10.5.4 Stratified Sampling to Calculate Value-at-Risk

10..5.4  Stratified Sampling to Calculate Value-at-Risk

Cárdenas et al. (1999) propose the following method of stratified sampling to calculate value-at-risk. Consider a portfolio (0p, 1P) with portfolio mapping 1P = θ(1R), where 1R conditional on information available at time 0, has distribution Nn(1|0μ,1|0Σ). We construct a quadratic remapping  = (1R) and set  = 0p. We wish to estimate the portfolio’s q-quantile of loss, which we denote ψ. The corresponding q-quantile of loss for the remapped portfolio is denoted si tilde. It is calculated using the methods of Section 10.3.

To estimate ψ, we stratify real numbersn into two regions, Ω1 and Ω2. Optimally, realizations 1r for which portfolio losses exceed the value-at-risk ψ should fall into one region, with the rest falling into the other region:

  • Ω1 = {1r : 0p – θ(1r) ≤ ψ};
  • Ω2 = {1r : 0p – θ(1r) > ψ}.

We will explain why this is optimal shortly. For now, we observe that the optimal stratification is impractical. Its definitions of Ω1 and Ω2 depend upon the portfolio’s value-at-risk ψ, which is what we are trying to estimate. As an alternative, we approximate the optimal stratification with one based upon the known value-at-risk si tilde of ( , ):

  • Ω1 = {1r :   – (1r) ≤ si tilde};
  • Ω2 = {1r :   – (1r) > si tilde}.

Let’s elaborate. To estimate a quantile of loss  (q), it is sufficient to estimate the corresponding quantile  (1 – q) of 1P, since


Directly specifying a stratified sampling estimator for values of  is difficult. We focus instead on devising a stratified sampling estimator for values of  . If we can estimate  (1p) for suitable values 1p based upon a single Monte Carlo analysis, we can estimate the quantile  (1 – q) based upon that same analysis.

Define the indicator function


For example, I(x > 3) equals 1 if x = 5 but it equals 0 if x = 2. A crude Monte Carlo estimator for  (1p) is


Since I( θ(1R) ≤ 1p) can only take on values 0 or 1, it makes sense to stratify with just two subregions, Ω0 and Ω1, of real numbersn such that Ω0 primarily contains values 1r for which the indicator function equals 0, and Ω1 primarily contains values 1r for which the indicator function equals 1.

With such a stratification, define 1R1 = 1R |1R ∈ Ω1 and 1R2 = 1R |1R ∈ Ω2. Our estimator becomes


We equally weight realizations by setting



for a suitable value m. The estimator becomes


This has standard error


If I( θ(1R1) ≤ 1p) always equals 1 and I( θ(1R2) ≤ 1p) always equals 0, the standard error is 0. It is this observation that motivated the optimal stratification described earlier. Because that stratification is impractical, we resort to the related stratification based upon the remapped portfolio. Formally, we stratify real numbers into two unbounded intervals:



This is illustrated based upon a hypothetical conditional PDF for   in Exhibit 10.15.

Exhibit 10.15: A stratification of real numbers into two intervals.

Based upon stratification


define a stratification




To estimate the quantile  (1 – q), generate realizations  and . Apply θ to all points  to obtain m = m1 + m2 realizations 1p[k] of 1P. Estimate  (1 – q) as that value 1p[k] such that (1 – q)m of the values are less than or equal to it.