Calculate and interpret loss elimination ratios, increased limits factors, and deductible factors.

A homeowner with a $2,000 wind deductible files a $1,200 hail claim and collects nothing. The pricing actuary needs to answer how much premium that deductible should eliminate across the entire book. Three ratios (LER, ILF, deductible factor) answer it.

The limited expected value is the workhorse.

$$E[X\wedge d]={\int }_0^d{S}_X(x)dx$$

This is the expected payment per loss when coverage is capped at $d$. It is also the expected dollar amount eliminated per loss when $d$ is an ordinary deductible. Same integral, two readings. The duality is why all three ratios hinge on it.

Loss Elimination Ratio (LER). With an ordinary deductible $d$, the insurer pays $(X-d{)}_{+}=\max (X-d,0)$ per loss. Per-loss accounting:

• Expected gross loss: $E[X]$
• Expected payment after deductible: $E[X]-E[X\wedge d]$
• Expected loss eliminated: $E[X\wedge d]$

The fraction killed by the deductible is the LER:

$$\text{LER}(d)=\frac{E[X\wedge d]}{E[X]}$$

LER lives in [0, 1]. At $d=0$ it is zero (no deductible, no elimination). As $d\to \infty$ it approaches 1 (the deductible eats every loss).

Common mistakes

• Computing LER as $d/E[X]$. The correct numerator is $E[X\wedge d]$, which is always less than $d$.
• Multiplying ILF($U$) by DF($d$) to combine modifications. The two effects do not commute; compute $(E[X\wedge U]-E[X\wedge d])/E[X\wedge B]$ directly.
• Assuming frequency drops with the deductible. Under the standard model, frequency is unchanged; severity carries the discount.

Bottom line

• LER(d) = E[X ∧ d] / E[X]: fraction of pure premium eliminated by an ordinary deductible $d$; always in [0, 1].
• Deductible factor = 1 − LER(d): fraction of premium retained after applying deductible $d$.
• ILF(U) = E[X ∧ U] / E[X ∧ B]: multiplier on base-limit premium, with ILF(B) = 1 and monotone increasing in $U$.
• Combined deductible-and-limit pricing = (E[X ∧ U] − E[X ∧ d]) / E[X ∧ B]; ILF and DF do not commute, so never multiply them.

Exam shortcut

When the problem gives a limited-expected-value table, compute every ratio directly from the table; do not re-derive from the density. Memorize three closed forms: exponential gives $\theta (1-e^{-d/\theta })$; Pareto gives $\frac{\theta }{\alpha -1}\left[1-{\left(\frac{\theta }{\theta +d}\right)}^{\alpha -1}\right]$; uniform on $[0,b]$ gives $d-d^2/(2b)$ for $d\le b$. Excess-layer shortcut: when asked for "premium for the $[d,U]$ layer," compute $E[X\wedge U]-E[X\wedge d]$ and skip the ratios entirely.

The full lesson (about 1,980 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 3b

Browse all free Exam ASTAM lessons or jump into free Exam ASTAM practice questions.