Sensitivity Testing Rate Indications

Why Test the Assumptions Behind an Indication?

A rate indication is a result conditional on assumptions about future losses, trend, expenses, exposures, and the current premium base. A single indicated change can conceal how much each assumption contributes. Sensitivity testing makes those dependencies visible: change one assumption at a time to isolate its effect, then test selected assumptions together to assess a plausible combined scenario.

The purpose is not to select whichever scenario produces a preferred rate. It is to understand whether the result is robust, identify the assumptions that matter most, and explain how the indication changes across segments. This tutorial uses deterministic expected values; uncertainty around estimates is distinct from sensitivity to chosen assumptions.

Definition: For segment $j$, let $E_j$ be the projected exposure, $L_j$ the expected loss cost per exposure before the sensitivity changes being considered, $T_j$ the loss trend factor, $f_j$ fixed expense per exposure, $v_j$ the variable expense ratio, and $P_j$ the current premium per exposure. Under the assumption that variable expenses are a proportion of premium and that no other premium provisions are included, the indicated premium per exposure is $$ R_j=\frac{L_jT_j+f_j}{1-v_j}, \qquad v_j<1. $$ The portfolio indication is $$ I=\frac{\sum_j E_jR_j}{\sum_j E_jP_j}-1. $$

The formula assumes that $L_jT_j$ represents the expected losses for the prospective rating period, that fixed expenses are appropriately expressed per exposure, and that the variable expense ratio is applied to indicated premium. If expenses are instead modeled differently, or if the indication includes other provisions, the numerator or denominator must reflect that structure. Do not apply the displayed formula mechanically to a different expense model.

Separate the Portfolio Indication From Segment Relativities

The overall indication is a premium-weighted comparison of indicated premium with current premium. It is not generally the simple average of segment indications. Define each segment's indication as $I_j=R_j/P_j-1$, and its share of current premium as $w_j=E_jP_j/\sum_k E_kP_k$. Then

I=\sum_j w_j I_j.

This identity follows by substituting the definitions: $$ \sum_j w_jI_j =\sum_j\frac{E_jP_j}{\sum_kE_kP_k}\left(\frac{R_j}{P_j}-1\right) =\frac{\sum_j E_jR_j}{\sum_k E_kP_k}-1. $$ Thus, a change in exposure mix can change the overall result by changing the weights, even when every segment's indicated rate and segment indication are unchanged.

A segment relativity compares indicated rates between segments. For two segments, define the relativity of segment B to segment A as $R_B/R_A$. A uniform multiplicative change to every segment's indicated rate leaves this ratio unchanged. However, additive fixed expenses, different segment trends, or different loss-level changes can change the ratio. A change in exposure mix alone does not change a segment relativity when segment rates themselves are held fixed.

Worked Example: Testing Trend and Variable Expense Assumptions

A hypothetical portfolio has 600 exposures in segment A and 400 in segment B. Before prospective trend, expected loss costs are $300 and $500 per exposure, respectively. Trend is compounded annually for two years. Fixed expense is $60 per exposure in each segment, the variable expense ratio is 20%, and current premiums are $500 and $800 per exposure. Calculate the indication using 5% annual trend, then test 7% annual trend and a 22% variable expense ratio separately. No interest discounting, mortality, or stochastic loss distribution is used; the losses and expenses are deterministic expected values for this calculation.

Method and baseline indication. The two-year trend factor at 5% is $T=(1.05)^2=1.1025$. Trended loss costs are $300(1.1025)=330.75$ in A and $500(1.1025)=551.25$ in B. The indicated rates are $R_A=(330.75+60)/0.80=488.4375$ and $R_B=(551.25+60)/0.80=764.0625$.

Current premium is $600(500)+400(800)=620{,}000$. Indicated premium is $600(488.4375)+400(764.0625)=293{,}062.50+305{,}625.00=598{,}687.50$. Therefore $I=598{,}687.50/620{,}000-1=-0.034375$, or -3.4375%.

Alternative trend. At 7% for two years, $T=(1.07)^2=1.1449$. The trended losses are $300(1.1449)=343.47$ and $500(1.1449)=572.45$. Keeping expenses and the variable ratio unchanged gives $R_A=(343.47+60)/0.80=504.3375$ and $R_B=(572.45+60)/0.80=790.5625$. Indicated premium is $600(504.3375)+400(790.5625)=302{,}602.50+316{,}225.00=618{,}827.50$. The indication is $618{,}827.50/620{,}000-1=-0.0018911$, or -0.1891%.

