Calculating the Lowest and Highest Possible Premiums

Contractual Limits and Possible Premiums

A retrospective premium responds to losses, but a contract may prevent the final premium from falling below a stated minimum or rising above a stated maximum. Those provisions create a contractual bracket. Calculating a premium for a particular loss outcome requires first calculating the unbounded premium, then applying both limits in the contractually specified way.

There is a useful distinction between the contractual bracket and the range of premiums that can actually result from the loss outcomes under consideration. A policy may specify a minimum of $70,000, but if the premium at zero losses is already $84,000, the contractual minimum cannot be attained under any nonnegative loss outcome in a plan whose premium increases with losses.

Definition: Let $P_{\min}$ be the contractual minimum final premium and $P_{\max}$ the contractual maximum final premium, with $P_{\min}\le P_{\max}$. Let $U(L)$ be the unbounded retrospective premium for aggregate losses $L$. The final premium is the unbounded premium restricted to the contractual bracket: it equals the minimum if $U(L)<P_{\min}$, equals $U(L)$ if $P_{\min}\le U(L)\le P_{\max}$, and equals the maximum if $U(L)>P_{\max}$.

For the standard linear retrospective formula used here, let $B$ denote the basic premium, $c$ the loss conversion factor, and $T$ the tax multiplier. We assume $B\ge 0$, $L\ge 0$, $c\ge 0$, and $T>0$. The unbounded premium and the final premium are

$$ U(L)=T(B+cL) $$

P(L)=\min\!\left(P_{\max},\,\max\!\left(P_{\min},\,U(L)\right)\right).

Here $cL$ is the converted loss component, and the tax multiplier is applied to the sum of the basic premium and converted losses. The stated $P_{\min}$ and $P_{\max}$ in this tutorial are final-premium limits: they are applied to the premium after the tax multiplier. A contract that defines its limits differently must be calculated according to that wording; the order cannot be inferred from the numbers alone.

Why the Bound Formula Works

The contract requires the final premium to be no less than $P_{\min}$ and no greater than $P_{\max}$. If the unbounded premium is below the permitted interval, the only way to satisfy the lower restriction is to set the final premium to $P_{\min}$. If it is within the interval, changing it would not be required by either restriction. If it is above the interval, the upper restriction requires setting it to $P_{\max}$. These three cases give the piecewise formula directly.

P(L)= \begin{cases} P_{\min}, & U(L)<P_{\min},\\ U(L), & P_{\min}\le U(L)\le P_{\max},\\ P_{\max}, & U(L)>P_{\max}. \end{cases}

Because $U(L)=T(B+cL)$ is nondecreasing in $L$ under the stated assumptions, the premium cannot decrease as losses increase. If aggregate losses can be any nonnegative amount and $c>0$, the lowest attainable premium is the final premium at zero losses, $P(0)$, and sufficiently large losses can reach the maximum. Thus, the actual range is from $P(0)$ to $P_{\max}$. The contractual minimum is the actual lowest premium only when $U(0)\le P_{\min}$. If losses are restricted to $0\le L\le L_{\max}$, the actual highest premium is $P(L_{\max})$; the contractual cap need not be reached. These range statements assume the loss outcomes can vary continuously over the stated interval. With discrete or otherwise restricted outcomes, only premiums corresponding to allowed loss values are attainable.

Key distinction: The contractual bracket is $[P_{\min},P_{\max}]$. For a specified set of possible losses, the attainable premium range is determined by applying the premium formula to the smallest and largest allowed losses. The two ranges need not be identical.

Worked Calculations

Worked Example: Premium Below, Within, and Above the Limits

A hypothetical plan has a basic premium of $60,000, a loss conversion factor of 0.50, and a tax multiplier of 1.10. Its final-premium minimum is $70,000 and its final-premium maximum is $120,000. Calculate the final premium at aggregate losses of $0, $40,000, and $120,000. Then state the attainable range if losses can take any nonnegative value.

