Question 1
1. Assume all rates are annualized with semi-annual compounding. Please be explicit about how you derive your results and round to four decimals after the comma. ?
$100 par of a 0.5-year 12%-coupon bond has a price of $104. $100 par of a 1-year 14%-coupon bond has a price of $108.
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 10%-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 14%-coupon ?bond be higher or lower than the 1-year par rate?
Question 2
2. 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 higher yield. Which bond has the higher coupon?
Question 3
3. Assume all rates are annualized with semi-annual compounding. Please be explicit about how you derive your results and round to four decimals after the comma.
Suppose that at time 0 you buy a 10%-coupon 20-year bond priced at par, and at time 0.5 you sell this bond at a yield of 12%.
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 4
4. Assume all rates are annualized with semi-annual compounding. Please be explicit about how you derive your results and round to four decimals after the comma.
The 0.5-year zero rate is 7% and the 1-year zero rate is 9%.
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 10%-coupon bond
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 10%-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 10%-coupon bond
d.Use dollar duration to estimate the change in value of $1,000 par of the 1-year 10%- coupon bond if all zero rates rise 100 basis points.
Question 5
Your liabilities have a market value of $1,120,000 and a duration of 7.5. 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
|
600
|
10
|
|
#2
|
200
|
3
|
a. 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 6
Suppose you have a short position in a 30-year 5%-coupon bond and a long position in a zero- coupon bond with exactly the same market value and duration. If all zero rates fall by 25 basis points, will your net position rise or fall in value? Explain.
Question 7
Assume all rates are annualized with semi-annual compounding. Please be explicit about how you derive your results and round to four decimals after the comma.
The current price of $1 par of a zero maturing at time 2 is $0.97
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 $0.92 ?
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?
Question 8
Assume all rates are annualized with semi-annual compounding. Please be explicit about how you derive your results and round to four decimals after the comma.
(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 $93,000.
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 0.97 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?