V1 0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.065 0.125 0.177 0.219 0.254 0.282 0.50 0.117 0.219 0.303 0.371 0.426 0.469 0.75 0.153 0.282 0.386 0.469 0.535 0.587 V2 1.00 0.177 0.323 0.439 0.531 0.603 0.660 1.25 0.193 0.349 0.472 0.569 0.645 0.705 1.50 0.203 0.366 0.493 0.593 0.672 0.733
10.17. (Spreadsheet Provided) The procedure for taking a random sample from a bivariate Student t-distribution is described on page 214. This can be used to produce Figure 10.5. For the second part of the question we sample U1 and U2 from a bivariate normal distribution where the correlation is 0.5 as described in Section 10.3. We then convert each sample into a variable with a Student t-distribution on a percentile-to-percentile basis. Suppose that U1 is in cell C1. The Excel function TINV gives a “two-tail” inverse of the t-distribution. An Excel instruction for determining V1 is therefore
=IF(NORMSDIST(C1)<0.5,-TINV(2*NORMSDIST(C1),4),TINV(2*(1-NORMSDIST(C1)),4)). The scatter plot shows that there is much less tail correlation when the normal copula is used for the t-distributions.
10.18. (Spreadsheet Provided) The WCDR with a 99.7% confidence level is from equation (10.12)
?N?1(0.012)??N?1(0.9997)?? N???1????
The table below gives the variation of this with the copula correlation.
21 / 40
Copula Correlation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 WCDR (%) 1.2 10.8 21.0 32.6 45.5 59.5 73.7 86.9 96.5 99.9 22 / 40
Chapter 11: Regulation, Basel II and Solvency II
11.19. The capital requirement is the current exposure plus an add-on amount multiplied by the counterparty risk weight multiplied by 8%. The add-on amount is to allow for a possibility that the exposure will increase prior to a default. To argue for a relationship between the add-on amount and the value of the transaction, consider two cases: 1. The value of the transaction is zero.
2. The value of the transaction is –$10 million
The current exposure is zero in both cases. In the first case any increase in the value of the
transaction will lead to an exposure. In the second case the transaction has to increase in value by more than $10 million before there is an exposure—and it might be very unlikely that this will happen. However, the capital required is the same in both cases.
11.20. Using Table 11.2 the credit equivalent amounts (in millions of dollars) for the three transactions are (a) 2 + 0.05 × 50 = 4.5 (b) 4 + 0.06 × 20 = 5.2 (c) 0.12 × 30 = 3.6
The total credit equivalent amount is 4.5+5.2+3.6 = 13.3. The risk weighted amount is 13.3 × 0.2 = 2.66. The capital required is 0.08 × 2.66 or $0.2126 million.
If netting applies, the current exposure after netting is in millions of dollars 2+4?5 =
1. The NRR is therefore 1/6 = 0.1667. The credit equivalent amount is in millions of dollars 1 + (0.4 + 0.6 × 0.1667)×(0.05 × 50 + 0.06 × 20 + 0.12 × 30) = 4.65 The risk weighted amount is 0.2×4.65 = 0.93 and the capital required is 0.08×0.93 = 0.0744. In this case the netting amendment reduces the capital by about 65%.
11.21. Using Table 11.2 the credit equivalent amount under Basel I (in millions of dollars) for the three transactions are (a) 3 + 0.005 × 100 = 3.5 (b) 0.01 × 150 = 1.5 (c) 7 + 0.01 × 50 = 7.5
The total credit equivalent amount is 3.5 + 1.5 + 7.5 = 12.5. Because the corporation has a risk weight of 100% the risk weighted amount is also 12.5. The capital required is 0.08 × 12.5 or $1.0 million.
If netting applies, the current exposure after netting is in millions of dollars 3?5+7 =5. The NRR is therefore 5/10 = 0.5. The credit equivalent amount is in millions of dollars
5 + (0.4 + 0.6 × 0.5)×(0.005 × 100 + 0.01 × 150 + 0.01 × 50) = 6.75 The risk weighted amount is also 6.75 and the capital required is 0.08 × 6.75 = 0.54. In this case the netting amendment reduces the capital by 46%.
Under Basel II when the standardized approach is used the corporation has a risk weight of 20% and the capital required is therefore one fifth of that required under Basel I or 0.2 × 0.54 or $0.108 million.
11.22. Under the Basel II advanced IRB approach
?= 0.12[1 + e?50×0.003 ] = 0.2233
b = [0.11852 ? 0.05478 × ln(0.003)]2 = 0.1907
23 / 40
MA?1?(3.0?2.5)?0.1907?1.53
1?1.5?0.1907and
?N?1(0.003?0.2233N?1(0.999)? WCDR?N???0.0720
1?0.2233??The RWA is
500 × 0.6 × (0.0720 ? 0.003) × 1.53 × 12.5 = 397.13
The total capital is 8% of this or $31.77 million. Half of this must be Tier I. Under both the Basel II standardized approach and under Basel I the risk weight is 100% and the total capital required is 8% of $500 or $40 million.
24 / 40
Chapter 12: Market Risk VaR: The Historical Simulation Approach
12.12. The standard error is
10.025?0.975
f(q)2000where f(q) is an estimate of the loss probability density at the VaR point. In this case the 0.975 point on the approximating normal distribution is NORMINV(0.975,0,6) = 11.76. f(q) is estimated as NORMDIST(11.76,0,6,FALSE) = 0.0097. The standard error is therefore
10.025?0.975?0.358
0.00972000
A 99% confidence interval for the VaR is 13 ? 2.576 × 0.358 to 13 + 2.576 × 0.358 or 12.077 to 13.923.
12.13. (Spreadsheet Provided)
(a) $230,897 (see Ranked Losses worksheet) (b) $339,308 (see Ranked Losses worksheet)
(c) $648,257 (see Ranked Losses Vol Adjusted Scenarios Worksheet)
(d) The values of ? and ? given by Solver are 30.69 and 0.684. The VaR with 99% confidence given by extreme value theory is $241,389
12.14. (Spreadsheet Provided) The VaR estimate for a 99% confidence level is $587,621. 12.15. (Spreadsheet Provided) The worst scenario (number 494) now has a weight of 0.01371, which is more than 0.01. As a result the 99% VaR is the loss associated with this scenario or $499,395.
12.16. (Spreadsheet Provided) The one-day 99% VaR changes to $619,992.
12.17 (Spreadsheet Provided)
(a) The fifteenth worst daily change in the NASDAQ during the period considered is about –5.39% and the 1-day 99% VaR is $538,938. (See Ranked Losses worksheet)
(b) When weights are assigned to each day and the daily changes are listed from the worst to the best, the weights we see that the VaR is $229,401. (See Ranked Losses with Weights worksheet) (c) In this case we use the EWMA updating scheme to update the variance and therefore the volatility. The volatility estimate for the day after March 10, 2006 is 0.755% per day. The volatility changes during the 1,500 day period as indicated in the chart below. Volatility was relatively low on March 10, 2006. Earlier observations are scaled down to reflect this. The resulting estimate of the 1-day 99% VaR is $160,985. (See Ranked Losses, Volatility Scaling worksheet)
25 / 40
风险管理与金融机构-约翰-第二版-答案
