Bahnemann

Question 1

Methods for Estimating Distribution Parameters

Answer 1

method of moments

maximum likelihood

minimum chi squared

minimum distance

Question 2

truncation

Answer 2

discarding; usually in case of claims below a deductible

Question 3

censoring

Answer 3

capping; usually in case of limit

Question 4

shifting

Answer 4

usually with straight deductible; for claims larger than deductible, they get reduced by the deductible amount

Question 5

since limits reduce volatility of severity compared to unlimited data

Answer 5

• may be interesting in computing the variability of losses in layer
• can use the coefficient of variation as a way to measure the variability for different distributions

Question 6

claim contagion parameter

Answer 6

accounts for claim counts not being independent of each other (where 1 claim encourages others to file a claim too)

-if claim counts have a Poisson, γ=0

Question 7

final rate for a policy needs to incorporate

Answer 7

all other expenses and profit as well as charge for risk

Question 8

risk charge

Answer 8

premium amount used to cover contingencies such as:

1. random deviations of losses from expected values (process risk)
2. uncertainty in selection of parameters describing the loss process (parameter risk)

Question 9

for purpose of pricing, instead of publishing full rates for every limit

Answer 9

insurers usually use relativities called ILFs to rate for a basic limit

Question 10

ILFs can be determined using

Answer 10

empirical data directly or can be obtained using a theoretical curve fit to empirical data with latter approach being more common for highest limits with little empirical loss data

Question 11

in determining ILFs appropriate for each limit, following assumptions are commonly made:

Answer 11

1. all UW expenses and profit are variable and don’t vary by limit
   - in practice, profit loads might be higher for higher limits since they are more volatile
2. frequency and severity are independent
3. frequency is same for all limits

Question 12

ILFs must be (with Bahnemann’s assumption of fx(l) is not equal to 0 )

Answer 12

increasing at a decreasing rate

Question 13

in terms of premium, premium for successive layers of coverage of constant width

Answer 13

will be decreasing

Question 14

checking that a set of ILFs satisfies above criteria for I’(l) and I’’(l)

Answer 14

performing a consistency test:

Per occurrence limit l

Increased limit factor I(l)

Marginal rate per $1k coverage I’()

Question 15

an exception to consistency test

Answer 15

could occur if one of the ILF assumptions was violate like if liability lawsuits were influenced by size of limit, then frequency would not be same for all limits so formulas would not hold

Question 16

one reason that a set of increased limits factors may fail this consistency test yet still generate actuarially reasonable prices.

Answer 16

Adverse selection, which could happen if insureds that expect higher loss potential are more inclined to buy higher limits.

Adverse selection, which could happen if liability lawsuits are influenced by the size of the limit.

Question 17

how the consistency test has both a mathematical interpretation and a practical meaning

Answer 17

The practical interpretation is that as the limit increases, there are less losses expected at higher layers, so rates should not increase more for higher limits than for lower limits.

mathematical interpretation is that I’(l) ≥ 0 and I’‘(l) ≤ 0

Question 18

there is more volatility (process risk) for policies with

Answer 18

with higher limits or higher attachment points, insurers will also want to charge a risk load for these policies

-to do this, need to include a risk charge ρ(l)

Question 19

risk charge ρ(l) options

Answer 19

old Miccolis aka variance method

old ISO aka std dev method

Question 20

risk load increases as

Answer 20

policy limit increases and is used to take into account the higher process risk for policies with higher limits

Question 21

deductibles typically reduce

Answer 21

coverage limit so layer of coverage with deductible d and limit l is (d,l] and not (d,d+l]

Question 22

3 types of deductibles:

Answer 22

straight

franchise

diminishing

Question 23

straight deductible

Answer 23

loss is truncated and shifted by d such that net losses

Xd = X-d for d

Question 24

franchise deductible

Answer 24

loss is truncated but not shifted by d such that

Xd=X for d

Question 25

diminishing deductible

Answer 25

• aka disappearing deductible
• loss below amount d is fully eliminated, deductible declines linearly from d to another larger D and loss above D is paid in full

Xd,D = D/(D-d) * (X-d) for d <d></d>

<p> = X for D

</p>

<p>-formula for LER even assuming alae is not additive is long, so calculate the loss eliminated at each size of loss level as a % of total losses</p>

</d>

Question 26

when comparing the 3 deductibles

Answer 26

straight eliminates the most loss, then diminishing and then franchise

Question 27

when pricing a layer of coverage and applying a risk load, it is not appropriate to

Answer 27

simply subtract risk loaded ILFs or premiums for layer since that would result in an incorrect risk load

while without risk loading we have

Pa,l=Pl-Pa=Pb*[I(a+l)-I(a)]

-these relationships do not hold when risk loading is applied

Question 28

Assuming retention R describe how inflation will affect the expected losses for the excess cover relative to unlimited ground up losses.

Answer 28

Inflation would affect excess losses more than total losses. Some losses that were below the retention will now get into the excess layer. Also, for losses that were already in the excess layer, the increase in losses due to inflation will affect the excess layer entirely.

Question 29

reason why it is desirable for a set of increased limits factors to pass this consistency test.

Answer 29

So long as partial losses are possible, expected losses will not increase as much as the increase in limits, so the rate per $1k of coverage should decrease as the limit increases.

Question 30

diagram for expected losses

Answer 30

y=loss size

x = F_X(x) = cumulative loss distribution

Question 31

why the loss cost for a given straight deductible policy can increase more than the ground-up severity trend.

Answer 31

For losses above the deductible, the trend is entirely in the excess layer.

Also, losses just under the deductible are pushed into the excess layer by the trend, creating new losses for the excess layer.

Question 32

How do the policy conditions alter the coefficient of variation of the claim-size variable

Answer 32

The policy restrictions restrict the variability of claims, such that the coefficient of variation will be lower with the policy restrictions than the coefficient of variation for the unlimited claim size variable

Question 33

Briefly describe one reason why increased limit factors might fail a consistency test and still produce reasonable rates

Answer 33

If adverse selection is occurring because higher risk insureds are more likely to purchase higher limits, then it would be reasonable to have factors that fail the consistency test (e.g., ILFs that increase at an increasing rate).