风险管理与金融机构-约翰-第二版-答案

Chapter 8: Value at Risk

8.12.

(a) A loss of $1 million extends from the 94 percentile point of the loss distribution to the 96 percentile point. The 95% VaR is therefore $1 million.

(b) The expected shortfall for one of the investments is the expected loss conditional that the loss is in the 5 percent tail. Given that we are in the tail there is a 20% chance than the loss is $1 million and an 80% chance that the loss is $10 million. The expected loss is therefore $8.2 million. This is the expected shortfall.

(c) For a portfolio consisting of the two investments there is a 0.04 × 0.04 = 0.0016 chance that the loss is $20 million; there is a 2 × 0.04 × 0.02 = 0.0016 chance that the loss is $11 million; there is a 2 × 0.04 × 0.94 = 0.0752 chance that the loss is $9 million; there is a 0.02 × 0.02 = 0.0004 chance that the loss is $2 million; there is a 2 × 0.2 × 0.94 = 0.0376 chance that the loss is zero; there is a 0.94 × 0.94 = 0.8836 chance that the profit is $2 million. It follows that the 95% VaR is $9 million.

(d) The expected shortfall for the portfolio consisting of the two investments is the expected loss conditional that the loss is in the 5% tail. Given that we are in the tail, there is a 0.0016/0.05 = 0.032 chance of a loss of $20 million, a 0.0016/0.05 = 0.032 chance of a loss of $11 million; and a 0.936 chance of a loss of $9 million. The expected loss is therefore $9.416.

(e) VaR does not satisfy the subadditivity condition because 9 > 1 + 1. However, expected shortfall does because 9.416 < 8.2 + 8.2.

8.13. The correct multiplier for the variance is

10 + 2 × 9 × 0.12 + 2 × 8 × 0.122 + 2 × 7 × 0.123 + . . . + 2 × 0.129 = 10.417 The estimate of VaR should be increased to 2?10.417/10 = 2.229

8.14. In this case p = 0.01, m = 15, n = 1000. Kupiec’s test statistic is

?2 ln[0.999985 × 0.0115] + 2 ln[(1 ? 15/1000)985 × (15/1000)15] = 2.19 This is less than 3.84. We should not therefore reject the model.

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Chapter 9: Volatility

9.18. The calculations are shown in the table below.

?ui?0.09471?ui2?0.01145 and an estimate of standard deviation of weekly returns is:

0.011450.094712??0.02884 1314?13The volatility per annum is therefore 0.0288452=0.2079 or 20.79%. The standard error of the estimate

0.2079?0.0393 2?14

or 3.9% per annum. Week, i Closing Stock Price ($) Price Relative = Si/Si-1 Daily Return, ui=ln(Si/Si-1) 1 30.2 2 32.0 1.05960 0.05789 3 31.1 0.97188 –0.02853 4 30.1 0.96785 –0.03268 5 30.2 1.00332 0.00332 6 30.3 1.00331 0.00331 7 30.6 1.00990 0.00985 8 33.0 1.07843 0.07551 9 32.9 0.99697 –0.00303 10 33.0 1.00304 0.00303 11 33.5 1.01515 0.01504 12 33.5 1.00000 0.00000 13 33.7 1.00597 0.00595 14 33.5 0.99407 –0.00595 15 33.2 0.99104 –0.00900

9.19. The proportional change in the price of gold is ?2/300 = ?0.00667. (a) Using the EWMA model the variance is updated to

0.94 × 0.0132 + 0.06 × 0.006672 = 0.00016153

so that the new daily volatility is 0.00016153 = 0.01271 or 1.271% per day. (b) Using GARCH (1,1) the variance is updated to 0.000002 + 0.94 × 0.0132 + 0.04 × 0.006672 = 0.00016264

so that the new daily volatility is 0.00016264 = 0.1275 or 1.275% per day.

9.20. (Spreadsheet Provided) The data give “best” values for ? higher than the 0.94 used by RiskMetrics. For AUD, BEF, CHF, DEM, DKK, ESP, FRF, GBP, ITL, NLG, and SEK they are 0.983, 0.967, 0.968, 0.960, 0.971, 0.983, 0.965, 0.977, 0.939, 0.962, and 0.989, respectively.

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For TSE, S&P, FTSE, CAC, and Nikkei, they are 0.991, 0.989, 0.958, 0.974, and 0.961, respectively. The spreadsheet shows results for AUD and TSE.

9.21.

(a) The long-run average variance, VL, is

?0.000002??0.0001

1????0.02The long run average volatility is 0.0001 = 0.01 or 1% per day.

