FIN 448: Midterm 2019
90 min
April 23, 2019
1. Do not open the exam until everyone gets a copy and starts at the same time.
3. Show your work. Significant partial credit will be given for correct approach.
4. If for some question you need a number that you couldn’t calculate from the previous
question simply plug in the variable and write down the formulas you would solve.
This will give you significant partial credit.
Points Max
Part 1 15
Part 2 16
Part 3 8
Part 4 8
Total 47
The following Simon Code of Academic Integrity will be enforced:
“Every Simon student is expected to be completely honest in all academic matters. Simon
students will not in any way misrepresent their academic work or attempt to advance their
academic position through fraudulent or unauthorized means. No Simon student will be
involved knowingly with another student’s violation of this standard of honest behavior.”
1
Part 1
True or False (3pts). Evaluate whether the following statements are true of false.
1. 1pt (True / False) Suppose that there is no inflation (and there will be no inflation in
the future). Then for a long-term investor long-term bonds are safer than short-term
bonds.
Solution. This statement is true. Without inflation long-term bonds move wealth
across time preserving its purchasing power. Rolling over short-term bonds exposes
you to the fluctuations of the interest rates, thus, introduces uncertainty about the
long-term wealth through the interest rate risk.
2. 1pt (True / False) Continuously compounded yield curve is higher than semi-annually
compounded yield curve (when all rates are positive).
Solution. This statement is false. The same price corresponds to a smaller interest
rate rate the higher is the frequency of compounding.
3. 1pt (True / False) Time series of repo rates is not enough to estimate the parameters
of the Vasicek Model.
Solution. This statement is true. You need bond prices to estimate the two risk-neutral
parameters of the model r¯∗ and γ∗.
2
Multiple Choice (4pts). Answer the following multiple choice questions. Explain your
choice in a few sentences. You will receive 50% of the point for the correct answer and 50%
points for the correct explanation.
1. 2pts Consider a newly issued 3-year FRN with quarterly payments and a newly is-
sued 3-year coupon paying bond with quarterly coupons being determined by today’s
forward rates. Compare their prices:
(a) FRN is more expensive than the coupon bond
(b) FRN has the same price as the coupon bond
(c) FRN is cheaper than the coupon bond
Solution. The correct answer is (b). They are both priced at par.
2. 2pts Consider again a newly issued 3-year FRN with quarterly payments and a newly
issued 3-year coupon paying bond with quarterly coupons being determined by today’s
forward rates. Compare their durations:
(a) Duration of the FRN is larger than duration the coupon bond
(b) Duration of the FRN is the same as duration the coupon bond
(c) Duration of the FRN is smaller than duration the coupon bond
Solution. The correct answer is (c). Duration of the FRN is 0.25 (0ime till the next
payment) but the duration of the coupon bond will be close to 3 because of the large
principal at the end.
3
1. 2pts Write down at least two equivalent statements of the expectation hypothesis.
Solution. Long spot rates are just the average of future expected spot rates. Forward
rates are equal to expected future spot rates. Expected holding period returns have to
be equal across maturities.
2. 2pts Name two reasons why banks were manipulating LIBOR.
Solution. To change the payoffs of derivatives linked to LIBOR and to appear strong
to their creditors during ’08-’09 crisis.
3. 2pts Suppose you test in the Treasury data whether 1-year forward rate f(0, 1, 2)
predicts 1-year spot rate r(1, 2) 1-year ahead. What coefficient b will you find in the
following regression? Why?
r(1, 2)− r(0, 1) = a+ b[f(0, 1, 2)− r(0, 1)] + ε
Solution. The coefficient will be close to 0. That’s because the forward rates do not
predict future spot rates, instead they predict excess returns.
4. 2pts Suppose that the 5-year LIBOR swap rate is 4%. What would be the price of a
5-year fixed coupon bond issued by the LIBOR quality bank with a 4% coupon rate?
Explain.
Solution. The price should be par since the swap rate is simply the par rate of the
LIBOR quality bonds.
