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FIN 448: Midterm
Creation date：2024-04-07 16:42:45
Belonging category：金融Finance
Tag FIN 448
Midterm

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FIN 448: Midterm 


1. Do not open the exam until everyone gets a copy and starts at the same time.
2. Write your answers in the space provided.
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.”
Clearly Print Your Name:
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
Short Answer (8pts). Answer the following questions.
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
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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