Sensitivity Analysis for Rate Assumptions

Why Sensitivity Analysis Matters

A rate indication is the output of a model built from assumptions. A small change in the trend rate, development factor, credibility weight, or expense provision can change the indicated rate. Sensitivity analysis makes those effects visible: it shows how much the rate moves when an assumption changes, and whether the direction of the movement is reasonable.

Sensitivity analysis does not establish which assumption is correct. It helps an actuary assess the consequences of an assumption, prioritize review of material inputs, and explain how a proposed rate responds to alternative choices. The calculations below use a hypothetical prospective indication; no scenario represents observed company or industry data.

Definition: A sensitivity analysis compares a rate indication under a baseline set of assumptions with indications calculated after changing one or more assumptions. A one-way sensitivity changes one assumption at a time, holding all others fixed. A scenario analysis changes several assumptions together to represent a coherent alternative set.

A Rate Indication to Sensitize

Suppose the experience loss cost $E$ is adjusted for development by a factor $D$ and for trend by a factor $T$. The resulting experience loss cost is $EDT$. Let $Z$ be the credibility assigned to that adjusted experience, and let $C$ be a complementary loss cost already stated at the prospective level. The indicated prospective loss cost is $L=ZEDT+(1-Z)C$.

Let $F$ be fixed expense per exposure, $v$ be the variable expense proportion of premium, and $p$ be the profit and contingency provision as a proportion of premium. Under this model, the indicated rate $R$ is:

R=\frac{ZEDT+(1-Z)C+F}{1-v-p}

This formula assumes that $D$ and $T$ adjust the experience loss cost, that $C$ is already prospective and needs no further adjustment, and that $v+p<1$. The variable expense and profit provisions are both percentages of the indicated premium, while $F$ is a dollar amount per exposure. The denominator must be positive. If the complement also requires development or trend adjustment, this formula must be modified rather than applying the experience adjustments to $C$ without justification.

Key convention: In this tutorial, changing one assumption means holding every other input fixed unless a scenario explicitly says otherwise. State which inputs are changed before interpreting a rate difference.

Worked Example: Establishing a Baseline and Changing Trend

A hypothetical portfolio has an experience loss cost of $E=\$600$ per exposure. Use development factor $D=1.10$, trend factor $T=1.05$, credibility $Z=0.60$, and prospective complement $C=\$700$. Fixed expense is $F=\$80$ per exposure, variable expense is $v=0.25$, and profit and contingency provision is $p=0.05$. Calculate the baseline rate and then recalculate it if $T$ increases to $1.08$, with all other assumptions unchanged.

Method. Calculate the adjusted experience loss cost, credibility-weight it with the complement, add fixed expense, and divide by the premium share remaining after variable expense and profit provisions.

Baseline adjusted experience. $EDT=600(1.10)(1.05)=660(1.05)=\$693.00$.

Baseline credible loss cost. $ZEDT+(1-Z)C=0.60(693)+0.40(700)=415.80+280.00=\$695.80$.

The denominator is $1-v-p=1-0.25-0.05=0.70$. The numerator, including fixed expense, is $695.80+80=\$775.80$. Therefore, $R=775.80/0.70=\$1,108.2857\ldots$, or \$1,108.29 per exposure, rounded to the nearest cent.

Trend alternative. With $T=1.08$, adjusted experience is $600(1.10)(1.08)=660(1.08)=\$712.80$. The credible loss cost is $0.60(712.80)+0.40(700)=427.68+280=\$707.68$. Adding fixed expense gives $\$787.68$, so the alternative rate is $787.68/0.70=\$1,125.2571\ldots$, or \$1,125.26, rounded to the nearest cent. The increase is $\$1,125.26-\$1,108.29=\$16.97$ per exposure, using unrounded rates for the subtraction.

Check. Trend increased the adjusted experience loss cost, while all other assumptions stayed fixed, so the indicated rate increased. The calculation avoids the common mistake of applying the new trend factor to the complement as well as to experience.

