B1. Distributions for Actuaries

Question 1

2 things covered by a risk charge

Answer 1

1. process risk: random deviations of losses from expected values ( greater if higher limits or higher attachment points)
2. parameter risk: uncertainty in the selection of the parameters describing the loss process

Question 2

ILF assumptions

Answer 2

-all underwriting expense and profit are variable but do not vary with limit (In practice, profit loads might be
higher for higher limits since they are more volatile)
-frequency is independent of severity
-frequency is the same for all limits (may be violated if there is adverse/ favorable selection) ***

Question 3

ILF methods

Answer 3

(1) empirical data
(2) theoretical curve

Empirical losses at higher limits may be volatile. Curve fitting can smooth out the volatility.

Gaps, intervals with no claims. Empirical losses may not reach maximum policy limits, so no factor can be calculated (free cover). Curve fitting can extrapolate losses to higher limits.

***Losses used in fitting the curve may develop further.
Immature losses from recent policy year. Curve fitting can take loss development into consideration

The credibility at the high end of the distribution is a concern. Curve fitting fits a curve that maximizes the likelihood of all reported losses. ***

Distribution bias, result from data on losses generated from different limits

cluster pts: maybe artificial

• appear at points other than policy limits
• case reserves are often rounded

Question 4

Practical interpretation of ILF consistency

Answer 4

If limits increase, rates should increase less for higher limits than for lower limits:

• expected loss do not increase as much as limit since most are partial losses
• less losses are expected to reach higher layers

Question 5

Why a set of ILF may fail consistency but still generate reasonable rates

Answer 5

Adverse selection:
-insureds that are more likely to have large losses buy higher limits
-**jury verdicts have higher awards when the insured has a higher limit **
( ILF INCREASE AT INCREASING RATE)

Favorable selection:

• insureds that are financially secure may protect themselves with higher limits
• brokers offering higher limits when they feel the insured is a good risk

Question 6

How to demonstrate a set of ILF is affected by anti-selection

Answer 6

by comparing the ILF calculated on a specific group vs the ILF calculate on the whole population

Question 7

Impact of inflation on loss layers

Answer 7

scenario 1 :
If losses are below retention:
-inflation pushes them in XS layer
- impact is = infinity
-XS inflation > inflation

If losses are above retention:
-inflation makes them bigger.
scenario 2 :
If capped by upper limit ( if there is one)
- XS inflation <= inflation 
scenario 3 : 
if not capped by upper limit 
- XS inflation vs inflation is unknown

if light tail scenario 1 is the most frequent

if heavy tail, scenario 2-3 are the most frequent
so impact of inflation on excesse losse is lower

Question 8

2 phenomena affecting XS ratios

Answer 8

1. different loss development for different sizes of loss

2. dispersion effect (variance of sev dist) affects the XS ratio

Question 9

Impact of simple dispersion

Answer 9

Simple dispersion: maintains the mean but increases the variance by a scaling (increases the CoV)

-increases XS ratio(for high limit) and alters XS ratio(for lower limit)

Question 10

Impact of gamma dispersion

Answer 10

Gamma dispersion: simple dispersion where the scaling is a gamma dist

• allows extreme values with small prob (leads to more variance, higher CoV)
• increases much more XS ratio( for high limit)

Question 11

Interpretation of gamma dispersion applied to pareto losses

Answer 11

• if alpha of pareto is smaller, heavier tail, higher dispersion effect
• if CoV of gamma is higher, higher variance, higher dispersion effect

Question 12

Claim contagion parameter

Answer 12

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

Question 13

Issue with fitting excess severity curves

Answer 13

Data usually thin and volatile at the higher amounts – difficult to see a pattern

Question 14

size method preferred :

Answer 14

when empirical data is not available and integrals need to be evaluated algebraically.

when calculating expected losses at one limt as it is more intuitive to explain

Question 15

layer method preferred:

Answer 15

when survival function is easy to integrate

when calculating expected losses at many limites

Question 16

ILF adjusted with risk load for pricing different limit vs risk load treated as variable expense in prm component

Answer 16

Accuracy:
Risk-adjusted ILF produces a more accurate premium. Varying the P/C load as a variable expense will require us to take in a different amount of fixed expenses.*
(P+𝑓)/(1−V). This is less accurate because fixed expenses should not vary by limit. Risk-adjusted ILF allows us to calculate p directly and keep fixed expenses the same.

Ease of calculation :
Determining a risk load may require more work because the k constant will need to be calibrated to the portfolio. Profit and contingency needs to be set judgmentally for each limit, which may also take time but is far less technically rigorous.

Clarity:
The varying P/C is likely more clear and transparent for those who don’t have a thorough
understanding of the portfolio. This is because it is explicitly defined in the premium calculation
rather than buried in the pure premium calculation (which itself is an input into the premium
calculation).