Why Compare Premiums Across Loss Scenarios?
A retrospective rating plan makes the insured’s final premium responsive to losses, subject to contractual limits. Calculating one scenario shows the premium for one outcome; comparing several scenarios reveals how much the premium changes as converted losses change, and whether a minimum or maximum premium reduces that change. The comparison is meaningful only when the plan terms are held fixed across the scenarios.
The essential sequence is to calculate the premium before bounds, then apply the minimum and maximum. In particular, zero converted losses do not necessarily mean a zero premium: the plan’s basic premium still contributes to the calculation, and a contractual minimum may raise the result further.
The word uncapped here means that neither the minimum nor maximum premium has been applied; it does not mean that losses themselves have been limited or converted in some particular way. The converted loss \(C\) is the value supplied to the premium formula under the stated plan terms. This comparison focuses on the premium outcomes given those values.
The formula’s order matters. First calculate \(P_{\mathrm{raw}}\); next raise it to \(P_{\min}\) if it is below the minimum; finally reduce it to \(P_{\max}\) if it is above the maximum. A valid plan has \(P_{\min}\leq P_{\max}\). With that condition, applying the bounds is equivalent to the min-max expression above.
A Justification for the Comparison
For fixed \(B\), \(k\), and \(T\), the uncapped premium is a straight-line function of converted losses. Subtracting the premium at one loss value from the premium at another gives the change in premium directly:
The basic premium cancels because it is identical in both scenarios. Thus, when \(T\) and \(k\) are positive, each additional unit of converted loss increases the uncapped premium by \(Tk\) units. Bounds alter that response only where the raw premium falls below the minimum or rises above the maximum.
To see why bounding cannot increase the difference between two outcomes, define \(g(x)=\min\{P_{\max},\max\{P_{\min},x\}\}\). This function is constant below the minimum and above the maximum, and increases one-for-one between them. Therefore, for \(x_1\leq x_2\), \(0\leq g(x_2)-g(x_1)\leq x_2-x_1\). Applied to two raw premiums, this shows that the bounded premium difference is no greater than the uncapped difference. It may be equal when both outcomes are between the bounds, or smaller when a bound applies.
Compare Outcomes Under the Same Plan
For a clear comparison, keep the plan terms fixed, calculate the raw premium separately for each scenario, and then apply the same bounds to every result. A compact table helps prevent the common error of comparing a bounded result in one scenario with an uncapped result in another.
Worked Example: A Zero-Loss Scenario and Two Higher-Loss Outcomes
A hypothetical plan has basic premium \(B=30.000\), loss conversion factor \(k=1.20\), tax multiplier \(T=1.10\), minimum premium \(P_{\min}=35.000\), and maximum premium \(P_{\max}=90.000\). Premiums and converted losses are in thousands of dollars. Calculate the uncapped and final premiums for \(C=0.000\), \(10.000\), and \(40.000\), and compare the results.
Method. For each scenario, substitute its converted loss in \(P_{\mathrm{raw}}=T(B+kC)\), then apply the minimum and maximum. The common mistake to avoid is treating zero converted losses as zero raw premium; the basic premium remains in the formula.
At \(C=0.000\). The loss-responsive component is \(kC=1.20(0.000)=0.000\). Thus, \(P_{\mathrm{raw}}=1.10(30.000+0.000)=33.000\). Since \(33.000<35.000\), the minimum applies, so \(P=35.000\).
At \(C=10.000\). The converted-loss component is \(1.20(10.000)=12.000\). The raw premium is \(1.10(30.000+12.000)=1.10(42.000)=46.200\). This is between \(35.000\) and \(90.000\), so \(P=46.200\).
At \(C=40.000\). The converted-loss component is \(1.20(40.000)=48.000\). The raw premium is \(1.10(30.000+48.000)=1.10(78.000)=85.800\). This is also between the bounds, so \(P=85.800\).
The final premiums are \(35.000\), \(46.200\), and \(85.800\) thousand dollars, respectively. From zero to \(10.000\) of converted losses, the raw premium rises by \(46.200-33.000=13.200\), while the final premium rises by \(46.200-35.000=11.200\); the floor changes the first outcome. From \(10.000\) to \(40.000\), neither outcome is bounded, and the premium rises by \(85.800-46.200=39.600\), exactly \(Tk(40.000-10.000)=1.10(1.20)(30.000)=39.600\). The calculation is consistent with the formula and the stated bounds.
Worked Example: Locating the Boundaries Before Comparing Scenarios
A hypothetical plan has \(B=24.000\), \(k=1.25\), \(T=1.08\), \(P_{\min}=30.000\), and \(P_{\max}=75.000\), with all monetary amounts in thousands of dollars. Calculate final premiums at converted losses \(C=0.000\), \(20.000\), \(40.000\), and \(50.000\). Then identify approximately where the minimum and maximum begin to apply.
Method. First simplify the raw-premium equation: \(P_{\mathrm{raw}}=1.08(24.000+1.25C)=25.920+1.350C\). The minimum applies when this expression is below \(30.000\); the maximum applies when it is above \(75.000\). Do not compare \(C\) directly with the premium bounds, because converted loss and premium are different quantities.
At \(C=0.000\), \(P_{\mathrm{raw}}=25.920+1.350(0.000)=25.920\). Since this is below \(30.000\), the final premium is \(30.000\).
At \(C=20.000\), \(P_{\mathrm{raw}}=25.920+1.350(20.000)=25.920+27.000=52.920\). This is between the bounds, so the final premium is \(52.920\).
At \(C=40.000\), \(P_{\mathrm{raw}}=25.920+1.350(40.000)=25.920+54.000=79.920\). This exceeds \(75.000\), so the final premium is \(75.000\).
