Severity, Frequency and Aggregate Models Flashcards
kth raw moment
uk’=E(x^k)
kth central moment
uk=E((x-u)^k)
Variance(X)
u2=sigma^2
Covariance(X,Y)
COV(X,Y)=E(XY)-E(X)E(Y)
Coefficient of Variation
CV=sigma/u
Skewness
u^3/sigma^3
Kurtosis
u^4/sigma^4
Moment generating function
Mx(x)=E(e^tx)
Derivative of the Moment generating function
Mx^n(0)=E(X^n) where Mx^n is the nth derivative
Probability generating function
Px(z)=E(z^x)
Derivative of the probability generating function
Px^n(1)=E(X(X-1)…(X-n+1))
Conditional probability
Pr(A/B)=Pr(B/A)Pr(A)/Pr(B)
Law of total probability
Pr(X=x)=E(Pr(X=x/y))
Law of total Expectation
Ex(x)=E(E(X/Y))
Parametric Distributions - Special Distribution Shortcuts X-d/X>d
Pareto (alpha,theta)= Pareto (alpha, theta+d)
Exponential (theta)= Exponential (theta) memoryless distribution
Uniform (a,b)=Uniform (0, b-d)
Zero-Truncated Distribution
pn^t=(1/(1-p0))*pn
Expected value of a truncated distribution
E((N^t)^k)=(1/(1-p0))*E(N^k)
Zero-Modified Distributions
pn^m = (1-p0^m)/(1-p0)*pn
Expected value of a zero-modified distribution
E((N^m)^k)=(1-po^m)/(1-p0)*E(N^k)
(a,b,0) class
pn/pn-1=a+b/n for n=1,2…
Bernoulli shortcut (Mixtures ans Splices)
x=(a with pr=q and b with pr=1-q) then Var(X)=(a-b)^2q(1-q)
Poisson-Gamma Mixture
if x/lambda - Poisson(lambda) and lambda- Gamma(alpha, theta) then X follows a negative binomial (r=alpha, beta=theta)
Policy Limits, u
E((Y^l)^k)=E((x^u)^k)=integral from 0 to u of kx^(k-1)S(x)dx or integral from 0 to u of x^kf(X)dx +u^k*S(u)
Increased Limit Factor ILF
ILF=E(x^u)/E(x^b) where u=increased limit and b=original limit
Ordinary deductible
Y^l=(X-d)+ = 0 when X= d
E(Y^l)=E((X-d)+)=E(X)-E(X^d)
E(Yl^k)= integral from d to infinity of (x-d)^kf(x)dx or k(x-d)^(k-1)S(x)dx
Loss Elimination Ratio
LER=E(x^d)/E(x)
Franchise deductible
Y^l = 0 when xd
E(Y^l)= E((X-d)+) +d*S(d)
Payment per payment
mean excess loss e(d) with ordinary deductible d
Y^p= E(Y^l)/S(d)
e(d)=E(X-d/X>d)= E((X-d)+)/S(d)
Special Cases for e(d) shortcuts
Exponential = e(d)=theta
Uniform(a,b) = (b-d) / 2
Pareto (alpha, theta) = (theta+d) / (alpha-1)
Single Parameter Pareto= d / (alpha-1)
Expected value and variance of an aggregate loss models - collective risk model
X and N must be independent
E(S)=E(N)*E(X)
Vas(S)=E(N)Var(X)+E(X)^2Var(N)
Impact of a deductible on claim frequency
original exposure n1
Poisson= lambda
Binomial =m,q
Neg. Binomal = r,beta
Exp. Mod.
Exposure n2
Poisson (n2/n1)lambda
Binomial (n2/n1)m,q
Neg. Binomial (n2/n1)r, Beta
Coverage Modification Pr(x>0)=v
Poisson lambdav
Binomial m,vq
Neg. Binomial r,v*beta
* coinsurance doesn’t affect the frequency ****
Negative Binomial and Exponential Compound Models
N follows a Binomial (r, beta/(1+beta))
X follows and exponential (Theta * (1+beta))
Compound Poisson Models
a collective risk where the frequency follows a poisson distribution
Value-at-Risk VaR
VaR=Fx^(-1)(p)
Tail-Value-at-Risk TVaR
TVaRp(x)= E(X/ X> VaRp(X))
= VaRp(X) + e(VaRp(x))
TVaR of a Normal Distribution
TVaRp(X)= u + sigma* (phi(Zp) / (1-p))
TVaR of a LogNormal Distribution
TVaRp(X)= E(X) * (phi(sigma - Zp) / (1-p))
Coherence
p(x) is coherent is it satisfies the properties
translation : p(x+c)=p(x) + c
positive homogeneity : p(cx)= c * p(x)
Subadditivity : p(x+y) = p(x) + p(y)
monotonicity : p(x)<p></p>
Tail Weight
Fewer positive raw moments = heavier tail
if lim S1(x)/S2(x) = infinity or f1(x)/f2(x)= infinity then numerator has a heavier tail
h(x) decreases with x than heavy tail
e(d) increases with d than heavy tail
h(x)
h(x)=f(x)/S(x)
Ultimate Formula for Insurance
E(Y^l)= alpha(1+r)(E(x^(m/(1+r))- E(X^(d/(1+r))
d=deductible alpha=coinsurance u=policy limit r=inflation rate m=maximum covered loss, which equals u/alpha +d
Variance of S
Var(S)=E((X-d)+)E(N)
The variance of S is the expected number of payments times the second moment
