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.
    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[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.
    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.

 

Welcome!

Enter your address for insights that will transform your risk management.

Click the link in the email I just sent you to confirm your address and start your subscription.

Defining Risk

Enter your email address to receive your copy and subscribe to Glyn's Risk Management Newsletter.

Click the link in the email I just sent you to download your paper and start your subscription.