Ruin Theory

[Jia Syuen]

Good days to those who're studying for CM2.

Want to ask your opinions for this chapter. Do we need to memorise all the formulae in this chapter or can you suggest which one should we memorise?

 

[Anna Bishop]

Hi Jia

You are right that this chapter is fairly formulae heavy :-(. Hopefully by the time you've practised past questions though, many of the formulae will start to embed themselves. This is what I recommend my tutorial students learn:


Definition of a Poisson process: N(t)

Definition of a compound Poisson process: S(t) = X1 + ... + XN(t) where N(t) is Poi(λt) and Xi is the amount of the ith claim. {Xi} are i.i.d.

Definition of the surplus process: U(t) = U + ct - S(t)

Definition of premium: c = (1+θ)λE[X]

Definition of the four ruin probabilities: Ψ(U), Ψ(U,t), Ψh(U), Ψh(U,t)

Know how the factors: U, t, θ, λ, μ, σ^2 affect the ultimate and finite probabilities of ruin

Be able to solve λ +cr = λMX(r) for the adjustment coefficient r

Lundberg’s inequality: Ψ(U) ≤ exp(rU)

Definition of insurer's expected profit = θλE[X]

Be able to solve λ +cnetr = λMY(r) for the adjustment coefficient in the presence of reinsurance

Definition of cnet = (1+θ)E(S) - (1+ξ)E[SR]

Definition of insurer's expected profit in presence of reinsurance = cnet - E[SI]

Derivation of formula for upper bounds on adjustment coefficient: R < 2θm1/m2 where m1 = E[X] and m2 = E[X^2]

Derivation of formula for lower bounds on adjustment coefficient: R > (1/M) ln (c/λm1) where claim amounts are continuously distributed on (0, M)

Distribution of time until the first claim for a Poisson process: T1 is Exp(λ)

Calculate probability of ruin on first claim, P(U(T1) < 0) where U(T1) = U + cT1 - X1


Hope this is a useful list

Anna

 

[Jia Syuen]

Hi Jia

You are right that this chapter is fairly formulae heavy :-(. Hopefully by the time you've practised past questions though, many of the formulae will start to embed themselves. This is what I recommend my tutorial students learn:


Definition of a Poisson process N(t)

Definition of a compound Poisson process: S(t) = X1 + ... + XN(t) where N(t) is Poi(λt) and Xi is the amount of the ith claim. {Xi} are i.i.d.

Definition of the surplus process: U(t) = U + ct - S(t)

Definition of premium: c = (1+θ)λE[X]

Definition of the four ruin probabilities: Ψ(U), Ψ(U,t), Ψh(U), Ψh(U,t)

Know how the factors: U, t, θ, λ, μ, σ^2 affect the ultimate and finite probabilities of ruin

Be able to solve λ +cr = λMX(r) for the adjustment coefficient r

Lundberg’s inequality: Ψ(U) ≤ exp(rU)

Definition of insurer's expected profit = θλE[X]

Be able to solve λ +cnetr = λMY(r) for the adjustment coefficient in the presence of reinsurance

Definition of cnet = (1+θ) λ E - (1+ξ) λ E[SR]

Definition of insurer's expected profit in presence of reinsurance = cnet - E[SI]

Derivation of formulae for upper bounds on adjustment coefficient: R < 2θm1/m2 where m1 = E[X] and m2 = E[X^2]

Derivation of formulae for lower bounds on adjustment coefficient: R > (1/M) ln (c/λm1) where claim amounts are continuously distributed on (0, M)

Distribution of time until the first claim for a Poisson process: T1 is Exp(λ)

Calculate probability of ruin on first claim, P(U(T1) < 0) where U(T1) = U + cT1 - X1


Hope this is a useful list

Anna

Thanks for your reply! It helps a lot.

 

[Anna Bishop]

You're welcome - sorry, couldn't get the formatting right for ages - kept putting a line through my maths! Think I've sorted it now though