4
Part 2 (16 pts)
Suppose that the semi-annually compounded Treasury yield curve today looks like
6mo 1y 18mo 2y
2% 2.25% 2.5% 3%
1. 4pts What is 12-18 months semi-annually compounded forward rate today, i.e., what
is f2(0, 1, 1.5)?
Solution. Using the forward discount factor we can write(
1 +
f2(0, 1, 1.5)
2
)−1
= F (0, 1, 1.5) =
B(0, 1.5)
B(0, 1)
=
(1 + 2.25%/2)2
(1 + 2.5%/2)3(
1 +
f2(0, 1, 1.5)
2
)−1
= 0.985217173684379
hence f2(0, 1, 1.5) ≈ 3.00%
2. 4pts Suppose that 6 months ago you bought a 18-24 months FRA with a forward rate
of 2% and notional \$100. What is the value of this FRA now?
Solution. The value is
1
(1 + 2.5%/2)2
· f2(0, 1, 1.5)− 2%
2
· 100 ≈ 0.4817
5
3. 4pts What is the current 18 months swap rate?
Solution. First, let’s compute the prices of ZCB’s:
B(0, 0.5) = (1 + 2%/2)−1 ≈ 0.9901, B(0, 1) = (1 + 2.25%/2)−2 ≈ 0.97787,
B(0, 1.5) = (1 + 2.5%/2)−3 ≈ 0.96341.
And the 18-month swap c rate solves
c
2
[B(0, 0.5) +B(0, 1) +B(0, 1.5)] +B(0, 1.5) = 1
so c ≈ 2.495%
4. 4pts What is the current price of the last payment of the 18 months swap with notional
\$100?
Solution. The last payment of the swap is 100 · r2(1,1.5)−2.495%
2
. Its price is the same as
the price of 100 · f2(0,1,1.5)−2.495%
2
since the difference between those is an FRA whose
price is 0. And the latter price is
1
(1 + 2.5%/2)3
· 100 · 3%− 2.495%
2
≈ 0.24326
6
Part 3 (8 pts)
Suppose that the price of a 6-month T-Bill is 99.005, the price of the 1-year 2% T-Note
is 99.496 and the price of a 1.5 year 4% T-Note is 101.442.
1. 4pts Bootstrap the 1-year continuously compounded spot rate.
Solution. The one-year ZCB solves
1 · 0.99005 + (1 + 100)B(0, 1) = 99.496 ⇒ B(0, 1) ≈ 0.97530
and the continuous spot rate is
r(0, 1) = −ln(B(0, 1)) ≈ 2.5%
2. 4pts Bootstrap the 1.5-year continuously compounded spot rate.
Solution. The 1.5-year ZCB solves
2 · 0.99005 + 2 · 0.97530 + (2 + 100) ·B(0, 1.5) = 101.442
so the price of the ZCB is
B(0, 1.5) ≈ 0.95599
and the continuous spot rate is
r(0, 1.5) = −ln(B(0, 1.5))/1.5 ≈ 3%
7
Part 4 (8 pts)
Suppose you have a portfolio which pays
6mo 1y 18mo 2y
200× r2(0,0.5)
2
200× r2(0.5,1)
2
100× r2(1,1.5)
2
100× r2(1.5,2)
2
where r2(0, T ) is the semi-annually compounded spot rate.
1. 8pts Replicate this portfolio using swaps and zero-coupon bonds. Suppose that swaps
have semi-annual payments and that the 1-year swap rate is 2%, 1.5-year swap rate is
2.5%, and 2-year swap rate is 3%.
Solution. Break the initial portfolio into
6mo 1y 18mo 2y
100× r2(0,0.5)−3%
2
100× r2(0.5,1)−3%
2
100× r2(1,1.5)−3%
2
100× r2(1.5,2)−3%
2
100× r2(0,0.5)−2%
2
100× r2(0,0.5)−2%
2
0 −0
100× 5%
2
100× 5%
2
100× 3%
2
100× 3%
2
The first line is a payoff of an 2-year swap with notional 100.
The second line is a payoff of a 1-year swap with notional 100.
The third line is a payoff of four different ZCBs.
8 