# Final Exam FIN 525 Fixed-Income Securities

Philip H. Dybvig
Washington University in Saint Louis
March, 2005

This is a closed-book examination. Answer all questions as directed. Mark your answers directly on the examination. There are no trick questions on the exam. There are some formulas from the course (including some you will not need) at the end of the exam. All cash flows and interest rates are annual. Good luck!

A. General Concepts Short Anwer: 20 points (Answer each question in no more than two sentences of ordinary length.)

1. When interest rates rise, what happens to most bond prices?
```
They fall.

```
2. If the spot interest rate rises 1% today, do we expect the 10-year forward rate to rise by more than 1%, by less than 1%, or by 1%?
```
The forward rate should typically fall by less than 1%, because current
news has less impact on the future than on the present.

```
3. What is the usual reason to buy an interest cap?
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The main use of a cap is to limit the risk of rising rates when borrowing
at a floating rate.

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4. Does a coupon bond's duration rise, fall, or stay the same when interest rates rise? Explain.
```
Rising rates implies later cash flows are relatively less important so that duration
falls.

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5. Is mortgage fallout higher or lower when rates rise? Explain why.
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Fallout is lower when rates rise because higher rates make it less attractive
to find a new loan elsewhere.

```
B. Basic rates and arbitrage 30 points

Today we can buy or sell a riskless claim paying \$100 a year from now for \$75. Or, we can buy a self-amortizing claim paying \$100 one year out and \$100 two years out for \$140.

1. What are the discount factors for one year out and two years out?
```
The one-year discount factor is given by the short bond:

75
D(0,1) = --- = 0.75.
100

The two-year discount factor is given by taking opposite positions in
the one-year and two-year bonds.  The net cash flow up-front is 140 - 75
= 65, the net cash flow one-year out is 0, and the net cash flow two
years out is 100.  Therefore,

65
D(0,2) = --- = 0.65.
100

```
2. What are the implied one- and two-year par coupon yields?
```
As always, the one-year par coupon yield is the same as the one-year
discount yield, which is

100
z(0,1) = --- - 1 = 33 1/3%
75

The two-year coupon yield can be given by the formula

1 - D(0,2)       1 - 0.65    0.35
c(0,2) = --------------- = ----------- = ---- = 25%,
D(0,1) + D(0,2)   0.75 + 0.65   1.40

or we note that going long 1.25 2-year bonds and short 1 1-year bond
gives net cash flows - \$100, \$25, and \$125, which are the payments from
a 2-year par-coupon bond yielding 25%.

```
3. Suppose you can buy or sell a two-period par coupon bond yielding 30%. Construct an arb with the original two claims.
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Go long \$100 worth of the par coupon bond yielding 30%, short 1.25
self-amortizing bonds, and long one short zero-coupon bond.  The
cash flows are:

0         1        2
-100        30      130
175      -125     -125
-75       100        0
-----     -----    -----
0         5        5

Other arbs are possible, depending on when you want the cash out.

```
C. Duration 20 points

A pension liability consists of three cash flows: \$625 million 5 years out, \$800 million 10 years out, and \$250 million 15 years out. The liability is funded by a single asset, a pure discount bond maturing in 10 years. The market value of the pension asset equals the market value of the liability, that is, the pension is fully funded in economic terms.

1. The discount factor is 0.8 for 5 years out, 0.5 for 10 years out, and 0.4 for 15 years out. What is the market value of the liability?
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0.8 x 625 + 0.5 x 800 + 0.4 x 250 = 500 + 400 + 100 = \$1,000 million
= \$1 billion

By assumption this is also the market value of the asset.

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2. What is the duration of the liability? What is the duration of the asset?
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liability:

500      400       100
---- 5 + ---- 10 + ---- 15 = 2.5 + 4 + 1.5 = 8 years
1000     1000      1000

asset:

1000
---- 10 = 10 years
1000

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3. According to the duration measure, will the reduction in market value be larger for the asset or for the liability if rates rise? Explain briefly.
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The asset has a longer duration so it is more sensitive to interest rate
risk.  Therefore, the duration measure predicts that a rise in interest
rates will cause the market value of the asset to fall more than the market
value of the liability.

```
D. Binomial Option Pricing 30 points

Assume that the interest rate starts at 4% and in each period and either increases by 2% or decreases by 2% (from 4% up to 6% or down to 2% would be the first move). The risk-neutral probabilities of ups and downs are all 1/2.

1. What is the price now of a discount bond with face of \$100 maturing one year from now?
```
This is a one-period discount bond with face of \$100 and an interest rate
equal to the initial rate of 4%.  Therefore, the price is

100
---- ~ \$96.15
1.04

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2. What is the price now of a discount bond with face of \$100 maturing two years from now?
```
interest tree:

8%
/
6%
/    \
4%        4%
\    /
2%
\
0%

discount bond price:

100
/
94.34
/       \
92.49           100
\       /
98.04
\
100

```
3. What is the price today of a two-year collar with a cap price of 5% and a floor price of 3%? The underlying notional is \$1,000.
```
cash flows:

30
/
10
/     \
0           0
\     /
-10
\
-30

price (pre-cash flow)

30
/
24.151
/         \
-0.267               0
\         /
-24.706
\
-30

calculations:

10 + (30 + 0)/2/1.06 = 24.151
-10 + (0 - 30)/2/1.02 = -24.706

(24.151 - 24.706)/2/1.04 = -.267

```