# Exercise Solution 10.6

- A primary mapping is given by
[s1]

[s2]

b,c. Computations are performed in a spreadsheet. We obtain portfolio remapping

[10.85]

where

[s3]

[s4]

[s5]

- A scatter plot is shown below. The approximation appears reasonable but not excellent.
- Our crude Monte Carlo estimator for
^{0}*std*(^{1}*P*) is[s6]

It is implemented in a spreadsheet.

- Our control variate Monte Carlo estimator for
^{0}*std*(^{1}*P*) is[s7]

where H is defined as in [s6] above, and

[s8]

The control variate Monte Carlo estimator is implemented in a spreadsheet.

- As described in Section 10.5.3, we stratify into
*w*= 16 disjoint subintervals ϑ_{j}based upon the conditional CDF of . For this purpose, we use the values of the CDF of indicated in Exhibit 10.17. Results are[s9]

[s10]

[s11]

︙

[s12]

Based upon stratification

[s13]

define stratification

[s14]

where

[s15]

for all

*j*.Define 16 random vectors

^{1}=**R**_{j}^{1}|**R**^{1}∈ Ω**R**_{j}. That is,^{1}equals**R**_{j}^{1}conditional on**R**^{1}being in Ω**R**_{j}. Define^{1}*P*= θ(_{j}^{1}) for all**R**_{j}*j*. Given samples for the^{1}, each of respective size**R**_{j}*m*, we define our stratified sampling Monte Carlo estimator for_{j}^{0}*std*(^{1}*P*) as[s16]

where probabilities

[s17]

can be calculated from the values of the CDF of indicated in Exhibit 10.17. Sample sizes

*m*are selected as suggested Section 10.5.3. Generating sample realizations for this estimator is difficult with a spreadsheet. The estimator is easy to code as a simple program._{i} - Each of the three estimators is applied ten times to obtain a total of 30 estimates for
^{0}*std*(^{1}*P*). Results are indicated below.

crude control

variatestratified

sampling1 39030 38376 38147 2 38351 38066 38475 3 39396 38375 37997 4 39140 38351 37963 5 37958 38308 38107 6 38379 38165 38321 7 39565 38455 37962 8 38350 38667 38295 9 38249 38615 38128 10 36993 38494 38184 mean 38541 38387 38158 stdev 690 166 151 - Based upon our results from part (i), we estimate that the crude, control variate and stratified sampling estimators have standard errors of 1.79%, 0.43%, and 0.40% (results are obtained by dividing
**stdev**results by**mean**results from the above table for each estimator). These results were obtained using a sample size of 1000 for each estimator. Since standard error is inversely proportional to the square root of sample size, we estimate that the three estimators will require respective sample sizes of 3209, 187, and 156 to have a 1% standard error. - Based upon a sample {
^{1}**R**^{[1]},^{1}**R**^{[2]}, … ,^{1}**R**^{[1000]}} for^{1}, we define a sample {**R**^{1}*P*^{[1]},^{1}*P*^{[2]}, … ,^{1}*P*^{[1000]}} for^{1}*P*with^{1}*P*^{[k]}= θ(^{1}**R**^{[k]}). Our crude Monte Carlo estimator estimates the 95% VaR as 89700 less the sample .05-quantile of {^{1}*P*^{[1]},^{1}*P*^{[2]}, … ,^{1}*P*^{[1000]}}. This is implemented as a spreadsheet. - Based upon a sample {
^{1}**R**^{[1]},^{1}**R**^{[2]}, … ,^{1}**R**^{[1000]}} for^{1}, we define a sample {**R**^{1}*P*^{[1]},^{1}*P*^{[2]}, … ,^{1}*P*^{[1000]}} for^{1}*P*with^{1}*P*^{[k]}= θ(^{1}**R**^{[k]}) and another sample {^{[1]},^{[2]}, … ,^{[1000]}} for with^{[k]}= (^{1}**R**^{[k]}). Let*H*be the sample .05-quantile estimator for the^{1}*P*^{[k]}, and let be the sample .05-quantile estimator for the^{[k]}. Our control variate estimator for 95% VaR then is[s18]

This is implemented as a spreadsheet.

- As described in Section 10.5.4, we stratify into 2 disjoint subintervals ϑ
:_{j}[s19]

[s20]

Based upon stratification

[s21]

define stratification

[s22]

where

[s23]

for each

*j*.Define two random vectors

^{1}=**R**_{j}^{1}|**R**^{1}∈ Ω**R**_{j}. That is,^{1}equals**R**_{j}^{1}conditional on**R**^{1}being in Ω**R**_{j}. Define samples and . Define sample {^{1}*P*^{[1]},^{1}*P*^{[2]}, … ,^{1}*P*^{[1000]}} with^{1}*P*^{[k]}= θ() for 1 ≤*k*≤ 50 and^{1}*P*^{[k]}= θ() for 51 ≤*k*≤ 1000. VaR is estimated as 89700 less the sample .05-quantile of the^{1}*P*^{[k]}. Generating sample realizations for this estimator is difficult with a spreadsheet. The estimator is easy to code as a simple program. - Each of the three estimators for 95% VaR is applied ten times to obtain a total of 30 estimates for
^{0}*std*(^{1}*P*). Results are indicated below.

crude control

variatestratified

sampling1 65391 67811 68682 2 71173 70749 68569 3 66265 67532 68698 4 71563 69005 67909 5 66527 68949 68085 6 70794 69455 68073 7 65592 68869 68248 8 69931 69199 67900 9 71113 68171 69095 10 70503 69756 67927 mean 68885 68950 68319 stdev 2328 855 374 - Based upon our results from part (m), we estimate that the crude, control variate and stratified sampling estimators have standard errors of 3.38%, 1.24%, and 0.55% (results are obtained by dividing stdev results by mean results from the above table for each estimator). These results were obtained using a sample size of 1000 for each estimator. Since standard error is inversely proportional to the square root of sample size, we estimate that the three estimators will require respective sample sizes of 11417, 1539, and 300 to have a 1% standard error.