## Practice Problem Answers 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.

The PV of the cash flow is:

PV = \$100/1.15+\$90/(1.15*1.1) = \$158.10,
T he NPV of the cash flow is:
NPV = PV-P = \$158.10-\$150 = \$8.10.
2. How do we profit from the discrepancy between the market price and PV?

As the cash flow is cheap, we should buy the cash flow and finance by spot borrowing and forward borrowing.

3. Construct an arb to convert the profit into a sure thing.

time (years from now) 0 1 2
buy the cash flow -150 100 90
spot borrowing 158.10 -181.82 0
forward borrowing 0 81.82 -90
total net cash flow 8.10 0 0

Strategy for constructing arb (match cash flow): We start from the year that has the minimal number of cash flows, which is usually the last year. Then we work back our way to the present by matching the cash flows year by year.

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. Compute the PV and NPV of the cash flow.

The PV is \$250*0.9+\$150*0.8 = \$345.

The NPVis \$345-\$350 = -\$5.

2. Can you construct an arb to convert the possible profit if any, into a sure thing?

As the price of the cash flow is high, according to our "deep theoretical insight", we should sell the cash flow, and buy the zero-coupon bonds.

time (years from now) 0 1 2
sell the cash flow 350 -250 -150
buy 1-year zero -225 250 0
buy 2-year zero -120 0 150
the total net cash flow 5 0 0

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 are the implied forward rates?

The implied forward rate f(0,1) = 1/D(0,1)-1 = 1/0.9-1 = 11.11%.

The implied forward rate f(0,2) = D(0,1)/D(0,2)-1 = 0.9/0.8-1 = 12.5%.

2. Is there any arbitrage opportunity?

The implied forward rate f(0,2) is lower than the actual borrowing and lending rate one year from now, i.e. 12.5% < 15%. Yes, there is an arbtrage opportunity. We should lend forward.

3. If there is, how do we construct an arb to convert the profit into a sure thing?

At the scale of \$1, the arb is

time (years from now) 0 1 2
lend forward 0 -1 1.15
buy 1-year zero -0.9 1 0
short 2-year zero 0.92 0 -1.15
the total net cash flow 0.02 0 0

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 are the implied forward rates?

In order to calculate the implied forward rate f(0,2), we need to compute the discount factors first.

D(0,1) = price of 1-year coupon/Face value of 1-year coupon bond = 100/110 = 0.9091.

D(0,2) can be computed by solve the following equation:

\$10*D(0,1)+\$110*D(0,2) = \$97.1.

D(0,2) = (\$97.1-\$10*D(0,1))/\$110 = 0.8.

The implied forward rate f(0,2) = D(0,1)/D(0,2)-1 = 0.9091/0.8-1 = 13.6%.

2. Is there any arbitrage opportunity?

The implied forward rate f(0,2) is higher than the actual borrowing and lending rate one year from now, i.e. 13.6% > 12%. Yes, there is an arbtrage opportunity. We should borrow forward.

3. If there is, how do we construct an arb to convert the profit into a sure thing?

At the scale of \$100, the arb is

time (years from now) 0 1 2
borrow forward 0 100 -112
short 1-year coupon bond 100.16 -110.18 0
buy 2-year coupon -98.87 10.18 112
the total net cash flow 1.29 0 0

If coupon bonds cannot be bought and sold fractionally, one can construct an arb as follows.

time (years from now) 0 1 2
borrow forward 0 98.2 -110
lend spot -1.64 1.8 0
short 1-year coupon bond 100 -110 0
buy 2-year coupon -97.1 10 110
the total net cash flow 1.26 0 0

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))

In order to compute the discount factors, we can construct portfolios that give us cash flow only at the time that we are interested in. However, we will use a different approach, which is more mechanical, and intuitive. We solve the following system of equations:

110*D(0,1) = 100

10*D(0,1)+110*D(0,2) = 105

10*D(0,1)+10*D(0,2)+110*D(0,3) = 107

This gives us D(0,1) = 0.9091, D(0,2) = 0.8719, D(0,3) = 0.8108.

2. Compute the par coupon bond yields for one, two and three years out.

one year par coupon bond yield:

y1 = (1-D(0,1))/D(0,1) = (1-0.9091)/0.9091 = 10%;

y2 = (1-D(0,2))/(D(0,1)+D(0,2)) = (1-0.8719)/(0.9091+0.8719) = 7.2%;

y3 = (1-D(0,3))/(D(0,1)+D(0,2)+D(0,3)) = (1-0.8108)/(0.9091+0.8719+0.8108) = 7.3%

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 zero-coupon yields for one-year, two-year and three-year zero-coupon bonds;

Using the approximation formula z(0,t) = (1-D(0,t))/t, the yields are

z(0,1) = (1-D(0,1))/1 = (1-0.9)/1 = 10%;

z(0.2) = (1-D(0,2))/2 = (1-0.8)/2 = 10%;

z(0,3) = (1-D(0,3))/3 = (1-0.7)/3 = 10%.

