GL, Auto & Umbrella Pricing: Increased Limits and Excess Ratemaking

Pricing general liability, commercial auto liability, and umbrella/excess layers all share one core problem: the basic limit loss experience used for ratemaking must be extended to whatever limit the policyholder actually buys, and umbrella layers sit entirely above that basic limit where direct experience is thin or nonexistent.

The increased limits ratemaking framework

Basic limits ratemaking (see Ratemaking Fundamentals) produces an indicated rate at a low, stable basic limit $B$ (e.g., 100,000 per occurrence), because losses capped at $B$ are more stable, more available historically, and less distorted by a handful of large claims. To price any higher limit $L$, an increased limits factor (ILF) rescales the basic limit rate:

$ILF(L) = \frac{E[X \wedge L]}{E[X \wedge B]},$

where $E[X \wedge L] = \int_0^L (1-F(x))\,dx$ is the limited expected value at $L$ — the average loss capped at $L$. ILFs are built from a fitted severity curve (commonly a mixed exponential, lognormal, or Pareto) fit to unlimited/censored claim data, since the very large losses that drive high layers are rare and require size-of-loss curves rather than raw layer averages.

A full ILF table is monotonically increasing and concave in $L$: each additional increment of limit costs progressively less per dollar of coverage, because the probability of a claim reaching very high layers shrinks quickly. ALAE (allocated loss adjustment expense) is typically loaded either pro-rata with indemnity or added on top of the limit (as in "pay all ALAE in addition"), which materially changes the ILF's tail — pro-rata ALAE amplifies the relative cost of higher limits less than a flat load.

Layer (excess) pricing from ILFs

The pure premium for the layer between attachment $L_1$ and exhaustion $L_2$ is

$E[X \wedge L_2] - E[X \wedge L_1],$

and the corresponding excess rate relative to the basic-limit rate $R_B$ is $R_{excess} = R_B \times (ILF(L_2) - ILF(L_1))$. This is exactly how umbrella and excess liability layers are priced when there is no credible direct experience for the layer: apply increased limits factors (or an excess loss cost formula built from the same severity curve) to the underlying (primary) rate.

Exposure rating vs. experience rating for umbrella/excess

Two philosophies compete for pricing a given excess layer:

• Exposure rating prices the layer from first principles using industry severity curves and the insured's exposure base (payroll, sales, vehicle count), independent of that insured's own loss history in the layer. This is essential when the account has never had a loss reach the layer — which is expected and desirable for a well-attaching excess layer, since an absence of losses is not evidence the layer is cheap.
• Experience rating uses the account's own limited (primary-layer) losses, developed and trended, then projects them into the excess layer using ILFs — i.e., "burn" the account's own basic-limits experience through the same layering formula used for exposure rating.

In practice, umbrella/excess pricing blends the two with credibility: $\hat R = Z \cdot R_{exp} + (1-Z) \cdot R_{table}$, where $Z$ reflects the volume and layer-relevance of the account's own experience (see Credibility Theory). Low, working excess layers (attaching close to the primary limit) get meaningful experience weight; high, remote umbrella layers get little to none because a large account can go a decade without a claim reaching a 10,000,000 attachment purely by chance.

Territory and class relativities

Base rates for GL and auto liability are refined by territory relativities (reflecting jurisdictional tort environment, venue, weather/road density for auto) and class relativities (reflecting hazard by classification code — e.g., artisan contractors vs. offices for GL, or vehicle use/radius for commercial auto). These relativities are typically derived via a multiplicative rating plan fit with a GLM (see GLMs and Predictive Modeling), holding other rating variables fixed, and are periodically re-based as loss patterns shift (e.g., litigation funding driving up certain territories' severity — "social inflation").

Worked ILF example

Suppose a fitted severity curve gives limited expected values (in USD):

ILFs (relative to the 100,000 basic limit):

• $ILF(300K) = 68{,}000 / 42{,}000 = 1.619$
• $ILF(500K) = 81{,}000 / 42{,}000 = 1.929$
• $ILF(1M) = 98{,}000 / 42{,}000 = 2.333$

If the basic-limit rate is $R_B = 900$, the 500,000-limit rate is $900 \times 1.929 = 1{,}736$. The excess layer 500,000 xs 500,000 (i.e., $L_1=500K$, $L_2=1M$) pure premium is $98{,}000 - 81{,}000 = 17{,}000$, and its rate is $R_B \times (ILF(1M)-ILF(500K)) = 900 \times (2.333-1.929) = 900 \times 0.404 = 364$. Note this layer rate (364) is much smaller than the primary layer rate (1,736 for the first 500,000) even though the layer is the same width — a direct consequence of the concave severity curve.

Pitfalls

• Using paid-claim-only ILF data without adjusting for ALAE treatment consistency between the basic limit and the layer being priced.
• Applying ILFs fit on an outdated severity curve without trending for inflation and social inflation, which disproportionately understates high-layer costs.
• Zero experience-rating an umbrella layer with any credibility just because a couple of large claims happened to hit — one or two losses in a thin layer carry very little statistical credibility.
• Ignoring policy limit censoring when fitting the severity curve — claims settled at the policy limit understate true severity and must be treated as censored observations.

Exam relevance

Increased limits ratemaking and excess/umbrella pricing are core topics on CAS Exam 8, building on limited expected value and layering concepts introduced in Exam 4/STAM-level severity distribution theory.

Further reading

• Palmer, An Introduction to Increased Limits Ratemaking, CAS Study Note.
• Miccolis, On the Theory of Increased Limits and Excess of Loss Pricing, PCAS.
• Werner & Modlin, Basic Ratemaking, Chapter on Special Classification.