Final Exam
FIN 525 FixedIncome Securities
Philip H. Dybvig
Washington University in Saint Louis
December, 2000
This is a closedbook 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.)

Name three examples of fixedincome securities.
corporate bond, Treasury STRIP, interest rate cap (many other possible answers)

Which is a better asset for a shortterm 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.

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.

What are i.o.'s in the world of mortgagebacked securities?
They are interestonly obligations that pay interest but not principal.

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

implied forward rate Suppose a twoyear
Treasury STRIP costs $92 per $100 of
face value and the oneyear 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

implied forward arb Suppose a twoyear
Treasury STRIP costs $80 per $100 of face value and the
oneyear 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.

replication with coupon bonds A
oneyear coupon bond with face of $100 and coupon rate of
5% is trading at par. A twoyear 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.

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

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

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 riskneutral probabilities of ups and downs are
1/2.

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

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)

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)