Method. For each loss amount, calculate $U(L)=1.10(60{,}000+0.50L)$, then restrict that amount to the contractual interval from $70,000 to $120,000.

At $L=$0$. Converted losses are $0.50\times $0=$0$. Therefore,

$$ U(0)=1.10($60{,}000+$0)=1.10($60{,}000)=$66{,}000. $$

This is below the $70,000 minimum, so the final premium is $70,000.

At $L=$40,000$. Converted losses are $0.50\times $40{,}000=$20{,}000$. Thus,

$$ U($40{,}000)=1.10($60{,}000+$20{,}000) =1.10($80{,}000)=$88{,}000. $$

Since $88,000 is between $70,000 and $120,000, neither limit binds. The final premium is $88,000.

At $L=$120,000$. Converted losses are $0.50\times $120{,}000=$60{,}000$. Therefore,

$$ U($120{,}000)=1.10($60{,}000+$60{,}000) =1.10($120{,}000)=$132{,}000. $$

This exceeds the $120,000 maximum, so the final premium is $120,000.

At zero losses, the final premium is the contractual minimum of $70,000. As losses increase without an upper bound and $c>0$, the unbounded premium increases and reaches the contractual cap. The attainable range is therefore $70,000 to $120,000. The results check against the three cases: the first premium is at the floor, the second is unadjusted, and the third is at the cap. The calculation avoids the mistake of applying the cap to converted losses rather than to the final premium.

Worked Example: The Contractual Minimum Is Not the Lowest Attainable Premium

A hypothetical account has a basic premium of $100,000, a loss conversion factor of 0.40, and a tax multiplier of 1.08. Its final-premium limits are $95,000 and $160,000. Losses can be any nonnegative amount. Find the lowest and highest attainable final premiums, and calculate the final premium at losses of $50,000 and $150,000.

Find the lower endpoint. With zero losses, the converted loss amount is $0, so

U(0)=1.08($100{,}000+0.40\times $0) =1.08($100{,}000)=$108{,}000.

The zero-loss unbounded premium already exceeds the $95,000 contractual minimum. The final premium at zero losses is therefore $108,000, not $95,000. Since $U(L)$ is nondecreasing, no nonnegative loss amount can produce a lower premium. The lowest attainable premium is $108,000.

At $L=$50,000$. Converted losses are $0.40\times $50{,}000=$20{,}000$, giving

$$ U($50{,}000)=1.08($100{,}000+$20{,}000) =1.08($120{,}000)=$129{,}600. $$

This is within the contractual limits, so the final premium is $129,600.

At $L=$150,000$. Converted losses are $0.40\times $150{,}000=$60{,}000$, so

$$ U($150{,}000)=1.08($100{,}000+$60{,}000) =1.08($160{,}000)=$172{,}800. $$

The unbounded premium exceeds the $160,000 cap; the final premium is $160,000. As losses can increase without an upper bound and the loss-responsive factor is positive, the cap is attainable. The actual premium range is $108,000 to $160,000. This confirms why the contractual minimum should not automatically be reported as the lowest possible premium: the zero-loss premium must also be checked.

Worked Example: A Restricted Loss Range Does Not Reach the Cap

A hypothetical plan has a basic premium of $90,000, a loss conversion factor of 0.50, a tax multiplier of 1.10, and final-premium limits of $80,000 and $160,000. Aggregate losses are restricted to the interval from $0 to $60,000. Find the attainable premium range and calculate the final premium at losses of $0, $30,000, and $60,000.

At the minimum possible losses. At $L=$0$,

U(0)=1.10($90{,}000+0.50\times $0) =1.10($90{,}000)=$99{,}000.

This is within the contractual bracket, so the lowest attainable premium is $99,000. The $80,000 floor does not bind.

At $L=$30,000$. Converted losses are $0.50\times $30{,}000=$15{,}000$. Thus,

$$ U($30{,}000)=1.10($90{,}000+$15{,}000) =1.10($105{,}000)=$115{,}500. $$

This is between the limits, giving a final premium of $115,500.

