Value-at-Risk

Chapter 3, Page 143
Exercise 35

Consider random variable , where

[3.123]

Calculate the mean, standard deviation and .25-quantile of X.

Solution

a. By [3.114]

[s1]

b. By [3.115]

	[s2]
	[s3]
	[s4]

c. By [3.117], we seek probabilities q₁ and q₂ such that

0.3 q₁ + 0.7 q₂ = .25

[s5]

By [3.118], those probabilities must satisfy

[s6]

Combining [s5] and [s6], we seek probabilities q₁ and q₂ such that

[s7]

This is a two-dimensional non-linear equation. It can be solved using Newton's method. If you are using Microsoft Excel, you can simply use the solver function (see spreadsheet). The solution is

[s8]

Applying [3.119], we obtain the .25-quantile of X as –1.598.