Using the exact formula z(0,t) = D(0.t)-1/t-1, the yields are

z(0,1) = D(0,1)-1-1 = 1/0.9-1 = 11.11%;

z(0,2) = D(0,2)-1/2-1 = 0.8-1/2-1 = 11.8%;

z(0,3) = D(0,3)-1/3-1 = 0.7-1/3-1 = 12.6%;

2. the implied forward interest rates;

f(0,1) = 1/D(0,1)-1 = 1/0.9-1 = 11.11%;

f(0,2) = D(0,1)/D(0,2)-1 = 0.9/0.8-1 = 12.5%;

f(0,3) = D(0,2)/D(0,3)-1 = 0.8/0.7-1 = 14.3%

3. the yield of a par coupon bond maturing three years from now.

The yield of a par coupon bond is

y = (1-D(0,T))/sumTs=1D(0,s)= 1-0.7/(.9+.8+.7) = 12.5%

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.)

To match the cash flows, we start from the last period. We need to buy two T-bond 3 to match the last period's cash flow. Similarly, we need 1.5 T-bond 2 to match the second to last period's cash flow. Finally we need 3 T-bond 1 to match month 6's cash flow (see the table below).

2. What is the market value of the pension liability?

The market value of the pension liability is the value of the portfolio.

MV = 3*\$100+1.5*\$98+2*\$103 = \$653.
time(months out) 0 6 12 18
pension fund cash flow -653 -320 -161 -208
TBond 1 -100*3 103*3 0 0
TBond 2 -98*1.5 2*1.5 102*1.5 0
TBond 3 -103*2 4*2 4*2 104*2

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.

The price is

P = \$10*0.9+\$10*0.85+\$110*0.8 = \$9+\$8.5+\$88 = \$105.5.

Duration is

D = (9*1+8.5*2+88*3)/105.5 = 2.75

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.

The Macayley duration is 1/2*10+1/4*20+1/4*30 = 17.5

sens(10) = (1-exp(-.125*10))/.125 = 5.7

sens(20) = (1-exp(-.125*20))/.125 = 7.3

sens(30) = (1-exp(-.125*30))/.125 = 7.8

sens of the bond = 1/2*5.7+1/4*7.3+1/4*7.8 = 6.6

The effective duration is -log(1-0.125*sens)/.125 = 14

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?

The interest tree is

```             |- 50%
|- 40% -|
30% -|       |- 30%
|- 20% -|
|-  10%
```

and the value of the bond is given by

```                 |- 100
|- 71.43 -|
59.52 -|         |- 100
|- 83.33 -|
|- 100
```

and the value of the option is given by

```       |- 0
3.20  -|
|- 8.33

```

The price of a two-year discount bond is \$59.52 and the price of the option is \$3.20, as immediate exercise is not profitable.

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.

The interest tree is

```             |- 50%
|- 40% -|
30% -|       |- 30%
|- 20% -|
|- 10%
```

and the value of the bond is given by

```                 |- 100
|- 71.43 -|
60.44 -|         |- 100
|- 83.33 -|
|- 100
```

and the value of the option is given by

```       |- 0
3.84  -|
|- 8.33

```

The price of a two-year discount bond is \$60.44 and the price of the option is \$3.84, as immediate exercise is not profitable.

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-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.

The interest tree is

```             |- 20%
|- 15% -|
10% -|       |- 10%
|-  5% -|
|-  0%
```

and the value of the two-year bond is given by

```                 |- 100
|- 86.96 -|
82.82 -|         |- 100
|- 95.24 -|
|- 100
```

and the value of the one-year bond is given by

```       |- 100
90.91 -|
|- 100

```

The discount factors computed from the model is

Dom(0,1) = 0.9091,

Dom(0,2) = 0.8282.

The discount factors from the economy are

D(0,1) = 1/1.11 = 0.9009,

D(0,2) = 1/(1.12)2= 0.7972. Therefore, the fudge factors are f1 = 0.9091/0.9009 = 1.0091,

f2 = (0.8282/0.9091)/(0.7972/0.9009) = 0.9110/0.8849 = 1.0295.

The adjusted interest rates are r1* = (1+r1)*f1-1 = (1+0.1)*1.0091-1 = 11%

r2* = (1+r2)*f2-1, it is 1.15*1.0295-1 = 18.4% for upstate, and 1.05*1.0295-1 = 8.1% for down state.

The adjusted interest rate process is

```     |-18.4%
11% -|
|- 8.1%
```

The above formula is just an approximation. We can compute the implied forward rates to verify.

The implied forward rates in the economy:

f(0,1) = 11%,

f(0,2) = D(0,1)/D(0,2)-1 = 0.9009/0.7972-1 = 13%.

The implied forward rates from the model are

fom(0,1) = 10%,

fom(0,2) = Dom(0,1)/Dom(0,2)-1 = 0.9091/0.8282-1 = 9.8%.

The difference in interest rate in first year is 1%, and 3.2% in the second year.

So the adjusted interest rate process is

```     |-18.2%
11% -|
|- 8.2%
```
Thus we are ensured that the results are indeed quite close.