Sensitivity to Projected Losses

Why Measure Sensitivity to Projected Losses?

An indicated rate depends on estimates, and projected losses are often a major component of the indication. If that projection changes, the indicated rate changes too. Measuring the effect helps an actuary explain how much of a revised indication is attributable to the new loss estimate, while separating that effect from changes in expenses, profit provisions, or other assumptions.

This analysis is a controlled comparison: change projected losses and hold the other inputs fixed. It does not establish whether a revised loss projection is more credible or preferable. It quantifies the rate consequence conditional on the revised projection. The calculations below use a simplified overall indication model; the earlier material on assembling an overall rate indication provides broader context for the provisions.

Definition: In this tutorial, $L$ is projected losses per exposure, $F$ is fixed expenses and any other fixed dollar provisions per exposure, $v$ is the variable expense provision as a fraction of premium, and $q$ is the profit and contingency provision as a fraction of premium. The indicated rate per exposure is $R$. The model assumes the exposure base and $F$, $v$, and $q$ are unchanged when projected losses are varied, and that $v+q<1$.

Under this model, premium must cover losses and fixed dollar provisions, as well as the variable provisions that are calculated as percentages of premium. Thus the indicated rate satisfies $R=L+F+(v+q)R$. Solving for $R$ gives the central relationship used throughout this tutorial.

R=\frac{L+F}{1-v-q}.

If inputs are aggregate amounts rather than per-exposure amounts, the same equation applies to the aggregate indicated premium, provided the same exposure base and provisions are used in both comparisons. Dividing by a fixed exposure count then converts the result to a per-exposure rate.

How the Indicated Rate Responds to Loss Changes

Write $c=v+q$, where $c$ is the combined premium-based provision. With $F$, $v$, and $q$ held fixed, a change in losses from $L$ to $L+\Delta L$ changes the indication to $R_{\text{new}}$:

R_{\text{new}}=\frac{L+\Delta L+F}{1-v-q}, \qquad R_{\text{new}}-R=\frac{\Delta L}{1-v-q}.

The dollar sensitivity is therefore the loss change divided by the share of premium remaining after variable provisions. Because $0<1-v-q\leq 1$ in the usual case of nonnegative provisions, the indicated-rate change is at least as large as the loss change in dollars. The extra amount is needed to fund the variable provisions on the additional premium.

The percentage effect is not generally the same percentage as the change in projected losses. Dividing the rate change by the original indicated rate gives:

\frac{R_{\text{new}}-R}{R} = \frac{\Delta L}{L+F}.

If losses increase by a proportion $x$, so that $\Delta L=xL$, the percentage change in the indication is $xL/(L+F)$. Since $F$ is held fixed, the rate change is usually a smaller percentage than the loss change. The result also shows why the loss share of the fixed-dollar portion of the indication matters: if losses are a larger share of $L+F$, the indication is more sensitive in percentage terms to a proportional loss change.

Deriving the Sensitivity

The finite-change formula follows directly from the indication equation, without an approximation. Starting with the revised and original indications, subtract the latter from the former:

\begin{aligned} R_{\text{new}}-R &=\frac{L+\Delta L+F}{1-v-q}-\frac{L+F}{1-v-q}\\ &=\frac{(L+\Delta L+F)-(L+F)}{1-v-q}\\ &=\frac{\Delta L}{1-v-q}. \end{aligned}

This derivation depends on the denominator and $F$ being the same in both indications. If the loss change is accompanied by a change in variable expenses, profit provisions, fixed expenses, or exposures, then subtracting the two full indications requires changing those inputs as well; the formula above would no longer isolate the total effect.

For a small change in losses, the derivative expresses the local sensitivity in dollars of indicated rate per dollar of projected losses:

\frac{\partial R}{\partial L}=\frac{1}{1-v-q}.

In this model the derivative is constant with respect to $L$, so the same dollar sensitivity applies to finite loss changes as well as small ones. The relative sensitivity, or elasticity of $R$ with respect to $L$, is $L/(L+F)$. This is a percentage comparison, not a dollar multiplier; it follows by multiplying the derivative by $L/R$.

Worked Applications

Worked Example: A Percentage Increase in Projected Losses

A hypothetical commercial property program has projected losses of $420 per exposure and fixed expenses of $80 per exposure. Variable expenses are 18% of premium, and the profit and contingency provision is 4% of premium. Calculate the original indication, then find the revised indication and its percentage change if projected losses increase by 5%. Hold all other assumptions fixed.

