# 1.7.5 Example: Australian Equities (Linear Remapping)

###### 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.