Clark

Question 1

Loglogistic G(x) where G(x) = 1/LDF_x

Answer 1

G(x|w,ø) = x^w/(x^w+ø^w)

Question 2

Loglogistic LDF_x

Answer 2

1+ø^w * x^-w

Question 3

Weibull G(x)

Answer 3

G(x|w,ø) = 1 - exp(-(x/ø)^w)

Question 4

Advantages to using parameterized curves to describe the emergence pattern

Answer 4

• Only have to estimate two parameters
• Can used data that is not from a triangle with evenly spaced evaluation data
• final pattern is smooth and does not follow random movements in the historical age-to-age factors

Question 5

u_AY:x,y (expected incremental loss dollars in accident year AY between ages x and y)

LDF Method

Answer 5

=ULT_AY * [G(y|w,ø) - G(x|w,ø)]

Question 6

u_AY;x,y Cape Cod Method

Answer 6

=Premium_AY * ELR * [G(y|w,ø) - G(x|w,ø)]

Question 7

Reasons Cape Cod Method is preferred

Answer 7

• data is summarized into a loss triangle with relatively few data points. LDF method requires an estimation of a number of parameters (one for each AY ultimate loss, as well as ø and w), it tends to be over-parameterized when few data points exist
• Cape Cod method has a smaller parameter variance. Process variance can be higher or lower than LDF method, but in general produces a lower total variance than LDF method.

Question 8

Variance/Mean (σ²)

Answer 8

1/(n-p) * _AY,tⁿΣ[(c_AY,t - u_AY,t)²/u_AY,t]

where n= # of data points

p= # of parameters

c_AY,t = actual incremental loss emergence

u_AY,t = expected incremental loss emergence

Question 9

over-dispersed Poisson mean and variance

Answer 9

E[c] = ^σ² = u

Var(c) = ^σ⁴ = uσ²

Question 10

Key advantages of over-dispersed Poisson distribution

Answer 10

1. Inclusion of scaling factors allows us to match the first and second moments of any distribution, allowing high flexibility
2. MLE estimation produces the LDF and Cape Cod estimates of ultimate losses, so the results can be presented in a familiar format

Question 11

log likelihood, l, of over-dispersed Poisson

Answer 11

=_iΣc_i * ln(u_i) - u_i

Question 12

MLE estimate for ULT_i

Answer 12

_tΣc_i,t/_tΣ[G(x_t)-G(x_t-1)]

estimate for each ULT_i is equivalent to LDF Ultimate

Question 13

MLE estimate for ELR

Answer 13

_i,tΣc_i,t/_i,tΣP_i*[G(x_t) - G(x_t-1)]

equvalent to Cape Cod Ultimate

Question 14

advantage of MLE function

Answer 14

works in the presence of negative or zero incremental losses

Question 15

Process Variance of R

Answer 15

σ² * Σu_AY;x,y

Question 16

Key Assumptions of Clark Model

Answer 16

1. Incremental losses are independent and identically distributed
2. The variance/mean scale parameter σ² is fixed and known
3. Variance estimates are based on an approximation to the Rao-Cramer lower bound