How Premium Bounds Change Loss Sensitivity

Why Premium Bounds Change Loss Sensitivity

In a retrospective rating plan, an increase in converted losses raises the unbounded premium when the plan’s other inputs are held fixed. A contractual minimum or maximum can change how much of that increase reaches the insured’s final premium. Below the minimum, an additional loss may have no premium effect; above the maximum, the same is true. If the increase moves the unbounded premium across a bound, only part of the increase may affect the final premium.

The key is to separate the change in the unbounded premium from the change in the bounded premium. This tutorial treats the plan parameters as fixed and examines an increase in the loss measure that is multiplied by the loss conversion factor. It does not require a probability distribution for losses.

Definition: Let $B$ be the basic premium, $f$ the loss conversion factor, $T$ the tax multiplier, and $C$ the converted-loss measure used in the premium formula. The unbounded premium is $U=T(B+fC)$. Given a minimum premium $P_{\min}$ and maximum premium $P_{\max}$, where $P_{\min}\le P_{\max}$, the bounded premium is $P=\min(P_{\max},\max(P_{\min},U))$. All quantities must use consistent monetary units.

Unless stated otherwise, assume $B$, $f$, $T$, $P_{\min}$, and $P_{\max}$ are fixed, with $f>0$ and $T>0$. An increase $\Delta C>0$ changes the unbounded premium by $Tf\Delta C$. The bounds then determine how much of that change appears in $P$. The relevant sensitivity is with respect to $C$; it is not automatically the sensitivity to a different loss measure that may need conversion first.

Marginal Sensitivity in Each Premium Region

When the unbounded premium is strictly between the bounds, neither bound affects the premium. A small increase in $C$ therefore raises the bounded premium at the same rate as the unbounded premium. When the unbounded premium is strictly below the minimum, the bounded premium remains at the floor for a sufficiently small increase. When it is strictly above the maximum, the bounded premium remains at the ceiling.

\frac{dP}{dC}= \begin{cases} 0, & U<P_{\min},\\ Tf, & P_{\min}<U<P_{\max},\\ 0, & U>P_{\max}. \end{cases}

At exactly $U=P_{\min}$ or $U=P_{\max}$, there is a kink: the one-sided sensitivity depends on the direction of change. For an increase from the minimum, the premium begins responding at rate $Tf$; for a decrease from the minimum, the floor prevents the premium from decreasing further. At the maximum, an increase is blocked by the ceiling, while a decrease from the maximum can lower the premium at rate $Tf$.

This is a local, or marginal, result. For a finite increase, the starting and ending positions matter. If the unbounded premium begins below the minimum and ends above it, the premium rises only after the floor has been reached. If it begins inside the bounds and ends above the maximum, the premium rises only until the ceiling is reached.

Why the Sensitivity Formula Holds

The justification follows directly from applying the bounds after calculating the unbounded premium. For an increase $\Delta C$, the new unbounded premium is $U+Tf\Delta C$. The bounded premium is obtained by replacing any value below the floor with $P_{\min}$, and any value above the ceiling with $P_{\max}$. Thus the finite premium change is the difference between the bounded new and old unbounded premiums:

\Delta P = \min\!\bigl(P_{\max},\max(P_{\min},U+Tf\Delta C)\bigr) - \min\!\bigl(P_{\max},\max(P_{\min},U)\bigr).

If both unbounded premiums lie strictly between the bounds, the inner and outer operations do not alter either value. Subtracting gives $\Delta P=(U+Tf\Delta C)-U=Tf\Delta C$. If both lie below the floor, each bounded value equals $P_{\min}$, so $\Delta P=0$. If both lie above the ceiling, each equals $P_{\max}$, again giving zero. If the interval crosses a bound, only the portion of the unbounded change that falls inside the permitted premium range affects the bounded premium.

Theorem: With $Tf>0$, the premium’s marginal sensitivity to converted losses is $Tf$ while the unbounded premium is inside the bounds, and zero while it is strictly below the minimum or strictly above the maximum. For a finite increase, calculate both bounded premiums and subtract; do not assume the entire increase has one marginal rate.

