Sensitivity to the Projected Loss Selection

Introduction

A rate indication depends on several inputs, including the projected losses used to represent the prospective period. If the selected projected loss estimate changes, the indication changes too. To understand that effect, compare selections while holding the premium base and every other input fixed. This isolates the sensitivity to projected losses rather than mixing it with changes in premium, expenses, or other assumptions.

The calculations below use a standard overall indication framework. The loss amounts are hypothetical and are assumed to be on the same coverage, expense, trend, and experience-period basis as the premium base. The examples are deterministic calculations: no interest rate, mortality assumption, or probability distribution is used.

Definition: A projected loss selection is the loss amount selected to represent losses expected for the prospective rating period, after the adjustments required for the indication have been made. In a sensitivity comparison, the premium base and all other indication assumptions are held constant while this selected loss amount varies.

The Indication Relationship

Let $L$ be the selected projected loss amount, $F$ the total provision for fixed expenses on the same basis, $P$ the premium at the current rate level, $V$ the variable expense ratio, and $Q$ the provision for profit and contingencies as a proportion of premium. Assume $P$ is positive, $V+Q<1$, and these quantities are consistent with one another. The indicated factor is the premium required under the assumptions divided by the current-rate-level premium base.

I(L)=\frac{L+F}{P(1-V-Q)}

This form treats fixed expenses as a total dollar amount and variable expenses and the profit-and-contingencies provision as proportions of premium. If fixed expenses are instead expressed per exposure, they must first be converted to a total on the same exposure basis. If the indication uses a different expense or profit structure, the equation must be adapted rather than applying this one mechanically.

The indicated percentage change is $(I(L)-1)\times 100\%$. An indicated factor of $1.04$, for example, means an indicated increase of 4%, not an increase of 104%. The sensitivity comparison concerns changes in $I(L)$; it does not, by itself, establish whether the selected losses are appropriate.

Why the Change Is Linear

When $P$, $F$, $V$, and $Q$ are fixed, the denominator in the indication equation is constant. For two possible loss selections $L_1$ and $L_2$, subtract their indicated factors:

\begin{aligned} I(L_2)-I(L_1) &=\frac{L_2+F}{P(1-V-Q)}-\frac{L_1+F}{P(1-V-Q)}\\ &=\frac{L_2-L_1}{P(1-V-Q)}. \end{aligned}

The fixed-expense provision cancels in the difference. Thus, for a fixed premium base and fixed provisions, a change in the selected projected losses changes the indicated factor in direct proportion to the change in losses. The direction is also immediate: because $P(1-V-Q)>0$, increasing $L$ increases $I(L)$. This result does not hold as a comparison of loss selection alone if the premium base or other assumptions also change.

Key result: With the premium base and all other inputs fixed, a loss-selection change of $\Delta L$ changes the indicated factor by $\Delta L/[P(1-V-Q)]$. To express the effect as a change in indicated percentage points, multiply that factor change by 100.

Worked Applications

Worked Example: Comparing Two Loss Selections

A hypothetical insurer uses a current-rate-level premium base of $12.00 million. The selected variable expense ratio is 12%, the profit-and-contingencies provision is 8% of premium, and fixed expenses total $0.60 million. Compare indications using projected losses of $8.04 million and $8.52 million. Keep the premium base and all provisions fixed.

Method and setup. Apply the overall indication equation separately to each loss selection. The common denominator is:

P(1-V-Q)=12.00(1-0.12-0.08)=12.00(0.80)=\$9.60\text{ million}.

For the $8.04 million selection, add fixed expenses to the projected losses and divide by the common denominator:

I(8.04)=\frac{8.04+0.60}{9.60} =\frac{8.64}{9.60}=0.9000.

The indicated percentage change is $(0.9000-1)\times100\%=-10.00\%$. For the $8.52 million selection:

I(8.52)=\frac{8.52+0.60}{9.60} =\frac{9.12}{9.60}=0.9500.

This corresponds to an indicated change of $(0.9500-1)\times100\%=-5.00\%$. The projected-loss selection increased by $0.48 million, so the indicated factor increased by:

\frac{8.52-8.04}{9.60} =\frac{0.48}{9.60}=0.0500.

Check and common mistake avoided. The direct difference, $0.9500-0.9000=0.0500$, agrees with the sensitivity formula. The indication moved upward by 5.00 percentage points, from a 10.00% decrease to a 5.00% decrease. It is incorrect to describe this as a 5% relative increase in the rate level; the calculation gives a change in indicated factor, or equivalently a 5-point change in the indicated percentage change.

Worked Example: Measuring Sensitivity per Dollar of Loss

For a hypothetical book, the current-rate-level premium base is $10.00 million, the variable expense ratio is 15%, the profit-and-contingencies provision is 5%, and fixed expenses are $0.40 million. A selected projected loss estimate is $6.80 million. Calculate the indication, then find the effect of increasing the selection by $200,000. Also find the factor change for each $100,000 change in projected losses.

Method and setup. First calculate the fixed denominator:

10.00(1-0.15-0.05)=10.00(0.80)=\$8.00\text{ million}.

The indication at $6.80 million of projected losses is:

I(6.80)=\frac{6.80+0.40}{8.00} =\frac{7.20}{8.00}=0.9000.

The indicated percentage change is $(0.9000-1)\times100\%=-10.00\%$. An increase of $200,000 is $0.20 million, so the new selection is $6.80+0.20=\$7.00$ million. Its indicated factor is:

I(7.00)=\frac{7.00+0.40}{8.00} =\frac{7.40}{8.00}=0.9250.

