From the perspective of a lender, explain how lending funds at LIBOR rate differs from lending funds at repo rate with respect to a general shortage of the underlying collateral.
Fixed Income and Credit Risk
Sample Exam
Abstract
This is a two-hour written closed book exam. Non-programmable calculators are permitted, however no substitutes, e.g. cell phones or related devices, are allowed on this exam. Only
calculators TI-30X II (B or S) and HP 10s are allowed. You are allowed to bring a single
one-sided sheet of A5 paper containing formulas for your exclusive use, no other materials are
permitted. This is a two-hour exam with no breaks.
1. You have 120 minutes to complete this final exam.
2. There are 3 problems on the exam. Please show your reasoning sufficiently for all problems to ensure full credit. You need to answer everything that follows “Question:” label.
3. Try to answer the questions in the space provided. If needed, use the additional space on
the other side of each page.
1
Problem 1
Grading: 5 points each. Please answer briefly the following 8 questions. No calculations are required, however your qualitative justification is especially welcome for each question in order to
receive full credit.
1) Interest rates. Question: From the perspective of a lender, explain how lending funds
at LIBOR rate differs from lending funds at repo rate with respect to a general shortage of the
underlying collateral.
2) Flat and Forward Volatilities. Imagine a graph of the forward volatility curve constructed
from a set of plain-vanilla quarterly caps prices. Question: Explain why the forward volatility is
equal to the flat volatility for 0.5 year maturity.
2
3) Forwards. You have just entered into a forward contract to deliver a zero-coupon bond one
year from now at T1 = 1 maturing at T2 = 2 at a predetermined forward price today K = $90. You
know that Z(0,1) = 0.99. The bank hedges its exposure to the interest rate changes due to this
forward, and uses a short-term bond and a long-term bond in its hedging portfolio. Question: If
the current forward rate today f(0,1,2) jumps down suddenly and gets lower, what direction the
value of the forward contract today will go?
4) Forward Risk-Neutral Measure. Consider an in-arrears forward contract on the 2-times
compounded LIBOR with delivery rate rK and notional N that matures at T and pays at maturity:
(N/2) (r2(T,T +0.5)rK). Question: How many zero-coupon bonds Z (0,T +0.5) this forward
contract is worth at t = 0 under the (T +0.5)-forward risk-neutral measure?
3
5) MBS Payments. The graph below demonstrates the total principal payments for the mortgage
pool M(t) with notional $100, weighted average coupon of 5.50%, pass-through rate of 5.00%,
and weighted average maturity of 6 months. In particular, the Blue curve and Dashed Red curve
correspond to two different levels of PSA for the same pass-through security. Question: Explain
which of the two curves corresponds to a lower PSA and why.
16.3
16.4
16.5
16.6
16.7
16.8
16.9
1 2 3 4 5 6
MBS Principal Payments
Blue Dashed Red
6) Prepayment Risk. In the context of MBS pass-through securities the key risk besides interest
rate risk is the prepayment risk. Prepayments will occur mainly when interest rates decline, so
PSA is likely to increase when interest rates decline. Question: Explain how changes in the PSA
affect convexity of MBS pass-through securities.
4
7) Physical Binomial Trees. Below you see the physical probability tree.
r2(0, 0.5) = 0.66%
r2(0.5, 1) = 0.82%
r2(0.5, 1) = 0.50%
0 0.5 1
When investors are risk-neutral, what would be the market price of risk l0 using a long-term
bond?
l0 = Z(0,0.5)E[Z(0.5,1)]Z(0,1)
Zu(0.5,1)Zd(0.5,1) .
8) Under the Merton model of default, the expected recovery rate is:
E
AT
D
AT < D
= E
AT
D
⇥
N(d1)
N(d2)
If the firms value increases, then its probability of default tends to decrease while the expected
recovery rate at default increases (Altman, Resti, Sironi 2004). Question: What would be the expected recovery rate under this calculation for the standard first passage model of default? (assume
that the default boundary is also D for comparability)
5
Problem 2
Topics: Binomial trees, options, defaultable bonds. Grading: 6 points each
A binomial tree is a powerful valuation tool for fixed income instruments. Below you are
given a physical probability binomial tree. The interest rate on the tree is annualized and quarterly
compounded rate. The physical probability of a upward move in the interest rate is Pr(upward) =
1/2.
Problem 2
Topics: Binomial risk-natural trees, swaptions, defaultable bonds. Grading: 6/100 points each (3
or 4 points out of 60 as described below). 18 points maximum for Problem 2.
The macroeconomic predictions for the economy result in the physical probability binomial
tree below. The interest rate on the tree is the annualized and quarterly compounded rate:
r4(0, 0.25) r4(0.25, 0.5)
12% 13%
9%
14%
10%
6%
r4(0.5, 0.75)
Risk-Natural Tree:
Risk-neutral probability of an
upward movement is p⇤:
15%
11%
7%
r4(0.75, 1.0)
3%
60% 70% 65% 60%
a) (4 points) Question: Calculate Z(0,0.5) and Z(0,0.75). Then calculate f4(0,0.5,0.75) and
describe terms of an FRA forward contract that will have zero value at inception (which rate will
be paid and when in exchange of which other rate?).
