Sensitivity Analysis For Exposure-Rated Layers

Why Sensitivity Analysis Matters

An exposure-rated layer indication depends on more than the premium base. It also depends on the expected ground-up loss, the shape of the severity curve, and the layer’s attachment and limit. Sensitivity analysis separates these effects: it shows which assumption changed, how the layer indication responds, and whether the direction and size of the change are reasonable.

The calculations here concern expected loss only. They do not add expenses, brokerage, profit provisions, or risk loads. The expected loss ratio and subject premium must refer to compatible coverage, periods, and premium bases.

Definition: Let $P_s$ be subject premium and $r$ the expected loss ratio on that premium basis. The implied expected ground-up loss is $P_s r$. Let $X$ be a nonnegative ground-up loss severity with finite positive mean. Its limited expected loss at limit $u$ is $L(u)=E[\min(X,u)]$, and its exposure curve is $G(u)=L(u)/E[X]$. For a layer attaching at $a$ and paying at most $w$ per loss, define its upper limit $b=a+w$. The layer’s expected loss indication is $P_s r [G(b)-G(a)]$, when the curve represents the same ground-up severity basis as the expected loss ratio.

How the Layer Factor Is Obtained

For a single loss $X$, the payment to a layer attaching at $a$ and having width $w$ is $\min((X-a)^+,w)$, where $(z)^+=\max(z,0)$. This payment also equals $\min(X,b)-\min(X,a)$, with $b=a+w$. Taking expectations gives the layer’s expected payment as $L(b)-L(a)$. Dividing by $E[X]$ gives the layer factor $G(b)-G(a)$. Multiplying that factor by expected ground-up losses, $P_s r$, produces the indicated expected layer loss.

This difference is essential: $G(b)$ alone represents losses limited at $b$, including losses below the attachment. Subtracting $G(a)$ removes the expected loss retained below the layer.

\begin{aligned} E[\min((X-a)^+,w)] &= L(a+w)-L(a),\\ q(a,w) &= G(a+w)-G(a),\\ I &= P_s r\,q(a,w). \end{aligned}

For a nonnegative severity, the limited expected loss also has a useful survival-function representation. Since $\min(X,u)=\int_0^u \mathbf{1}_{\{X>x\}}\,dx$, taking expectations gives $L(u)=\int_0^u S_X(x)\,dx$. Thus, for a continuous severity distribution, $G'(u)=S_X(u)/E[X]$. This result helps explain the direction of changes to attachment and limit.

Worked Example: Changing the Severity-Curve Shape

A hypothetical insurer has subject premium of $1,500,000 and an expected loss ratio of 0.62. Compare the expected loss indications for a $1,000,000 excess of $1,000,000 layer under two severity models with the same mean. Model A is exponential with mean $1,000,000. Model B is Lomax with shape parameter $\alpha=3$ and scale parameter $\lambda=$2,000,000$. Assume the same ground-up expected loss base applies to both models. Calculate all dollar amounts to the nearest cent and the percentage difference to two decimal places.

Step 1: Determine ground-up expected loss. The premium and expected loss ratio imply $P_s r=$1,500,000(0.62)=$930,000$. The layer attaches at $a=$1,000,000$ and has upper limit $b=$2,000,000$.

Step 2: Calculate the exponential layer factor. For an exponential severity with mean $\theta$, $G(u)=1-e^{-u/\theta}$. With $\theta=$1,000,000$, $G(b)-G(a)=e^{-1}-e^{-2}=0.3678794-0.1353353=0.2325442$. The expected layer loss is $$930,000(0.2325442)=$216,266.07$.

Step 3: Calculate the Lomax layer factor. For a Lomax severity with $\alpha>1$, scale $\lambda$, and survival function $S_X(x)=(1+x/\lambda)^{-\alpha}$, the mean is $\lambda/(\alpha-1)$ and $G(u)=1-(1+u/\lambda)^{1-\alpha}$. Here the mean is $2,000,000/(3-1)=$1,000,000$, matching Model A. The layer factor is $(1+1,000,000/2,000,000)^{-2}-(1+2,000,000/2,000,000)^{-2}=1/2.25-1/4=0.1944444$. The expected layer loss is $$930,000(0.1944444)=$180,833.33$.