Worked Applications

Worked Example: Premium Increase Entirely Within the Bounds

A hypothetical plan has $B=\$40.00$ thousand, $f=1.10$, $T=1.20$, $P_{\min}=\$60.00$ thousand, and $P_{\max}=\$100.00$ thousand. Converted losses increase from $C=\$20.00$ thousand to $\$23.00$ thousand. Find the bounded premium change and the marginal increase per additional thousand dollars of $C$.

Method. First calculate the loss-responsive premium rate $Tf$:

$$ Tf=1.20(1.10)=1.32. $$

Thus each additional thousand dollars of $C$ adds $\$1.32$ thousand to the unbounded premium. The increase in converted losses is $\Delta C=\$23.00-\$20.00=\$3.00$ thousand, so the unbounded premium increases by $1.32(3.00)=\$3.96$ thousand.

At the starting loss level, the unbounded premium is $U_0=1.20[40.00+1.10(20.00)]=1.20(62.00)=\$74.40$ thousand. At the ending loss level, it is $U_1=1.20[40.00+1.10(23.00)]=1.20(65.30)=\$78.36$ thousand. Both amounts are between $\$60.00$ thousand and $\$100.00$ thousand, so neither bound changes either premium. The bounded increase is $\$78.36-\$74.40=\$3.96$ thousand. The marginal increase is $\$1.32$ thousand per additional thousand dollars of $C$.

Check and trap avoided. The difference $78.36-74.40=3.96$ agrees with $Tf\Delta C=1.32(3.00)=3.96$. Do not confuse the increase in $C$, which is $\$3.00$ thousand, with the resulting premium increase, which is $\$3.96$ thousand.

Worked Example: An Increase That Moves the Premium Above the Minimum

A hypothetical plan has $B=\$50.00$ thousand, $f=1.25$, $T=1.20$, a minimum of $\$72.00$ thousand, and a maximum of $\$100.00$ thousand. Converted losses increase from $\$6.00$ thousand to $\$10.00$ thousand. Calculate the bounded premium increase and show how much of the loss increase occurs before the premium leaves the floor.

Method. The unbounded sensitivity is $Tf=1.20(1.25)=1.50$: each additional thousand dollars of $C$ increases $U$ by $\$1.50$ thousand. Initially, $U_0=1.20[50.00+1.25(6.00)]=1.20(57.50)=\$69.00$ thousand. Since this is below the minimum, the starting bounded premium is $P_0=\$72.00$ thousand.

At the ending loss level, $U_1=1.20[50.00+1.25(10.00)]=1.20(62.50)=\$75.00$ thousand. This is above the minimum and below the maximum, so $P_1=\$75.00$ thousand. Therefore the bounded premium increases by $\$75.00-\$72.00=\$3.00$ thousand.

The unbounded premium must rise by $\$3.00$ thousand to move from $\$69.00$ thousand to the $\$72.00$ thousand floor. At $\$1.50$ thousand per thousand dollars of $C$, that takes $3.00/1.50=\$2.00$ thousand of additional $C$. The total increase in $C$ is $\$4.00$ thousand, leaving $\$2.00$ thousand after the floor is reached. That remaining amount raises the premium by $1.50(2.00)=\$3.00$ thousand.

Check and trap avoided. The final premium change is $\$3.00$ thousand, not $Tf\Delta C=1.50(4.00)=\$6.00$ thousand. The first $\$2.00$ thousand of the loss increase raises the unbounded premium only up to the floor; it does not raise the bounded premium.

Worked Example: An Increase That Reaches the Maximum

A hypothetical plan has $B=\$50.00$ thousand, $f=1.00$, $T=1.20$, a minimum of $\$60.00$ thousand, and a maximum of $\$96.00$ thousand. Converted losses rise from $\$27.50$ thousand to $\$31.50$ thousand. Find the bounded premium increase, and then find the effect of a further $\$2.00$ thousand increase in $C$.

Method. The unbounded rate is $Tf=1.20(1.00)=1.20$. Initially, $U_0=1.20[50.00+1.00(27.50)]=1.20(77.50)=\$93.00$ thousand. This is within the bounds, so $P_0=\$93.00$ thousand.

