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Quantitative Risk Management

Coursework

The objective of the coursework is to allow you a ﬁrst hand appreciation of some of the key issues in measuring risk. You will need to acquire daily data on three assets (stock, index funds, commodities for instance), going back at least 5 years and ending on 16 Feb 2022. Let us refer to these assets as A, B and C. Please note that the dates for all three should coincide. For example, if asset A is available for each trading day between 5 Jan 2012 to 16 Feb 2022, then data for B and C should also be available on those dates, and vice versa.

1. Designate one of your assets as asset A.

(a) Using the data for asset A calculate the simple daily returns [use simple returns throughout this coursework].

(b) Examine and describe the key statistical features of your sample of returns.

In particular, comment on the signiﬁcance of the values of the

i. mean

ii. standard deviation

iii. skewness

iv. kurtosis

for the management of risk.

(c) Suppose now that you have, on the 16th of February 2022, a portfolio con- sisting only of asset A. This portfolio is valued at £1 Million.

i. What would this portfolio have been worth at the beginning of the data period?

ii. Calculate VaR and ES for 17th February 2022 using a one day holding period, and a conﬁdence level of 95%, using each of the following meth- ods (present your results in a single table):

A. Basic Historical Simulation

B. Age-weighted Historical Simulation

C. Hull-White

D. Parametric, using an appropriate distribution, without volatility ad- justment

E. Parametric, using an appropriate distribution, with volatility adjust- ment

iii. Discuss the extent to which, for a given risk measure, there is variation of measured risk across the different approaches.

iv. Discuss the extent to which, for a given approach, there is variation in measured risk between the two different risk measures.

2. Suppose on the 16th February you were also to possess holdings of Asset B worth 1 Million. Use the Hull-White method to calculate 1 day ahead VaR using a 95% conﬁdence level for:

(a) your holdings of asset A (you may have done this already)

(b) your holdings of asset B

Is it the case that VaR(A) + VaR(B) < VaR(A+B)? Brieﬂy comment.

3. Now suppose that on 16th February 2022 you were considering how to manage the composition of your portfolio. You wish to retain at least a 20% holding of asset A in your portfolio, and you wish to combine your holding of A with either B or C (but not both), while keeping the portfolio’s total value equal to £2 million. Furthermore, you wish to minimise portfolio risk1.

(a) In order to minimise portfolio risk subject to the conditions described, should

you combine asset A with asset B, or with asset C? Explain.

(b) Compare the expected proﬁt of your minimum risk portfolio, with the ex-

pected proﬁt you could have achieved by investing all £2 million in the asset which has the higher expected return. Do you feel this sacriﬁce is worth the reduction in risk?

The deadline for submission is as notiﬁed in the module outline. Please see Moodle for further discussion of useful approaches to this topic, and hints about R code.