a) The company wants to completely eliminate its exposure to foreign exchange rate risk.
(2 marks)
b) The company wants to eliminate any downside risk if the exchange rate increases, while keeping the upside opportunity to gain if the exchange rate declines.
(2marks)
c) The company is happy to sell the upside opportunity to gain when the exchange rate decreases, while keeping the downside risk if exchange rate increases.
(2 marks)
2) Assume that the stock price follows a geometric Brownian motion, where the current price S0 = 100, expected return μ = 0 and volatility σ = 0.80. The risk-free rate r = 0.02 p.a. For each of the following scenarios, find the worst-case value of the portfolio in 6 months such that there is only 2.5% chance of the actual value being lower.
a) An investor owns 10,000 shares of this stock.
(2 marks)
b) An investor owns a call option on 100,000 shares of the stock with a strike price of $80 and a maturity of 2 years.
(2 marks)
c) An investor owns a put option on 100,000 shares of the stock with a strike price of $120 and maturity of 2 years.
(2 marks)
3) Suppose that the parameters in a GARCH (1,1) model are α = 0.013, β = 0.975 and VL = 0.0001.
a) If the previous volatility was σn-1 = 2% per day and return was un-1=3%, what is your current estimate of the volatility σn?
(1 mark)
b) What is the volatility forecast in 60 days?
(1 mark)
c) What volatility should be used to price 60-day option?
(1 mark)
d) Suppose that there is an event that increase the current volatility from 2% per day to 3% per day. Estimate by how much the event increases the volatility to price 60-day options.
(1 mark)
e) Given the GARCH (1,1) model and values of the parameters, what are the contributions (weights) of the most recent squared return (un)2 and the previous squared return (un-1)2 to the next day variance (σn+1)2, respectively?
(2 marks)
4) Suppose that from historical simulation, you have obtained n = 500 loss scenarios, the following table contains the five worst loss scenarios and the estimated daily volatilities.
a) Assume each scenario is equally likely, find the one-day 99% value-at-risk (VaR) and expected shortfall (ES).
(1 mark)
b) Assume exponential weighting, that is, the weight for scenario i is given by λn-i (1-λ)/(1-λn), where λ = 0.99. Find the one-day 99% VaR and ES.
(2 marks)
c) Suppose that we have estimated the daily volatility σi for each scenario i, and the current volatility is estimated to be σn+1 = 206.4 basis points (bps) per day. Find volatility-adjusted losses for each scenario, then calculate the one-day 99% VaR and ES. Assume exponential weighting for each scenario.
(2 marks)
d) Suppose we back-test the one-day 99% VaR and observe 10 exceptions. Should we reject the model at the 5% significance level? Use Kupiec’s two-tailed test.
(1 mark)
5) A bank has sold a European put option on 100,000 shares of a non-dividend paying stock. The current stock price is 50, expected return is zero, strike price is 25, volatility is 45% per annum, time to maturity is nine months. The risk-free rate is zero. Assume the stock price follows a geometric Brownian motion.
a) What is the 10-day 99% VaR using linear approximation with delta?
(2 marks)
b) What is the exact 10-day 99% VaR?
(2 marks)
c) How do you explain the difference between the VaRs estimated in (a) and (b)? Hint: you should refer to the skewness of the loss distribution.
(1 mark)
d) If the strike price becomes lower than 25, do you expect the difference between the VaRs estimated in (a) and (b) to increase or decrease? Explain.
(1 mark)
6) (a) Suppose that the assets of bank consist of $500 million of loans to BBB-rated corporations. The probability of default (PD) for the corporations is estimated as 0.5%and the loss given default (LGD) is 60%. The average maturity is 2.5 years. Calculate the risk-weighted assets (RWA) under the Basel II internal rating based (IRB) approach.
(3 marks)
(b) Basel III introduced requirements involving two liquidity ratios that are designed to ensure that banks can survive liquidity pressures. What are the two ratios? Briefly explain how they are measured.
