Derive Premium Sensitivities To LCFs And Caps

Why Premium Sensitivity Depends on the Active Cap

A retrospective premium can change when the loss conversion factor or a loss cap changes. The size of that change is not always constant: a claim cap may begin limiting another claim, or an aggregate cap may stop the converted loss component from increasing at all. Sensitivity analysis makes these effects explicit. It answers a local question—how fast does premium change near the current parameters?—and helps explain finite differences between two parameter choices.

The order in which the plan applies its parameters matters. This tutorial uses a specified plan formula in which each claim is first subject to a per-claim cap, the resulting limited losses are multiplied by the loss conversion factor, and an aggregate cap is then applied to that converted loss component. A plan that applies its aggregate cap before the conversion factor has a different formula and may have different sensitivities.

Definition: Let $x_1,\ldots,x_n$ be nonnegative claim losses, $c$ a per-claim cap, $k>0$ the loss conversion factor (LCF), $A$ an aggregate cap on the converted loss component, and $B$ a fixed basic premium. Define $$ S(c)=\sum_{j=1}^{n}\min(x_j,c), \qquad P(k,c)=B+\min\{A,\ kS(c)\}. $$ Here $S(c)$ is the total of the per-claim limited losses, and $P(k,c)$ is the premium under this specified formula.

All monetary quantities in the formula must use the same unit. For example, if $x_j$, $c$, $A$, and $B$ are stated in thousands of dollars, the premium calculation also produces thousands of dollars. The aggregate cap $A$ is held fixed when taking sensitivities unless a problem explicitly says otherwise. The calculations below concern this stated formula; they do not include minimum or maximum premium provisions.

Sensitivities in a Smooth Region

First suppose no individual claim equals the per-claim cap. The claims with $x_j>c$ are currently limited by $c$; claims with $x_j<c$ are not. Write $$ N(c)=\#\{j:x_j>c\}. $$ As long as $c$ does not cross a claim amount, the set of limited claims stays the same. Increasing $c$ by a small amount therefore increases $S(c)$ by that amount for each of the $N(c)$ limited claims.

There are two smooth regions for the premium formula. If the aggregate cap is not binding, then $kS(c)<A$, so $P=B+kS(c)$. If it is binding, then $kS(c)>A$, so $P=B+A$. Away from the equality $kS(c)=A$, the partial derivatives are:

\begin{aligned} kS(c)<A:\quad &\frac{\partial P}{\partial k}=S(c), &&\frac{\partial P}{\partial c}=kN(c),\\[4pt] kS(c)>A:\quad &\frac{\partial P}{\partial k}=0, &&\frac{\partial P}{\partial c}=0. \end{aligned}

The derivative with respect to $k$ measures the premium change per one-unit increase in the LCF, holding the cap fixed. The derivative with respect to $c$ measures the premium change per dollar (or other stated currency unit) increase in the per-claim cap, holding the LCF fixed. When the aggregate cap is binding, a small increase in either parameter leaves the capped loss component at $A$, so neither parameter changes premium locally.

Threshold condition: The aggregate cap is reached when $kS(c)=A$. For fixed $c$, the LCF threshold is $k^*=A/S(c)$, provided $S(c)>0$. For fixed $k$, a cap threshold satisfies $kS(c^*)=A$. The function $S(c)$ is piecewise linear, so a cap threshold can be solved within the interval between adjacent claim amounts. At a threshold, or when $c=x_j$ for any claim, an ordinary two-sided derivative may not exist; use the relevant one-sided changes instead.

Why These Derivatives Hold

The justification follows directly from the formula. On an interval of cap values that contains no claim amount, each limited claim contributes $c$ to $S(c)$, while each unlimited claim contributes the constant $x_j$. Thus, for a small change $\Delta c$ that remains in that interval, $$ S(c+\Delta c)-S(c)=N(c)\Delta c. $$ If the aggregate cap is not binding throughout the change, premium is $B+kS(c)$. Holding $k$ fixed gives $\Delta P=kN(c)\Delta c$, hence $\partial P/\partial c=kN(c)$. Holding $c$ fixed and changing $k$ gives $\Delta P=S(c)\Delta k$, hence $\partial P/\partial k=S(c)$. If instead $kS(c)>A$ throughout a sufficiently small change, premium is the constant $B+A$, and both local derivatives are zero. At equality, a change in a direction that crosses from one branch to the other cannot be described by either branch’s derivative alone.