Method. Use the overall rate indication with $L=420$, $F=80$, $v=0.18$, and $q=0.04$. First calculate the denominator:

$$ 1-v-q=1-0.18-0.04=0.78. $$

The original indication is:

R=\frac{420+80}{0.78} =\frac{500}{0.78} =641.025641\ldots \approx \$641.03.

A 5% increase in losses is $\Delta L=0.05(420)=\$21.00$, so revised losses are $420+21=\$441.00$. The revised indication is:

R_{\text{new}}=\frac{441+80}{0.78} =\frac{521}{0.78} =667.948718\ldots \approx \$667.95.

The dollar increase is $667.948718\ldots-641.025641\ldots=\$26.923077\ldots$, or $\$26.92$. The sensitivity formula gives the same result: $21/0.78=\$26.923077\ldots$. The percentage increase in the indication is $26.923077\ldots/641.025641\ldots=0.042=4.2\%$. This also equals $5\%\times420/(420+80)=4.2\%$.

Check. The revised indication exceeds the original because losses increased, and the 4.2% indicated increase is less than the 5% loss increase because fixed expenses did not increase. The common mistake avoided is applying the 5% loss change to the entire indication: that would give $641.025641\ldots(1.05)=\$673.08$, rather than the correctly calculated $\$667.95$.

Worked Example: Separating the Indicated Rate From the Rate Change

A hypothetical liability program has projected losses of $390 per exposure, fixed expenses of $90 per exposure, variable expenses of 20% of premium, and a profit and contingency provision of 5% of premium. The current rate is $600 per exposure. Find the indicated rate and indicated rate change relative to the current rate. Then calculate both again after projected losses increase by 8%, holding the other inputs fixed.

Method. The combined premium-based provision is $0.20+0.05=0.25$, so the denominator is $1-0.25=0.75$. The original indication is:

R=\frac{390+90}{0.75} =\frac{480}{0.75} =\$640.00.

Relative to the current rate of $600, the indicated rate change is:

\frac{640-600}{600} =\frac{40}{600} =0.066666\ldots \approx 6.67\%.

The 8% increase in projected losses is $0.08(390)=\$31.20$, making revised losses $390+31.20=\$421.20$. The revised indication is:

R_{\text{new}}=\frac{421.20+90}{0.75} =\frac{511.20}{0.75} =\$681.60.

The indication increased by $\$681.60-\$640.00=\$41.60$, which also equals $31.20/0.75$. Relative to the current rate, the revised indicated rate change is:

\frac{681.60-600}{600} =\frac{81.60}{600} =0.136 =13.60\%.

Thus the increase in the indicated rate change is $13.60\%-6.67\%=6.93$ percentage points, using unrounded values for the subtraction: $41.60/600=6.9333\ldots$ percentage points. It is not an 8-percentage-point increase. The 8% refers to projected losses; the effect on the rate change is measured against the current rate and reflects the fixed expense amount and the variable provisions.

Check. The revised indication of $681.60 is above both the original indication and the current rate. The common mistake avoided is confusing a percentage change in the loss estimate with a percentage-point change in the indicated rate change.

Worked Example: Aggregate Losses and a Rate Sensitivity Check

A hypothetical program covers 1,200 exposures. Its projected aggregate losses are $720,000, and aggregate fixed expenses are $96,000. Variable expenses are 22% of premium, and the profit and contingency provision is 3% of premium. Calculate the aggregate indicated premium and the indicated premium after projected aggregate losses increase by 10%. Also find the percentage increase in the indication.

Method. Here $L=\$720{,}000$, $F=\$96{,}000$, and $1-v-q=1-0.22-0.03=0.75$. The original aggregate indicated premium is:

R=\frac{720{,}000+96{,}000}{0.75} =\frac{816{,}000}{0.75} =\$1{,}088{,}000.

The loss increase is $0.10(720{,}000)=\$72{,}000$, so revised aggregate losses are $\$792{,}000$. The revised indicated premium is:

R_{\text{new}}=\frac{792{,}000+96{,}000}{0.75} =\frac{888{,}000}{0.75} =\$1{,}184{,}000.

The indicated premium rises by $\$1{,}184{,}000-\$1{,}088{,}000=\$96{,}000$. The direct sensitivity calculation agrees: $\$72{,}000/0.75=\$96{,}000$. Its percentage increase is $\$96{,}000/\$1{,}088{,}000=0.088235294\ldots$, or 8.8235% to four decimal places.