We are given annualized interest rates that are compounded quarterly:
Z(0,0.5)
0.94399
=
1+ 0.12
4
1
0.58444
0.6
✓
1+
0.13
4
◆1
+
0.394498
0.4
✓
1+
0.09
4
◆1
Z(0,0.75)
0.91834
=
1+ 0.12
4
1
0.6
1+ 0.13
4
1
+
0.969011
0.7
✓
1+
0.14
4
◆1
+ 0.3
✓
1+
0.4
4
◆1
+ 0.4
1+ 0.09
4
1
0.7
✓
1+
0.1
4
1◆
+ 0.3
✓
1+
0.06
4
◆1
0.97849
6
a) The market price of risk l0 obtained from the zero-coupon bond maturing at T = 1 can be
computed as: (where P(t,T) is the market price at t of the bond maturing at T , Pu(t,T) if short
rate went up, Pd(t,T) if short rate went down)
l0 = Z(0,0.25)⇥E[P(0.25,0.5)]P(0.5)
Pu(0.25,0.5)Pd(0.25,0.5)
Question: Use the tree to compute Pu(0.25,0.5), Pd(0.25,0.5), and E[P(0.25,0.5)] where the
expectation is taken with respect to the physical probability of short rate movement. What should
be the r2(0,0.5) today in order for the market price of risk l0 to be negative?
6
b) Question: Calculate the expected change in quarterly interest rates over the next 3 months
from r4(0,0.25) to r4(0.25,0.5) under physical measure. Then calculate the same expected change
in interest rates under the risk-neutral measure. To do so, use the term structure above, which
contains valuable information about investors’ risk aversion. Under which of the two measures the
expected change is larger?
c) Consider a simple Merton model:
ri+1, j = ri, j + µ ⇥D+s
p
D where p = 1/2
ri+1, j+1 = ri, j + µ ⇥Ds
p
D
Question: Find the parameters of the Merton model µ and s that correspond to the physical
binomial tree you work with.
7
d) You calibrate the tree to the current term structure of rates and find the two risk-neutral
probabilities for the two steps of the tree:
Step 0 1
R-N prob. 60% 70%
Consider an in-arrears interest rate call option (caplet) on the quarterly compounded rate
r4(…) maturing at T = 0.5. The option has the following parameters:
rk = 0.66% the strike rate (annualized, semiannually compounded)
N = $100 the notional amount
Question: Compute the market price of the caplet using the binomial tree above. How does
this caplet differ from a standard caplet (not in-arrears)? Argue qualitatively which of the two
caplets will have greater market price today?
8
e) Now consider a defaultable zero-coupon bond that matures at T = 0.5. The risk-neutral
rate of default is l⇤ = 0.1, while the risk-neutral loss rate is L⇤ = 0.7 (30% of the pre-default
market-value is recovered if default occurs).
Question: Compute the market value of the defaultable zero-coupon bond D(0,0.5) and the
credit spread d(0,0.5):
d(t,T) = logD(t,T)logZ(t,T)
T t
9
Problem 3
Topics: Fundamental pricing equation, forward risk-neutral measure. Grading: 6 points each
Think of a one-factor interest rate model, where all the risk is captured in random variations of
the short (overnight) rate rt:
drt = 0.05⇥dt +dBt under physical measure
There is an interest rate derivative, the value of which is Vt = V(t,rt)—can depend on time t and
the prevailing level of the short rate rt. Ito Lemma tells us the following about the changes in this ˆ
value:
dVt =
✓∂V
∂t
◆
⇥dt +
✓∂V
∂ rt
◆
⇥drt +
1
2
✓∂ 2V
∂ r2
t
◆
⇥dr2
t
a) The dollar capital gain of the interest rate derivative above can be written in the form below.
Question: Fill in the missing terms in this equation:
dVt = (…)⇥dt + (…)⇥dBt
Then answer the following: What is the expected dollar return overnight Et[dVt] on this derivative?
10
b) If investors were risk-neutral, they would have cared only about the expected dollar returns
and would not care about their random variability. Question: Use no arbitrage principle to argue
that the following equation must hold:
rtVt =
✓∂V
∂t
◆
+
✓∂V
∂ rt
◆
m(t,rt) + 1
2
✓∂ 2V
∂ r2
t
◆
s
2(t,rt)
Without any additional derivations argue how different the equation would be if investors were
risk-averse?
c) Using the Feynman-Kac Theorem the value of the interest rate derivative paying g(T,rT ) at
maturity T, which satisfies the above equation, is:
V(0,r0) = E
exp✓
Z T
0
r(s)ds◆
g(T,rT )
Question: Apply this Feynman-Kac formula to a zero coupon bond Vt = Z(t,T) and simplify as
much as you can the resulting expression for Z(0,T). Without any additional derivations argue
how your result would be different if investors were risk-averse.
11
d) Now return back to an arbitrary derivative Vt = V(t,rt) and consider its normalized value in
terms of zero-coupon bonds with maturity T:
Vet = V(t,rt)
Z(t,T)
Question: Rewrite the equation in b) for…
