# City, University of LondonMSc Financial Mathematics SMM272

## Alternative Timed Assessment

· 案例展示

Academic excellence for business
and the professions
Cass Business School
MSc Financial Mathematics
MS Mathematical Trading and Finance
MSc Quantitative Finance
Alternative Timed Assessment
Resit

 Module Code Module Title SMM272 Risk Analysis August 2020 3hrs 00mins

Division of Marks: All questions carry equal marks
Instructions to students: Answer THREE questions
This paper contains FIVE questions and comprises TEN pages including the title
page
Any other additional materials: Cambridge Statistical Tables Extract (2 pages at
the end of the paper)
Internal Examiner: Dr Gianluca Fusai
External Examiner: Professor Stephen Hall
Conventions
� Always assume that a year is 250 days long.
� Always makes clear the assumptions underlying your computations.
2 of 10
Question 1
1. [25%] In a RiskMetrics EWMA model the decay factor is λ = 0:8:
What is the weight assigned to an observation that is 21 days old?
Explain what does it means that the EWMA is a conditional volatility
model. Briefly state the main advantages and limitations of the EWMA
model.
2. [25%] The following information is given: an initial 2x2 covariance
matrix with diagonal elements 5 and 10, a smoothing parameter λ = 0:8
and the following sequence of daily returns on two stocks
Days Stock 1 Stock 2
1 -3 +2
2 -1 0
3 0 +1
Determine the EWMA covariance matrix for the fourth day.
3. [25%] Using the estimated covariance matrix for the fourth day and
given portfolio weights of 0.2 and 0.8 for the two stocks, compute the
portfolio VaR at a generic confidence level α. Finally compute the
Marginal VaR of the two stocks and the incremental VaR, assuming
a 1% increase of the weight of the first stock and a simultaneous 1%
reduction of the weight of the second stock.
4. [25%] The following information is given: the sample size is T = 250,
the sample estimate of the population mean is 0, the sample estimate
of the population daily variance is 0:22=250, the sample estimate of the
population skewness is -0.2, and the sample estimate of the population
kurtosis is 5, according to the Jarque-Bera test can you accept the null
hypothesis of Gaussian returns at a 90% probability level? [Recall that
if the sample of size T is extracted from a Gaussian population: i)
the asymptotic distribution of the sample skewness is Gaussian with
zero mean and variance 6=T, ii) the asymptotic distribution of the
sample kurtosis is Gaussian with mean 3 and variance 24=T , iii) the
two estimators are independent].
3 of 10
Question 2
1. [25%] Let us suppose we have the following sample of daily log-returns:
+4, -4, +2, -3, +2. Assume that the population is Gaussian, estimate
the 10 days VaR at 90% confidence level. Then determine the horizon
(in days) at which the VaR is maximized.
2. Consider the following 2x2 covariance matrix of daily log-returns of two
stocks
� 1 2 2 13 �
(a) [25%] Perform a Cholesky decomposition.
(b) [25%] Suppose you have two uniform random numbers, 0.1056
and 0.5 for stock 1 and stock 2 respectively. Simulate the logreturns of the two stocks. The two stock prices have initial values
of \$100 and \$200 respectively. Also simulate the profit and loss
for the individual stocks as well as the portfolio profit and loss if
you hold 6 units of the first stock and 4 units of the second.
(c) [25%] Assume that you also hold a derivative written on the two
stocks. The theta of the derivative is 0, the deltas are 0.5 and -0.7,
and the Hessian matrix containing the Gammas of the option has
elements
� 1 4 4 2 � :
Use the simulated P&L of the two stocks from the previous question and write the expression (no computation is required) which
you should use to simulate the P&L of the derivative position via
the delta-Gamma Monte Carlo technique.
4 of 10
Question 3
1. [25%] Given the estimated values of the skewness and kurtosis are -0.1
and 6 respectively, and given a (daily) sample mean of 0.01 and sample
variance of 0:22, you have decided to fit the following model to the data
r(t) = µ + σ�v(t); (1)
where the error terms �v(t) are i.i.d with a Student’s-t Distribution with
v degrees of freedom. In particular, �v(t) has the following moments
(expected value, variance, skewness and kurtosis)
E(�v) = 0; V (�v) = v
v - 2
; Skew(�v) = 0; Kurt(�v) = 3 + 6
v - 4
; v > 4
Explain how to fit the model parameters (µ; σ; v) to the sample information using the method of moments.
