MATH377: Financial and Actuarial Modelling in R Tutorial 10
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MATH377: Financial and Actuarial Modelling in R
Tutorial 10
Exercise 1. Use the upper and unbiased methods with a step of size 0.1 to find discretizations of a Lognormal distribution with parameters µ = − 1.5 and σ = 2 over the interval (0, 8). Plot the cdf of the discretizations against the original cdf.
Exercise 2. Consider S = X1 + · · · + XN , where N is a random variable taking values 0, 1, 2, 3 with probabilities 0.3, 0.3, 0.1 and 0.2, respectively, and X is a random variable taking values 1, 2 and 4 with probabilities 0.2, 0.5 and 0.3, respectively.
b(a) Find the pf of the random sum S .
Exercise 3. Let N be a Poisson distributed random variable with mean 5 and X a discrete random variable with pf fX (1) = fX (2) = fX (4) = 1/4 and fX (5) = fX (7) = fX (9) = 1/12. For the aggregate loss S = X1 + · · · + XN , use Panjer’s recursion to find the probabilities P(S = 8) and P(S = 25).
Exercise 4. Let N be a discrete random variable with pf p0 = 0.5, p1 = 0.4, and p3 = 0.1, and X be a discrete random variable with pf fX (1) = 0.9 and fX (10) = 0.1. Find P(S/E(S) > 3), where
S = X1 + · · · + XN .
Exercise 5. Consider S = X1 + · · · + XN such that N is Poisson distributed with mean 10 and X ∼ Gamma(2, 1). Approximate the cdf of S by:
a) Panjer’s recursion with a discretized Gamma distribution on (0, 22) with the unbiased method and a step of 0.5.
b) Panjer’s recursion with a discretized Gamma distribution on (0, 22) with the upper method and a step of 0.5.
c) Panjer’s recursion with a discretized Gamma distribution on (0, 22) with the lower method and a step of 0.5.
e) Normal approximation.
Plot the cdf’s above in a single graphic.
Exercise 6. Consider a classical risk model with premium rate c = 2.99, individual loss amount distribution given by P(X = 1) = 0.2, P(X = 2) = 0.3 and P(X = 3) = 0.5, and Poisson rate λ = 1. Determine the adjustment coefficient.
Exercise 7.
a) Calculate the adjustment coefficient for a classical risk model with c = 3, λ = 4, and individual loss size with density
f(x) = e −2x + 3e −3x , x > 0 .
b) Find a bound for the ruin probability if u = 2.
2022-05-11