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.

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

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 $E[X \wedge m] = E[\min(X, m)]$.

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

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: $d^{*} = d/(1+r)$, $u^{*} = u/(1+r)$. Then one formula, $\alpha(1+r)[E[X \wedge u^{*}] - E[X \wedge d^{*}]]$, 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 $d'$ collapses to $E[S \wedge d'] - E[S \wedge d]$. 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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