## Practice Problems for Lecture 6

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%.

1. Compute the PV and NPV of the cash flow.
2. How do we profit from the discrepancy between the market price and PV?
3. Construct an arb to convert the profit into a sure thing.

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.

1. Is there any arbitrage opportunity?
2. If there is, how do we construct an arb to convert the profit into a sure thing?

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.

1. What is the implied forward rate?
2. Is there any arbitrage opportunity?
3. If there is, how do we construct an arb to convert the profit into a sure thing?

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.

1. What is the implied forward rate?
2. Is there any arbitrage opportunity?
3. If there is, how do we construct an arb to convert the profit into a sure thing?

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):

• Bond A is currently trading at a price of 107, has a face value of 100 and 10% coupon and three years to maturity.
• Bond B is currently trading at a price of 105, has a face value of 100 and 10% coupon and two years to maturity.
• Bond C is currently trading at a price of 100., has a face value of 100 and 10% coupon and 1 year to maturity.
1. Compute the zero-coupon discount factors for one, two and three years out (i.e. find D(0,1), D(0,2) and D(0,3))
2. Compute the par coupon bond yields for one, two and three years out.

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:

1. the zeor-coupon yields for one-year, two-year and three-year zero-coupon bonds;
2. the implied forward interest rates;
3. the yield of a par coupon bond maturing three years from now.

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.

1. What portfolio of the three Treasury bonds below would immunize the liability? (Match the cash flows.)
2. What is the market value of the pension liability?
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.

1. What is the price at each node of a discount bond with face value of \$100 maturing two periods from the start?
2. What is the value at each node of an American call option on the discount bond (with face of \$100 maturing two periods from now) with a strike price of \$75 and maturity one year from now?

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.

1. What is the price of a one-year discount bond in this original model? the two two-year discount bond?
2. Suppose the one-year discount rate in the economy is 11% and the two-year discount rate is 12%. Compute the fudge factors and draw the tree for the adjusted interest rate process.