# 22.4 THE LINEAR MODEL

The examples we have just considered are simple illustrations of the use of the linear model for calculating VaR or ES. Suppose that we have a portfolio worth `P` consisting of `n` assets with an amount `α _{i}` being invested in asset $i\text{\hspace{0.17em}}\left(1\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}i\text{\hspace{0.17em}}\le n\right)$. Define Δ

`x`as the return on asset

_{i}`i`in one day. The dollar change in the value of the investment in asset

`i`in one day is

`α`

_{i}Δ

`x`

_{i}and

**(22.2)**

where Δ`P` is the dollar change in the value of the whole portfolio in one day.

In the example considered in the previous section, $10 million was invested in the first asset (Microsoft) and $5 million was invested in the second asset (AT&T), so that (in millions of dollars) ${\alpha}_{1}=10$, ${\alpha}_{2}=5$, and

If we assume that the Δ`x`_{i} in ...

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