Step 4: Compare. Model B’s indication is $$216,266.07-$180,833.33=$35,432.74$ lower than Model A’s. Relative to Model A, the reduction is $35,432.74/216,266.07=16.38\%$, rounded to two decimal places.

The method is to recompute the layer factor from each full severity curve, then apply the same expected ground-up loss. The common mistake is to assume that equal means imply equal layer factors; the calculation shows that the shape of the severity distribution matters even when the means agree.

Sensitivity to Attachment and Limit

For a fixed layer width $w$, increasing the attachment shifts both the attachment and upper limit upward. Since $G'(u)=S_X(u)/E[X]$, the derivative of the layer factor with respect to attachment is $[S_X(a+w)-S_X(a)]/E[X]$, which is nonpositive because survival probabilities do not increase with the loss threshold. By contrast, increasing the width while holding attachment fixed increases the factor: its derivative with respect to width is $S_X(a+w)/E[X]$, which is nonnegative.

These directions assume a fixed severity curve and a nonnegative loss. The size of the change depends on the probability mass and expected loss in the affected part of the curve. It is not generally appropriate to apply a fixed percentage change to the layer indication when attachment or width changes.

Worked Example: Stressing Attachment and Layer Width

A hypothetical reinsurance indication uses subject premium of $2,000,000, an expected loss ratio of 0.60, and exponential ground-up severity with mean $\theta=$1,000,000$. Compare a $1,000,000 excess of $1,000,000 layer with (i) a $1,000,000 excess of $1,500,000 layer and (ii) a $1,500,000 excess of $1,000,000 layer. Treat the first amount in each description as the layer width. Round dollar indications to cents.

Step 1: Establish the loss base. Expected ground-up loss is $$2,000,000(0.60)=$1,200,000$. For an exponential severity, the factor for attachment $a$ and upper limit $b$ is $e^{-a/\theta}-e^{-b/\theta}$.

Step 2: Calculate the baseline layer. For attachment $1,000,000 and upper limit $2,000,000, the factor is $e^{-1}-e^{-2}=0.3678794-0.1353353=0.2325442$. The indication is $$1,200,000(0.2325442)=$279,052.99$.

Step 3: Raise the attachment, holding width fixed. The revised layer attaches at $1,500,000 and ends at $2,500,000. Its factor is $e^{-1.5}-e^{-2.5}=0.2231302-0.0820850=0.1410452$. The indication is $$1,200,000(0.1410452)=$169,254.19$, lower than the baseline.

Step 4: Increase width, holding attachment fixed. The layer still attaches at $1,000,000 but now ends at $2,500,000. Its factor is $e^{-1}-e^{-2.5}=0.3678794-0.0820850=0.2857944$. The indication is $$1,200,000(0.2857944)=$342,953.33$, higher than the baseline.

The direction of each result agrees with the fixed-curve sensitivity: raising attachment reduces the factor, while extending the limit increases it. A common mistake is to use $G($2,000,000)$ for the baseline layer. That would give $1-e^{-2}=0.8646647$, which includes expected losses below the $1,000,000 attachment; the correct layer factor is $G($2,000,000)-G($1,000,000)=0.2325442$.

Sensitivity to Severity Parameters, Premium, and Loss Ratio

An indication can be written as $I=P_s r q$, where $q$ is the layer factor. If the curve and layer remain fixed, the indication is directly proportional to both premium and expected loss ratio. For example, multiplying premium by 1.08 and changing the expected loss ratio from 0.55 to 0.60 multiplies the indication by $1.08(0.60/0.55)$, not by the sum of the two percentage changes. If the curve also changes, calculate the new $q$ and use the full product rather than combining the effects as if they were independent additive adjustments.

Severity-parameter changes can have less intuitive effects. For an exponential severity with mean $\theta$, the layer factor is $q=e^{-a/\theta}-e^{-b/\theta}$. Its derivative with respect to $\theta$ is $[a e^{-a/\theta}-b e^{-b/\theta}]/\theta^2$. This derivative need not have the same sign for every layer and parameter value: increasing severity scale shifts losses upward, but it can move expected loss into or beyond the layer.

