FIXED INCOME SECURITIES
Sitting: 1
Date of exam: June 17, 2011
Time: 2.00 PM
Duration: 4 hours
READ THESE INSTRUCTIONS FIRST
1. The exam takes place in a computer room where computers have internet connection. You can
prepare your answers to the questions asked on these computers or on your own laptop if you
prefer to do so.
2. As explained before, time pressure is real during the exam. The exam schedule is as follows:
 1.45pm: Students enter the exam room, student ID’s are checked.
 1.55pm: Students ready for exam, computers logged on.
 2.00pm: Students receive multiple choice questions.
 2.30pm: Deadline for handing in multiple choice questions and receiving open questions.
 3.00pm: Deadline for handing in open questions and receiving a problem set.
 5.55pm: Students finalise their work on the problem set in a single Excel-sheet.
 6.00pm: End of exam.
3. You are allowed to use all means at your disposition during the exam, the only condition being that
you do not communicate with any other person inside or outside the exam room, i.e. your exam
must be an individual effort. If you break this rule your exam will be voided.
4. Mobile phones should be switched off and must not be kept on the desk at any time.
5. Take into account that no two students have identical question sets. Minor and major, subtle and
less subtle, obvious and not so obvious differences between exam sets will make any attempt to
exchange answers or answer sets traceable. Although question sets are non-identical they are
equivalent.
6. You will receive - in the order explained above - the following question sets:
 Four multiple questions with five alternatives:
i. Each correct answer is worth one point (out of twenty). There is no correction for
guessing.
ii. Clearly indicate the alternative of your choice with an arrow pointing at the alternative.
iii. Only one of the alternatives is correct.
 Two open questions:
iv. Each correct answer is worth two points (out of twenty).
v. The answer to an open question is a structured and original text. It is not merely a
reference to or a copy of one or a number of slides.
 A problem set consisting of three problems:
vi. The total weight of the problem set is seven points (out of twenty). The value of each
correctly solved problem is indicated in the problem set.
vii. Write your answers to the problems in the reserved area on the problem set pages.
viii. A numerically correct answer to the exercises is a priori considered 100% correct. In order
to prevent fraud you must show in detail how you obtained these answers. Do this by
mailing your self made solution file named “your full name.xls” or “your full name.xlsx” to
bart.vinck@hubrussel.be before 6.00pm today (and before leaving the exam room).
ix. Avoid disaster and save your intermediate results regularly.
x. Do not round off intermediate results.
7. All paper used during the exam must be handed in before leaving the exam room.
GOOD LUCK!
11 NAME & STUDENT NUMBER: (IN FULL, USE PRINT WRITING)
MULTIPLE CHOICE QUESTIONS (4 points)
The four questions underneath are multiple choice questions. Indicate your choice by an
arrow () in front of the option of your choice. No points will be subtracted for a wrong
answer. Only one of the alternatives is 100% correct.
MPC 11 (1 point)
In a CMO-structure with a floating rate tranche (FR) and an inverse floating rate
tranche (IFR), the interest rate conditions for both tranches are set in such a way that
a. the total interest paid to the FR tranche is equal to the total interest paid to the
IFR tranche.
b. the interest “saved” on other tranches is equally spread over the FR and IFR
tranches.
c. an investor would a priori be indifferent to investing in the FR tranche
compared to the IFR tranche.
d. the interest paid to the FR in excess of the reference rate is covered by the
prepayments to the IFR tranche.
e. it is possible to obtain a fixed coupon payment by combining adequate
proportions of the FR tranche and the IFR tranche.
MPC 14 (1 point)
If the surplus (modified) duration of a financial institution is positive,
a. it benefits from rising interest rates only when the leverage is
sufficiently low.
b. it can benefit from falling interest rates only when the duration gap is
substantially different from zero.
c. it benefits from falling interest rates only when the leverage is sufficiently high.
d. it can benefit from rising as well as from falling interest rates, depending on
leverage.
e. it runs into trouble if the duration gap is too high.
MPC 06 (1 point)
In a fixed for floating interest rate swap, the fixed rate payer
a. benefits more from volatility in interest rates than the floating rate payer.
b. takes a prudent position because paying fixed rates is less risky than paying
floating rates.
c. in normal circumstances has a comparative advantage on floating rate
borrowing compared to his counterparty.
d. runs less reinvestment risk than the floating rate payer.
e. cannot make a profit unless the floating rate payer makes a loss.
MPC 20 (1 point)
An essential difference between the mean reverting model and the arbitrage free
model lies in the fact that:
a. the mean reverting model is not arbitrage free, the arbitrage free model is.
b. the jump probabilities in the mean reverting model are constant throughout
the tree, in the arbitrage free model they aren’t.
c. the mean reverting model uses the principle of backward induction, the
arbitrage free model doesn't.
d. the mean reverting model is stochastic, the arbitrage free model is
deterministic.
e. spot curves produced by the mean reverting model are unrealistic, those
produced by the arbitrage free model are realistic.
