B1. Bahnemann Flashcards
E[Xa]
Formula, what is it, what is a?
E[Xa] = e(a) = (E[X] - E[X;a]) / (1 - Fx(a))
Expected value of X that exceeds a given that X > a
Expected excess severity
a = attachment point
E[Xa;l]
Formula, what is it, what is a, what is l?
E[Xa;l] = (E[X;a+l] - E[X;a]) / (1 - Fx(a))
Expected value of X in layer [a, a+l] given that X > a (X hits layer)
Expected excess severity in layer [a, a+l]
a = attachment point | l = limit
Fxa(x) / fxa(x)
Formula, what is it, what is a?
Fxa(x) = [Fx(x+a) - Fx(a)] / [1 - F(a)]
fxa(x) = fx(x+a) / [1 - Fx(a)] … take d/dx
P(Xa < x | x exceeds a)
Xa ~ amount that exceeds a given that x > a
a = attachment point
Coefficient of Variation (CV) / E=[Xa^2;l]
All formulas
CV(X) = sqrt(Var[Xa;l]) / E[Xa;l)
Var = E[X^2] - E[X]
E[Xa^2;l] = [ E[X^2;a+l] - E[X^2;a] - 2a (E[X;a+l) - E[X;a]) ] / [1 - Fx(a)]
E[Xa;l] = (E[X;a+l) - E[X;a]) / [1 - Fx(a)]
Excess Claim Counts
Assume binomial distribution, formula for E[X] * Var[X]
E[Na] = p * E[N]
Var[Na] = p^2 * Var[N] + p(1-p) * E[N]
Both = pλ if poisson
Aggregate Claims in Layer
Formula EV & Variance
E[S] = E[Na]E[Xa;l] = E[N] (E[X;a+l] - E[X;a]) = Expected Excess Count * Expected Excess Severity = Freq * Sev
* E[Na] = p * E[N] where p = 1 - Fx(a)
* E[Xa;l] = (E[X;a+l] - E[X;a]) / (1 - Fx(a))
Var[S] = E[N] (E[X^2;a+l] - E[X^2;a]) - 2aE[S] + yE[S]^2
* y = claim contagion, which accounts for claim counts not being independent
* If poisson, y = 0 (independent)
w
Increased Limit Factor
Formula including everything
ILF(l) = Rate(limit) / Rate(basic limit)
= (E[X;l] + ε)(1+u)+p(l) / (E[X;b] + ε)(1+u)+p(b)
* Expected limited severity
* ALAE
* ULAE
* Risk load
Excess Severity Distribution Graph
5 distributions
X axis = attachment point
Y axis = excess severity
Exponential - constant
Pareto - linear
Weibull - r shape
Lognormal - u shape going up
Gamma - L shape going down
Inflation Impact to Freq/Sev/Agg Loss
Formula
E[Xa;l] * T = E[TXa;l] = T(E[X;a+l/t] - E[X;a/t])/(1-F(a/t))
1-F(a) * Tn = 1-F(a/t)
Agg loss = Tn * T = T(E[X;a+l/t] - E[X;a/t])/(1-F(a/t)) / (E[X;a+l]-E[X;a])
Premium
Formula for basic and in layer
Basic prem = exposure * (freq * sev * 1+ALAE% + fixed expenses) / (1- variable expense)
Layer Prem = basic prem * [ILF(a+l) - ILF(a)] = Prem(a+l) - Prem(a)
Consistency
ILF(l) = (E[X,l] + ε) / (E[X;b] + ε)
ILF’(l) = (1 - F(l)) / (E[X;b] + ε) ≥ 0: ILF always increasing
ILF’‘(l) ≤ 0: Marginal rate increasing at decreasing rate
Risk Load
Old Miccolis (Variance) & Old ISO Approach & Pricing a Layer
All formulas
p(l) = kVar[S]/E[N] = k(E[X^2;l]+δ(E[X;l])^2)
* δ = Var[N]/E[N] - 1 … (=0 if poisson)
p(l) = kSD[S]/E[N] = k’ * sqrt(E[X^2;l]+δ(E[X;l])^2)
* k’ = k / sqrt(E[N])
p(a,l) = kVar[S]/E[N] = k(E[X^2;a+l] - E[X^2;a]) - 2a(E[X;a+l] - E[X;a])
Straight / Franchise Dedictible Pure Prem Premium and LER
Pure Prem = freq * (E[X;b] - E[X;d] + [1-F(d)] * ε) * (1+ULAE)
Pure Prem = freq * (E[X;b] - E[X;d] + [1-F(d)] * (d+ε)) * (1+ULAE)
Also insurer needs to pay deductible for X>d
LER = (E[X;d]+F(d)ε) / E[X;b] + ε
LER = (E[X;d]+F(d)ε-d[1-F(d)]) / E[X;b] + ε
No longer eliminating deductible for X>d
Premium and Pure Prem for Deductible + Limit in a Policy
Prem = Basic Prem * [ILF(l) - LER(d)]
Pure Prem = Basic Pure Prem * [ILF(l) - LER(d)]
Pure Premium and Inflation Impact with Policies with Deductible and Inflation
Pure Prem = Pure Prem = freq * (E[X;l/t] - E[X;d/t] + [1-F(d/t)] * ε) * (1+ULAE)
Inflation impact = pure prem (w/ inflation) / pure prem (without)
freq and ULAE will cancel out