Worked Example: Separating Premium and Loss-Ratio Effects

A hypothetical portfolio has subject premium of $2,400,000 and an expected loss ratio of 0.55. Ground-up severity is exponential with mean $\theta=$1,500,000$. The layer attaches at $500,000 and has width $1,500,000. First calculate its expected loss indication. Then calculate the revised indication if subject premium increases by 8% and the expected loss ratio changes to 0.60, with the severity curve and layer unchanged. Round dollar amounts to cents.

Step 1: Find the layer factor. The upper limit is $2,000,000. Therefore, $q=e^{-500,000/1,500,000}-e^{-2,000,000/1,500,000}=e^{-1/3}-e^{-4/3}=0.7165313-0.2635971=0.4529342$.

Step 2: Calculate the initial indication. Ground-up expected loss is $$2,400,000(0.55)=$1,320,000$. Thus the layer indication is $$1,320,000(0.4529342)=$597,873.11$.

Step 3: Update the premium and loss base. Revised subject premium is $$2,400,000(1.08)=$2,592,000$. Revised ground-up expected loss is $$2,592,000(0.60)=$1,555,200$. The curve and layer have not changed, so the factor remains 0.4529342. The revised indication is $$1,555,200(0.4529342)=$704,403.23$.

Step 4: Check the proportional effect. The revised-to-initial ratio is $(2,592,000/2,400,000)(0.60/0.55)=1.08(1.0909091)=1.1781818$. Applying it to the initial indication gives $$597,873.11(1.1781818)=$704,403.23$, matching the direct calculation. The common mistake is to leave the loss ratio unchanged when only the premium adjustment is noticed, or to add percentage changes instead of multiplying the two factors.

Practical Interpretation and Exam Traps

A sensitivity calculation is most useful when it distinguishes a change in the risk from a change in the exposure base. A revised premium may reflect a different volume of exposure, while a revised expected loss ratio may reflect a different estimate of ground-up loss cost relative to premium. A revised curve or layer changes the fraction of ground-up loss assigned to the layer. Keeping these components visible makes the result auditable.

For a fixed curve and layer, $I$ scales linearly with $P_s$ and $r$. But a curve-shape or severity-scale change is not generally a linear adjustment. As a check, for an exponential distribution with mean $1,000,000, a $1,000,000 excess of $1,000,000 factor is $e^{-1}-e^{-2}=0.2325$. If the mean increases to $2,000,000, the factor is $e^{-0.5}-e^{-1}=0.2387$, slightly higher. At a mean of $4,000,000, it is $e^{-0.25}-e^{-0.5}=0.1723$, lower. Thus “heavier severity” does not by itself determine the direction of a particular layer-factor change.

Before accepting a scenario result, check that the exposure curve, expected loss ratio, and premium use consistent bases; that the upper limit equals attachment plus layer width; and that the layer factor is a difference of limited expected loss ratios. These checks catch many errors before they affect a rate indication.

Check Your Understanding

For each hypothetical scenario, calculate the requested expected layer loss or sensitivity. Round dollar amounts to cents and percentages to two decimal places.

Subject premium is $800,000 and the expected loss ratio is 0.50. Severity is exponential with mean $2,000,000. What is the expected loss for a $1,000,000 excess of $500,000 layer? (Answer: $122,573.68)
Subject premium is $1,600,000 and the expected loss ratio is 0.50. Severity is exponential with mean $1,000,000. Calculate the expected loss for a $1,000,000 excess of $2,000,000 layer. (Answer: $68,438.57)
Subject premium is $600,000 and the expected loss ratio is 0.50. Severity is Lomax with shape $\alpha=4$ and scale $\lambda=$1,500,000$. What is the expected loss for a $1,000,000 excess of $500,000 layer? (Answer: $89,062.50)
Subject premium is initially $1,000,000 with expected loss ratio 0.50. Severity is exponential with mean $1,000,000, and the layer is $1,000,000 excess of $1,000,000. What is the revised indication if premium rises by 10% and the expected loss ratio becomes 0.55, with all other assumptions unchanged? (Answer: $140,689.22)
For exponential severity with mean $1,000,000, compare the factor for a $1,000,000 excess of $500,000 layer with the factor for a $1,000,000 excess of $1,000,000 layer. By what percentage does the factor decrease when the attachment is raised, holding width fixed? (Answer: 39.35%)

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