(b) From equation (9.14) the expected variance in 20 days is

0.0001 + 0.9820(0.0152 ? 0.0001) = 0.000183

The expected volatility per day is therefore 0.000183 = 0.0135 or 1.35%. Similarly the expected volatilities in 40 and 60 days are 1.25% and 1.17%, respectively.

(c) In equation (9.15) a = ln(1/0.98) = 0.0202. The variance used to price 20-day options is

??1?e?0.0202?20252?0.0001?(0.0152?0.0001)??0.051

0.0202?20??so that the volatility is 22.61%. Similarly, the volatilities that should be used for 40- and 60-day options are 21.63% and 20.85% per annum, respectively. (d) From equation (9.14) the expected variance in 20 days is

0.0001 + 0.9820(0.022 ? 0.0001) = 0.0003

The expected volatility per day is therefore 0.0003 = 0.0173 or 1.73%. Similarly the expected volatilities in 40 and 60 days are 1.53% and 1.38% per day, respectively.

(e) When today’s volatility increases from 1.5% per day (23.81% per year) to 2% per day (31.75% per year) the equation (9.16) gives the 20-day volatility increase as

1?e?0.0202?2023.81??(31.75?23.81)?6.88

0.0202?2022.61

or 6.88% bringing the volatility up to 29.49%. Similarly the 40- and 60-day volatilities increase to 27.37% and 25.70%.

9.22. (Spreadsheet Provided) In EWMA the best fit ? are 0.931 for TSE and 0.909 for S&P. For GARCH(1,1) and TSE, ?= 0.0000040, ? = 0.1393 and ?= 0.7928. For GARCH(1,1) and S&P, ? = 0.0000024, ? = 0.0895 and ? = 0.8816.

9.23.

(a) The 99% VaR is

N?1(0.99)10??1?14.14

N(0.95)or $14.14 million.

(b) The probability that the loss is greater than x is Kx-?. We know that ? = 3 and K × 10-3 = 0.05. It follows that K = 50 and value of x that is the 99% VaR is given by

50x-3 = 0.01 or

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x = (5000)1/3 = 17.10

The 99% VaR using the power law is $17.10 million.

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Chapter 10: Correlations and Copulas

10.15. The proportional change in the price of gold is ?2/300 = ?0.00667. Using the EWMA model the variance is updated to

0.94 × 0.0132 + 0.06 × 0.006672 = 0.00016153

so that the new daily volatility is 0.00016153 = 0.01271 or 1.271% per day.

Using GARCH (1,1), the variance is updated to

0.000002 + 0.94 × 0.0132 + 0.04 × 0.006672 = 0.00016264

so that the new daily volatility is 0.00016264 = 0.1275 or 1.275% per day.

The proportional change in the price of silver is zero. Using the EWMA model the variance is updated to

0.94 × 0.0152 + 0.06 × 0 = 0.0002115

so that the new daily volatility is 0.0002115 = 0.01454 or 1.454% per day.

Using GARCH (1,1), the variance is updated to

0.000002 + 0.94 × 0.0152 + 0.04 × 0 = 0.0002135

so that the new daily volatility is 0.0002135 = 0.01461 or 1.461% per day.

The initial covariance is 0.8×0.013×0.015 = 0.000156. Using EWMA the covariance is updated to

0.94 × 0.000156 + 0.06 × 0 = 0.00014664 so that the new correlation is 0.00014664/(0.01454×0.01271) = 0.7934. Using GARCH (1,1) the covariance is updated to

0.000002 + 0.94 × 0.000156 + 0.04 × 0 = 0.00014864 so that the new correlation is 0.00014864/(0.01461 × 0.01275) = 0.7977.

For a given ? and ?, the ? parameter defines the long run average value of a variance or a covariance. There is no reason why we should expect the long run average daily variance for gold and silver should be the same. There is also no reason why we should expect the long run average covariance between gold and silver to be the same as the long run average variance of gold or the long run average variance of silver. In practice, therefore, we are likely to want to allow ? in a GARCH(1,1) model to vary from market variable to market variable. (Some instructors may want to use this problem as a lead-in to multivariate GARCH models.)

10.16. (Spreadsheet Provided) The probability that V1 < 0.25 is 1 – e?1.0×0.25 = 0.221. The

probability that V2 < 0.25 is 1 – e?2.0×0.25 = 0.393. These are transformed to the normal variates –0.768 and –0.270. Using the Gaussian copula model the probability that V1 < 0.25 and V2 < 0.25 is M(?0.768,?0.270,?0.2) = 0.065. The other cumulative probabilities are shown in the table below and are calculated similarly.

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