###### 10.5.3 Stratified Sampling to Estimate Standard Deviation of Loss

Stratified sampling can also dramatically reduce the standard error of a Monte Carlo estimator for various PMMRs. A quadratic remapping guides us both in specifying a stratification and in selecting sample sizes *m _{i}* for each subregion of the stratification. Our approach depends upon the PMMR to be estimated. Below, we consider standard deviation of loss and then value-at-risk.>

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

^{1}

*R**N*(

_{n}^{1|0}

**μ**,

^{1|0}

**Σ**). Construct a quadratic remapping = (

^{1}

**), and set =**

*R*^{0}

*p*. To estimate

^{0}

*std*(

^{1}

*L*), we note

[10.50]

so it is sufficient to estimate ^{0}*var*(^{1}*P*). We stratify into *w* disjoint subintervals ϑ* _{j}* based upon the conditional PDF of as follows. Since is a quadratic polynomial of a joint-normal random vector, we may apply the methods of Section 10.3 to calculate its .01 and .99 quantiles, (.01) and (.99). Set

[10.51]

[10.52]

Define intervening subintervals ϑ* _{j}*, each of length

[10.53]

so

[10.54]

A stratification of size *w* = 8 is illustrated based upon a hypothetical conditional PDF for in Exhibit 10.14. In practice, stratification sizes of between 15 and 30 may be appropriate.

*w*= 8 constructed as described in the text.

Based upon stratification

[10.55]

define a stratification

[10.56]

where

[10.57]

Specifically, Ω* _{ j}* is the set of realizations

^{1}

**for**

*r*^{1}

**such that corresponding portfolio values**

*R*^{1}

*p*= (

^{1}

*) are in ϑ*

**r***. In mathematical parlance, each set Ω*

_{j}*is the*

_{ j}**preimage**under of the set ϑ

*.*

_{j}Define *w* random vectors ^{1}*R*_{ j} = ^{1}** R** |

^{1}

**∈ Ω**

*R**. That is,*

_{j}^{1}

*R*_{ j}equals

^{1}

**conditional on**

*R*^{1}

**being in Ω**

*R**. Define*

_{j}^{1}

*P*

_{ j}= θ(

^{1}

*R*_{ j}) for all

*j*. Then

^{1}

*P*is a mixture, in the sense of Section 3.11, of the

^{1}

*P*

_{ j}. Applying [3.128] and [3.129],

[10.58]

[10.59]

Given samples {, , … , } for the ^{1}*R*_{ j} of respective sizes *m _{j}*, we define an estimator for

^{0}

*std*(

^{1}

*P*):

[10.60]

where

[10.61]

Since is conditionally a quadratic polynomial of a joint-normal random vector, we can apply the methods of Section 10.3 to value its conditional CDF, so values *p _{j}* can be calculated. With exceptions of

*m*

_{1}and

*m*, all

_{w}*m*are set equal to each other. Specifically,

_{j}[10.62]

where the *m _{j}* sum to

*m*. The formula for

*m*

_{1}and

*m*is reasonable based upon empirical analyses.

_{w}Generate realizations simultaneously for all *j* by generating realizations ^{1}*r*^{[k]} for ^{1}** R**, and allocating each to one of the

^{1}

*R**according to which set ϑ*

_{j}_{j}the corresponding realization (

^{1}

*r*^{[k]}) falls in. Stop when you have sufficient realizations for each

^{1}

*R**. For some*

_{j}^{1}

*R**, you will have more than enough realizations, but extras can be discarded.*

_{j}