Question

With the following information:

Maturity (year)	ZCB Price
0.25	0.988
0.5	0.971
0.75	0.953
1	0.933

(a) Instead of using an FRA directly, what positions in zero coupon bonds would you use to synthetically create the FRA borrower position?

A. Long the 6 months zero coupon bonds and short the 9 months zero coupon bonds

B. Short the 6 months zero coupon bonds and long the 9 months zero coupon bonds.

C. Long the 6 months zero coupon bonds and short the 3 months zero coupon bonds.

D. Short the 6 months zero coupon bonds and long the 3 months zero coupon bonds.

(b) Suppose you observed the FRA you required has an interest rate of 2% effectively for 3 months. What positions in bonds would you take to capture the arbitrage profit? (Assume you would take an appropriate position for the FRA)

A. Long the 6 months zero coupon bonds and short the 9 months zero coupon bonds.

B. Short the 6 months zero coupon bonds and long the 9 months zero coupon bonds.

C. Long the 6 months zero coupon bonds and short the 3 months zero coupon bonds.

D. Short the 6 months zero coupon bonds and long the 3 months zero coupon bonds.

(c) What is your arbitrage profit per dollar borrowed/lend on the maturity day?

Answer 1

Part (a)

The correct answer is option B. Short the 6 months zero coupon bonds and long the 9 months zero coupon bonds.

Instead of using an FRA directly, what positions in zero coupon bonds would you use to synthetically create the FRA borrower position.

FRA borrower position is long FRA. The borrower goes long on on an FRA to hedge his position and a lender is thus short
the FRA.

The payoff from the long FRA can be replicated by taking following positions in two zero coupon bonds of different maturity, but both bonds having same face value and notional principal

• Short position in a zero-coupon bond maturing at beginning of FRA period
• Long position in the zero-coupon bond maturing at the end of FRA and

Hence, in our case the equivalent position will be :

• Short position in 6 months zero-coupon bond
• Long position in 9 months zero coupon bond

Hence,The correct answer is option B. Short the 6 months zero coupon bonds and long the 9 months zero coupon bonds.

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(b) Suppose you observed the FRA you required has an interest rate of 2% effectively for 3 months. What positions in bonds would you take to capture the arbitrage profit? (Assume you would take an appropriate position for the FRA)

The correct answer is option A. Long the 6 months zero coupon bonds and short the 9 months zero coupon bonds.

By using the ZCBs, we can obtain the following interest rate effectively for 3 months = P_0.5 / P_0.75 - 1 = 0.971 / 0.953 -1 = 0.018887723 = 1.89% < 2.00% = FRA interest rate of 2 months. Hence, the position we should take is long FRA and short the synthetic position we had in part (a) above. That is we reverse what we did in part (a).

Hence, our position now should be:

• Long position in 6 months zero-coupon bond
• Short position in 9 months zero coupon bond

Hence the correct answer is option A. Long the 6 months zero coupon bonds and short the 9 months zero coupon bonds.

(c) What is your arbitrage profit per dollar borrowed/lend on the maturity day?

Arbitrage profit per dollar borrowed/lend on the maturity day = (FRA interest rate - forward interest rate locked using zero coupon bonds) = 2% - 1.89% = 0.11% = $ 0.11123