1.7.5 Example: Australian Equities (Linear Remapping)
As an alternative solution, let’s approximate θ with a linear polynomiale 
based upon the gradient3 of θ. We must choose a point at which to take the gradient. A reasonable choice is 0E(
), which is the expected value of , conditional on information available at time 0. Let’s assume 0E() = 
. We define
[1.40]
Our approximation 
= () of the portfolio mapping 
is an example of a portfolio remapping. We obtain = (.20.080, 10.800, 1.150, 0.3892) from Exhibit 1.5 and evaluate
[1.41]
[1.42]
Our remapping [1.40] is
[1.43]
[1.44]
[1.45]
The portfolio remapping is represented schematically as
[1.46]
The upper part of the schematic is precisely schematic [1.34] indicating the original portfolio mapping . The lower part indicates the remapping 
. In such schematics, vertical arrows indicate approximations. approximates 
.
Because [1.45] is a linear polynomial, we can apply [1.11] to obtain the conditional standard deviation 1|
of :
[1.47]
Assume is conditionally normal with this conditional standard deviation 1| and conditional mean 1|
= 
. The .05-quantile of a normal distribution occurs 1.645 standard deviations below its mean, so
[1.48]
and the .95-quantile of portfolio loss is
[1.49]
The portfolio’s 1-day 95% GBPvalue-at-risk is approximately GBP 5,876. This result compares favorably with our previous result of GBP 5,925, which we obtained with the Monte Carlo method.