1. Suppose the market price is $150 for a security paying $100 a year from now and $90 two years from now. Further suppose that the spot one-year rate is 15% and the forward rate for lending from one year out to two years out is 10%.
2. Consider the following scenario. A marketed claim is a riskless self-amortizing loan that pays $250 a year from now and $150 two years from now and is priced at $350 in the market. A zero-coupon bond maturing one year from now costs 90¢ per dollar of face value, while a zero-coupon bond maturing two years from now costs 80¢ per dollar of face value.
3. Suppose we can borrow and lend forward one year from now at 15%, and that the discount factor is 90% one year out and 80% two years out.
4. Suppose we can borrow and lend forward one year from now at 12%, and that a 10% coupon bond maturing one year from now is priced at $100 and a 10% coupon bond maturing two years from now is priced at $97.1. Coupons are paid annually and this year's coupon payment has already been made.
5. Assume you observe the following three coupon bond prices and remaining cash flows (coupons are paid annually and this year's coupon has already been paid):
6. Suppose a zero-coupon bond maturing one year from now costs 90¢ per dollar of face value, a zero-coupon bond maturing two years from now costs 80¢ per dollar of face value, and a zero-coupon bond maturing three years from now costs 70¢ per dollar of face value.
Calculate:
7. You are managing the final years of a pension fund. There are three remaining dates at which lump-sum payments will be made to beneficiaries: $320 million 6 months from now, $161 million 12 months from now, and $208 million 18 months from now.
| time(months out) | 0 | 6 | 12 | 18 |
|---|---|---|---|---|
| TBond 1 | -100 | 103 | 0 | 0 |
| TBond 2 | -98 | 2 | 102 | 0 |
| TBond 3 | -103 | 4 | 4 | 104 |
8. Suppose a coupon bond with a face value of 100, maturing 3 years from now, has a coupon rate of 10% (paid annually). What are its price and duration if the market discout factors are 0.9, 0.85, 0.8 for 1-year, 2-year and 3-year bonds.
9. Assuming the sensitivity of discount bond prices to a shock is given by sens(t-s)=(1-exp(-.125(t-s)))/.125 (as we have been assuming), compute the Macauley duration and the effective duration of a bond which pays 1/4 of its value at 30 years out, 1/4 of the value 20 years out and 1/2 of its value at 10 years out. Either use the graph on the lecture note to obtain an approximate value, or use the formulas from the lecture note to perform a more exact computation.
10. Consider a two-period binomial model in which the short riskless interest rate starts at 30% and moves up or down by 10% each period (i.e., up to 40% or down to 20% at the first change). The artificial probability of each of the two states at any node is 1/2.
11. As interest rate might follow a mean-reverting process, the assumption of constant artificial probabilities at each node might not be reasonable. Under the assumption of mean-reverting process, the probability of interest rete going up is a decreasing function of the level of interest rate. Assume the probability of going up pu is 0.6 when interest rate r is 20%, 0.4 when r is 30%, and 0.2 when r is 40%. Solve the previous problem under the new assumption.
12. Consider a two-year binomial model. Start with an original model in which the short riskless interest rate starts at 10% and moves up or down by 5% each period (i.e., up to 15% or down to 5% at the first change). The artificial probability of each of the two states at any node is 1/2.