Question 1 (25 marks)

1.1 Suppose that the gain from a portfolio during six months is normally distributed with a mean of $3.5 million and a standard deviation of $12 million. Calculate and interpret the VaR of the portfolio with a 99% confidence level. (5)

1.2 A fund manager announces that the fund's one-month 95% expected shortfall is 5.6% of the size of the portfolio being managed. You have an investment of R750,000 in the fund. How do you interpret the portfolio manager's announcement? (3)

1.3 Suppose we estimate the one-day 95% VaR from 1,200 observations as 6 (in millions of dollars). By fitting a standard distribution to the observations, the probability density function of the loss distribution at the 95% point is estimated to be 0.02. What is the standard error of the VaR estimate? (2)

1.4 Suppose that a one-day 95% VaR is estimated as $15 million from 2,000 observations. The one-day changes are approximately normal with mean zero and standard deviation $5.5 million. Estimate a 99% confidence interval for the VaR estimate. (5)

1.5 Consider a position consisting of a $120,000 investment in asset A and a $120,000 investment in asset B. Assume that the daily volatilities of both assets are 1% and that the coefficient of correlation between their returns is 0.4. What are the five-day 95% VaR and ES for the portfolio? (5)

1.6 Explain why the linear model can provide only approximate estimates of VaR for a portfolio containing options. (5)

Question 2 (25 marks)

2.1 A stock price has an expected return of 15% and a volatility of 25%. It is currently $56.

2.1.1 What is the probability that it will be greater than $85 in two years? (4)

2.1.2 What is the stock price that has a 5% probability of being exceeded in two years? (2)

2.2 A binary option pays off $150 if a stock price is greater than $40 in three months. The current stock price is $35 and its volatility is 35%. The risk-free rate is 4% and the expected return on the stock is 10%.

2.2.1 What is the value of the option? (6)

2.2.2 What is the real-world probability that the payoff will be received? (4)

2.3 An investor owns 12,000 shares of a particular stock. The current market price is R100. What is the "worst case" value of the portfolio in six months? For the purposes of this question, define the worst case value of the portfolio as the value which is such that there is only a 1% chance of the actual value being lower. Assume that the expected return and volatility of the stock price are 8.5% and 23%, respectively. (4)

2.4 You are a risk manager at World Finance Ltd, the CEO has asked you to provide a brief report to him on the definition of traffic light options and what their drawbacks will be for the organisation. Correctly reference two academic articles that deals with traffic light options for the CEO report. (5)

Question 3 (25 marks)

3.1 Briefly discuss the lessons that can be learned from past risk management mistakes. (10)

3.2 Discuss whether you will classify COVID-19 as a black swan event or not. (15)

You should include the following:

Unknown and unknowable risks

Definition of a black swan

The attributes required to be classified as a black swan

Reasons for your classification

List of references