Worked Applications

Worked Example: Sensitivities Below the Aggregate Cap

A hypothetical plan has basic premium $B=\$12{,}000$, LCF $k=1.10$, three claim losses of $\$20{,}000$, $\$45{,}000$, and $\$80{,}000$, a per-claim cap $c=\$50{,}000$, and aggregate cap $A=\$200{,}000$. Calculate the premium and both local sensitivities. Then calculate the exact premium change if the per-claim cap rises to $\$55{,}000$, with all else unchanged.

Method. Apply the per-claim cap to each loss, multiply their sum by the LCF, and check whether the aggregate cap binds.

Step 1: Calculate limited losses. They are $\min(20{,}000,50{,}000)=20{,}000$, $\min(45{,}000,50{,}000)=45{,}000$, and $\min(80{,}000,50{,}000)=50{,}000$. Therefore, $S(50{,}000)=\$115{,}000$.

Step 2: Check the aggregate cap and calculate premium. The converted amount is $1.10(\$115{,}000)=\$126{,}500$, which is below $A=\$200{,}000$. Thus, $P=\$12{,}000+\$126{,}500=\$138{,}500$.

Step 3: Calculate local sensitivities. The derivative with respect to the LCF is $S(c)=\$115{,}000$ per unit of $k$. Only the $\$80{,}000$ claim exceeds the current cap, so $N(c)=1$ and $\partial P/\partial c=1.10$. An increase of one dollar in the cap therefore increases premium locally by $\$1.10$.

Step 4: Check the finite cap increase. At $c=\$55{,}000$, the limited losses are $\$20{,}000$, $\$45{,}000$, and $\$55{,}000$, summing to $\$120{,}000$. The converted amount is $1.10(\$120{,}000)=\$132{,}000$, still below the aggregate cap. The revised premium is $\$12{,}000+\$132{,}000=\$144{,}000$; the increase is $\$5{,}500$. This also equals the local rate $\$1.10$ per dollar times the $\$5{,}000$ cap increase, since the cap remains between the $\$45{,}000$ and $\$80{,}000$ claim amounts.

Common mistake avoided: Counting all three claims when differentiating with respect to the cap would give $1.10(3)$. Only claims above the cap respond to a small cap increase, so the correct derivative is $1.10(1)$.

Worked Example: Finding the Aggregate-Cap Threshold

A hypothetical plan has $B=\$8{,}000$, claim losses of $\$30{,}000$, $\$60{,}000$, and $\$90{,}000$, LCF $k=1.20$, and aggregate cap $A=\$100{,}000$. At a per-claim cap of $\$20{,}000$, calculate premium and local sensitivities. Then find the cap at which the aggregate cap first binds as $c$ increases from $\$20{,}000$.

Step 1: Calculate the capped loss total at $\$20{,}000$. Each claim exceeds the cap, so $S(20{,}000)=3(\$20{,}000)=\$60{,}000$. The converted amount is $1.20(\$60{,}000)=\$72{,}000$, below the aggregate cap. Premium is therefore $\$8{,}000+\$72{,}000=\$80{,}000$.

Step 2: Calculate local sensitivities. All three claims exceed the current cap, so $\partial P/\partial c=1.20(3)=3.60$. Also, $\partial P/\partial k=S(c)=\$60{,}000$ per unit of $k$.

Step 3: Solve for the cap threshold. The threshold satisfies $1.20S(c^*)=\$100{,}000$, so $S(c^*)=\$83{,}333.33$. For $c$ between $\$20{,}000$ and $\$30{,}000$, all three claims remain above the cap and $S(c)=3c$. Hence $3c^*=\$83{,}333.33$, giving $c^*=\$27{,}777.78$.