The indication therefore increases by $0.9250-0.9000=0.0250$, which is 2.50 percentage points. For each $100,000 change, or $0.10 million, the factor changes by:

\frac{0.10}{8.00}=0.0125.

Check and common mistake avoided. Two increments of $100,000 produce $2(0.0125)=0.0250$, matching the $200,000 result. A frequent error is to use the $10.00 million premium base alone as the denominator. The correct denominator is $8.00 million because the variable expense and profit provisions reduce the portion of premium available for losses and fixed expenses.

Worked Example: Indication From a Weighted Loss Selection

A hypothetical review team considers two projected loss estimates, $7.20 million and $7.60 million, and assigns them weights of 40% and 60%, respectively. The current-rate-level premium base is $10.00 million, the variable expense ratio is 10%, the profit-and-contingencies provision is 5%, and fixed expenses are $0.56 million. Find the weighted projected-loss selection and its indicated factor. Check the result against the weighted indications from the two individual selections.

Method and setup. The weighted selection is the sum of each estimate multiplied by its weight. The weights total 100%:

L_{\mathrm{sel}}=0.40(7.20)+0.60(7.60) =2.88+4.56=\$7.44\text{ million}.

The common denominator is:

10.00(1-0.10-0.05)=10.00(0.85)=\$8.50\text{ million}.

Using the weighted loss selection, the indicated factor is:

I(7.44)=\frac{7.44+0.56}{8.50} =\frac{8.00}{8.50}=0.941176\ldots

Rounded only at the final step to four decimal places, the indicated factor is 0.9412, corresponding to an indicated decrease of 5.88%. To check, calculate the individual factors:

I(7.20)=\frac{7.20+0.56}{8.50} =\frac{7.76}{8.50}=0.912941\ldots

I(7.60)=\frac{7.60+0.56}{8.50} =\frac{8.16}{8.50}=0.960000.

Weighting those factors gives $0.40(0.912941\ldots)+0.60(0.960000)=0.941176\ldots$, which agrees with the factor from the weighted loss selection. This agreement follows from the indication being linear in $L$ when its other inputs are fixed.

Check and common mistake avoided. The loss weights must be applied to the loss estimates, or equivalently to their indicated factors under these fixed assumptions. They must not be applied to the loss estimates and then added to fixed expenses a second time; fixed expenses enter the indication once.

Conditions for a Valid Sensitivity Comparison

A clean comparison changes only the loss selection. Before interpreting the result, confirm that the selected loss alternatives are suitable for the same indication setup. In particular:

The premium base is fixed and expressed at the same current rate level in both calculations.
The losses use a consistent coverage, exposure, and experience-period basis, including any required adjustments.
The treatment of loss adjustment expenses is consistent across selections.
The variable expense ratio, profit-and-contingencies provision, and fixed-expense amount do not change between calculations.
The loss selection is substituted once; adjustments already reflected in that selection are not applied again.

If one alternative reflects a different coverage mix or adjustment basis, the difference between indications cannot be attributed solely to loss selection. Reconcile the inputs first. Likewise, the linear formula measures the arithmetic effect of substituting an estimate; it does not establish whether that estimate is reasonable or appropriately supported.

Common exam trap: Do not divide a loss change by $P$ when the indication equation has a denominator of $P(1-V-Q)$. Also distinguish a factor change from a percentage change. A change from 0.90 to 0.95 is an increase of 0.05 in the factor and 5 percentage points in the indicated percentage change, from -10% to -5%.

Sensitivity analysis is useful because it shows how strongly an indication responds to an uncertain or judgmental loss selection without obscuring that effect with other changes. The resulting comparison is a diagnostic, not a substitute for selecting a well-supported projected loss estimate.

Check Your Understanding

For each problem, hold the premium base and every other stated assumption fixed. Amounts are hypothetical; use millions of dollars where stated.

A book has a $8.00 million premium base, a 10% variable expense ratio, a 5% profit-and-contingencies provision, and $0.20 million in fixed expenses. Projected losses are $5.08 million. Find the indicated factor and indicated percentage change. (Answer: factor 0.7765; indicated change -22.35%.)
A $15.00 million premium base has a 14% variable expense ratio, a 6% profit-and-contingencies provision, and $0.90 million in fixed expenses. By how much does the indicated factor change if selected projected losses increase by $0.30 million? (Answer: factor increases by 0.0250, or 2.50 percentage points.)
For a $10.00 million premium base, the variable expense ratio is 20%, the profit-and-contingencies provision is 5%, and fixed expenses are $0.50 million. Compare indicated factors using projected losses of $7.10 million and $7.50 million. (Answer: 1.0133 and 1.0667, respectively; the factor increases by 0.0533, or 5.33 percentage points.)
Two loss estimates, $6.20 million and $7.00 million, receive weights of 25% and 75%. The premium base is $9.00 million, the variable expense ratio is 10%, the profit-and-contingencies provision is 10%, and fixed expenses are $0.40 million. Find the weighted loss selection and indicated factor. (Answer: weighted loss selection $6.80 million; factor 1.0000.)
A $20.00 million premium base has a 15% variable expense ratio, a 5% profit-and-contingencies provision, and $0.80 million in fixed expenses. What projected loss amount produces an indicated factor of 1.05? (Answer: $16.00 million.)

What You'll Learn