Problem Set 5
Investment Theory
Instructions: This problem set is due on 10/3 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file.
Bond Pricing
Problem 1 Your company aims to raise $10 million by issuing 20-year zero-coupon bonds. With a yield to maturity of 6% per year, compounded annually, what should be the total face value of the bonds to achieve this goal?
Problem 2 Suppose a seven-year, $1,000 bond with an 8% coupon rate and semiannual coupons is trading with a yield to maturity of 6.75%.
- Is this bond currently trading at a discount, at par, or at a premium? Explain.
- If the YTM of the bond suddenly rises to 7% (APR with semiannual compounding), what price will the bond trade for?
Problem 3 Suppose that General Motors Acceptance Corporation issued a bond with 10 years until maturity, a face value of $1,000, and a coupon rate of 7% (annual payments). The yield to maturity on this bond when it was issued was 6%.
- What was the price of this bond when it was issued?
- Assuming the yield to maturity remains constant, what is the price of the bond immediately before it makes its first coupon payment?
Problem 4 Your company currently has $1,000 par, 6% coupon bonds with 10 years to maturity and a yield-to-maturity of 5% per year with semi-annual compounding. If you want to issue new 10-year coupon bonds at par, what coupon rate do you need to set? Assume that for both bonds, the next coupon payment is due in exactly six months.
Problem 5 Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with an 8% coupon if it is currently selling at par and the probability distribution of its YTM a year from now is as follows:
| Economy | Probability | YTM | Price | HPR |
|---|---|---|---|---|
| Boom | 0.2 | 11% | ||
| Normal | 0.5 | 8% | ||
| Recession | 0.3 | 7% |
Assume that the entire 8% coupon is paid at the end of the year rather than every 6 months over a principal of $100.
Forward Rates
Problem 6 Below is a list of prices for $1,000 par zero-coupon bonds of various maturities.
| Bond | Maturity (years) | Price ($) |
|---|---|---|
| Z(1) | 1 | 930 |
| Z(2) | 2 | 850 |
| Z(3) | 3 | 770 |
| Z(4) | 4 | 700 |
- Compute the zero-coupon rates for years 1, 2, 3 and 4. Express the rates per year with annual compounding.
- Consider an 8% coupon $1,000 par bond (denoted by B) paying annual coupons and expiring in 4 years. Compute the no-arbitrage price of the bond and its yield-to-maturity.
- If the expectations hypothesis holds, what is your forecast for the 3-year interest rate (per year compounded annually) expected next year?
- If bond B was trading today for $985, is there an arbitrage opportunity that can be exploited? If so, explain how an investor would exploit such a strategy, i.e. indicate which securities the investor would buy or sell, as well as the quantities.
Problem 7 Below is a list of zero-coupon rates expressed per year with annual compounding for various maturities:
| Maturity (years) | 1 | 2 | 3 |
|---|---|---|---|
| Zero Rate | 10% | 9% | 8% |
- Compute the prizes of zero-coupon bonds ($1,000 face value) with maturities 1, 2 and 3 years.
- Compute the current forward rates (per year with annual compounding) from years 1 to 2, from 2 to 3 and from 1 to 3.
- Consider an 8.5% coupon ($1,000 face value) bond paying annual coupons and expiring in 3 years. Compute the no-arbitrage price of the bond.
- If at the end of the first year the yield curve flattens out at 10% for all maturities, what will be the 1-year holding-period return (per year with continuous compounding) on the coupon bond?
- If the coupon bond described in c. was instead trading today for $1,000, is there an arbitrage opportunity? If so, explain how an investor would exploit such a strategy, i.e. indicate which securities the investor would buy or sell, as well as the quantities.
Interest Rate Risk
Problem 8 An insurance company must make payments to a customer of $10 million in 5 years and $25 million in 30 years. The yield curve is flat at 8% per year with annual compounding.
- What is the present value and duration of its obligation?
- If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?
- Suppose you buy a zero-coupon bond with value and duration equal to your obligation, and that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of the bond and that of your insurance obligation?
Problem 9 An insurance company must make payments to a customer of $20 million in 10 years and $40 million in 20 years. The yield curve is flat at 6% per year with annual compounding.
What is the present value and duration of its obligation?
If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what zero-coupon bond must it purchase? Specify the maturity and face value of the bond.
Suppose that instead of using a single zero-coupon bond as in part b, the insurance company plans to use the following bonds to immunize its exposure to interest rate risk:
Bond Duration (years) B_{5} 5 B_{30} 30 Determine the total amount to invest in each bond that the insurance company must buy to immunize its obligation.