MAY 2020 EXAMINATION DIET
SCHOOL OF MATHEMATICS & STATISTICS
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 th
e 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) =