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Average Basic Limit Severity (ABLS) vs ILFs
- ABLS = E[g(x;L)]
- I'(k) = [1 - F(k)] / ABLS
- F(x) → 1 as x → ∞
- I'(k) → 0 as k → ∞
- I(k) must approach a constant as k → ∞
- past some large limit, there is no additional charge
- I'(k) must decrease monotonically
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Using ILFs to price XS layers
PP = E(N)(E[g(x;s)] - E[g(x;r)]) = E(N)ABLS(I(s) - I(r))
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ILF separate trending
- could trend separately avg severity of ABLS and of ILF
- however results suggest lower inflationary factor for large losses → counterintuitive
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Including process risk in P
- Miccolis prefers variance
- Premium = E(Y) + λVar(Y)
- λ chosen judgmentally
- Risk adj P = E[g(x;k)] + λE[g(x;k)2]
- E[h(x;r,j)2] = E[g(x;s)2] - E[g(x;r)2] - 2jE[h(x;r,j)]
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Difficulties in creating a severity distribution from empirical data
- have to consider development on any set of clms
- data often comes from policies w different limits → bias
- credibility of the dist at high end might be low
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Examples of parameter risk
- catastrophes such as hurricanes, tornadoes, eq, ...
- change in mix of business
- small insurers face sampling errors
- incorrect ratemaking data
- claims practices
- uw practices
- social attitudes
- judicial or legislative climate
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2 dimensional ILF consistency test
- Test 1: marginal premium per $1,000 should decr as limit incr
- Test 2: for any 2 occurence limits, diff in ILF must not decr as aggregate limit incr
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Anti-selection
- adverse: high limit = worse experience because of (1) insd who expect high loss buy high limits and (2) lawsuits influenced by limit
- favorable: high limit = better experience because of (1) financially secure buy high limit because they have more assets and (2) insr is willing to insure good risks at higher limtis
- in some case affects ILF
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ILFs vs coinsurance
- if insure x%, P > x% PP
- E[g(x;x%k)] > x%E[g(x;k)]
- P = full value exposure units x coins % x coins factor x rate
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Rosenberg's consistency test
- A1<2 = aggregate limit, O1<2 = occurence limit
- A1 to A2 adds at least as much exposure at O2 as for O1
- O1 to O2 adds a least as much exposure at A2 as for A1
- for any 2 aggregate limits, diff in factors must not decrease as occurence limit grows
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