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.
|
Asset
|
Market Value
|
Duration
|
|
#1
|
100
|
2
|
|
#2
|
200
|
10
|
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.
|
Asset
|
Market Value
|
Duration
|
|
#1
|
100
|
2
|
|
#2
|
200
|
10
|
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?