Assume all rates are annualized with semi-annual compounding.

Question 1. $100 par of a 0.5-year 8%-coupon bond has a price of $101.

$100 par of a 1-year 10%-coupon bond has a price of $104.

a. What is the price of $1 par of a 0.5-year zero?

b. What is the price of $1 par of a 1-year zero?

c. Suppose $100 of a 1-year 6%-coupon bond has a price of $99. Is there an arbitrage opportunity? If so, how?

d. What is the 0.5-year zero rate?

e. What is the 1-year zero rate?

f. What is the 1-year par rate, i.e., what coupon rate would make the price of a 1-year coupon bond equal to par?

g. Considering the shape of the yield curve, should the yield on the 1-year 10%-coupon bond be higher or lower than the 1-year par rate?

Question 2. Suppose that at time 0 you buy a 6%-coupon 30-year bond priced at par, and at time 0.5 you sell this bond at a yield of 8%.

a. What is your time 0.5 payoff per $1 of initial investment?

b. What is the rate of return on your investment (annualized, with semi-annual compounding)?

Question 3. Suppose the yield curve is upward-sloping and there is no arbitrage. Two ordinary fixed coupon bonds, bond A and bond B, have the same maturity, but bond A has a lower yield. Which bond has the higher coupon?

Question 4. The 0.5-year zero rate is 10% and the 1-year zero rate is 12%.

a. What is the price of:

i. $1 par of a 0.5-year zero?

ii. $1 par of a 1-year zero?

iii. $100 par of a 1-year 12%-coupon bond, in the absence of arbitrage?

b. What is the dollar duration of:

i. $1 par of a 0.5-year zero?

ii. $1 par of a 1-year zero?

iii. 100 par of a 1-year 12%-coupon bond?

c. What is the duration of:

i. $1 par of a 0.5-year zero

ii. $1 par of a 1-year zero?

iii. $100 par of a 1-year 12%-coupon bond?

d. Use dollar duration to estimate the change in value of $1,000,000 par of the 1-year 12%- coupon bond if all zero rates rise 50 basis points.

Question 5. Your liabilities have a market value of $1,000,000 and a duration of 6. You want to immunize your position by constructing a portfolio of two assets below that has the same market value and duration as your liabilities.

Question 6. Your liabilities have a market value of $1,000,000 and a duration of 6. You want to immunize your position by constructing a portfolio of two assets below that has the same market value and duration as your liabilities.

Write down equations that determine the number of units of each asset in the portfolio. Use notation N1 and N2 to represent the number of units of asset #1 and #2, respectively.

b. Solve the equations for N1 and N2.

Question 7. (Part I) At time 0, Investor A enters into a forward contract, at no cost, to buy, at time 2, $100,000 par of a zero maturing at time 3. The forward price this investor locks in to pay at time 2 is $92,000.

a. What forward rate does this investor lock in at time 0, through this forward contract, for lending from time 2 to time 3?

(Part II) At time 1, the spot price of $1 par of a zero maturing at time 2 is 0.96 and the spot price of $1 par of a zero maturing at time 3 is 0.93.

a. At time 1, what is the forward price an investor could lock in to pay, at time 2, for $100,000 par of a zero maturing at time 3?

b. What is the value, at time 1, of Investor A's position in the forward contract from Part I?

Question 8. The current price of $1 par of a zero maturing at time 2 is $0.90

a. What is the 2-year spot rate?

b. What is the dollar duration of $1 par of the 2-year zero?

The current price of $1 par of a zero maturing at time 3 is $0.84

c. What is the 3-year spot rate?

d. What is the dollar duration of $1 par of the 3-year zero?

You can enter into a forward contract today to buy, at time 2, $1 par of a zero maturing at time 3. The price you would pay at time 2 is the forward price. The cost today of entering into this contract is zero.

e. Construct a portfolio of 2- and 3-year zeroes that synthesizes this forward contract.

f. What is the no arbitrage forward price?

g. What is the dollar duration of the forward contract?