2. [25%] What are the limitations of the assumed model with respect to
empirical stylized facts?
3. [25%] If you use delta-VaR for a portfolio of options, briefly explain
which of the following statements is always correct?
(a.) It necessarily understates VaR because it uses a linear approximation.
(b.) It can sometimes overstate VaR.
(c.) It performs most poorly for a portfolio of deep in-the-money options.
(d.) It performs most poorly for a portfolio of deep out-of-the-money
options.
4. [25%] Risk-neutral default probability and real-world (or physical)
probability are used in the analysis of portfolios containing derivatives.
Which one of the following statements on their uses is correct? Briefly
explain your answer
(a.) Real-world probability should be used in scenario analyses of potential future losses, and real-world probability should also be used
in valuing derivatives.
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(b.) Real-world probability should be used in scenario analyses of potential future losses, but risk-neutral probability should be used
in valuing derivatives.
(c.) Risk-neutral probability should be used in scenario analyses of
potential future losses, and risk-neutral probability should be used
in valuing derivatives.
(d.) Risk-neutral probability should be used in scenario analyses of
potential future losses, but real-world probability should be used
in valuing credit derivatives.
6 of 10
Question 4
1. [25%] Your counterparty in a bilateral contract has issued two zerocoupon bonds (zcb) with time to maturity of 1 and 2 years. The risk
free discount factors for those maturities are 0.98 and 0.95, respectively
and the prices of the two defaultable zcb are 0.97412 and 0.9272 for the
1 year and 2 year zcb. Compute the 1 and 2 year cumulative survival
probabilities of your counterparty. Assume a recovery rate of 0.4.
2. [25%] The following Tables provide the time evolution of the market
fair value of a long position in a bilateral contract at times 0, 1 and
2 (left panel) and the corresponding risk-neutral probabilities of each
state of the world (right panel). For example, 3=20 in the second panel
is the cumulative probability that the contract will have a value of 4
(see first panel) in two period time. Using the survival probabilities,
computed in the previous Question, calculate the Expected Exposure
of the contract due to the counterparty default.
Period 0 1 2
4
95
98
0 0
-
285
196
-3
Fair value of the contract over time .
Period 1 2
3
20
12
20
13
20
8
20
4
20
Risk neutral probabilities .
3. [25%] Compute the Credit Value Adjustment (CVA) of the derivative
contract, assuming independence of all relevant random variables and
that default can only occur at time periods 1 and 2.
4. [25%] We want to backtest a statistical VaR model using the Z-Test.
In particular, let j be the number of 99% VaR violations observed in
a sample of size T = 250. Do you reject the model if you observe 4
violations? In general, how many violations would justify in rejecting
the model.
7 of 10
Question 5
1. [25%] Explain under what assumptions the square-root rule for scaling
the volatility at different time horizons is valid.
2. [25%] Consider the log-return over a trading day. The expected daily
log return is 0.1% and the standard deviation is 2%. Assuming that
log-returns are iid and Gaussian and that a year contains 250 trading
days, compute the expected annual linear return.
3. [25%] Kupiec’s POF (probability of failure) Test is based on the null
hypothesis that the empirically determined probability matches the
given probability i.e.
H0 : α = ^ α = 1 - j
n
Discuss why and how the following Log-Likelihood ratio test statistic
is used to test the accuracy of a risk model:

 LRuc = -2 �j ln �1-α� + (n - j) ln �α αb�� :

1 - αbIn the above formula n is the number of observations, j is the number
of exceptions, α is the confidence level, αb is the observed 95% VaR violation frequency. Suppose for a given model, you observe 4 violations in
250 comparisons. Check whether you would reject the null hypothesis
that the model is accurate on the basis of this test. Discuss the main
limitations of the Kupiec’s test.
4. [25%] Define the concept of a risk measure, and describe the properties
that a risk measure needs to satisfy in order to be a coherent risk
measure. Given the Profit and Loss discrete probability distribution in
Table (1), compute the Expected Shortfall in monetary terms at the
confidence level of 90%.

