[SOLVED] MTH 223 Mathematical Risk Theory Tutorial 1

MTH 223: Mathematical Risk Theory

Tutorial 1

Formula Sheet

• Gamma with parameters α > 0 and β > 0 : A random variable X is said to have a gamma distribution denoted by X ∼ G(α, β) if X has the following probability density function (p.d.f):

The gamma function Γ(α) is defined by

with Γ(α + 1) = αΓ(α) for α > 0 and Γ(n + 1) = n! for n = 0, 1, 2, …

1. In this course, we often encounter the computation of integrals such as dt and dt. The following formula can be used to expedite the computation:

Also note that the Gamma function is defined by

(a) Prove (1.1) by induction principle.

(b) Using (1.1) to show Γ(n + 1) = n ! for n = 0, 1, . . .. Therefore,

(c) Consider a loss r.v. X ∼ GAM(α, θ) with α = 3 and θ = 2 so that it has a pdf

Compute Var(X).

2. Assume a random loss X is subject to a normal distribution with mean µ, standard deviation σ, and pdf

Let φ(x) and Φ(x) respectively denote the pdf and the cdf of the stan-dard normal distribution. Show the following

(a) VaRp(X) = µ + σΦ
−1(p);

(b) TVaRp(X) =

3. Assume a random loss X has a Pareto distribution with scale parameter θ > 0, shape parameter α > 1, and cdf F(x) = , x > 0. Find VaRp(X) and TVaRp(X) for p ∈ (0, 1).

4. Suppose that the distribution of X is continuous on (x0,∞) where −∞ < x0 < ∞ (this does not rule out the possibility that Pr (X ≤ x0) > 0 with discrete mass points at or below x0). For x > x0, let f(x) be the pdf, h(x) be the hazard rate function, and e(x) be the mean excess loss function. Demonstrate that

and hence E[X | X > x] is nondecreasing in x for x > x0.

5. Assume that a stock index at the end of one year is X which has a Pareto distribution with cdf

The return of a one year guaranteed investment linked to the index is Y = max{X, 100}. Denote the distribution function of the return by FY (y).

(a) Calculate FY (y) for all y ∈ (−∞,∞).

(b) Calculate the mean of the return.

(c) Calculate the median of the return.

(d) Calculate the variance of the return.