Exercise Solution 10.6

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 real numbers 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.
    crude control
    variate
    stratified
    sampling
    1 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
  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[1], 1R[2], … , 1R[1000]} for 1R, we define a sample {1P[1]1P[2], … , 1P[1000]} 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 {1P[1]1P[2], … , 1P[1000]}. This is implemented as a spreadsheet.
  8. Based upon a sample {1R[1]1R[2], … , 1R[1000]} for 1R, we define a sample {1P[1]1P[2], … , 1P[1000]} for 1P with 1P[k] = θ(1R[k]) and another sample { [1],  [2], … ,  [1000]} 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 real numbers 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 {1P[1]1P[2], … , 1P[1000]} 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.
    crude control
    variate
    stratified
    sampling
    1 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
  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.

 
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