# Final Exam FIN 525 Fixed-Income Securities

Philip H. Dybvig
Washington University in Saint Louis
December, 2000

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 one sentence of ordinary length.)

1. Name three examples of fixed-income securities.

corporate bond, Treasury STRIP, interest rate cap (many other possible answers)
2. Which is a better asset for a short-term fund to be used to pay salaries and other expenses, Treasury Bonds or AAA corporate bonds? Why? Assume the bonds have the same maturity and similar risk characteristics.

The Treasury bond is better because of its higher liquidity.
3. Which is more sensitive to interest rate risk, a standard coupon bond or an inverse floater of the same maturity?

The inverse floater is more sensitive to interest rate risk since when interest rates go up, not only are future cash flows discounted more (as with a standard coupon bond) but also the cash flows are smaller.
4. What are i.o.'s in the world of mortgage-backed securities?

They are interest-only obligations that pay interest but not principal.
5. Why might smoothing the forward rate curve be a good idea?

Smoothing the forward rate curve can remove apparent mispricing that is not economically significant due to transaction costs and the spread.
B. Basic rates and arbitrage 30 points
1. implied forward rate Suppose a two-year Treasury STRIP costs \$92 per \$100 of face value and the one-year Treasury STRIP costs \$96.25 per \$100 of face value. What is the implied forward rate for borrowing and lending from one year out to two years out?

```D(0,1) = 96.25/100 = .9625

D(0,2) = 92/100 = .9200

D(0,1)       .9625
f(0,2) = ------ - 1 = ----- - 1 ~ 4.62%
D(0,2)       .9200
```

2. implied forward arb Suppose a two-year Treasury STRIP costs \$80 per \$100 of face value and the one-year Treasury STRIP costs \$88 per \$100 of face value. Then the implied forward rate for borrowing and lending from one year out to two years out is 10%. If we can also borrow and lend forward from one year to two years at 12%, what is the arbitrage?

Since the implied rate is smaller than the actual rate, we should lend long and borrow implicitly through futures trading.
```                      0       1       2
lend long                   -100     112
short 2yr STRIP     89.6            -112
long 1yr STRIP     -88       100
---------------------
1.6       0       0
```
This is at a scale of lending long 100, the same arb can be done at any scale.
3. replication with coupon bonds A one-year coupon bond with face of \$100 and coupon rate of 5% is trading at par. A two-year coupon bond with face of \$200 and a coupon rate of 4.5% is also trading at par. A company would like to pay you \$880,000 to assume a pension liability that is expected to cost \$418,000 in two years and \$543,000 in one year. Is this a good deal? (The first coupon of each coupon bond will be paid one year from now.)

```coupon bonds themselves:

0       1       2
1yr coupon bond     -100     105
2yr coupon bond     -200       9     209

all in \$1,000s        0       1       2
assume pension liab  880    -543    -418
buy 200 2yr bonds   -400      18     418
buy 500 1yr bonds   -500     525
----------------------------------------
-20       0       0

Assuming the pension liability for \$880,000 is a bad deal

```
C. Duration 20 points

A portfolio includes three discount bonds: one paying \$3 million 5 years from now, one paying \$2.5 million 10 years from now, and one paying \$2 million 15 years from now.

1. What is the duration of the portfolio when the term structure of forward rates is flat at 6%?
```
5                10              15
5 * 3/1.06  + 10 * 2.5/1.06   + 15 * 2/1.06
duration =  ---------------------------------------------
5               10            15
3/1.06    +   2.5/1.06    +   2/1.06

= 8.43 years

```
2. What is the duration of the portfolio when the term structure of forward rates is flat at 7%?
```
5                10              15
5 * 3/1.07  + 10 * 2.5/1.07   + 15 * 2/1.07
duration =  ---------------------------------------------
5               10            15
3/1.07    +   2.5/1.07    +   2/1.07

= 8.29 years

```
3. How does duration change with interest rates? Explain why.
```
Duration falls when rates rise.  A larger rate discounts later cash flows
more and makes the later cash flows relatively less important.  The
relatively larger weight on the earlier cash flows implies a smaller
duration.

```
D. Binomial Option Pricing 30 points

Assume that the interest rate starts at 6% and in each period and either increases by 2% or decreases by 2% (from 6% up to 8% or down to 4%). The risk-neutral probabilities of ups and downs are 1/2.

1. What is the price now of a discount bond with face of \$100 maturing one year from now?
```
\$94.3 = 100/1.06

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

10%
/
8%
/    \
6%         6%
\    /
4%
\
2%

discount bond prices

100
/
92.6
/      \
\$89.0          100
\      /
96.2
\
100

(92.6 = 100/1.08, 96.2 = 100/1.04, 89.0 = 0.5 * (92.6 + 96.2)/1.06)

```
3. What is the price today of an interest rate floor with a strike of 7% and two periods to maturity? The underlying notional is \$100 (so the cash flow is 2 if the rate is 5%).
```
cash flows

0
/
0
/   \
0       1
\   /
3
\
5

Values

0
/
0.46
/      \
\$2.99          1
\      /
5.88
\
5

```
(0.46 = 0.5 * 1/1.08, 5.88 = 3 + 0.5*(1 + 5)/1.04, 2.99 = 0.5*(0.46 + 5.88)/1.06)