1. You are given the following continuously compounded forward rates:0,2 0 = 6.5%,2,3 0 = 8.0%, 2,4 0 = 8.0%, and2,5 0 = 8.0%. You also know that Bond A, which matures at the end of year 5 and pays $10 at the end of year 1 through year 5, has a face value of $100. Its price is $109.76782.

(a) Compute the forward rate1,3 0 .

(b) Assume the price of a 1-year zero remains the same as in the previous part. If the forward rate 2,3 0 is now 7.4% instead of 8.0%, is there an arbitrage opportunity? What would be your investment strategy and how large of a profit can you make buying or selling one unit of Bond A? Assume you can buy or sell zero coupons.

2. Today is period 0. Consider the following default-free securities. A two year zero coupon bond, which pays $1000 in period 2, currently costs $869.36. A three-year annuity that pays $1000 at the end of period 1, 2, and 3, and has a current price of $2,604.94.

(a) A forward contract that matures in period 2 is also available. As you know, the current price is zero, but the buyer of the contract must pay $1837.02 at the maturity of the contract and in return receives a certain amount in period 3. What is the continuously compounded forward rate implicit in this contract if you know that 1,2 0 = 7.5%? What will be the payoff at period 3?

(b) A growing bond G pays $1000 in period 1, $2000 in period 2, and $3000 in period 3. The cost of this bond is $5100. Provide an investment strategy that will let you make an arbitrage profit of $28.67 at period zero. Provide a different strategy that will give you an arbitrage profit of $32.98 at period 2.

3. Assume you wish to take a long position in a forward contract with a 2-year maturity. However, this forward contact is not for delivery of a zero-coupon bond on the maturity of the forward contract. On the maturity of the forward contract, this forward contract delivers a four-year coupon bond with annual coupon payments of $80.00 and a face value of $1000.00.

The current annualized forward rates with continuous compounding are the following:

0,1 0 = 4 % 1,2 0 = 5% 2,3 0 = 8% 3,4 0 =10% 4,5 0 =12% 5,6 0 =11%

(a) Based on the above forward rates, determine the appropriate price that the long position should pay for this four-year coupon-bearing bond. As in all forward contracts, the long party will not pay the delivery price until the maturity date of the forward contract.

(b) What is the dollar duration for this particular forward contract, which delivers coupon-bearing bond?

(c) Now assume you have a balance sheet that contains only a single security. This single security is a six-year zero with face value of $4050.75. The balance sheet has no liabilities. Without selling this six-year zero, how many forward contracts would you go long or short to obtain a delta of zero for your net equity. (You are to use the forward contract, which delivers the four-year coupon-bearing bond).

4. Throughout this question we will focus on the characteristics of a long term bond with constant coupon payments. This security will be used in all three parts that follow.

(a) When the continuously compounded yield on a 1 year zero coupon bond is

5.02%, this long term security is worth $49.5883. When the continuously compounded yield on a 1 year zero coupon bond is 4.98%, this long term security is worth $49.9866.

What is the approximate dollar delta for this long term security when the continuously compounded yield on a 1 year zero is 5.00%?

(b) When the continuously compounded yield on a 1 year zero coupon bond is

5.02%, this long term security is worth $49.5883. When the continuously compounded yield on a 1 year zero coupon bond is 5.06%, this long term security is worth $49.1932.

What is the approximate dollar delta for this long term security when the continuously compounded yield on a 1 year zero is 5.04%?

(c) Based on the information in Parts a and b above, what is the approximate gamma value for this long term security when the continuously compounded yield on a 1 year zero is 5.02%?