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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.

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:

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

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

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

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

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

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

A portfolio manager holds 100,000 shares of IPRD Company (which is trading today for $9 per share) for a client. The client informs the manager that he would like to liquidate the position on the last day of the quarter, which is 2 months from today. To hedge against a possible decline in price during the next two months, the manager enters into a forward contract to sell the IPRD shares in 2 months. The risk-free rate is 2.5%, and no dividends are expected to be received during this time. However, IPRD has a historical dividend yield of 3.5%. The forward price on this contract is closest to:

A) $905,175.

B) $901,494.

C) $903,712.

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The historical dividend yield is irrelevant for calculating the no-arbitrage forward price because no dividends are expected to be paid during the life of the forward contract. In the absence of an arbitrage opportunity, the value of should be 0.
Therefore, FP = S0(1 + Rf)T
903,712 = 900,000(1.025)2/12

JoeyDVivre

The forward price in a 90-day forward contract on a non-dividend-paying stock currently (at contract initiation) selling for $55 when the 90-day risk-free rate is 5% is closest to:

A) $55.67.

B) $54.32.

C) $52.38.

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