Short Term FAM

E[X ^ u]

Policy Limit Insurer Expected Lost E[Y^L]

E[X ^ u] / E[X ^ b]

ILF

E[(X-d)+] = E[X] - E[X ^ d]

Ordinary Deductible Insurer Expected Loss E[Y^L]

E[X ^ d] / E[X]

LER

E[(X-d)+] + d*S(d)

Franchise Deductible E[Y^L]

E[Y^L]/S(d)

E[Y^P]

u/alpha + d

Max loss covered formula (m)

alpha(1+r) [ E[X ^ m/(1+r)] - E[X ^ d/(1+r)] ]

Ultimate Formula E[Y^L]

(n+r-1 choose n) (beta/(1+beta)^n * (1/(1+beta))^r

Negative Binomial PDF

a + b/n

Pn/Pn-1

Poisson

a = 0, b= +

Negative Binomial

a = + , b= +

Binomial

a= -, b= +

Binomial

mu > sigma²

Negative Binomial

mu < sigma²

(1/(1-Po) * Pn

Zero Truncated P^Tn

(1-P^m)/(1-Po) * Pn

Zero Modified P^Mn

E[N]E[X]

E[S]

E[N]Var[X] + Var[N](E[X])²

Var[S]

mu = E[S], var=Var[S]

Normal Approx Aggregation

Pr(S<=5.5)

Pr(S<=5)

Pr(S>=4.5)

Pr(S>=5)

Pr(4.5<S<5.5)

Pr(S=5)

Pr(S<=4) = Pr(S<=4.5)

Pr(S<5)

Pr(S>=6) = Pr(S>=5.5)

Pr(S>5)

E[S] - E[S ^ d]

Aggregate Stop Loss E[(S-d)+]

Fx^(-1)p

VaRx(p)

E[X|X>VaRx(p)] = integral VaRx(p) to infinity (x * fx) / Pr(X>x)

TVaRx(p)

Translation invariance, positive homogeneity, Subadditivity, monotinicity

Coherence

subadditivity

What makes VaR(p) not coherent

Tail Weight

Fewer the amounf of positive raw moments, the greater the ______

MLE

1.) L(theta)
2.) ln L(theta) = l(theta)
3.) l’(theta)
4.) l’(theta) = 0

Pr(x) / Pr(X>d)

Left Truncation

Complete Data

Can match moments, the MLE of certain parameters can be found by matching fitted moments when there’s ______

[ Z(1-p)/2 / k]² * CVs²

n(e)

[ Z(1-p)/2 / k]² * (sigma²N/muN + CVx²)

nc

nc = ne * muN

nc in terms of ne

Zxbar + (1-Z)M = M + Z(xbar - M)

Partical Credibility

sqrt(n/ne) = sqrt(n’/nc)

Z

L^ult = ELR * P^e

Ultimate Loss in terms of Expected Loss Ratio

Earned Premium

P^e

L^ult - L^p

Reserve (R)

Li,k / Li,k-1

Chain Ladder age to age factor fi,k

i

accident year

k

development year

Li,k * fi^ult

Ultimate Loss in terms of Chain Ladder

L^ult(LR) * (1 - 1/f^ult(CL))

Bornheutter Ferguson Method Reserved Formula

w * Rcl + (1-w)Rlr

Bornheutter Ferguson Method Reserved Formula Other

1/f^ult

w

Loss + Expenses + Profit

Premium

V * P + Ef

Expenses

Qt * P

Profit

Li = Lp + Ri - Ri-1

Calendar Year Loss

Li = Lp + R

Policy Year Loss

1 - V - Qt

Permissible Loss Ratio

Variable Expense/Premium

V

((LR + F) / (1-V -Qt)) -1

Loss Ratio Indicated Average Change

(Lbar - Efbar) / (1-V-Qt)

Pure Premium or Loss Cost Method Indicated Avg Rate