Value-at-Risk

Chapter 3, Page 113
Exercise 3

Consider a continuous random variable Z with PDF

[3.13]

a. Calculate the mean of Z.

b. Calculate the variance of Z.

c. Calculate the standard deviation of Z.

d. Calculate the skewness of Z.

e. Calculate the kurtosis of Z.

f. Calculate a .10 quantile of Z.

g. Calculate a .875 quantile of Z.

Solution

a. By [3.4]

	[s1]
	[s2]
	[s3]
	[s4]
	[s5]

b. By [s5] and [3.7],

	[s6]
	[s7]
	[s8]
	[s9]
	[s10]
	[s11]
	[s12]

c. The standard deviation of a random variable is simply the square root of its variance:

[s.13]

d. By [3.8]

[s.14]

We obtain from item (c) above.

	[s.15]
	[s.16]
	[s.17]
	[s.18]
	[s.19]
	[s.20]

Accordingly

[s.21]

e. By [3.9]

[s.22]

We obtain from item (c) above.

	[s.23]
	[s.24]
	[s.25]
	[s.26
	[s.27]
	[s.28]

Accordingly

[s.29]

f. To determine the .10-quantile of Z, we construct the CDF of Z. Formally, we do so by integrating its PF , which is given by [3.13]. However, since is so simple, is easily obtained by inspection:

[s.30]

A .10-quantile is any value z for which:

[s.31]

This equation has the single solution z = 1.2.

g. A .875-quantile is any value z for which:

[s.32]

This equation has the single solution z = 2.75.