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MT4539 Quantitative Risk Management

创建日期：2024-04-11 14:15:31

所属分类：数学mathematics

标签： MT4539

Quantitative Risk Management

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MODULE CODE: MT4539
MODULE TITLE: Quantitative Risk Management
EXAM DURATION: 2 hours
EXAM INSTRUCTIONS: Attempt ALL questions.
The number in square brackets shows the
maximum marks obtainable for that
question or part-question.
Your answers should contain the full
working required to justify your solutions.
PERMITTED MATERIALS: Non-programmable calculator
YOU MUST HAND IN THIS EXAM PAPER AT THE END OF THE EXAM
PLEASE DO NOT TURN OVER THIS EXAM PAPER UNTIL YOU ARE
INSTRUCTED TO DO SO.
MT4539 May 2020, Page 1 of 8
Risk management
1. (a) An investor holds a portfolio consisting of 2 assets. The number of shares
she holds for asset i (i = 1, 2) at time point t, denoted by ni,t, and the
prices of asset i at time points t and t + 1, denoted by Pi,t and Pi,t+1,
respectively, are summarised in the following table:
Asset i 1 2
ni,t 10 20
Pi,t 10 7.5
Pi,t+1 12 8
For this question assume that the shares are infinitely divisible so that
n1,t+1 and n2,t+1 can be real numbers.
(i) The weight of asset i is given by
wi,t =
ni,tPi,t
PP,t
,
where PP,t is the price of the portfolio. Provide a verbal interpreta-
tion of the term weight.
(ii) Show that if at time point t + 1 the number of shares remains
unchanged, i.e. n1,t = n1,t+1 = 10 and n2,t = n2,t+1 = 20, the weight
of asset 1 in the portfolio will change.
(iii) Naive diversification is a strategy that assigns equal weights to each
asset. Determine the number of shares n1,t+1 and n2,t+1 of asset 1
and asset 2, respectively, so that the following two conditions are
simultaneously fulfilled:
the value of the portfolio at time t+ 1 is 280 and
the investor follows the naive diversification strategy.
Given the number of shares at time t, describe the transactions the
investor needs to make in order to achieve this.
MT4539 May 2020, Page 2 of 8
(iv) Determine the number of shares n1,t+1 and n2,t+1 of asset 1 and
asset 2, respectively, so that the following two conditions are simul-
taneously fulfilled:
the value of the portfolio at time t+ 1 is 280 and
the weights of the portfolio remain unchanged, i.e.
w1,t = w1,t+1 and w2,t = w2,t+1.
Given the number of shares at time t, describe the transactions the
investor needs to make in order to achieve this.
[8]
(b) Let X be a continuous random variable and Y its transformation such that
Y = g(X), where g(·) is strictly monotonic decreasing. Let y↵ and x1↵ be
the ↵quantile and the (1 ↵)quantile of Y and X, respectively. Show
that:
y↵ = g(x1↵)
[2]
MT4539 May 2020, Page 3 of 8
2. (a) Consider a loss X, which follows a uniform distribution with parameters
-1 and 1, i.e. X ⇠ Unif(1, 1). The density of X is given by
fX(x) =
⇢
1
2 for x 2 [1, 1]
0 otherwise
Assume the time horizon is fixed.
(i) Give the cumulative distribution function (CDF) of X, i.e. FX(x),
defined for all real numbers x. [1]
(ii) Give the Value-at-Risk (VaR) as a function of the confidence level
↵. [1]
(iii) The Expected Shortfall (ES) at confidence level ↵ is defined as:
ES[↵] =
R 1
↵ VaR[u]du
1 ↵
Give the ES for the loss X as a function of the confidence level ↵,
simplifying the expression as far as possible. [2]
(iv) Using your results from parts 2 (a) (ii) and 2 (a) (iii), show that
ES[↵] VaR[↵]. [1]
(v) For a continuous loss X, the corresponding ES can be equivalently
defined as:
ES[↵] = E(X|X > VaR[↵]) =
Z 1
1
xf(x|X > VaR[↵])dx
Calculate the ES using this formula and thus show that the result
is the same as in part 2 (a) (iii).
[3]
