is B)

The forward price (FP) of a fixed income security is the future value of the spot price net of the present value of expected coupon payments during the life of the contract. In a formula:

FP = (S₀ ? PVC) × (1 + R_f)^T

A 6% coupon translates into semiannual payments of $30. With a risk-free rate of 5% and 183 days until the next coupon we can find the present value of the coupon payments from:

PVC = $30 / (1.05)^183/365 = $29.28.

With 195 days to maturity the forward price is:

FP = ($1,071.77 – $29.28) × (1.05)^{195 / 365} = $1,070.02.

(Study Session 16, LOS 60.c)

Q7. Suppose that the price of the forward contract for the Treasury bond was negotiated off-market and the initial value of the contract was positive as a result. Which party makes a payment and when is the payment made?

A)   The short pays the long at the maturity of the contract.

B)   The long pays the short at the maturity of the contract.

C)   The long pays the short at the initiation of the contract.

Correct answer is C)

If the value of a forward contract is positive at initiation then the long pays the short the value of the contract at the time it is entered into. If the value of the contract is negative initially then the short pays the long the absolute value of the contract at the time the contract is entered into. (Study Session 16, LOS 60.a)

Q8. Suppose that instead of a forward contract on the Treasury bond, a similar futures contract was being considered. Which one of the following alternatives correctly gives the preference that an investor would have between a forward and a futures contract on the Treasury bond?

A)   The futures contract will be preferred to the forward contract.

B)   The forward contract will be preferred to the futures contract.

C)   It is impossible to say for certain because it depends on the correlation between the underlying asset and interest rates.

Correct answer is B)        

The forward contract will be preferred to a similar futures contract precisely because there is a negative correlation between bond prices and interest rates. Fixed income values fall when interest rates rise. Borrowing costs are higher when funds are needed to meet margin requirements. Similarly reinvestment rates are lower when funds are generated by the mark to market of the futures contract. Consequently the mark to market feature of the futures contract will not be preferred by a typical investor. (Study Session 16, LOS 60.a)

Q9. Based on the information given, what initial price should Lewis recommend for the 3 × 9 FRA?

A)   4.96%.

B)   5.66%.

C)   4.66%.

Correct answer is A)

The price of an FRA is expressed as a forward interest rate. A 3 × 9 FRA is a 180-day loan, 90 days from now. The current annualized 90-day money market rate is 3.5% and the 270-day rate is 4.5%. The actual (unannualized) rates on the 90-day loan (R90) and the 270-day loan (R270) are:

R90 = 0.035 × (90 / 360) = 0.00875

R270 = 0.045 × (270 / 360) = 0.03375

The actual forward rate on a loan with a term of 180 days to be made 90 days from now (written as FR (90, 180)) is:

Annualized = 0.02478 × (360 / 180) = 0.04957 or 4.96%.

(Study Session 16, LOS 60.c)

Q10. Based on the information given and assuming a notional principal of $10 million, what value should Lewis place on the 3 × 9 FRA at time of settlement?

A)   $38,000 paid from short to long.

B)   $37,218 paid from long to short.

C)   $19,000 paid from long to short.

Correct answer is B)

The value of the FRA at maturity is paid in cash. If interest rates increase then the party with the long position will receive a payment from the party with a short position. If interest rates decline the reverse will be true. The annualized 180-day loan rate is 4.96%. Given that annualized interest rates for a 180-day loan 90 days later are expected to drop to 4.2%, a cash payment will be made from the party with the long position to the party with the short position. The payment is given by:

The present value of the FRA at settlement is:

38,000 / {1 + [0.042 × (180 / 360)]} = 38,000 / 1.021 = $37,218

(Study Session 16, LOS 60.c)

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