JoeyDVivre

Calculate the no-arbitrage forward price for a 90-day forward on a stock that is currently priced at $50.00 and is expected to pay a dividend of $0.50 in 30 days and a $0.60 in 75 days.  The annual risk free rate is 5% and the yield curve is flat.

A) $50.31.

B) $48.51.

C) $49.49.

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The present value of expected dividends is: $0.50 / (1.0530 / 365) + $0.60 / (1.0575 / 365) = $1.092
Future price = ($50.00 − 1.092) × 1.0590 / 365 = $49.49

JoeyDVivre

An index is currently 965 and the continuously compounded dividend yield on the index is 2.3%. What is the no-arbitrage price on a one-year index forward contract if the continuously compounded risk-free rate is 5%.

A) 991.1.

B) 991.4.

C) 987.2.

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The futures price FP = S0 e-δT (eRT)
= S0 e(R-δ)T
= 965e(.05-.023)
= 991.4

JoeyDVivre

Jim Trent, CFA has been asked to price a three month forward contract on 10,000 shares of Global Industries stock. The stock is currently trading at $58 and will pay a dividend of $2 today. If the effective annual risk-free rate is 6%, what price should the forward contract have? Assume the stock price will change value after the dividend is paid.

A) $56.85.

B) $58.85.

C) $56.82.

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One method is to subtract the future value of the dividend from the future value of the asset calculated at the risk free rate (i.e. the no-arbitrage forward price with no dividend).
FP = 58(1.06)1/4 – 2(1.06)1/4 = $56.82
This is equivalent to subtracting the present value of the dividend from the current price of the asset and then calculating the no-arbitrage forward price based on that value.

JoeyDVivre

The value of the S&P 500 Index is 1,260.  The continuously compounded risk-free rate is 5.4% and the continuous dividend yield is 3.5%.  Calculate the no-arbitrage price of a 160-day forward contract on the index.

A) $562.91.

B) $1,310.13.

C) $1,270.54.

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FP = 1,260 × e(0.054 − 0.035) × (160 / 365) = 1,270.54

JoeyDVivre

A stock is currently priced at $110 and will pay a $2 dividend in 85 days and is expected to pay a $2.20 dividend in 176 days. The no arbitrage price of a six-month (182-day) forward contract when the effective annual interest rate is 8% is closest to:

A) $110.06.

B) $110.00.

C) $110.20.

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In the formulation below, the present value of the dividends is subtracted from the spot price, and then the future value of this amount at the expiration date is calculated.
(110 – 2/1.0885/365 – 2.20/1.08176/365) 1.08182/365 = $110.06
Alternatively, the future value of the dividends could be subtracted from the future value of the stock price based on the risk-free rate over the contract term.

JoeyDVivre

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.

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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 – 35/1.05182/365) 1.059/12 = $1,001.84

JoeyDVivre

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) $0.00750 $0.00212

C) $0.00764 $0.00037

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

kmf229

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) −$15,495.

B) −$15,154.

C) −$15,280.

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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.

kmf229

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 L10(110) is 6.15% and L10(20) is 6.05%?

A) $767.40.

B) $700.00.

C) $745.76.

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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

kmf229

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:

• A forward contract on a U.S. Treasury bond
• A forward rate agreement (FRA)
• 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) $1,035.12.

C) $1,070.02.

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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 + Rf)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 54.c)

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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 long pays the short at the initiation of the contract.

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

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

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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 54.a)

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

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

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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 54.a)

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

A) 5.66%.

B) 4.96%.

C) 4.66%.

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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) = 0.03375The 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 54.c)

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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) $37,218 paid from long to short.

B) $38,000 paid from short to long.

C) $19,000 paid from long to short.

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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 54.c)

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Based on the information given, what initial price should Lewis recommend for a forward contract on Swiss Francs based on a discrete time calculation?

A) $0.8656.

B) $1.1552.

C) $1.0053.

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The value of a forward currency contract is given by:
Where F and S are quoted in domestic currency per unit of foreign currency. Substituting:

(Study Session 16, LOS 54.c)