Alternative variable expense. Return to 5% trend and use $v=0.22$. The segment rates become $390.75/0.78=500.9615$ and $611.25/0.78=783.6538$. Indicated premium is approximately $600(500.9615)+400(783.6538)=300{,}576.92+313{,}461.54=614{,}038.46$. The indication is $614{,}038.46/620{,}000-1=-0.0096154$, or -0.9615%.

The baseline B-to-A relativity is $764.0625/488.4375=1.5643$. Under 7% trend it is $790.5625/504.3375=1.5675$. The common variable expense ratio cancels from the relativity when applied to both segments, but fixed expense does not cancel from the sum of loss and fixed expense. This is why the different trend scenarios slightly change the relativity.

Check. Each scenario uses the same exposure counts and current premiums, so only its stated assumption changes. The indicated premium rises from $598,687.50 to $618,827.50 under higher trend, and to $614,038.46 under higher variable expense; both remain below the $620,000 current premium.

Common mistake avoided: Do not treat a 22% variable expense ratio as a 22% addition to losses. With a ratio applied to premium, the required premium divides losses plus fixed expenses by $1-v$.

Exposure Assumptions Change Portfolio Weights

Exposure assumptions deserve a separate sensitivity test. A change in total exposure that multiplies every segment's exposure by the same factor cancels from the portfolio indication. A change in the distribution of exposures across segments generally does not, because segments can have different current and indicated premiums. Be precise about the scenario: changing exposure counts is not the same as changing a segment's indicated rate or relativity.

Worked Example: Separating an Exposure-Mix Effect

A hypothetical two-segment program has expected loss costs of $500 in A and $900 in B, no fixed expenses, and a 10% variable expense ratio. Current premiums are $600 in A and $1,000 in B. Compare a portfolio with 70 A exposures and 30 B exposures to one with an even 50/50 mix. No interest discounting, mortality, or stochastic distribution is used; the stated loss costs are deterministic expected values.

Method. The indicated rates are $500/0.90=555.5556$ in A and $900/0.90=1{,}000$ in B. For the 70/30 mix, current premium is $70(600)+30(1{,}000)=72{,}000$. Indicated premium is $[70(500)+30(900)]/0.90=(35{,}000+27{,}000)/0.90=68{,}888.89$. Thus the indication is $68{,}888.89/72{,}000-1=-0.04321$, or -4.321%.

For a 50/50 mix, current premium is $50(600)+50(1{,}000)=80{,}000$. Indicated premium is $[50(500)+50(900)]/0.90=(25{,}000+45{,}000)/0.90=77{,}777.78$. The indication is $77{,}777.78/80{,}000-1=-0.02778$, or -2.778%.

The indication changes by $ -2.778\%-(-4.321\%)=1.543$ percentage points because the portfolio mix changes. The segment relativity is $1{,}000/555.5556=1.800$ under either mix. The variable expense divisor is common to both segments and cancels in their ratio.

Check. Both indicated rates are below their respective current premiums, so both segment indications are negative. The 50/50 portfolio places more weight on segment B, whose indication is $1{,}000/1{,}000-1=0\%$, rather than segment A's negative indication; its overall result is therefore less negative. The relativity remains 1.800 because neither segment rate changed.

Common mistake avoided: Do not average the two segment indications equally for the 70/30 portfolio. Its overall indication must use current-premium weights, not exposure percentages alone.

Test Loss-Level Assumptions and Relativity Changes

Trend changes expected losses over time; a loss-level sensitivity changes the assumed expected loss itself. The distinction is useful when an indicated result depends on assumptions such as the selected loss estimate or adjustments to its level. If these changes differ by segment, they can alter both the overall indication and segment relativities. A common multiplicative loss change can also affect relativities when fixed expenses are present, because fixed expenses are not multiplied with losses.

Worked Example: Differential Loss-Level Changes

A hypothetical portfolio has 100 exposures in each of two segments. Prospective loss costs are $400 in A and $600 in B; fixed expense is $40 per exposure, variable expense is 20%, and current premiums are $600 and $800. Test a loss-level increase of 10% in A and a decrease of 5% in B, holding every other assumption fixed. No interest discounting, mortality, or stochastic distribution is used; the loss costs are deterministic expected values.

