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%.

- Compute the PV and NPV of the cash flow.
- How do we profit from the discrepancy between the market price and PV?
- 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.

- Is there any arbitrage opportunity?
- 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.

- What is the implied forward rate?
- Is there any arbitrage opportunity?
- 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.

- What is the implied forward rate?
- Is there any arbitrage opportunity?
- 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.

- 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))
- 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:

- the zeor-coupon yields for one-year, two-year and three-year zero-coupon bonds;
- the implied forward interest rates;
- 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.

- What portfolio of the three Treasury bonds below would immunize the liability? (Match the cash flows.)
- 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*.

- What is the price at each node of a discount bond with
face value of
*$100*maturing two periods from the start? - 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 p_{u} 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*.

- What is the price of a one-year discount bond in this original model? the two two-year discount bond?
- 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.