# 10.2 Linear Transformation Procedures

Linear transformations were pioneered by Markowitz (1952) and Roy (1952). Consider a portfolio (^{0}*p*, ^{1}*P*) with linear portfolio mapping

[10.1]

By [3.28], given a conditional covariance matrix ^{1|0}**Σ** for ^{1}** R**, the conditional standard deviation of

^{1}

*P*is

[10.2]

This defines an transformation that can be used to evaluate PMMRs such as ^{0}*std*(^{1}*P*), ^{0}*var*(^{1}*P*) or ^{0}*std*(^{1}*L*). On its own, a standard deviation is not sufficient to determine a quantile, so the transformation does not support value-at-risk metrics.

For the transformation to be applicable for value-at-risk, we make additional assumptions. Various approaches are possible, but standard assumptions specify a value for ^{0}*E*(^{1}*P*) and assume ^{1}*P* is conditionally normal. This fully specifies a conditional distribution for ^{1}*P*, making it possible to value any PMMR. By [3.95], value-at-risk measured as a *q*-quantile-of-loss is evaluated as1

[10.3]

[10.4]

[10.5]

where *Z* ~ *N*(0,1). Specific values for (*q*) corresponding to commonly used value-at-risk metrics are (see Exhibit 3.16):

- (.90) = 1.282 for 90% VaR;
- (.95) = 1.645 for 95% VaR;
- (.975) = 1.960 for 97.5% VaR;
- (.99) = 2.326 for 99% VaR.

If our value-at-risk horizon is short—say a day or a week—it may be reasonable to assume ^{0}*E*(^{1}*P*) = ^{0}*p*. In this case, [10.5] simplifies to

[10.6]

This solution is widely used. Because [10.6] does not depend upon the value of ^{0}*p*, there is no need to calculate ^{0}*p*.

An assumption that ^{1}*P* is conditionally normal may be justified in various ways. If we assume ^{1}** R** is joint-normal, by linearity condition [10.1],

^{1}

*P*will be normal. If we can’t assume

^{1}

**is joint-normal, we may still assume**

*R*^{1}

*P*is normal based upon the central limit theorem. As long as

^{1}

*P*has exposures diversified across multiple key factors that have modest correlations, conditions for invoking a central limit theorem should—at least approximately—be met.

###### Exercises

Measure value-at-risk as 1-day 95% USD VaR. Consider a portfolio invested in four stocks. Holdings are 100, 250, –200, and 500 shares. Current market values for the stocks are USD 30, USD 45, USD 60, and USD 20 per share. Let ^{1}** S** represent tomorrow’s accumulated value (price plus any dividends since time 0) for the respective stocks. Assume

^{1}

**is conditionally joint-normal with covariance matrix**

*S*[10.7]

Assume tomorrow’s expected accumulated value for each stock is its current value. Calculate the portfolio’s 1-day 95% USD VaR by performing the following steps:

- Specify
^{1}*P*as a mapping of^{1}. Is the mapping function a linear polynomial?*S* - Calculate
^{0}*p*and^{0}*E*(^{1}*P*). - Based upon your result from part (a) and the covariance matrix for
^{1}, calculate the conditional standard deviation*S*^{0}*std*(^{1}*P*). - Based upon your results for parts (b) and (c), calculate the portfolio’s value-at-risk.