10.4  Monte Carlo Transformation Procedures

If a portfolio mapping is neither linear nor quadratic, we may apply a function remapping to make it such. If doing so would entail too large an approximation error, we resort instead to transformations that employ numerical methods of integration. Because of the high dimensionality of many value-at-risk analyses, we focus on the Monte Carlo method. Numerical transformations based upon the Monte Carlo method were applied to PMMRs as early as Lietaer (1971).

Consider a portfolio (⁰p,¹P) with mapping ¹P = θ(¹R). Based upon a sample {¹R^[1], ¹R^[2], … , ¹R^[m]} for ¹R, we define a sample {¹P^[1], ¹P^[2], … , ¹P^[m]} for ¹P with ¹P^[k] = θ(¹R^[k]). Value-at-risk—or any reasonable PMMR—may be estimated by applying a suitable sample estimator to {¹P^[1], ¹P^[2], … , ¹P^[m]}. The result is a crude Monte Carlo estimator for the portfolio’s value-at-risk.

To apply the estimator, we need a realization {¹r^[1], ¹r^[2], … , ¹r^[m]}. The ¹r^[k] may be pseudorandom vectors constructed as described in Section 5.8. They may also be constructed directly from historical data for key factors, as described in Section 11.2. In the former case, the transformation is called a Monte Carlo transformation. In the latter case, it is called an historical transformation. The following discussion applies to both. We address issues specific to historical transformations in Section 10.6.

10.4.1 Monte Carlo Standard Error

In implementing Monte Carlo transformations, an important issue is how large a sample size m to use. The standard error of the Monte Carlo analysis depends upon several factors:

• sample size m;
• the value-at-risk metric—or more generally, the PMMR—being estimated;
• the conditional distribution of ¹P, which depends upon both the conditional distribution of ¹R and the portfolio’s composition.

As with all Monte Carlo analyses, standard error is inversely proportional to the square root of sample size m but is unrelated to the dimensionality n of the problem. Below, we explore the effects of chosen PMMR and portfolio composition on standard error.

A simple way to estimate the standard error of a Monte Carlo analysis is to perform the same analysis multiple times—using different pseudorandom vectors each time—and take the standard deviation of the results. For a given portfolio and PMMR, employ a Monte Carlo transformation procedure with sample size m = 1000 to approximate a value for the PMMR. Repeat the analysis 50,000 times, each time employing different pseudorandom vectors, to yield 50,000 independent approximations of the PMMR. Taking their sample standard deviation provides an estimate of the standard deviation of a single Monte Carlo analysis for that portfolio/PMMR pair based upon a sample size of m = 1000.

10.4.2 Empirical Analysis of Standard Error for Value-at-Risk

By performing the aboves analysis for various portfolio/PMMR pairs, we may obtain a sense of how standard error varies for different portfolios and PMMRs. Exhibit 10.5 indicates the results of such an analysis based upon five portfolios and three PMMRs—for a total of 15 portfolio/PMMR pairs.

Exhibit 10.5: Results of a Monte Carlo analysis of standard error for Monte Carlo transformation procedures. Monte Carlo transformation procedures employing a crude Monte Carlo estimator and sample size 1000 were applied to each of 15 portfolio/PMMR pairs a total of 50,000 times each. Standard errors were estimated for each portfolio/PMMR pair by taking the sample standard deviation of the 50,000 results for each pair. The estimated standard errors are presented as a percentage of the PMMR’s correct value for each portfolio/PMMR pair. Our first column lists the portfolios considered. The second column lists estimated standard errors for each assuming a standard-deviation of portfolio value metric. The third and fourth columns indicate estimated standard errors for 90%value-at-risk and 99%value-at-risk.

We have given the five portfolios descriptive names. Their actual compositions are unimportant. What matters is their conditional PDFs for ¹P.4 These are graphed in Exhibit 10.6. The first three portfolios are most typical of what is encountered in applications. Portfolio (a) might correspond to diversified or undiversified linear exposures. It might also correspond to highly diversified nonlinear exposures. Portfolios (b) and (c) might correspond to a handful of positive or negative gamma positions. They might also correspond to delta and gamma exposure to a single underlier. Portfolios (d) and (e) might correspond to delta-hedged gamma exposure to a single underlier. These two portfolios are somewhat stylized. We include them in our analysis to illustrate how extreme standard errors might arrise. For all portfolios, ⁰E(¹P) = ⁰p.

Exhibit 10.6: Conditional PDFs for ¹P are indicated for the five portfolios considered in the analysis of Exhibit 10.5.

Another way to look at our results is to determine, based upon [5.38], for each portfolio/PMMR pair the sample size that would achieve a 1% standard error. Results of this analysis are presented in Exhibit 10.7.

Exhibit 10.7: Sample sizes required for a 1% standard error are indicated for the 15 portfolio/PMMR pairs. Values are obtained by applying [5.38] to the results of Exhibit 10.5.

To estimate standard deviation of portfolio value, a sample size of 8000 should ensure a standard error less than 1% for most diversified portfolios. Value-at-risk metrics require larger samples. For 90%value-at-risk or 99%value-at-risk, consider sample sizes of 30,000 or 45,000, respectively.

Even if a portfolio mapping function θ is simple, performing such large numbers of valuations can be computationally expensive. If θ is more complicated, run times may become prohibitive. The worst situation occurs if a portfolio holds exotic derivatives, mortgage-backed securities, or other instruments that must be valued using numerical techniques such as binomial trees or the Monte Carlo method. If θ is a primary mapping, a single portfolio valuation ¹p^[k] = θ(¹r^[k]) might require minutes of processing time. The thousands of valuations required to estimate value-at-risk would take days.