The threshold lies in the assumed interval, so the calculation is consistent. Below this cap the aggregate cap does not bind; above it, the converted amount reaches or exceeds $\$100{,}000$. Premium is $\$108{,}000$ at the threshold ($\$8{,}000+\$100{,}000$) and is locally flat for further cap increases once the aggregate cap binds.

Common mistake avoided: Solving the threshold as though only one claim were limited would use $S(c)=c+\$30{,}000+\$60{,}000$, which is not the correct expression below $\$30{,}000$. In that interval, all three claims are capped, so $S(c)=3c$.

Worked Example: Decomposing a Change in Both Parameters

A hypothetical plan has fixed basic premium $B=\$10{,}000$ and three claim losses of $\$30{,}000$, $\$60{,}000$, and $\$70{,}000$. The aggregate cap is high enough not to bind in either calculation. The original parameters are $k=1.05$ and $c=\$60{,}000$; the revised parameters are $k=1.08$ and $c=\$55{,}000$. Calculate the original and revised premiums, the total change, and a sequential decomposition that changes the cap first and the LCF second.

Step 1: Original premium. At a $\$60{,}000$ cap, the limited losses are $\$30{,}000$, $\$60{,}000$, and $\$60{,}000$. Their sum is $\$150{,}000$. The converted component is $1.05(\$150{,}000)=\$157{,}500$, so the original premium is $\$10{,}000+\$157{,}500=\$167{,}500$.

Step 2: Revised premium. At a $\$55{,}000$ cap, the limited losses are $\$30{,}000$, $\$55{,}000$, and $\$55{,}000$, summing to $\$140{,}000$. The converted component is $1.08(\$140{,}000)=\$151{,}200$. The revised premium is $\$10{,}000+\$151{,}200=\$161{,}200$.

Step 3: Calculate the total change. The revised premium less the original premium is $\$161{,}200-\$167{,}500=-\$6{,}300$.

Step 4: Decompose by changing the cap first. At the old LCF $1.05$, reducing the cap changes the limited-loss total from $\$150{,}000$ to $\$140{,}000$. Its premium effect is $1.05(\$140{,}000-\$150{,}000)=-\$10{,}500$.

Step 5: Then change the LCF. At the new limited-loss total of $\$140{,}000$, increasing the LCF from $1.05$ to $1.08$ changes premium by $(1.08-1.05)(\$140{,}000)=\$4{,}200$. The effects sum to $-\$10{,}500+\$4{,}200=-\$6{,}300$, matching the direct premium difference.

Common mistake avoided: The old limited-loss total is $\$150{,}000$, not $\$160{,}000$: the $\$70{,}000$ claim is capped at $\$60{,}000$. The cap effect and LCF effect must also use the stated sequence; changing parameters in another order can allocate the interaction differently, although the total premium change remains the same.

Worked Example: One-Sided Sensitivities at a Claim Amount

A hypothetical plan has $B=\$5{,}000$, claim losses of $\$30{,}000$ and $\$50{,}000$, LCF $k=1.10$, and an aggregate cap that does not bind. Find the left- and right-hand sensitivities to the per-claim cap at $c=\$30{,}000$, and calculate the exact premium change when the cap rises from $\$30{,}000$ to $\$31{,}000$.

Just below $\$30{,}000$, both claims exceed the cap, so $N(c)=2$ and the left-hand derivative is $1.10(2)=2.20$. Just above $\$30{,}000$, the $\$30{,}000$ claim is no longer capped, while the $\$50{,}000$ claim remains capped; thus $N(c)=1$ and the right-hand derivative is $1.10$. The two-sided derivative does not exist at this claim amount.

At $c=\$30{,}000$, the limited-loss total is $\$60{,}000$. At $c=\$31{,}000$, it is $\$30{,}000+\$31{,}000=\$61{,}000$. The increase in premium is $1.10(\$1{,}000)=\$1{,}100$, consistent with the right-hand sensitivity because the cap is being increased from the threshold.