 P &Lipi -5 -4 -3 -2 -1 0 1 2 3 4 50.03 0.05 0.08 0.1 0.12 0.25 0.15 0.1 0.06 0.04 0.02

Table 1: Top row: possible realizations of the P&L. Bottom row: corresponding probabilities pi.
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 NORMAL CUMULATIVE DISTRIBUTION FUNCTION x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.62.72.82.93.03.13.23.33.43.53.63.73.83.9 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7580 0.7611 0.7642 0.7673 0.7703 0.7734 0.7764 0.7794 0.7823 0.78520.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83890.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86210.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88300.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90150.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91770.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93190.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94410.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95450.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96330.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97060.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97670.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98170.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98570.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98900.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99160.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99360.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99520.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99640.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99740.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99810.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99860.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99900.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.99930.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.99950.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.99970.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.99980.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.99980.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99990.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99990.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99991.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.00002

9 of 10
CHI-SQUARED PERCENTAGE POINTS

 ν 0.1% 0.5% 1.0% 2.5% 5.0% 10.0% 12.5% 20.0% 25.0% 33.3% 50.0% 12345678910 0.000 0.000 0.000 0.001 0.004 0.016 0.025 0.064 0.102 0.186 0.4550.002 0.010 0.020 0.051 0.103 0.211 0.267 0.446 0.575 0.811 1.3860.024 0.072 0.115 0.216 0.352 0.584 0.692 1.005 1.213 1.568 2.3660.091 0.207 0.297 0.484 0.711 1.064 1.219 1.649 1.923 2.378 3.3570.210 0.412 0.554 0.831 1.145 1.610 1.808 2.343 2.675 3.216 4.3510.381 0.676 0.872 1.237 1.635 2.204 2.441 3.070 3.455 4.074 5.3480.598 0.989 1.239 1.690 2.167 2.833 3.106 3.822 4.255 4.945 6.3460.857 1.344 1.646 2.180 2.733 3.490 3.797 4.594 5.071 5.826 7.3441.152 1.735 2.088 2.700 3.325 4.168 4.507 5.380 5.899 6.716 8.3431.479 2.156 2.558 3.247 3.940 4.865 5.234 6.179 6.737 7.612 9.342

CHI-SQUARED PERCENTAGE POINTS

 ν 60.0% 66.7% 75.0% 80.0% 87.5% 90.0% 95.0% 97.5% 99.0% 99.5% 99.9% 12345678910 0.708 0.936 1.323 1.642 2.354 2.706 3.841 5.024 6.635 7.879 10.8281.833 2.197 2.773 3.219 4.159 4.605 5.991 7.378 9.210 10.597 13.8162.946 3.405 4.108 4.642 5.739 6.251 7.815 9.348 11.345 12.838 16.2664.045 4.579 5.385 5.989 7.214 7.779 9.488 11.143 13.277 14.860 18.4675.132 5.730 6.626 7.289 8.625 9.236 11.070 12.833 15.086 16.750 20.5156.211 6.867 7.841 8.558 9.992 10.645 12.592 14.449 16.812 18.548 22.4587.283 7.992 9.037 9.803 11.326 12.017 14.067 16.013 18.475 20.278 24.3228.351 9.107 10.219 11.030 12.636 13.362 15.507 17.535 20.090 21.955 26.1259.414 10.215 11.389 12.242 13.926 14.684 16.919 19.023 21.666 23.589 27.87710.473 11.317 12.549 13.442 15.198 15.987 18.307 20.483 23.209 25.188 29.588

ν are the degree of freedom
3
10 of 10

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