Online ExamFull Course Code
E261 A SPR24
Course Title
Fixed Income Securities
Faculty
Suleyman Basak
Date of Exam
17 March 2024
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Document Name: Online Exams Cover Sheet (September 2021)
Version: 2021/1.0
Approved by: Academic Policy Committee
Date:
Fixed Income Securities – Spring 2024
Final Examination
Prof. Süleyman Başak
Instructions
• This is an open-book take-home exam which is being released to you online on Canvas and you are required to submit it through Canvas. The exam is an individual
assignment. You may not discuss its content or the solutions with anyone.
• Please write your LBS student id number legibly on the top of the first page of your
solutions.
• There are 4 questions on this examination with a total of 100 points.
• The examination lasts for 2.5 hours, including the time to upload your solutions on
Canvas.
• Although you have access to your notes and other materials, it is highly recommended
that you use your crib sheet, and a calculator, to solve the questions.
• To receive full credit, you MUST show all the relevant algebraic steps in deriving
your solutions. Partial credit is given to answers that are numerically incorrect but
that show a correct understanding of the solution method. If the question is not clear,
state your assumptions and if they are reasonable, you will be given credit.
• Honour Code. Please note that by taking this exam, you comply with following
honour code:
“I confirm that I have had no prior knowledge of the content of this exam and the
answers to the exam are all my own work. I also confirm that I will not knowingly
disclose any information from this exam to others. I also understand that during the
exam, I am not permitted in any way to communicate with any person.”
1
Questions
1. Forwards, Arbitrage, and Pricing – 20 points
Today is year 0. Consider three bonds A, B and C. Bond A pays $9,231 in year 1,
and has a current price of $8,693.43. Bond B pays $200 in year 2 and $200 in year 3,
and has a current price of $346.01. Bond C pays $100 in year 3, and has a current
price of $86.07.
What is the implied continuously compounded forward rate between years 1 and 2
(i.e., 0 ṙ1,2 )?
2. Dollar Duration, Dollar Convexity, and Risk Management – 30 points
After his retirement, Jerry Basak decides to invest $100 million of his fortune in
fixed income markets. He hires you as his personal consultant. The current time
t = 0 (continuously compounded) zero-coupon yield curve is flat at 15% across all
maturities.
Consider the following securities:
• Security A – a 3-year zero-coupon bond with face value $100.
• Security B – a 3-year bond with face value $100 and annual coupon 10%.
Suppose you invest Jerry’s money in Securities A and B with the objective of currently
achieving a (fully) dollar duration-hedged portfolio (i.e., ∆$ = 0). Moreover, suppose
you believe that over the next instant, there is:
• 60% probability that there will be a parallel upwards shift in the (continuously
compounded) zero-coupon term structure to 25%.
• 40% probability that there will be a parallel downwards shift in the (continuously
compounded) zero-coupon term structure to 10%.
(a) (15 points) What is the trading strategy in Securities A and B that will achieve
the dollar duration-hedged portfolio at t = 0?
(b) (7 points) What is your estimate of the expected value of the dollar durationhedged portfolio over the next instant?
(c) (8 points) What is the expected value of the dollar duration-hedged portfolio over
the next instant as predicted by the combined dollar duration-convexity model?
2
3. Exotic Bond Options and Risk Measurement – 25 points
Consider the following CIR tree of the stochastic evolution of zero prices with a face
value of $1.
1 d2 = 0.93
1 d3 = 0.88
3 d4 = 0.93
d
=
0.95
1 d4 = 0.83 H
0 1
3 d5 = 0.88
H
1 d5 = 0.79
2 d3 = 0.95
H
0 d2 = 0.90
HH d = 0.90
0 d3 = 0.86 H
2 d4 = 0.86 HH
HH
2 5
0 d4 = 0.82
d
=
0.97
H
HH d = 0.97
HH 1 2
0 d5 = 0.78
d
=
0.93
1 3
H
3 4
H
3 d5 = 0.93
d
=
0.89
d
=
0.98
1 4
2 3
HH
d
=
0.95
1 d5 = 0.85
H 2 4
HH
2 d5 = 0.91
H
HH d = 0.99
3 4
3 d5 = 0.96
t=0
t=1
t=2
HH
HH
H 4 d5 = 0.95
H
HH
H
H 4 d5 = 0.98
H
HH
H
H 4 d5 = 0.99
t=3
An innovative investment bank has decided to offer a new security called a Başaklet.
