Tutorials › CAS Exam 8 - Advanced Ratemaking › The Retrospective Premium Formula

Home  /  Tutorials  /  CAS Exam 8 / Retrospective Rating Structure

CAS Exam 8 · Retrospective Rating Structure · Tutorial 592 of 989

The Retrospective Premium Formula

Calculate the standard retrospective premium from its components, then apply the minimum and maximum in the correct order.

Intermediate 10 min read

What You'll Learn

  • Identify the inputs to the standard retrospective premium formula
  • Calculate converted losses using a plan’s loss conversion factor
  • Apply the tax multiplier to the correct subtotal
  • Distinguish an unbounded indicated premium from the final premium
  • Apply the minimum and maximum after the indicated premium is calculated
  • Recognize common formula and bounds-ordering errors

Purpose and Structure of the Formula

A retrospective premium links part of an insured’s premium to losses observed during a specified experience period, while retaining contractual adjustments and limits. The standard formula is compact, but a calculation can go wrong if the components are combined in the wrong order. In particular, the premium bounds apply to the calculated premium, not to one of its components.

The formula below uses amounts supplied or defined by the plan. It is a contractual calculation: the claim amounts are treated as observed inputs, and no claim-frequency or claim-severity distribution is needed to calculate the premium. The examples use hypothetical policies and amounts.

Definition: In the standard retrospective premium formula, the basic premium is the plan’s specified fixed premium component; converted losses are the loss amount adjusted by the plan’s loss conversion factor; the tax multiplier is the plan’s multiplier applied to the sum of basic premium and converted losses; and the minimum and maximum premiums are the contractual floor and ceiling applied to the indicated premium.

Let \(B\) be the basic premium, \(L\) the loss amount recognized by the plan, \(c\) the loss conversion factor, \(t\) the tax multiplier, \(m\) the minimum premium, and \(M\) the maximum premium. The plan’s terms determine exactly which losses enter \(L\) and the values of its factors and bounds. The standard calculation is:

$$ \text{Converted losses}=cL $$
$$ \text{Unbounded indicated premium}=t(B+cL) $$

The final premium is the indicated premium restricted to the interval from \(m\) to \(M\). This standard ordering assumes \(m\leq M\) and that both bounds apply to the premium after the tax multiplier.

$$ P_{\mathrm{retro}}=\min\!\left(\max\!\left[t(B+cL),m\right],M\right) $$

If the indicated premium is below \(m\), the final premium is \(m\). If it is between \(m\) and \(M\), the final premium is unchanged. If it is above \(M\), the final premium is \(M\). A plan’s governing wording controls if it specifies a different sequence or defines its bounds on a different basis.

Why the Formula Has This Order

The formula can be understood as two operations. First, the plan converts losses and combines them with the basic premium. This produces the subtotal to which the tax multiplier applies. Second, the contract limits the resulting premium. The bounds are not another loading on the subtotal; they constrain the premium produced by the first operation.

The conversion step follows directly from the definition of the loss conversion factor. If the recognized losses are \(L\), then applying a factor \(c\) gives \(cL\). Adding the fixed component gives \(B+cL\). Applying the specified multiplier to that subtotal gives \(t(B+cL)\). To restrict any value \(x\) to the interval \([m,M]\), first raise it to at least \(m\), then lower it to at most \(M\):

$$ \max(x,m)= \begin{cases} m, & x<m,\\ x, & x\geq m, \end{cases} \qquad \min(\max(x,m),M)= \begin{cases} m, & x<m,\\ x, & m\leq x\leq M,\\ M, & x>M. \end{cases} $$

Substituting \(x=t(B+cL)\) gives the stated bounded formula. The cases also show why both bounds should be applied: using only a minimum leaves an excessive premium uncapped, while using only a maximum can leave a premium below the contractual floor.

Worked Calculations

Worked Example: Indicated Premium Within Both Bounds

A hypothetical policy has a basic premium of $18,000. The plan recognizes $42,500 of losses, uses a loss conversion factor of 1.10, and specifies a tax multiplier of 1.08. Its minimum and maximum premiums are $25,000 and $70,000. Calculate the final retrospective premium.

Method. Convert the recognized losses, add the basic premium, apply the tax multiplier, and then test the result against both bounds. The loss amounts are observed totals; no probability distribution, interest rate, or mortality assumption is used.

Step 1: Convert losses.

$$ cL=1.10(42{,}500)=46{,}750 $$

Step 2: Add the basic premium.

$$ B+cL=18{,}000+46{,}750=64{,}750 $$

Step 3: Apply the tax multiplier.

$$ t(B+cL)=1.08(64{,}750)=69{,}930 $$

The indicated premium of $69,930 is above the $25,000 minimum and below the $70,000 maximum. Therefore, neither bound changes it, and the final retrospective premium is $69,930.

Check and common mistake. The result lies within the stated interval. Multiplying only the converted losses by 1.08 and then adding the basic premium would give \(46{,}750(1.08)+18{,}000=68{,}490\), which is incorrect: the formula applies the multiplier to the sum, not just to converted losses.

