Evaluate the effects of the following coverage modifications: deductibles, policy limits, maximum covered loss, coinsurance, and stop loss reinsurance.

Coverage modifications turn raw ground-up losses into what the insurer actually pays. Master the four levers (deductible, limit, coinsurance, inflation) and the stop-loss adjustment, and most ASTAM severity questions collapse into a difference of limited expected values.

Building the modified payment. Start with ground-up loss $X$. Apply, in order: inflation, ordinary deductible $d$, maximum covered loss $u$, coinsurance $\alpha$. The per-loss random variable splits into three regions.

$$Y^L=\{\begin{array}{ll}0& X<d\\ \alpha (X-d)& d\le X\le u\\ \alpha (u-d)& X>u\end{array}$$

Here $u$ is the maximum covered loss (ground-up dollars) and $\alpha (u-d)$ is the policy limit (payment dollars). Exam wording flips between the two, so always check which scale the question quotes.

Expected per-loss payment. Use the limited expected value $E[X\wedge m]=E[\min (X,m)]$.

$$E[Y^L]=\alpha (E[X\wedge u]-E[X\wedge d])$$

Common mistakes

• Confusing $u$ with the policy limit. $u$ is ground-up max loss; payment cap is $\alpha (u-d)$.
• Inflating the deductible upward instead of deflating. The correct move divides $d$ by $(1+r)$ before plugging into $E[X\wedge \cdot ]$.
• Putting $\alpha$ in the per-payment denominator. The denominator is $1-F(d)$ only.

Bottom line

• Per-loss $Y^L=\alpha [(X\wedge u)-(X\wedge d)]$ with maximum payment $\alpha (u-d)$; per-payment $Y^P=Y^L/[1-F(d)]$.
• Maximum covered loss $u$ lives on the loss scale; policy limit $\alpha (u-d)$ lives on the payment scale. Don't conflate.
• Inflation by $r$: deflate $d$ and $u$ inside the LEVs and multiply outside by $\alpha (1+r)$; a fixed $d>0$ makes payment growth strictly exceed loss growth (leveraged inflation).
• Franchise deductible pays the full $X$ once $X>d$, adding $d[1-F(d)]$ over the ordinary case.

Exam shortcut

Write deflated thresholds first: $d^{\ast }=d/(1+r)$, $u^{\ast }=u/(1+r)$. Then one formula, $\alpha (1+r)[E[X\wedge u^{\ast }]-E[X\wedge d^{\ast }]]$, handles deductible, limit, coinsurance, and inflation in one stroke. For Pareto, chant: LEV is $\theta /(\alpha -1)$ times one minus the ratio $(\theta /(\theta +m){)}^{\alpha -1}$. Plug both thresholds, subtract, scale. Stop-loss layer between $d$ and $d^{\prime }$ collapses to $E[S\wedge d^{\prime }]-E[S\wedge d]$. Never compute $E[S]$ for a finite layer; it cancels. For lognormal at retention $d$, write three z-values up top: $z_0=(\ln d-\mu )/\sigma$, $z_1=z_0-\sigma$, $z_2=z_0-2\sigma$.

The full lesson (about 4,734 words, 32 min read) adds 3 worked examples, all 12 common mistakes, a self-check, free in the app.

Learning objectives

• 3a

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