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Options, Futures, and Other Derivatives

John C. Hull

Chapter 22

Valueat risk and expected shortfall - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider a position consisting of a $$\$ 100,000$$ investment in asset $$\mathrm{A}$$ and a $$\$ 100,000$$ investment in asset B. Assume that the daily volatilities of both assets are $$1 \%$$ and that the coefficient of correlation between their returns is 0.3 . Estimate the 5 -day $$99 \% \mathrm{VaR}$$ and ES for the portfolio assuming normally distributed returns.

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01:07

Problem 2

Describe three ways of handling instruments that are dependent on interest rates when the model-building approach is used to calculate VaR. How would you handle these instruments when historical simulation is used to calculate VaR?

Nick Johnson
Nick Johnson
Numerade Educator

Problem 3

A financial institution owns a portfolio of options on the U.S. dollar-sterling exchange rate. The delta of the portfolio is 56.0 . The current exchange rate is 1.5000 . Derive an approximate linear relationship between the change in the portfolio value and the percentage change in the exchange rate. If the daily volatility of the exchange rate is $0.7 \%$, estimate the 10 -day $99 \% \mathrm{VaR}$.

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Problem 4

Suppose you know that the gamma of the portfolio in the previous question is 16.2. How does this change your estimate of the relationship between the change in the portfolio value and the percentage change in the exchange rate?

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03:19

Problem 5

Suppose that the daily change in the value of a portfolio is, to a good approximation, linearly dependent on two factors, calculated from a principal components analysis. The delta of a portfolio with respect to the first factor is 6 and the delta with respect to the second factor is -4 . The standard deviations of the factors are 20 and 8 , respectively. What is the 5-day $90 \% \mathrm{VaR}$ ?

Shu Naito
Shu Naito
Numerade Educator
01:40

Problem 6

Suppose that a company has a portfolio consisting of positions in stocks and bonds. Assume that there are no derivatives. Explain the assumptions underlying (a) the linear model and (b) the historical simulation model for calculating $V a R$.

Jennifer Stoner
Jennifer Stoner
Numerade Educator

Problem 7

Explain how a forward contract to sell foreign currency is mapped into a portfolio of zero-coupon bonds with standard maturities for the purposes of a VaR calculation.

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Problem 8

Explain the difference between value at risk and expected shortfall.

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01:56

Problem 9

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

Hannah Vigran
Hannah Vigran
Numerade Educator
01:31

Problem 10

Some time ago a company entered into a forward contract to buy $£ 1$ million for $$\$ 1.5$$ million. The contract now has 6 months to maturity. The daily volatility of a 6-month zero-coupon sterling bond (when its price is translated to dollars) is $0.06 \%$ and the daily volatility of a 6-month zero-coupon dollar bond is $0.05 \%$. The correlation between returns from the two bonds is 0.8 . The current exchange rate is 1.53. Calculate the standard deviation of the change in the dollar value of the forward contract in 1 day. What is the 10-day $99 \%$ VaR? Assume that the 6-month interest rate in both sterling and dollars is $5 \%$ per annum with continuous compounding.

Nick Johnson
Nick Johnson
Numerade Educator
00:53

Problem 11

The text calculates a VaR estimate for the example in Table 22.9 assuming two factors. How does the estimate change if you assume (a) one factor and (b) three factors.

Milo Kessler
Milo Kessler
Numerade Educator
02:16

Problem 12

A bank has a portfolio of options on an asset. The delta of the options is -30 and the gamma is -5 . Explain how these numbers can be interpreted. The asset price is 20 and its volatility is $1 \%$ per day. Adapt Sample Application E in the DerivaGem Application Builder software to calculate VaR.

Nick Johnson
Nick Johnson
Numerade Educator
02:16

Problem 13

Suppose that in Problem 22.12 the vega of the portfolio is -2 per $1 \%$ change in the annual volatility. Derive a model relating the change in the portfolio value in 1 day to delta, gamma, and vega. Explain without doing detailed calculations how you would use the model to calculate a VaR estimate.