Worked Example: Maximum Premium Is Binding

A hypothetical account has a basic premium of $24,000, recognized losses of $61,000, a loss conversion factor of 1.15, and a tax multiplier of 1.06. The contractual minimum is $35,000 and the maximum is $92,000. Find the final premium.

Method. Calculate the unbounded indicated premium first, then apply the ceiling. The loss amount is an observed total; no probability distribution, interest rate, or mortality assumption is needed.

Step 1: Calculate converted losses.

$$ 1.15(61{,}000)=70{,}150 $$

Step 2: Add the basic premium.

$$ 24{,}000+70{,}150=94{,}150 $$

Step 3: Apply the tax multiplier.

$$ 1.06(94{,}150)=99{,}799 $$

The indicated premium of $99,799 exceeds the $92,000 maximum. It also exceeds the $35,000 minimum, so the minimum does not affect the result. Applying the maximum gives a final premium of $92,000.

Check and common mistake. The final amount equals the ceiling, as required when the indicated premium is above it. A tempting but incorrect approach is to cap the pre-multiplier subtotal at $92,000 and then multiply: \(92{,}000(1.06)=97{,}520\). That still exceeds the maximum because the bound is applied to the premium after the tax multiplier.

Worked Example: Minimum Premium Is Binding

A hypothetical policy has a basic premium of $12,500, recognized losses of $5,600, a loss conversion factor of 1.25, and a tax multiplier of 1.10. The minimum premium is $24,000 and the maximum is $60,000. Calculate the amount due under the retrospective formula.

Method. Calculate the indicated premium and compare it with each bound. The losses are observed amounts; no probability distribution, interest rate, or mortality assumption is used.

Step 1: Convert losses.

$$ 1.25(5{,}600)=7{,}000 $$

Step 2: Add the basic premium.

$$ 12{,}500+7{,}000=19{,}500 $$

Step 3: Apply the tax multiplier.

$$ 1.10(19{,}500)=21{,}450 $$

The indicated premium of $21,450 is below the $24,000 minimum, so the floor applies. Since $24,000 is also below the $60,000 maximum, the final retrospective premium is $24,000.

Check and common mistake. The final premium is at least the minimum and no greater than the maximum. Applying the minimum to the pre-multiplier subtotal and then multiplying would give \(24{,}000(1.10)=26{,}400\), but that incorrectly treats the minimum as a subtotal bound. It is a bound on the calculated premium.

Interpreting the Inputs and Avoiding Traps

The formula is straightforward only after the inputs have been identified correctly. Use the values and definitions specified by the plan rather than inferring them from labels alone.

  • Basic premium: use the plan’s stated amount. Do not treat it as an additional loss or multiply it separately from the converted-loss subtotal.
  • Recognized losses: \(L\) means the loss measure specified by the plan. If the contract defines an adjusted loss amount, use that amount; do not silently substitute a different measure.
  • Loss conversion factor: apply \(c\) to the recognized losses to obtain converted losses. Do not apply it to the basic premium.
  • Tax multiplier: use the multiplier supplied by the plan. Do not assume it is a tax rate or automatically replace it with one plus a tax rate.
  • Premium bounds: compare the unbounded indicated premium with both bounds, then use the contractual floor or ceiling if needed. Do not apply the bounds to losses or to the subtotal before multiplying.

A reliable exam calculation labels each intermediate amount: recognized losses, converted losses, pre-multiplier subtotal, unbounded indicated premium, and final bounded premium. This makes both the arithmetic and the order of operations visible. It is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Key takeaway: Calculate \(cL\), add \(B\), multiply the entire subtotal by \(t\), and then restrict the result to the interval from \(m\) to \(M\). The minimum and maximum apply to the calculated premium, in that order.

Check Your Understanding

For each hypothetical plan, calculate the final retrospective premium using the standard formula. Assume each stated loss amount is the amount recognized by the plan.

  1. A plan has \(B=\)$15,000, \(L=\)$20,000, \(c=1.10\), and \(t=1.05\). Its minimum is $18,000 and maximum is $50,000. Find the final premium. (Answer: $38,850)
  2. A plan has \(B=\)$32,000, \(L=\)$50,000, \(c=1.20\), and \(t=1.05\). Its bounds are $40,000 and $90,000. Find the final premium. (Answer: $90,000)
  3. A plan has \(B=\)$20,000, \(L=\)$16,000, \(c=1.25\), and \(t=1.10\). Its minimum is $46,000 and maximum is $70,000. Find the final premium. (Answer: $46,000)
  4. A plan has \(B=\)$30,000, \(L=\)$40,000, \(c=1.10\), and \(t=1.08\), with a $25,000 minimum and an $80,000 maximum. Calculate the unbounded indicated premium and then the final premium. (Answer: indicated premium $79,920; final premium $79,920)
  5. For a plan with \(B=\)$21,000, \(L=\)$45,000, \(c=1.20\), and \(t=1.04\), the minimum is $30,000 and the maximum is $85,000. Calculate the converted losses, the pre-multiplier subtotal, and the final premium. (Answer: converted losses $54,000; subtotal $75,000; final premium $78,000)

Previous

Next

Back to Tutorials
Clinical Biostats