Problem Set 5
Investment Theory
Instructions: This problem set is due on 10/14 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 The yield to maturity of a $1,000 bond with a 7.0% coupon rate, semiannual coupons, and two years to maturity is 7.6% APR, compounded semiannually. What is its price?
Problem 3 Suppose a ten-year, $1,000 bond with an 8.0% coupon rate and semiannual coupons is trading for $1,034.74.
- What is the bond’s yield to maturity (expressed as an APR with semiannual compounding)?
- If the bond’s yield to maturity suddenly changes to 9.0% APR, what will be the bond’s price?
Problem 4 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 5 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 6 Your company currently has $1,000 par, 6% coupon bonds with 10 years to maturity and a price of $1,078. 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 7 Consider a Treasury bond with maturity 30 years paying semiannually a coupon rate of 5% per year on a principal of $1,000.
- If the current YTM of the bond is 6%, compute the price of the security.
- Suppose that 6 months have passed, and the YTM of the bond is now 5.8%. Compute the new price of the security.
- If you buy the security at the price computed in a., hold it for 6 months and sell it at the price computed in b., what is your HPR expressed as an annual rate with annual compounding?
Problem 8 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 9 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 10 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 11 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 12 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.