Calculate limited expected value.

Reinsurance treaties, policy limits, and deductibles all carve a loss distribution into capped pieces. The limited expected value is the single calculation that prices each piece.

Definition. The limited loss variable is $X\wedge u=\min (X,u)$. It pays the actual loss when $X\le u$ and pays the cap $u$ when $X>u$. The limited expected value is its mean.

KEY: $E[X\wedge u]$ is the expected ground-up loss that lies below the limit $u$. Anything above $u$ collapses to $u$ before averaging.

Two integral forms. For a non-negative continuous loss with density $f$, cdf $F$, and survival $S=1-F$, the direct form sums the contributions below the cap and adds the cap mass.

$$E[X\wedge u]={\int }_0^{u}xf(x)dx+uS(u)$$

The survival form drops the density entirely and is usually faster for parametric work.

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

Discrete losses. Replace integrals with sums.

$$E[X\wedge u]=\sum _{x\le u}xp(x)+uP(X>u)$$

Common mistakes

• Forgetting the cap mass term. Writing ${\int }_0^{u}xf(x)dx$ and stopping omits $uS(u)$. The cap contribution is often the larger of the two pieces when $u$ sits in the body of the distribution.
• Subtracting in the wrong order. Layer cost is $E[X\wedge u]-E[X\wedge d]$, not the reverse. A negative answer is the diagnostic.
• Treating the policy maximum as the upper LEV argument when there is a deductible. With ordinary deductible $d$ and maximum payment $L$, the insurer pays $\min (X,d+L)-\min (X,d)$, so the upper LEV is $d+L$, not $L$.

Bottom line

• Limited expected value: $E[X\wedge u]=E[\min (X,u)]$, the expected loss after capping at limit $u$.
• Survival form (easiest for parametric): $E[X\wedge u]={\int }_0^{u}S(x)dx$ for non-negative $X$.
• Layer formula: expected payment in layer $(d,u]$ equals $E[X\wedge u]-E[X\wedge d]$; stop-loss $E[(X-d{)}_{+}]=E[X]-E[X\wedge d]$.
• Loss elimination ratio: $\text{LER}(d)=E[X\wedge d]/E[X]$, growing from 0 to 1 as $d$ increases.

Exam shortcut

When a problem gives you ground-up severity, a deductible, and a maximum covered loss, write the answer as $E[X\wedge (d+L)]-E[X\wedge d]$ before doing any algebra. For exponential losses, memorize that $E[X\wedge \theta ]=\theta (1-e^{-1})\approx 0.632\theta$; spotting this value short-circuits estimation questions.

The full lesson (about 1,871 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A3

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