(b) Give two disadvantages of using the variance as a risk measure. [2]
MT4539 May 2020, Page 4 of 8
3. Consider a portfolio with a price Pt (in ) and a continuous return rt at time
point t. It is known that at time point T (today) PT = 10,000 and rT = 0.1.
For all calculations below please round to 4 decimal places.
(a) Assume that rt can be modeled with the following Constant Mean Model
(CMM)
rt = 0.05 + "t, with "t
iid⇠ N(0, 0.08)
The 2-period return rt+2(2) is given as:
rt+2(2) = rt+1 + rt+2
Specify for this CMM model the distribution of rT+2(2) at time point T .
Using this result calculate the exact Value-at-Risk (VaR) for this portfolio
for time horizon h = 2 at significance level ↵ = 0.99, in other words
calculate VaRT [h = 2,↵ = 0.99]. You might find the following output
useful:
> qnorm(0.01)
[1] -2.3263
> qnorm(0.05)
[1] -1.6448
> qnorm(0.95)
[1] 1.6448
> qnorm(0.99)
[1] 2.3263
[3]
(b) Assume now that rt follows an AR(1) process given as
rt = 0.05 + 0.2rt1 + "t, with "t
iid⇠ N(0, 0.08)
(i) Show that for the 2-day-ahead return rT+2 it holds:
rT+2 = 0.06 + 0.04rT + 0.2"T+1 + "T+2
[1]
(ii) Give the distribution of the 2-period return rT+2(2) = rT+1 + rT+2
conditional on the information until time point T . Specify the pa-
rameters of the distribution. [4]
(iii) Using this AR model, calculate the exact VaR for this portfolio for
time horizon h = 2 at significance level ↵ = 0.99, in other words
VaR[h = 2,↵ = 0.99]. [2]
MT4539 May 2020, Page 5 of 8
4. (a) Let X1, X2, ..., XT be independently and identically distributed random
variables with existing mean µ and variance 2. Consider the estimator
µˆ = 1T
PT
t=1Xt. Define Mean Square Error (MSE) consistency and show
that µˆ is an MSE consistent estimator for the mean µ. [3]
(b) Consider the triangular kernel:
K(z) =
⇢
1 |z| for 1  z  1
0 otherwise
Show that this kernel fulfils the honesty condition. [2]
(c) Three distributions - Normal, standardised-t and skewed standardised-t -
are fit to 757 observations of continuous returns of the Royce-Rolls Hold-
ings stock. An abbreviated R output of the model fits is given below:
> # Normal distribution
> fit1<-fitdistr(ret,"normal")
> # Standard t-distribution
> fit2<-stdFit(ret)
> # Skewed standard t-distribution
> fit3<-sstdFit(ret)
> AIC1<--2*fit1$loglik + 2*2
> AIC2<- 2*fit2$objective + 2*3
> AIC3<- 2*fit3$minimum + 2*4
> AIC1;AIC2;AIC3
[1] -3861.705
[1] -4048.132
[1] -4046.806
(i) Based on the given Akaike Information Criterion (AIC) decide which
model provides the best fit.
(ii) Which of the stylized facts discussed in the lecture is most relevant
to the result from part (c) (i)?
MT4539 May 2020, Page 6 of 8
(iii) The R output below gives the result of the Jarque-Bera test run on
the normal pseudo-residuals (zs) from the model with the best fit.
Explain how the normal pseudo-residuals are obtained providing a
verbal interpretation and comment on the adequacy of the model
using the R output below.
> jarque.bera.test(zs)
Jarque Bera Test
data: zs
X-squared = 4.1569, df = 2, p-value = 0.1251
[5]
MT4539 May 2020, Page 7 of 8
5. (a) A company has invested 1,000,000 in the Dow Jones Industrial Index
(DJI). The quants in the company have decided that a GJR-GARCH(1,1)
with drift and GED distributed innovations given as
rt = µ+ "t = µ+ t⌘t with ⌘t
iid⇠ GED(⌫) and
2t = ↵0 + (↵1 + 1It1)"
2
t1i + 1
2
t1 with
It1 =
⇢
1 if "t1 < 0,
0 if "t1 0 and ↵0 > 0, ↵1 + 1 0, 1 0
is an adequate model for the continuous returns of the index.
The model was fitted (in R) to a data set with observations of the contin-
uous returns of DJI. The estimates of the model are:
µˆ ↵ˆ0 ↵ˆ1 ˆ1 ˆ1 ⌫ˆ

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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