This security is issued at t = 0 and matures at t = 1.
A Başaklet gives its holder the right (the option) to pick the best performing security
over the next period, out of 1-year, 3-year and 5-year zeros.
In particular, the random payoff to one unit of a Başaklet at t = 1, is the highest
payoff to a $1 investment in either the 1-year zeros, the 3-year zeros or the 5-year
zeros purchased at t = 0.
(a) (7 points) Order the 1-year, 3-year and 5-year zeros in terms of their “riskiness”,
from the riskiest to the least risky. Explain your ranking.
Hint: You may want to express the prices and payoffs of the zeros as a per dollar
($1) investment in the various zero coupon bonds.
(b) (12 points) What is the price of one unit of a Başaklet at t = 0? Where does
the riskiness of a Başaklet fall in the ranking above in part (a)? Explain your
answer.
(c) (6 points) What is the Cox-Ingersoll-Ross delta ∆CIR of one unit of a Başaklet
at t = 0?
3
t=4
4. Hurricane Bonds, Risk Measurement and Reinsurance – 25 points
The following is a CIR tree of the stochastic evolution of zero prices (with face value
$1) with T = 2 years and h = 1 year.
0 d1 = 0.95
0 d2 = 0.90
0 d3 = 0.86
1 d2 = 0.93
1 d3 = 0.88
XXX
XXX
XXX
1 d2 = 0.97
X
1 d3 = 0.93
t=0
t=1
XX
XXX
XXX
XX
XXX
XXX
XXX
X
2 d3 = 0.91
2 d3 = 0.95
2 d3 = 0.98
t=2
Today is year 0. IPG Insurance Company has just issued a 3-year Hurricane bond
whose coupon payments are indexed to the annually compounded one year market
rate plus 400 basis points. The market rate is the annually compounded one year
interest rate set at the previous time period. The annual coupons, which are based on
the face value of $1 million, are guaranteed. The principal repayment, however, is tied
to the occurrence of major hurricanes. If IPG does not experience a hurricane which
causes more than $1 billion in losses in the next 3 years, you will receive the entire
face value of $1 million back. Otherwise, your recovery will depend on the severity
of the hurricane. The following table presents the distribution of possible hurricane
losses three years later.
Event Hurricane Loss Probability Recovery value(∗)
A
2 billion
0.005
0%
B
1.5 billion
0.01
50%
C
1.25 billion
0.005
75%
D
< 1 billion
0.98
100%
(*) Recovery value is expressed as the fraction of the principal which you will get back
at maturity.
(a) (7 points) Determine an optimistic upper bound for the price of the Hurricane
bond today. In other words, how much would the bond be worth today if you
knew for sure that no hurricanes would occur over the next 3 years?
(b) (6 points) Determine a pessimistic lower bound for the price of the Hurricane
bond today. In other words, how much would the bond be worth if you knew for
sure hurricane event A would occur?
(c) (6 points) A risk neutral reinsurance company is willing to sell you a reinsurance
contract, which will guarantee repayment of your principal 3 years from today.
What is the price of this reinsurance contract today? Recall, risk-neutrality
implies the company only cares about expected values.
4
(d) (6 points) You have decided to purchase the reinsurance contract from part (c),
but rather than paying for it upfront, you have agreed to issue the reinsurance
company a coupon bond with a 3-year maturity and a face value equal to 80%
of the price of the reinsurance contract today. What will the coupon rate, x%,
have to be for the reinsurer to agree? Assume the reinsurer believes your bond
is default free.
5
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