11 NAME & STUDENT NUMBER: (IN FULL, USE PRINT WRITING)
HOME WORKS (5 points)
Please leave this section blank:
I : …………………
II : ……………………
OPEN QUESTIONS (4 points)
Please do NOT use numerical examples to make your point.
OQ 18 (2 points)
Can binomial models ever be arbitrage free? And mean reverting models?
http://www.macs.hw.ac.uk/~andrewc/papers/ajgc33.pdf
It depends a bit on how you interprete arbitrage-free and what the framework is
you are working in.
In principle, both models are arbitrage free; they are even no-arbitrage models.
They behave that way ‘by definition’: when setting up the model, you calibrate
it to market data. This initial calibration to observed prices makes that there is
no arbitrage opportunity at the outset.
However, if you use e.g. the one-factor binomial tree model, you only have 3
variables to play with. When you want to match 4 bonds (4 ‘observations in the
market’), you cannot perfectly match those, giving rise to arbitrage
opportunities. The same holds for the mean reverting tree model, although you
have one extra variable there (so you can ‘perfectly’ match 4 observations, but
not 5).
More complex models with more variables allow you to match more
observations. But then, this is only at the start. The next day, when rates
change, you would need to recalibrate the model. Since there are many factors
influencing the price of bonds in the market (...), reality will overtake the model
(eventually), creating (theoretic?) arbitrage opportunities – or the need to
recalibrate your model frequently.
OQ 11 (2 points)
What is wrong with using the simple yield to maturity concept in order to determine
the relative value of different bonds? (How) is the problem solved?
The bond yield is an endogeneous instrument: Price and yield are related to the same
instrument (from the same issuer):
n
P  c   (1  y ) t  (1  y )  n
t 1
This is of course not ideal if you want to compare different bonds (different
maturities). Therefore we would need a more exogeneous expression that takes into
account the ‘price’/’yield’ of other (though similar) bonds in the market. This is done
by following formula:
n
n
t 1
t 1
P  c   (1  rt ) t  (1  rn )  n  c   d t  d n
In this case, the price takes into account the spot rates / spot rate curve based on the
different instruments in the market (via bootstrapping, estimations, ...). It shows that a
change eg in the spot rate of a bond with a lower or higher maturity (can) also effect
the price of another bond.
Looking at the formula, it is clear this is most important for coupon bonds – or zero
bonds which are not held to maturity. If you buy a zero bond and hold it until maturity,
the
11 NAME & STUDENT NUMBER: (IN FULL, USE PRINT WRITING)
PROBLEM SET (7 points)
BP 11 (3 points)
Today is June 17, 2011.
Consider the portfolio of option free straight bullet bonds underneath.
number nominal
1
€ 50,000
2
€ 35,000
3
€ 40,000
4
€ 70,000
5
€ 65,000
maturity
next
coupon
21/02/201 21/02/2012
3
25/11/201 25/11/2011
5
25/07/201 25/07/2011
8
07/05/202 07/05/2012
0
14/09/202 14/09/2011
2
coupon price
rate
5.375% 104.46
7.500% 114.77
5.125% 108.86
5.750% 113.91
5.000% 102.46
All bonds are denominated in euro, pay annual coupons and are redeemed at par.
a. (1 point) Compute market value, yield to maturity, modified duration and
convexity of this portfolio.
market value
yield to maturity
modified
duration
convexity
Consider the following government bonds
number maturity
A
B
next
coupon
28/09/201 28/09/2011
2
28/03/202 28/03/2012
6
coupon price
rate
5.000% 108.86
4.500% 100.25
b. (2 points) Use these two bonds to construct a duration and convexity hedge
for the bond portfolio.
nominal position in A
nominal position in B
MI 11 (2 points)
Consider the portfolio of liabilities underneath.
number
size
1
€ 60,000
2
3
due
two years from
now
€ 25,000 three years from
now
€ 80,000 four years from
now
promised
yield
1.75%
2.50%
2.75%
Consider the following zero bonds available in the market.
number
A
B
C
maturity
18 months
42 months
60 months
YTM
1.500%
2.600%
3.200%
Immunise the portfolio of liabilities using the zero bonds. Do this in such a way that
the initial profit of the financial institution is maximal but without creating net short
positions.
nominal position in A
nominal position in B
nominal position in C
net profit for the
financial institution
TS 10 (2 points)
Use a mean reverting interest rate tree (0r1 = 3.50%,  = 3.5%,  = 0.75%, k = 2) to
compute the price of a risk free instrument with the following pay off structure:
o 10 at time 1 if the spot rate is between 2% and 5%, and 0 otherwise,
o 10 at time 2 if the spot rate is between 2% and 5%, and 0 otherwise,
o 11 at time 3 if the spot rate is between 2% and 5%, and 0 otherwise,
o 112 at time 4 if the spot rate is between 2% and 5%, and 100 otherwise.
price of the instrument