(3 marks)
7) (a) Suppose a three-year corporate bond provides a coupon of 7% per year payable semi-annually and has a yield of 6% per annum (expressed with semi-annual compounding). The yields for all maturities on risk-free bonds are 4% per annum (expressed with semi-annual compounding). Assume that defaults can take place at the end of each year (immediately before a coupon payment) and recovery rate is 45%. Estimate the default probabilities assuming the unconditional default probabilities are the same on each possible default date.
(3 marks)
(b) Assume that value of a company’s assets is V0 = 25 million and the volatility of asset value is σv = 0.18 per annum. The debt that will have to be repaid in two year is D = 17 million. The risk-free rate is 2% per annum, continuously compounded. Use Merton’s model to find the following.
i. The equity value E0 and equity volatility σE
(1 mark)
ii. The market value of debt D0, and the expected loss from default
(1 mark)
iii. The probability of default and the recovery rate
(1 mark)
8) (a) A bank has purchased a put option on 100,000 shares of a stock with current price S0 = 28, strike K = 30, volatility σ = 0.8 per annum, and 2 years to maturity. Assume the risk-free rate is 3% p.a. with continuous compounding. What is no-default value of the option? If the bank’s counterparty has a 2% chance of default at the end of each year and a recovery rate of R = 0.5, what is the CVA and credit-risk-adjusted value of the option?
(2 marks)
(b) In part (a), if the bank purchased a 1-year zero-coupon bond with a face value of 10 million from the same counterparty, what is the value of the zero-coupon bond after credit adjustment?
(1 mark)
(c) Suppose that bank has entered into a forward contract to buy 1 million ounces of gold from a mining company in two years for 1500 per ounce. The current forward price for the contract is 1778 per ounce. There is a 3% chance for the mining company to default in the middle of the second year (assume no other default times are possible). The risk-free rate is 3% p.a., continuously compounded. The volatility of the forward price of gold when forward contract expires in two years is 25% per annum. What is the default-free value of the forward contract? What is the value of the forward contract after credit adjustment? Assume the recovery rate is 50%.
(3 marks)
9) (a) The following losses are ranked from highest to lowest for 500 scenarios in a historical simulation. Suppose that three stress scenarios are considered. They lead to losses ($000s) 850 and 950. The probabilities assigned to the scenarios are 0.8% and 0.2%, respectively. The total probability of the stressed scenarios is, therefore 1%.
Assume exponential weighting is used (λ = 0.99) for the historical scenarios. Compute the 1-day 97.5% expected shortfall.
(3 marks)
(b) Suppose that we use Monte-Carlo simulation to determine the loss distribution for operational risk. Moreover, the loss-frequency is modeled by the Poisson distribution with λ = 3.5 (average number of losses per year).
i. Let n be the number of losses, calculate the probability for n = 0, n ≤ 1 and n ≤ 2, respectively.
(2 marks)
ii. If a random number between 0 and 1 is drawn, say 0.152, how many losses occurred in this simulation trial?
(1 mark)
10) (a) Suppose that a bank’s sole business is to lend in two regions of the world. The lending in each region has the same characteristics, spread between cost of funds and interest charges is 2.5%, administrative costs are 0.7% and expected losses are 1%. Lending to Region A is four times as great as lending to Region B. The correlation between loan losses in the two regions is 0.75. Assume the economic capital in each region per $100 of loans is $4. Estimate the total RAROC.
(3 marks)
(b) A trader wishes to unwind a position of 100 million units in an asset over ten days. Suppose the bid-offer spread p (measured in dollars) as a function of daily trading volume is p(q) = 0.1+0.05e0.03q, and cost of liquidation is 0.5[p(q1)q1 + … + p(q10)q10] + λ [σ2(x1)2 + … + σ2(x10)2]1/2, where σ=0.2 is the standard deviation of price change per day, λ = 2.326, and xi is the number of units held on Day i. Compute cost of liquidation if the trader liquidates 10 million units each day.
(2 marks)
(c) Explain in your own words what is risk culture and why is it important.
(1 mark)
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