Severity, Frequency and Aggregate Models

Question 1

kth raw moment

Answer 1

uk’=E(x^k)

Question 2

kth central moment

Answer 2

uk=E((x-u)^k)

Question 3

Variance(X)

Answer 3

u2=sigma^2

Question 4

Covariance(X,Y)

Answer 4

COV(X,Y)=E(XY)-E(X)E(Y)

Question 5

Coefficient of Variation

Answer 5

CV=sigma/u

Question 6

Skewness

Answer 6

u^3/sigma^3

Question 7

Kurtosis

Answer 7

u^4/sigma^4

Question 8

Moment generating function

Answer 8

Mx(x)=E(e^tx)

Question 9

Derivative of the Moment generating function

Answer 9

Mx^n(0)=E(X^n) where Mx^n is the nth derivative

Question 10

Probability generating function

Answer 10

Px(z)=E(z^x)

Question 11

Derivative of the probability generating function

Answer 11

Px^n(1)=E(X(X-1)…(X-n+1))

Question 12

Conditional probability

Answer 12

Pr(A/B)=Pr(B/A)Pr(A)/Pr(B)

Question 13

Law of total probability

Answer 13

Pr(X=x)=E(Pr(X=x/y))

Question 14

Law of total Expectation

Answer 14

Ex(x)=E(E(X/Y))

Question 15

Parametric Distributions - Special Distribution Shortcuts X-d/X>d

Answer 15

Pareto (alpha,theta)= Pareto (alpha, theta+d)

Exponential (theta)= Exponential (theta) memoryless distribution

Uniform (a,b)=Uniform (0, b-d)

Question 16

Zero-Truncated Distribution

Answer 16

pn^t=(1/(1-p0))*pn

Question 17

Expected value of a truncated distribution

Answer 17

E((N^t)^k)=(1/(1-p0))*E(N^k)

Question 18

Zero-Modified Distributions

Answer 18

pn^m = (1-p0^m)/(1-p0)*pn

Question 19

Expected value of a zero-modified distribution

Answer 19

E((N^m)^k)=(1-po^m)/(1-p0)*E(N^k)

Question 20

(a,b,0) class

Answer 20

pn/pn-1=a+b/n for n=1,2…

Question 21

Bernoulli shortcut (Mixtures ans Splices)

Answer 21

x=(a with pr=q and b with pr=1-q) then Var(X)=(a-b)^2q(1-q)

Question 22

Poisson-Gamma Mixture

Answer 22

if x/lambda - Poisson(lambda) and lambda- Gamma(alpha, theta) then X follows a negative binomial (r=alpha, beta=theta)

Question 23

Policy Limits, u

Answer 23

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)

Question 24

Increased Limit Factor ILF

Answer 24

ILF=E(x^u)/E(x^b) where u=increased limit and b=original limit

Question 25

Ordinary deductible

Answer 25

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

Question 26

Loss Elimination Ratio

Answer 26

LER=E(x^d)/E(x)

Question 27

Franchise deductible

Answer 27

Y^l = 0 when xd

E(Y^l)= E((X-d)+) +d*S(d)

Question 28

Payment per payment

mean excess loss e(d) with ordinary deductible d

Answer 28

Y^p= E(Y^l)/S(d)

e(d)=E(X-d/X>d)= E((X-d)+)/S(d)

Question 29

Special Cases for e(d) shortcuts

Answer 29

Exponential = e(d)=theta

Uniform(a,b) = (b-d) / 2

Pareto (alpha, theta) = (theta+d) / (alpha-1)

Single Parameter Pareto= d / (alpha-1)

Question 30

Expected value and variance of an aggregate loss models - collective risk model

Answer 30

X and N must be independent

E(S)=E(N)*E(X)

Vas(S)=E(N)Var(X)+E(X)^2Var(N)

Question 31

Impact of a deductible on claim frequency

Answer 31

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 ****

Question 32

Negative Binomial and Exponential Compound Models

Answer 32

N follows a Binomial (r, beta/(1+beta))

X follows and exponential (Theta * (1+beta))

Question 33

Compound Poisson Models

Answer 33

a collective risk where the frequency follows a poisson distribution

Question 34

Value-at-Risk VaR

Answer 34

VaR=Fx^(-1)(p)

Question 35

Tail-Value-at-Risk TVaR

Answer 35

TVaRp(x)= E(X/ X> VaRp(X))

= VaRp(X) + e(VaRp(x))

Question 36

TVaR of a Normal Distribution

Answer 36

TVaRp(X)= u + sigma* (phi(Zp) / (1-p))

Question 37

TVaR of a LogNormal Distribution

Answer 37

TVaRp(X)= E(X) * (phi(sigma - Zp) / (1-p))

Question 38

Coherence

p(x) is coherent is it satisfies the properties

Answer 38

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>

Question 39

Tail Weight

Answer 39

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

Question 40

h(x)

Answer 40

h(x)=f(x)/S(x)

Question 41

Ultimate Formula for Insurance

Answer 41

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

Question 42

Variance of S

Answer 42

Var(S)=E((X-d)+)E(N)

The variance of S is the expected number of payments times the second moment