10.2 Linear Transformation Procedures
Linear transformations were pioneered by Markowitz (1952) and Roy (1952). Consider a portfolio (0p, 1P) with linear portfolio mapping
[10.1]
By [3.28], given a conditional covariance matrix 1|0Σ for 1R, the conditional standard deviation of 1P is
[10.2]
This defines an transformation that can be used to evaluate PMMRs such as 0std(1P), 0var(1P) or 0std(1L). 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 0E(1P) and assume 1P is conditionally normal. This fully specifies a conditional distribution for 1P, 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%value-at-risk;
- (.95) = 1.645 for 95%value-at-risk;
- (.975) = 1.960 for 97.5%value-at-risk;
- (.99) = 2.326 for 99%value-at-risk.
If our value-at-risk horizon is short—say a day or a week—it may be reasonable to assume 0E(1P) = 0p. In this case, [10.5] simplifies to
[10.6]
This solution is widely used. Because [10.6] does not depend upon the value of 0p, there is no need to calculate 0p.
An assumption that 1P is conditionally normal may be justified in various ways. If we assume 1R is joint-normal, by linearity condition [10.1], 1P will be normal. If we can’t assume 1R is joint-normal, we may still assume 1P is normal based upon the central limit theorem. As long as 1P 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% USDvalue-at-risk. 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 1S represent tomorrow’s accumulated value (price plus any dividends since time 0) for the respective stocks. Assume 1S is conditionally joint-normal with covariance matrix
[10.7]
Assume tomorrow’s expected accumulated value for each stock is its current value. Calculate the portfolio’s 1-day 95% USDvalue-at-risk by performing the following steps:
- Specify 1P as a mapping of 1S. Is the mapping function a linear polynomial?
- Calculate 0p and 0E(1P).
- Based upon your result from part (a) and the covariance matrix for 1S, calculate the conditional standard deviation 0std(1P).
- Based upon your results for parts (b) and (c), calculate the portfolio’s value-at-risk.