Nick Johnson
Nick Johnson
Numerade Educator
01:18

Problem 14

The one-day $99 \% \mathrm{VaR}$ is calculated for the four-index example in Section 22.2 as $$\$ 253,385$$. Look at the underlying spreadsheets on the author's website and calculate:
(a) the one-day $95 \% \mathrm{VaR}$, (b) the one-day $95 \% \mathrm{ES}$, (c) the one-day $97 \% \mathrm{VaR}$, and
(d) the one-day $97 \%$ ES.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:39

Problem 15

Use the spreadsheets on the author's website to calculate the one-day $99 \% \mathrm{VaR}$ and ES. employing the basic methodology in Section 22.2, if the four-index portfolio considered in Section 22,2 is equally divided between the four indices.

Niamat Khuda
Niamat Khuda
Numerade Educator
03:19

Problem 16

A company has a position in bonds worth $$\$ 6$$ million. The modified duration of the portfolio is 5.2 years. Assume that only parallel shifts in the yield curve can take place and that the standard deviation of the daily yield change (when yield is measured in percent) is 0.09 . Use the duration model to estimate the 20-day $90 \% \mathrm{VaR}$ for the portfolio. Explain carefully the weaknesses of this approach to calculating VaR. Explain two alternatives that give more accuracy.

Shu Naito
Shu Naito
Numerade Educator

Problem 17

Consider a position consisting of a $$\$ 300,000$$ investment in gold and a $$\$ 500,000$$ investment in silver. Suppose that the daily volatilities of these two assets are $1.8 \%$ and $1.2 \%$, respectively, and that the coefficient of correlation between their returns is 0.6 . What is the 10-day $97.5 \% \mathrm{VaR}$ and ES for the portfolio? By how much does diversification reduce the $\mathrm{VaR}$ ? Assume normally distributed returns.

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02:16

Problem 18

Consider a portfolio of options on a single asset. Suppose that the delta of the portfolio is 12 , the value of the asset is $$\$ 10$$, and the daily volatility of the asset is $2 \%$. Estimate the 1-day $95 \% \mathrm{VaR}$ for the portfolio from the delta. Suppose next that the gamma of the portfolio is -2.6 . Derive a quadratic relationship between the change in the portfolio value and the percentage change in the underlying asset price in one day. How would you use this in a Monte Carlo simulation?

Nick Johnson
Nick Johnson
Numerade Educator
01:31

Problem 19

A company has a long position in a 2-year bond and a 3-year bond, as well as a short position in a 5 -year bond. Each bond has a principal of $$\$ 100$$ and pays a $5 \%$ coupon annually. Calculate the company's exposure to the 1-year, 2-year, 3-year, 4-year, and 5-year rates. Use the data in Tables 22.7 and 22.8 to calculate a 20-day $95 \%$ VaR on the assumption that rate changes are explained by (a) one factor, (b) two factors, and (c) three factors. Assume that the zero-coupon yield curve is flat at $5 \%$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 20

A bank has written a call option on one stock and a put option on another stock. For the first option the stock price is 50 , the strike price is 51 , the volatility is $28 \%$ per annum, and the time to maturity is 9 months. For the second option the stock price is 20 , the strike price is 19 , the volatility is $25 \%$ per annum, and the time to maturity is 1 year. Neither stock pays a dividend, the risk-free rate is $6 \%$ per annum, and the correlation between stock price returns is 0.4 . Calculate a 10 -day $99 \% \mathrm{VaR}$ :
(a) Using only deltas
(b) Using the partial simulation approach
(c) Using the full simulation approach.

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01:09

Problem 21

Use equation (22.1) to show that when the loss distribution is normal, VaR with $99 \%$ confidence is almost exactly the same as ES with $97.5 \%$ confidence.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 22

Suppose that the portfolio considered in Section 22.2 has (in $$\$000s$$) 3,000 in DJIA, 3,000 in FTSE, 1,000 in CAC 40 and 3,000 in Nikkei 225. Use the spreadsheet on the author's website to calculate what difference this makes to the one-day $99 \% \mathrm{VaR}$ and ES that is calculated in Section 22.2.

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