At \(C=50.000\), \(P_{\mathrm{raw}}=25.920+1.350(50.000)=25.920+67.500=93.420\). This also exceeds the maximum, so the final premium is \(75.000\). The final premium sequence is therefore \(30.000\), \(52.920\), \(75.000\), and \(75.000\) thousand dollars.
To find the converted-loss level where the minimum stops applying, set the raw premium equal to the minimum: \(25.920+1.350C=30.000\), so \(C=4.080/1.350=3.0222\) (approximately). To find where the maximum starts applying, solve \(25.920+1.350C=75.000\), giving \(C=49.080/1.350=36.3556\) (approximately). Below the first threshold the floor applies; above the second threshold the ceiling applies. At the thresholds themselves, the raw premium equals the corresponding bound. The scenario results agree with these thresholds: zero is below the minimum threshold, \(20.000\) is between thresholds, and \(40.000\) and \(50.000\) exceed the maximum threshold.
Worked Example: Comparing the Range of Premium Outcomes
Use the hypothetical plan from the first worked example: \(B=30.000\), \(k=1.20\), \(T=1.10\), \(P_{\min}=35.000\), and \(P_{\max}=90.000\), with monetary amounts in thousands of dollars. Compare premiums at \(C=0.000\), \(5.000\), \(45.000\), and \(55.000\). What are the uncapped and bounded ranges across these scenarios?
Method. Calculate the raw premium and final premium for each loss value using the same plan terms. The common mistake to avoid is calculating a range from raw premiums and then describing it as the range of final premiums.
At \(C=0.000\), \(P_{\mathrm{raw}}=1.10(30.000+1.20(0.000))=33.000\), so the minimum applies and \(P=35.000\).
At \(C=5.000\), \(P_{\mathrm{raw}}=1.10(30.000+6.000)=39.600\). This is within the bounds, so \(P=39.600\).
At \(C=45.000\), \(P_{\mathrm{raw}}=1.10(30.000+54.000)=92.400\). This exceeds the maximum, so \(P=90.000\).
At \(C=55.000\), \(P_{\mathrm{raw}}=1.10(30.000+66.000)=105.600\). This also exceeds the maximum, so \(P=90.000\).
The smallest and largest raw premiums are \(33.000\) and \(105.600\), so the uncapped range is \(105.600-33.000=72.600\) thousand dollars. The smallest and largest final premiums are \(35.000\) and \(90.000\), so the bounded range is \(90.000-35.000=55.000\) thousand dollars. The bounds reduce the range by \(72.600-55.000=17.600\) thousand dollars across these selected scenarios. This range comparison describes these four outcomes; it does not by itself calculate a probability distribution or variance for premium.
Interpreting Premium Variability
Across a set of loss scenarios, a lower final-premium range than raw-premium range indicates that the contractual bounds have reduced the spread of those outcomes. A minimum can raise low-premium scenarios, a maximum can lower high-premium scenarios, and either bound can make additional changes in converted losses produce no change in final premium while it remains binding. Between the bounds, the premium responds to converted losses at the uncapped rate \(Tk\).
This interpretation should be kept precise. A list of scenarios supports comparisons such as the difference between two premiums or the range across the listed outcomes. It does not specify how likely each loss scenario is. To calculate a variance or other probability-weighted measure of premium variability, one would also need a probability model for converted losses. The bounds make the final-premium function nonlinear, so one cannot generally calculate its variance simply by multiplying the variance of converted losses by \((Tk)^2\).
Check Your Understanding
For each problem, calculate or interpret the outcomes using the stated plan terms. Monetary amounts are in thousands of dollars unless specified otherwise.
- A hypothetical plan has \(B=18.000\), \(k=1.40\), \(T=1.05\), \(P_{\min}=22.000\), and \(P_{\max}=60.000\). Find the final premiums for \(C=0.000\), \(10.000\), and \(30.000\). (Answer: \(22.000\), \(33.600\), and \(60.000\) thousand dollars.)
- A plan has \(B=40.000\), \(k=0.90\), \(T=1.10\), \(P_{\min}=45.000\), and \(P_{\max}=100.000\). Calculate the raw and final premium at \(C=0.000\), and the final premiums at \(C=20.000\) and \(50.000\). (Answer: At \(C=0.000\), raw premium \(44.000\) and final premium \(45.000\); at \(C=20.000\), \(63.800\); at \(C=50.000\), \(93.500\) thousand dollars.)
- A plan has \(B=25.000\), \(k=1.20\), \(T=1.04\), \(P_{\min}=28.000\), and \(P_{\max}=80.000\). At what converted-loss values does the raw premium equal the minimum and maximum? (Answer: \(C\approx1.603\) and \(C\approx43.269\) thousand dollars, respectively.)
- A plan has \(B=32.000\), \(k=1.10\), \(T=1.05\), \(P_{\min}=35.000\), and \(P_{\max}=85.000\). For \(C=4.000\), \(24.000\), and \(54.000\), calculate the final premiums and the difference between the first and last final premiums. (Answer: \(38.220\), \(61.320\), and \(85.000\) thousand dollars; the difference is \(46.780\) thousand dollars.)
- A plan has \(B=20.000\), \(k=1.50\), \(T=1.10\), \(P_{\min}=25.000\), and \(P_{\max}=70.000\). Compare final premiums at \(C=8.000\) and \(C=18.000\), and state whether either bound applies. (Answer: \(35.200\) and \(51.700\) thousand dollars; neither bound applies, and the premium difference is \(16.500\) thousand dollars.)