2.1 Suppose that each of two investments has a 3% chance of a loss of R15 million, a 2% chance of a loss of R1.5 million and a 95% chance of a profit of R1.5 million. They are independent of each other. Calculate the expected shortfall (ES) when the confidence level is 95%. (4)

2.2 If an analyst wants to compare the expected future value of a stock index in real-world terms with the expected future value in the risk-neutral world, which of these is expected to have a higher value? Please explain your answer. (2)

2.3 A financial institution owns a portfolio of options dependent on the US Dollar Sterling exchange rate. The delta of the portfolio with respect to percentage changes in the exchange rate is 7.3. If the daily volatility of the exchange rate is 0.5% and a linear model is assumed, calculate the estimated 10-day 95% VaR for the portfolio. (4)

2.4 Suppose we estimate the one-day 97.5% VaR from 1,100 observations (in millions of dollars) as 5. By fitting a standard distribution to the observations, the probability density function of the loss distribution at the 97.5% point is estimated to be 0.08. Calculate the standard error of the VaR estimate in dollar terms. (Round your calculations to 8 decimal places) (4)

2.5 List two ways of handling interest-rate-dependent instruments when the model-building approach is used to calculate VaR. (2)

2.6 A binary option pays off R240 if a stock price is greater than R50 in six months. The current stock price is R45, and its volatility is 25%. The risk-free rate is 7% and the expected return on the stock is 11%. Calculate the value of the option. (5)

2.7 The probability that the loss from a portfolio will be greater than R20 million in one month is estimated to be 5%. What is the one-month 99% VaR assuming the change in the value of the portfolio is normally distributed with zero mean? (4)