Pareto Distribution
F(x) = 1− [β/(x+β)]^α
E[x] = β / (α - 1)
Var[x] = αβ^2 / [(α-1)^2 (α-2)]
E[X;x] =
β/(α-1) × {1− [β/(x+β)]^(α-1)}
e_x(X) = (x+β) / (α - 1)
Exponential Distribution (CDF, Mean, Var, mean limited at X, excess mean above x)
F(x) = 1− e^(-x/β)
E[x] = β
Var[x] = β
E[X;x] = β * 1− e^(-x/β)
e_x(X) = β
Formula: expected value with limit and deductible
E[X_a, l] = { E[X;a+l] - E[X;a] } /
{ 1-F(a) }
Loss below top layer - loss eliminated at bottom layer, rebased by the probability of the claim surviving past a.
Excess severity function characteristics (e_a(X)):
Pareto
Lognormal
Weibull
Exponential
Gamma
Pareto = linear increasing, heavy tail
Lognormal = dips down then increases slowly at an increasing rate
Weibull = slowly increasing at a decreasing rate like ln()
Exponential = flat
Gamma = decreasing tail
Deductible types
Reduction vs Excess (default) vs Franchise Deductible (removes small claims, doesn’t reduce large ones)
“$500 deductible with a $50k claim size limit” - reduces
“Underlying limit of 1000 and a layer limit of 5000” - underlying means NOT reduced
“Straight deductible” - reduces
Define Coefficient of Variation
CV = SD/mean
Inflation adjustments
Formulas moved back in time 1 year.
Severity pricing, Deductible = a/(1+inflation)
Then adjust forward in time at the end.
Frequency / Severity / PP formula differences
Frequency uses F(x), probability of claim not exceeding a point, or landing in an interval.
Severity uses E[x], counting loss dollars instead of probabilities.
The expected limited loss E[X;x] and the excess loss function e_a(x) are for severities.
Variance of severity in a layer:
E[X^2_a; L]
Used for finding the X^2 expectation WITHIN a layer a to L. Helps calculate variance of layer SEVERITY.
= { E[X2; a+L] - E[X2; a]
- 2a(E[X; a+L] - E[X; a]) } / (1-F(a))
= expected value of X2 in layer - expected value of x in layer × 2a, divided by the usual denominator
Variance of claim counts (freq) in a layer:
Var[N_a]
Used for finding the variance WITHIN a layer above deductible a. Helps calculate variance of layer FREQUENCY.
p = 1-F(a)
Var[N_a] = p^2 Var[N]
+ p(1-p)E[N]
Variance of aggregate losses paid in a layer:
Var[Loss paid in layer a to a+l]
Var = λ (E[X^2;a + l] - E[X^2;a])
- 2aE[S] + γ(E[S]^2)
λ = freq mean
γ = contagion parameter
X = severity distribution
S = freq × sev in layer, total expected loss
Base and Excess Layer Pricing
Base Layer = EEXP × (loss × ALAE + FE) / (1-V-Q)
ILFs = expected loss relative to the base layer
Excess layer premium = Base Premium × (ILF(top)-ILF(bottom))
ILF consistency
ILFs must increase at a decreasing rate per unit of coverage.
(ILF change) / (top-bottom)
Risk Load (2 approaches)
ST DEV method
Var method
SEE FORMULA SHEET
If Poisson, then δ=0
