1.

Show that (5.7) is valid.

2.

What are the results in the table in case of a dividend i = 2% and ? = 5%? Calculate the variance premium as well as the exponential premium.

1.

Take the derivative of (5.6) and set zero.

2.

Portfolio premium = 49.17; optimal u = 104.21; optimal R = 0.0287; premiums for A and B are 5.72 and 1.0287 (variance premium) and 5.90 and 1.0299 (exponential premium).

1.

Let X ~ exponential(1). Determine the premiums (a) (e) and (h) (j).

2.

[ ?] Prove that ?[ X; ?] = log(E[ e ^?X])/ ? is an increasing function of ?, by showing that the derivative with respect to ? is positive (see also Example 1.3.1).

3.

Assume that the total claims for a car portfolio has a compound Poisson distribution with gamma distributed claims per accident. Determine the expected value premium if the loading factor equals 10%.

4.

Determine the exponential premium for a compound Poisson risk with gamma distributed individual claims.

5.

Calculate the variance premium for the claims distribution as in Exercise 5.3.3.

6.

Show that the Esscher premium equals , where ?x is the cgf of X.

7.

What is the Esscher transformed density with parameter h for the following densities: exponential( ?), binomial( n, p) and Poisson( ?)?

8.

Show that the Esscher premium for