The first increase in $C$ is $\$31.50-\$27.50=\$4.00$ thousand, so $U$ rises by $1.20(4.00)=\$4.80$ thousand. The new unbounded premium is $U_1=\$93.00+\$4.80=\$97.80$ thousand. It exceeds the maximum, so $P_1=\$96.00$ thousand. The bounded premium increase is therefore $\$96.00-\$93.00=\$3.00$ thousand.

The premium reaches its $\$96.00$ thousand ceiling after an unbounded increase of $\$3.00$ thousand. At a rate of $\$1.20$ thousand per thousand dollars of $C$, this occurs after $3.00/1.20=\$2.50$ thousand of the $\$4.00$ thousand increase. The remaining $\$1.50$ thousand of $C$ does not increase the bounded premium. After the further $\$2.00$ thousand increase, the unbounded premium is $1.20[50.00+31.50+2.00]=\$100.20$ thousand, but the bounded premium remains $\$96.00$ thousand. That further increase changes the premium by zero.

Check and trap avoided. The first increase would add $\$4.80$ thousand without a cap, but only $\$3.00$ thousand fits between the starting premium and the maximum. The later increase has no effect because the premium is already at the ceiling.

Reading the Premium Response Correctly

The marginal rate $Tf$ is not the same as the average response over every finite loss increase. In the minimum-crossing example, the premium rises by $\$3.00$ thousand for a $\$4.00$ thousand increase in $C$, an average of $\$0.75$ thousand per thousand dollars of $C$. In the maximum-crossing example, it rises by $\$3.00$ thousand for the same-sized increase, also an average of $\$0.75$ thousand per thousand dollars of $C$. In both cases, the uncapped marginal rate is $\$1.50$ or $\$1.20$, respectively; the lower average arises because part of the increase is blocked by a bound.

Position of the unbounded premium	Effect of a small increase in converted losses	Reason
Below the minimum	No premium increase, while the premium remains at the floor	The minimum replaces the lower unbounded amount
Between the bounds	Increase of $Tf$ per unit of $C$	Neither bound changes the unbounded premium
Above the maximum	No premium increase, while the premium remains at the ceiling	The maximum replaces the higher unbounded amount
Crossing a bound	Only the part of the unbounded increase within the bounds affects the premium	The bounded premium changes regime during the increase

A useful exam discipline is to calculate the unbounded premium at both ends of the stated loss change, then apply the minimum and maximum separately to each result. This procedure handles increases that remain in one region and those that cross one or both bounds. It also avoids a common error: applying the uncapped rate to the full loss change even when the premium starts at a bound or reaches one during the change.

The distinction matters when explaining who bears loss variability. Inside the bounds, the insured’s premium responds to converted losses at the rate $Tf$. At either bound, additional loss variability does not change the premium as long as the premium remains constrained there. This conclusion describes the premium formula’s response, not the size or probability of the losses themselves.

Key takeaway: First find the unbounded response $Tf\Delta C$; then apply the minimum and maximum to the starting and ending premiums. Use $Tf$ as the marginal sensitivity only while the premium is inside the bounds. At a bound, the response may be zero, or partial if the change crosses into the responsive region.

Check Your Understanding

For each hypothetical plan, calculate the change in the bounded premium. Unless stated otherwise, premiums and converted losses are in thousands of dollars.

A plan has $B=30$, $f=1.20$, $T=1.10$, $P_{\min}=50$, and $P_{\max}=90$. Converted losses increase from $C=10$ by $2$. What is the premium change? (Answer: $0.00$ thousand dollars.)
A plan has $B=40$, $f=1.50$, $T=1.10$, and bounds of 55 and 80. If $C$ increases from 10 to 13, what is the bounded premium increase? (Answer: $4.95$ thousand dollars.)
A plan’s unbounded premium is currently 88, its maximum is 90, and $Tf=1.25$. If $C$ increases by 4, what is the bounded premium increase? (Answer: $2.00$ thousand dollars.)
A plan has a minimum premium of 70. Its current unbounded premium is 68, and $Tf=1.00$. If $C$ increases by 5, what is the bounded premium increase? (Answer: $3.00$ thousand dollars.)
A plan’s current unbounded premium is 74, its bounds are 60 and 90, and $Tf=1.40$. If $C$ increases by 10, what is the bounded premium increase? (Answer: $14.00$ thousand dollars.)

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