As a check using per-exposure amounts, projected losses are $\$720{,}000/1{,}200=\$600$, and fixed expenses are $\$96{,}000/1{,}200=\$80$. The original indication is $(600+80)/0.75=\$906.6667$ per exposure, and the revised indication is $(660+80)/0.75=\$986.6667$ per exposure. Each has the corresponding aggregate amount when multiplied by 1,200.

Check. The indication increases by less than 10% because fixed expenses remain unchanged; specifically, the percentage increase is $10\%\times720{,}000/(720{,}000+96{,}000)=8.8235\%$. The common mistake avoided is treating a percentage loss shock as the same percentage change in premium.

Common Mistakes and Scope of the Result

Applying the loss percentage to the entire indication. A percentage shock to losses is not automatically the same percentage shock to the indicated rate. With fixed expenses, the percentage effect is scaled by $L/(L+F)$.
Forgetting the variable-provision denominator. The dollar rate impact is $\Delta L/(1-v-q)$, not simply $\Delta L$, when those provisions are percentages of premium.
Mixing dollars and percentage points. The indicated rate change relative to current rates is a ratio with the current rate as its denominator. Report a change in that ratio in percentage points when comparing two indicated rate changes.
Changing more than losses while using a one-input sensitivity. The formula isolates losses only when fixed expenses, premium-based provision percentages, exposure counts, and other relevant assumptions are unchanged.
Confusing projected losses per exposure with aggregate losses. Either basis works, but all terms in the numerator must use the same basis. Divide or multiply by the unchanged exposure count only after the basis is clear.

For example, suppose an indication is $1,000, with $L=\$600$, $F=\$200$, and $v+q=0.20$, so $(600+200)/0.80=\$1,000$. If losses rise by 12.5%, the loss increase is $0.125(600)=\$75$, and the actual indicated-rate increase under the model is $75/0.80=\$93.75$. Applying 12.5% directly to the entire original indication gives $1{,}000(1.125)=\$1{,}125$, an increase of $\$125$, not $\$25$. That incorrect shortcut overstates the modeled increase by $\$125-\$93.75=\$31.25$.

Key takeaway: With the exposure base, fixed expenses, and premium-based provisions held constant, a loss change of $\Delta L$ changes the indicated rate by $\Delta L/(1-v-q)$. For a proportional loss change, the percentage impact on the indication also depends on the share of losses in the fixed-dollar portion, $L/(L+F)$.

This sensitivity calculation is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. It explains the mathematical consequence of a changed loss projection; selecting the projection itself and evaluating changes in other assumptions require separate judgment and analysis.

Check Your Understanding

For each problem, hold all assumptions other than projected losses fixed unless stated otherwise. Round dollar answers to the nearest cent and percentages to four decimal places where needed.

A hypothetical program has projected losses of $360 per exposure, fixed expenses of $90, variable expenses of 15% of premium, and profit and contingency of 5% of premium. Find the original indication and the revised indication after losses increase by 12%. (Answer: Original indication $562.50; revised indication $616.50.)
A book has 800 exposures, aggregate projected losses of $480,000, and aggregate fixed expenses of $64,000. Variable expenses are 20% of premium and profit and contingency are 4%. Find the original indicated rate per exposure and the revised rate if aggregate losses increase by 5%. (Answer: Original $894.74 per exposure; revised $934.21 per exposure.)
Projected losses are $450 per exposure and fixed expenses are $150. Variable expenses plus profit and contingency equal 25% of premium. If losses increase by 6%, what is the percentage increase in the indicated rate? (Answer: 4.5000%.)
Projected losses are $560 per exposure and fixed expenses are $140. Variable expenses plus profit and contingency equal 20% of premium. What is the largest proportional increase in projected losses that results in an indicated-rate increase of no more than 7%? (Answer: 8.7500%.)
A hypothetical program has projected losses of $520 per exposure, fixed expenses of $80, and variable expenses plus profit and contingency of 20% of premium. Its current rate is $700. Find the original indicated rate change relative to current rates, then the revised indicated rate change if losses rise by 4%; state the increase in the indicated rate change in percentage points. (Answer: Original indicated rate $750.00 and rate change 7.1429%; revised indicated rate $776.00 and rate change 10.8571%; increase 3.7143 percentage points.)

What You'll Learn