###### 10.5.5 Selective Valuation of Realizations

The computationally most expensive task in estimating value-at-risk with the Monte Carlo method is performing *m* valuations ^{1}*p*^{[k]} = θ(^{1}*r*^{[k]}). As we have seen, variance reduction can dramatically reduce the number of valuations that must be performed. Cárdenas *et al*. (1999) propose a complementary technique.

For estimating a standard deviation of ^{1}*P*, the precise value of every realization ^{1}*p*^{[k]} is important. Every one contributes to sample standard deviation. For estimating a quantile of ^{1}*P*, the precise value of only one realization ^{1}*p*^{[k]} is important—the one equal to the quantile being estimated. Unfortunately, we only find out which one that is after we have valued all the ^{1}*p*^{[k]}! We can avoid having to value every ^{1}*p*^{[k]} by employing a quadratic remapping = (^{1}** R**) to identify realizations

^{1}

*r*^{[k]}for which

^{1}

*p*

^{[k]}clearly exceeds the quantile. Since those values

^{1}

*p*

^{[k]}are unimportant, we may approximate them with values = (

^{1}

*r*^{[k]}). We now formalize the technique.

Consider a portfolio (^{0}*p*, ^{1}*P*) with portfolio mapping ^{1}*P* = θ(^{1}** R**), where

^{1}

*R**N*(

_{n}^{1|0}

**μ**,

^{ 1|0}

**Σ**). We construct a quadratic remapping = (

^{1}

**). Set =**

*R*^{0}

*p*. 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 . It is calculated using the methods of Section 10.3.

We stratify into *w* disjoint subintervals ϑ* _{j}* based upon the conditional PDF of as follows:

[10.76]

[10.77]

[10.78]

︙

This is illustrated for a hypothetical conditional PDF for in Exhibit 10.16.

*w*subintervals as described in the text.

Based upon stratification

[10.79]

define a stratification

[10.80]

where

[10.81]

The 1 – *q* quantile of is in Ω_{1} by construction. Based upon the approximation ≈ ^{1}*P*, we expect the 1 – *q* quantile of ^{1}*P* to be in Ω_{1} as well; but we cannot be sure. Generate a realization {^{1}**r**^{[1]}, ^{1}**r**^{[2]}, … ,^{1}**r**^{[m]}} and calculate corresponding values = (^{1}**r**^{[k]}). Based upon these, sort the ^{1}*r*^{[k]} into regions Ω* _{ j}*. For only those in regions Ω

_{1}and Ω

_{2}, calculate

^{1}

*p*

^{[k]}= θ(

^{1}

*r*^{[k]}). For the rest, approximate

^{1}

*p*

^{[k]}with . Based upon the (exact or approximate) values

^{1}

*p*

^{[k]}, estimate the value-at-risk of (

^{0}

*p*,

^{1}

*P*).

The purpose of the region Ω_{2} is to play a buffer role to protect against the possibility that the approximation ≈ ^{1}*P* is poor. If the approximation is good, all realized losses ^{1}*l*^{ [k]} = ^{0}*p* – ^{1}*p*^{[k]} for that region should be less than the estimated value-at-risk. If this is the case, then you are done. If not, improve your value-at-risk estimate as follows.

For all realizations ^{1}*r*^{[k]} in region Ω_{3}, calculate exact portfolio values ^{1}*p*^{[k]} = θ(^{1}*r*^{[k]}). Based upon all values ^{1}*p*^{[k]} (which are now exact in regions Ω_{1}, Ω_{2}, and Ω_{3} but approximate in the other regions) estimate value-at-risk again. Letting Ω_{3} play a buffer role, apply the same test as before. If all realized losses for Ω_{3} are less than the new value-at-risk estimate, you are done. Otherwise, repeat the same procedure again, but with Ω_{4} playing the buffer role. Continue in this manner until an acceptable value-at-risk estimate is obtained.

Technically, this is not a variance reduction technique, but by dramatically reducing the number of portfolio valuations ^{1}*p*^{[k]} = θ(^{1}*r*^{[k]}) that must be performed, it has the same effect.