Common mistake avoided: Applying the left-hand derivative of $2.20$ to the increase from $\$30{,}000$ gives an incorrect $\$2{,}200$ change. The finite increase starts at the kink and moves to the right, where only one claim responds to the cap.

Practical Checks and Exam Traps

Sensitivity formulas are useful only after the active branch of the premium formula has been identified. Before differentiating or estimating a finite change, check the limited-loss total, the converted amount, and whether the aggregate cap binds both before and after the proposed change. A local derivative can predict a finite change exactly only while the calculation remains in the same smooth region.

Do not count claims below the per-claim cap. A small cap increase affects only claims currently above the cap.
Do not continue a smooth-region derivative through a threshold. At a claim amount, the number of capped claims may change; at $kS(c)=A$, the aggregate-cap branch may change.
Respect the order of operations. This formula caps each raw claim before applying $k$, then applies $A$ to the converted total. A cap on converted losses or a cap applied before conversion must be represented differently.
Keep units consistent. A sensitivity of $1.10$ means $\$1.10$ of premium per $\$1$ of cap in the uncapped aggregate region, not $\$1.10$ per thousand dollars unless the cap itself is measured in thousands.
Separate exact changes from local approximations. For a change crossing a threshold, recompute the premium from the formula rather than extending the old derivative.

Key takeaway: In a smooth region below the aggregate cap, $\partial P/\partial k=S(c)$ and $\partial P/\partial c=kN(c)$. Once the aggregate cap binds, both local sensitivities are zero. At cap-trigger thresholds, identify the direction of change and use the appropriate one-sided calculation or evaluate the premium directly.

Check Your Understanding

For each problem, use the stated order of operations and identify whether the aggregate cap binds before applying a sensitivity.

A plan has $B=\$8{,}000$, $k=1.15$, claims of $\$25{,}000$, $\$40{,}000$, and $\$70{,}000$, per-claim cap $\$50{,}000$, and a nonbinding aggregate cap. Find premium and both local sensitivities. What is the exact premium increase if the per-claim cap rises by $\$2{,}000$? (Answer: premium $=\$140{,}250$; $\partial P/\partial k=\$115{,}000$; $\partial P/\partial c=1.15$; increase $=\$2{,}300$.)
A plan has losses $\$20{,}000$, $\$35{,}000$, and $\$50{,}000$, $k=1.25$, per-claim cap $\$30{,}000$, and aggregate cap $\$120{,}000$. Find $S(c)$, the premium’s loss component, and both local sensitivities. (Answer: $S(c)=\$80{,}000$; loss component $=\$100{,}000$; $\partial P/\partial k=\$80{,}000$; $\partial P/\partial c=2.50$.)
A plan has losses $\$10{,}000$, $\$40{,}000$, and $\$80{,}000$, per-claim cap $\$20{,}000$, and aggregate cap $\$90{,}000$. Find the LCF at which the aggregate cap is reached. (Answer: $S(c)=\$50{,}000$, so $k^*=\$90{,}000/\$50{,}000=1.80$.)
A plan has two losses, $\$30{,}000$ and $\$50{,}000$, $k=1.10$, and a nonbinding aggregate cap. Compare the left- and right-hand cap sensitivities at $c=\$30{,}000$. (Answer: left-hand sensitivity $=2.20$; right-hand sensitivity $=1.10$; the two-sided derivative does not exist.)
A plan has $B=\$6{,}000$, losses $\$20{,}000$, $\$50{,}000$, and $\$70{,}000$, $k=1.20$, per-claim cap $\$60{,}000$, and aggregate cap $A=\$110{,}000$. Find premium and local sensitivities. (Answer: $S(c)=\$130{,}000$, so $kS(c)=\$156{,}000>A$; premium $=\$116{,}000$; both local sensitivities are $0$.)

What You'll Learn