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# Exercise Solution 10.6

1. 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]

1. A scatter plot is shown below. The approximation appears reasonable but not excellent. 2. Our crude Monte Carlo estimator for 0std(1P) is

[s6]

It is implemented in a spreadsheet.

3. Our control variate Monte Carlo estimator for 0std(1P) is

[s7]

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

[s8]

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

4. As described in Section 10.5.3, we stratify into w = 16 disjoint subintervals ϑj 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 1Rj = 1R|1R ∈ Ωj. That is, 1Rj equals 1R conditional on 1R being in Ωj. Define 1Pj = θ(1Rj) for all j. Given samples for the 1Rj, each of respective size mj, we define our stratified sampling Monte Carlo estimator for 0std(1P) as

[s16]

where probabilities

[s17]

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

5. Each of the three estimators is applied ten times to obtain a total of 30 estimates for 0std(1P). Results are indicated below.
crudecontrol
variate
stratified
sampling
1390303837638147
2383513806638475
3393963837537997
4391403835137963
5379583830838107
6383793816538321
7395653845537962
8383503866738295
9382493861538128
10369933849438184
mean385413838738158
stdev690166151
6. 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.
7. Based upon a sample {1R, 1R, … , 1R} for 1R, we define a sample {1P1P, … , 1P} for 1P with 1P[k] = θ(1R[k]). Our crude Monte Carlo estimator estimates the 95% VaR as 89700 less the sample .05-quantile of {1P1P, … , 1P}. This is implemented as a spreadsheet.
8. Based upon a sample {1R1R, … , 1R} for 1R, we define a sample {1P1P, … , 1P} for 1P with 1P[k] = θ(1R[k]) and another sample { , , … , } for with [k] = (1R[k]). Let H be the sample .05-quantile estimator for the 1P[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.

9. As described in Section 10.5.4, we stratify into 2 disjoint subintervals ϑj:

[s19] [s20] Based upon stratification

[s21]

define stratification

[s22]

where

[s23]

for each j.

Define two random vectors 1Rj = 1R|1R ∈ Ωj. That is, 1Rj equals 1R conditional on 1R being in Ωj. Define samples and . Define sample {1P1P, … , 1P} with 1P[k] = θ( ) for 1 ≤ k ≤ 50 and 1P[k] = θ( ) for 51 ≤ k ≤ 1000. VaR is estimated as 89700 less the sample .05-quantile of the 1P[k]. Generating sample realizations for this estimator is difficult with a spreadsheet. The estimator is easy to code as a simple program.

10. Each of the three estimators for 95% VaR is applied ten times to obtain a total of 30 estimates for 0std(1P). Results are indicated below.
crudecontrol
variate
stratified
sampling
1653916781168682
2711737074968569
3662656753268698
4715636900567909
5665276894968085
6707946945568073
7655926886968248
8699316919967900
9711136817169095
10705036975667927
mean688856895068319
stdev2328855374
11. 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.