Frequency and Severity Models

Core Severity Models

Compare Exponential, Gamma, and lognormal claim models through moments, quantiles, and layer costs.

Core Severity Models

Let $X\ge0$ be one ground-up claim amount at a common price level. A severity model must match the support, centre, variability, and tail relevant to the decision.

1. Exponential: a transparent benchmark

With scale $\theta>0$ ,

f(x)=\frac{1}{\theta}e^{-x/\theta},\qquad \bar F(x)=e^{-x/\theta},\qquad x\ge0.

E[X]=\theta,\qquad \operatorname{Var}(X)=\theta^2.

Its memoryless property,

P(X>s+t\mid X>s)=P(X>t),

is convenient but often unrealistic: exceeding a large threshold supplies no information about further excess. Use Exponential as a benchmark, not a default.

2. Gamma: flexible centre with exponential-type tail

With shape $k>0$ and scale $\theta>0$ ,

f(x)=\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}, \qquad E[X]=k\theta, \qquad \operatorname{Var}(X)=k\theta^2.

$k=1$ recovers the Exponential distribution.
$k>1$ has density zero at the origin and a positive mode.
coefficient of variation is $1/\sqrt{k}$ , so shape directly controls relative dispersion.

Gamma regression is useful when mean severity changes with claim or policy characteristics while positive support is preserved.

3. Lognormal: multiplicative variation and a heavier tail

If $\log X\sim N(\mu,\sigma^2)$ , then

E[X]=e^{\mu+\sigma^2/2},

\operatorname{Var}(X)= \left(e^{\sigma^2}-1\right)e^{2\mu+\sigma^2},

Q(p)=\exp\!\left(\mu+\sigma\Phi^{-1}(p)\right).

The logarithm turns multiplicative effects into additive ones. The lognormal tail is heavier than Gamma or Exponential but lighter than a power-law Pareto tail.

4. Same mean, different decision

Choose parameters so all three models have mean £5,000:

Model	Parameters	SD	99% quantile
Exponential	$\theta=5{,}000$	£5,000	£23,026
Gamma	$k=4,\ \theta=1{,}250$	£2,500	£12,556
Lognormal	CV $=1.5$ ; $\mu=7.9279,\ \sigma=1.0857$	£7,500	about £34,665

An expected-loss calculation sees no difference. A capital or reinsurance decision does.

Equal means, unequal tails

5. Contracts transform severity

Ordinary deductible

With deductible $d$ , insurer payment per loss is

Y=(X-d)_+.

Unconditional expected payment is

E[Y]=E[(X-d)_+]=\int_d^\infty\bar F_X(x)\,dx.

For Exponential $X$ with mean $\theta$ ,

E[(X-d)_+]=\theta e^{-d/\theta}.

At $\theta=£5{,}000$ and $d=£2{,}000$ , expected insurer payment per underlying loss is $5{,}000e^{-0.4}\approx£3{,}352$ .

Limit

With payment limit $u$ applied to the post-deductible amount,

Y=\min\{(X-d)_+,u\}.

Values recorded at $u$ may be right-censored: the true loss exceeded the observable payment. Treating them as exact claims understates the tail.

Conditional payment severity

The mean payment given that a payment occurs is

E[X-d\mid X>d] =\frac{E[(X-d)_+]}{P(X>d)}.

Do not confuse this with the unconditional payment per exposure or per ground-up loss.

6. Choose a model by consequence

Task	Most important checks
ordinary claim budgeting	mean, calibration by segment, out-of-sample error
deductible pricing	probability above deductible and payment mean
high-layer reinsurance	survival and mean excess near attachment
capital	aggregate tail, dependence, parameter/model uncertainty

Practice

For Exponential mean £4,000, find $P(X>£10{,}000)$ .
A Gamma model has shape 9 and mean £4,500. Find scale and coefficient of variation.
Why can a model fit a histogram well but price a high layer badly?

Answers

$e^{-10{,}000/4{,}000}=e^{-2.5}\approx0.0821$ .
Scale $=4{,}500/9=£500$ ; CV $=1/\sqrt9=1/3$ .
Most histogram mass is in the body; few tail observations determine high-layer cost. Tail diagnostics and uncertainty must be assessed separately.

Continue to Pareto, Weibull, and extreme-value models.

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Frequency and Severity Models

Select loss models from the insurance question, observation process, and tail—not from a distribution list.

Tail Models and Extreme Values

Model large losses with Weibull, Pareto, and threshold exceedances while making tail uncertainty visible.