Why the Sensitivity Formulas Work

The rate formula can be written $R=N/q$, where $N=ZEDT+(1-Z)C+F$ is the numerator and $q=1-v-p$ is the premium share available for losses and fixed expense. This representation makes it possible to calculate the effect of a small change in one input while holding the rest fixed.

Differentiate with respect to trend.
Only the experience term $ZEDT$ changes with $T$, so $\partial N/\partial T=ZED$, and $\partial R/\partial T=ZED/q$.

Differentiate with respect to development.
Only $ZEDT$ changes with $D$, giving $\partial R/\partial D=ZET/q$.

Differentiate with respect to credibility.
Rewrite the credible loss cost as $C+Z(EDT-C)$. Thus $\partial R/\partial Z=(EDT-C)/q$.

Differentiate with respect to expenses.
Since $F$ is in the numerator, $\partial R/\partial F=1/q$. Since $v$ and $p$ reduce the denominator, $\partial R/\partial v=\partial R/\partial p=N/q^2$.

These derivatives are exact local rates of change for the stated formula. For example, $\partial R/\partial T$ gives the change in dollars per exposure for a one-unit increase in $T$; a change of $0.01$ in $T$ has approximately $0.01(\partial R/\partial T)$ dollars of effect. For this model, the relationship is linear in $T$, so that particular one-way change is exact, not merely approximate. The derivatives with respect to $v$ and $p$ describe local change in the denominator and are generally approximations for finite changes.

Worked Example: Ranking One-Way Sensitivities

Use the baseline from the previous example. Estimate the effect of increasing (a) $T$ by $0.01$, (b) $D$ by $0.01$, (c) $Z$ by $0.01$, and (d) $v$ by 0.01. Treat 0.01 as one percentage point for the proportion assumptions $Z$ and $v$.

Method. Evaluate each partial derivative at the baseline, then multiply by the stated change. The baseline values are $q=0.70$, $N=\$775.80$, and $EDT=\$693$.

(a) Trend. $\partial R/\partial T=ZED/q=0.60(600)(1.10)/0.70=396/0.70=\$565.7143$ per unit of $T$. For an increase of $0.01$, the rate increases by approximately $565.7143(0.01)=\$5.6571$, or \$5.66.

(b) Development. $\partial R/\partial D=ZET/q=0.60(600)(1.05)/0.70=378/0.70=\$540$ per unit of $D$. An increase of $0.01$ increases the rate by $\$5.40$.

(c) Credibility. $\partial R/\partial Z=(EDT-C)/q=(693-700)/0.70=-10$ dollars per unit of $Z$. An increase of $0.01$ changes the rate by $-10(0.01)=-\$0.10$. The sign is negative because adjusted experience is below the complement; assigning more credibility to that lower value reduces the indicated loss cost.

(d) Variable expense. $\partial R/\partial v=N/q^2=775.80/(0.70)^2=775.80/0.49=\$1,583.2653$ per unit of $v$. The local effect of a 0.01 increase is approximately $\$15.83$.

At the baseline, the largest of these four estimated one-percentage-point effects is the variable expense change, followed by trend and development; the credibility change has a small effect in the opposite direction. This ranking depends on the sizes of the assumed changes and on the baseline inputs. The variable-expense derivative is local: an exact finite-change calculation should recompute the denominator. For example, changing $v$ from $0.25$ to $0.26$ gives $775.80/0.69=\$1,124.35$, an increase of about $\$16.06$, rather than the local estimate of $\$15.83$.

Check. Each derivative has the expected sign: greater trend or development raises the rate, while greater variable expense reduces the premium share in the denominator and raises the rate. Credibility’s sign depends on whether experience is above or below the complement.

Comparing Combined Scenarios

One-way sensitivities help isolate effects, but assumptions may change together. A scenario analysis recalculates the entire indication under a specified alternative. The difference from baseline is useful for understanding the combined consequence; it should not be attributed to just one assumption.

When explaining a combined change, a sequential bridge can show the effect of changing assumptions one at a time in a chosen order. The contribution assigned to each step can depend on that order because the formula includes interactions. The total difference between the baseline and final scenario does not depend on the order.

