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Differences btwn Tables M & L
- Table L only contains California experience
- Table L losses limited to per accident limit
- Table L smoothes LR by replacing death cases w avg → small effect because <6% and already close to avg
- Table L has no overlap btwn charge for limiting total losses & individual losses
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Table L charge φ*(r) components
- diff btwn limited & unlimited losses
- limited losses in XS of r*EULR
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Table L formulae
- k = loss elimination ratio = [E - E(A*)] / E
- E(L) / E = 1 - φ*(r) + Ψ*(r)Ψ*(r) = φ*(r) + r - 1I* = [φ*(r) - Ψ*(r)]E
- E(L) = E - I*
- R = BP + cL (T incl in B & c)
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Table L balanced plan
- E(R) = P(1 - D), D = expense gradation
- B = 1 - D - (cE / P) + (cE / P) [φ*(rG) - Ψ*(rH)]
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Table L balance equations
- φ*(rH) - φ*(rG) = (P - PD - H) / cE
- rG - rH = (G - h) / cE, E = unlimited
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Incremental charge for per accident limitation
- should be between 0 and LER
- depends on overlap btwn per acc & overall limit
- size of overlap influenced by (1) per acc limit (2) ELR (3) P (4) r
- Δφ(r) = φ*(r) - φ(r)
- B = 1 - D - cE/P + cE/P[φ(rG) + Δφ(rG) - Ψ(rH) - Δφ(rH)]
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Table L construction
- replace death losses by avg
- cap at limit
- LER based on agg experience
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Properties of Table L
- as r → ∞, φ → k
- as P → ∞, φ → k if r ≥ 1 - k or 1 - r if r < 1- k
- as r → 0, φ → 1
- as P → 0, φ → 1
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Properties of φ(r)
- monotone decreasing function of the entry ratio
- monotone decreasing function of premium
- CA φ*(r) higher than countrywide due to higher loss variation
- diff btwn table L & M much smaller than LER would imply due to overlap
- Δφ(r) that does not vary by P results in overcharging small and undercharging large risks
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Table L limitations
- can't be used for alternate loss limits
- needs to be updated failry regularly for chg in Δφ(r), chg in aggregate loss dist due to fixed limit
- using limited table is much more flexible
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Use of Lemma 1
- E(L) = E(A) * [1 - φ*(r2) + Ψ*(r1)]
- use when plans don't balance
- use to determine P impact of updating EL size ranges
- aggregate impact = weighted by cpy size
- weight by prob of moving n grps
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