Baseline. Indicated rates are $(400+40)/0.80=550$ in A and $(600+40)/0.80=800$ in B. Current premium is $100(600)+100(800)=140{,}000$, and indicated premium is $100(550)+100(800)=135{,}000$. The baseline indication is $135{,}000/140{,}000-1=-3.5714\%$. The baseline B-to-A relativity is $800/550=1.4545$.

Apply the loss-level changes. A's loss cost becomes $400(1.10)=440$; B's becomes $600(0.95)=570$. The new indicated rates are $(440+40)/0.80=600$ and $(570+40)/0.80=762.50$. Indicated premium is $100(600)+100(762.50)=136{,}250$. The new indication is $136{,}250/140{,}000-1=-0.026786$, or -2.6786%. The new relativity is $762.50/600=1.2708$.

The overall indication becomes less negative even though B's indicated rate falls, because A's rate rises by more in dollars than B's rate falls. The relativity falls from 1.4545 to 1.2708 because the segment loss assumptions moved in different directions.

Check. Current premium is unchanged at $140,000, while indicated premium increases by $1,250 from the baseline $135,000. The indication must therefore increase by $1{,}250/140{,}000=0.8929$ percentage points, from -3.5714% to -2.6786%, as calculated.

Common mistake avoided: Do not apply a loss-level adjustment to current premiums unless the scenario explicitly changes the current rate level. Here, the sensitivity changes expected losses only.

Organize and Interpret a Sensitivity Analysis

A useful analysis distinguishes isolated effects from combined scenarios. First define a baseline and document its assumptions. Then vary one assumption at a time while holding the others fixed; this identifies which inputs are driving the result. Finally, combine assumptions only when the combined scenario is meaningful, and label it clearly. Changes can interact: for example, a higher loss estimate has a larger dollar effect when trend is also higher.

Specify the baseline.
Record loss costs, trend periods and rates, expense assumptions, exposure counts, and current premiums. Confirm that all inputs use consistent time periods and units.

Change one assumption at a time.
Recalculate segment indicated rates, segment indications, the portfolio indication, and any relevant relativities. This isolates the effect of the selected assumption.

Test exposure mix separately.
Reweight unchanged segment rates with the new exposure counts. Check whether the observed portfolio change comes from a different mix rather than a different segment indication.

Compare results and explain the drivers.
Report changes in percentage points for indications and as ratios for relativities. If multiple assumptions change together, do not attribute the entire result to one of them.

Sensitivity results are conditional calculations, not evidence that an alternative assumption is more likely. A strong analysis identifies which inputs materially affect the overall indication, which affect only the mix-weighted portfolio result, and which change the relationship between segments. This supports a clear explanation of the indication without confusing assumption sensitivity with statistical uncertainty or rate implementation.

Key takeaway: Recalculate both the total and the segment results. Trend, loss-level, and expense changes can alter indicated rates; exposure mix changes the weights used to combine segment results. A portfolio indication can move while segment relativities remain constant, and relativities can change when segment assumptions or additive expenses differ.

Check Your Understanding

For each problem, use the stated expense and exposure assumptions. Round indicated rates and percentages only as needed for the final answer.

A hypothetical single segment has a $250 loss cost, 6% annual trend for three years, fixed expense of $35 per exposure, a 25% variable expense ratio, and a current premium of $500. Find the indicated rate and indication. No discounting, mortality, or stochastic distribution is used. (Answer: indicated rate $443.67; indication -11.27%)
Two segments have indicated rates of $420 and $780 and current premiums of $500 and $900. There are 60 exposures in the first segment and 40 in the second. Calculate the portfolio indication and the second-to-first indicated-rate relativity. (Answer: -14.55%; relativity 1.8571)
Using the segment rates and current premiums in question 2, change the exposure mix to 40 in the first segment and 60 in the second. Calculate the new indication and state whether the relativity changes. (Answer: -14.05%; relativity remains 1.8571)
Two segments each have 50 exposures. Their loss costs are $320 and $480, fixed expense is $40 per exposure, the variable expense ratio is 20%, and current premiums are $500 and $700. Apply a 10% loss increase to the first segment and a 5% loss decrease to the second. Calculate the new portfolio indication and second-to-first relativity. (Answer: -7.50%; relativity 1.2653)
Two segments have indicated rates of $450 and $750, current premiums of $600 and $900, and no additional premium provisions. Compare the portfolio indication with 80 exposures in the first segment and 20 in the second against a 50/50 mix. Give both indications and the relativity. (Answer: 80/20 indication -22.73%; 50/50 indication -20.00%; relativity 1.6667 in both mixes)

What You'll Learn