At the maximum possible losses. At $L=$60,000$, converted losses are $0.50\times $60{,}000=$30{,}000$. Therefore,

$$ U($60{,}000)=1.10($90{,}000+$30{,}000) =1.10($120{,}000)=$132{,}000. $$

The result is below the $160,000 maximum, so the final premium is $132,000. Because the premium increases with losses over the stated interval, the actual attainable range is $99,000 to $132,000—not the full contractual bracket of $80,000 to $160,000. The endpoint calculations verify that neither contractual bound is reached within the allowed loss range.

Common Exam Trap: Applying a Bound Before the Tax Multiplier

A minimum or maximum stated as a final-premium provision is compared with the premium after the tax multiplier. Applying a floor to the pre-tax amount and then multiplying it by taxes can incorrectly raise the final premium. Conversely, clipping the final result using the wrong order can also change whether a limit binds.

Worked Example: Check the Final Premium Before Applying a Floor

A hypothetical plan has a basic premium of $70,000, losses of $20,000, a loss conversion factor of 0.75, and a tax multiplier of 1.10. The final-premium minimum is $85,000 and the maximum is $180,000. Calculate the final premium.

Converted losses are $0.75\times $20{,}000=$15{,}000$. The pre-tax amount is $ $70{,}000+$15{,}000=$85{,}000$. Apply the tax multiplier:

$$ U=1.10($70{,}000+$15{,}000) =1.10($85{,}000)=$93{,}500. $$

Since $85,000 is less than $93,500 and $93,500 is less than $180,000, neither limit binds. The final premium is $93,500.

Common mistake. Setting the premium to the $85,000 minimum because the pre-tax amount equals $85,000 would give the wrong answer. The minimum is a final-premium floor, and the correctly calculated final premium is already above it. The comparison must be made after applying the tax multiplier.

Calculation Checklist

Confirm what the limits apply to.
Check whether the stated minimum and maximum are final-premium amounts and whether the contract specifies a different calculation order.

Calculate the unbounded premium.
Convert losses using the stated loss conversion factor, add the basic premium, and apply the tax multiplier according to the plan formula.

Apply both bounds.
Raise an amount below the minimum to the minimum; leave an amount within the interval unchanged; reduce an amount above the maximum to the maximum.

Check attainability if asked for a range.
Evaluate the premium at the smallest and largest allowed losses. Do not assume the contractual minimum or maximum is attainable.

These calculations matter because a contract can limit how much of the loss variability is reflected in the final premium. For a particular loss outcome, the premium is determined by the unbounded calculation and the two contractual restrictions. For an overall range, the allowed loss outcomes matter as well. Stating both the contractual bracket and the attainable range makes clear whether a limit actually affects the scenario.

Check Your Understanding

For each problem, calculate the requested final premium or attainable range. Treat the stated minimum and maximum as final-premium limits.

A hypothetical plan has a basic premium of $40,000, loss conversion factor 0.50, tax multiplier 1.05, minimum $50,000, and maximum $100,000. Losses are $10,000. Calculate the final premium. (Answer: $50,000)
A plan has a basic premium of $55,000, loss conversion factor 0.40, tax multiplier 1.10, minimum $65,000, and maximum $120,000. Losses are $25,000. Calculate the final premium. (Answer: $71,500)
A plan has a basic premium of $90,000, loss conversion factor 0.30, tax multiplier 1.05, minimum $80,000, and maximum $125,000. Losses are $100,000. Calculate the final premium. (Answer: $125,000)
A plan has a basic premium of $70,000, loss conversion factor 0.20, tax multiplier 1.10, minimum $70,000, and maximum $110,000. Losses may be any nonnegative amount. Find the lowest and highest attainable premiums. (Answer: $77,000 to $110,000)
A plan has a basic premium of $50,000, loss conversion factor 0.50, tax multiplier 1.20, minimum $55,000, and maximum $100,000. Losses are restricted to the interval from $0 to $40,000. Find the attainable premium range. (Answer: $60,000 to $84,000)

What You'll Learn