Risk Management and Financial Engineering
代做finiance | 代写经济 – 这道题目是Risk Management and Financial Engineering方面
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Risk Management and Financial Engineering 2018/2019: Module name BS
Author: CJH
MSc Risk Management and Financial Engineering Examinations
2018/
For internal Students of Imperial College of Science Technology and Medicine. This paper also forms part of the examination for the Associateship.
Empirical Finance: Methods and Applications (B
(BS1033)
Tuesday 12th March; 14:00-16:
CLOSED BOOK
Instructions
Only college approved calculators may be used. There are 5 long answer questions:
Problem 1: Two parts, 15 marks total Problem 2: Three parts, 20 marks total Problem 3: One part, 20 marks total Problem 4: Two parts, 20 marks total Problem 5: Three parts, 25 marks total
Answer only the number of questions and any sub questions required as specified above. If additional questions are answered, questions will be marked in the order attempted unless a question attempt is clearly crossed out.
Candidate Name:
CID Number:
Suppose we see excess returnsxitonmassets (i= 1, 2 ,,m) over T time periods (t= 1, 2 ,,T).
We may write these together as a vector at timet:
xt=
x 1 t
x 2 t
..
.
xmt
.
Let =Cov(xt) be the covariance of asset returns.
(a) In general, how many unique elements are contained in ?(5 marks)
(b) Suppose these returns are driven by the following two factor model:
xit=i+ 1 if 1 t+ 2 if 2 t+it.
Assume the factors are uncorrelated and have equal variances^2 f. In other words, the covariance
matrix off 1 tandf 2 tis given by:
f=
(
f^20
0 ^2 f
)
.
Suppose the covariance matrix ofitfor alliis given by:
=
12 0 0
0 22 0
..
.
..
.
..
.
0 0 ^2 m
.
You may assume thatCov(fkt,it) = 0 for anyk,i,tandt, and that bothfktanditare
uncorrelated over time. express in terms of f, and anything else you need (please clearly
define any notation you use). How many unique parameters are contained in this formulation?
(10 marks)
Page 2 of 6
Risk Management and Financial Engineering 2018/2019: Module name BS
Author: CJH
Suppose we are interested in predicting some outcome variableyiwith a vector ofpexplanatory
variablesXi, whereXiis given by:
Xi=
x 1 i
x 2 i
..
.
xpi
.
The matrix containing theseXifor alliis can be written as:
X=
x 11 x 21 xp 1
..
.
..
.
..
.
..
.
x 1 i x 2 i xpi
..
.
..
.
..
.
..
.
x 1 n x 2 n xpn
.
You may assume thatyiand allxkihave been standardized to have mean 0 and variance 1. Consider
the following two minimization problems:^1
min
[N
i=
(yiXi)^2 +
p
j=
^2 j
]
(1)
min
[N
i=
(yiXi)^2 +
p
j=
|j|
]
. (2)
(a) Suppose we set= 0 in each of the above. Please providethat solves the minimization
problems (1) and (2).(5 marks)
(b) Describe generally in a sentence or two why we might we be interested in the solution to either (1) or (2) with >0. Additionally, describe at least one advantage of (2) over (1).(10 marks)
(c) Suppose a researcher sets= 0 and estimates parameters forp= 912 different explanatory
variables. The researcher finds that this explains 99.1% of the variation in the data used to
estimate the parameters. As a result, the researcher claims that they will be able to almost
perfectly predictyiout of sample. Discuss this claim (a few sentences or short paragraph should
be sufficient).(5 marks)
(^1) HereX idenotes the transpose ofXiand= 0 1 .. . p . Page 3 of 6 Risk Management and Financial Engineering 2018/2019: Module name BS Author: CJH
Consider two random variablesYandX. Show that^2
V ar(Y) =V ar(E[Y|X]) +E[V ar(Y|X)].
(^2) Recall that the variance of any random variableZisV ar(Z) =E[Z (^2) ]E[Z] (^2) .HereE[Y|X] denotes the conditional expectation ofYgivenXandV ar(Y|X) denotes the conditional variance ofYgivenX. Page 4 of 6 Risk Management and Financial Engineering 2018/2019: Module name BS Author: CJH
At the end of 1991, the US state of Delaware passed a new law (the reform) significantly streamlining
corporate bankruptcy proceedings. This reform reduced costs and time of litigation associated
with filing for bankruptcy. Researchers believe this may have impacted leverage choices of firms in
Delaware.
To evaluate this hypothesis, they collect data for firms in both Delaware and surrounding states
before and afterthe reformwent into place. LettingLeverageitbe the debt-to-equity ratio for firm
iin yeart, the researchers run the following difference-in-difference regression:
Leverageit= 0 + 1 DiTt+ 2 Di+ 3 Tt+vit.
HereDiis a dummy variable equal to 1 if firmiis located in Delaware, and equal to 0 otherwise.Tt
is a variable equal to 1 if yeartis afterthe reform(e.g. 1992 and later), and equal to 0 otherwise.
A few facts to keep in mind:
The average debt-to-equity ratio for firms in Delaware beforethe reformwas 1.5.
The average debt-to-equity ratio for firms in Delaware afterthe reformwas 1.8.
The average debt-to-equity ratio for firms in surrounding states beforethe reformwas 1.7.
The average debt-to-equity ratio for firms in surrounding states afterthe reformwas 1.9.
(a) Compute 0 OLS, 1 OLS, 2 OLS, andOLS 3 , the difference-in-difference coefficients based upon the
above specification.(10 marks)
(b) Discuss any assumptions necessary for this approach to recover the causal effect of bankruptcy l aws on leverage. What parameter represents this effect? Suggest an explicit reason why one of the assumptions you mentioned might fail.(10 marks)
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Risk Management and Financial Engineering 2018/2019: Module name BS
Author: CJH
Problem 5: 25 Points
Suppose we are interested in estimating the coefficients 0 , 1 , and 2 in the following linear model:
yi= 0 + 1 x 1 i+ 2 x 2 i+vi.
While we observexi, we are unable to observeyientirely. Instead, we seeyi, whereyiis given by:
yi=
clifyi< cl
yiifclyich
chifyi> ch
with constantscl< ch.LetviN(0,^2 ) be a normal random variable with probability density
functionf(z|x 1 i,x 2 i) =^1 (z) and cumulative distribution functionF(z|x 1 i,x 2 i) = (z). Here
() represents the probability density function of a standard normal random variable and () is
the corresponding cumulative distribution function.
(a) What is the name for data that is restricted in this manner?(5 points)
(b) What is the probability distribution function ofyigivenx 1 i,x 2 iand the parameters 0 , 1 , 2 , and:g(yi|x 1 i,x 2 i; 0 , 1 , 2 ,)?(10 points)
(c) Suppose instead we saw
yi=
kifyi< cl
yiifclyich
kifyi> ch
What is the probability distribution function ofyigivenx 1 i,x 2 iand the parameters 0 , 1 , 2 ,
andnow?(10 points)
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Risk Management and Financial Engineering 2018/2019: Module name BS
Author: CJH