Worked Example: A Combined Alternative Scenario

Continue with the same hypothetical portfolio and baseline rate. Consider a scenario with $T=1.08$, $D=1.14$, $Z=0.70$, $F=\$90$, and $v=0.27$. Keep $E=\$600$, $C=\$700$, and $p=0.05$. Calculate the alternative rate and construct a sequential bridge, changing assumptions in the order $T,D,Z,F,v$.

Alternative rate. Adjusted experience is $600(1.14)(1.08)=684(1.08)=\$738.72$. The credible loss cost is $0.70(738.72)+0.30(700)=517.104+210=\$727.104$. Adding fixed expense gives $727.104+90=\$817.104$. The denominator is $1-0.27-0.05=0.68$. Therefore, $R=817.104/0.68=\$1,201.6235\ldots$, or \$1,201.62 per exposure, rounded to the nearest cent.

Sequential bridge. The baseline rate is $\$1,108.2857\ldots$, calculated before rounding.

1. Change $T$ to 1.08, leaving other inputs at baseline. The rate is $\$1,125.2571\ldots$, an increase of $\$16.9714$.

2. Change $D$ to 1.14 as well. Adjusted experience is now $600(1.14)(1.08)=\$738.72$. With $Z=0.60$, credible loss cost is $0.60(738.72)+0.40(700)=443.232+280=\$723.232$. Adding $F=\$80$ and dividing by $0.70$ gives $803.232/0.70=\$1,147.4743$, an increase of $\$22.2171$ from the preceding step.

3. Change $Z$ to 0.70. The credible loss cost becomes $0.70(738.72)+0.30(700)=\$727.104$. The rate is $(727.104+80)/0.70=\$1,153.0057$, an increase of $\$5.5314$.

4. Change $F$ to $\$90$. The rate becomes $(727.104+90)/0.70=\$1,167.2914$, an increase of $\$14.2857$.

5. Change $v$ to 0.27. The rate is the final scenario rate, $\$1,201.6235$, an increase of $\$34.3321$.

The sequential increases sum to approximately $\$93.3377$, matching the total change from the unrounded baseline, $1,201.6235-1,108.2857=\$93.3378$ subject to displayed intermediate rounding. Using full precision gives the same total. The size assigned to each step is specific to this change order; another order can allocate interaction effects differently.

Check. The overall rate rises because the selected scenario increases trend, development, credibility, fixed expense, and variable expense. In this case, adjusted experience exceeds the complement, so increased credibility also raises the rate.

Common Traps and Practical Interpretation

Changing more than the stated assumption. In a one-way sensitivity, hold all other inputs fixed. If trend changes, do not also change the complement unless the scenario explicitly says to do so.
Misreading a percentage-point change. Moving $v$ from 0.25 to 0.26 is a 0.01 absolute increase, or one percentage point. It is not a 1% relative increase in 0.25.
Applying premium loads to the wrong base. Under the stated model, variable expense and profit provisions are proportions of premium, so they appear in the denominator. Fixed expense per exposure is a dollar amount and appears in the numerator. Simply adding $v$ and $p$ as dollar amounts is incorrect.
Ignoring the direction of credibility. More credibility does not always increase the rate. The sign depends on whether $EDT$ is greater or less than $C$.
Overinterpreting a local sensitivity. A derivative describes the effect near the baseline. For larger changes, recalculate the full indication; this is especially important when the denominator changes.
Rounding before comparing. Retain full precision through the calculation and round the final rates or differences as requested. Subtracting already-rounded rates can create a one-cent discrepancy.

A sound sensitivity summary reports the baseline, identifies each changed input and its alternative value, states the resulting rate and dollar or percentage change, and explains the direction. If assumptions change together, label the result as a scenario rather than presenting it as the effect of any one input. This disciplined presentation is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical indication, retain full precision until the final answer. Report rates to the nearest cent.

Use $E=\$600$, $D=1.10$, $T=1.05$, $Z=0.60$, $C=\$700$, $F=\$80$, $v=0.25$, and $p=0.05$. Recalculate the rate if $T$ increases to 1.07 and all other assumptions remain fixed. (Answer: \$1,119.60 per exposure)
Using those same baseline assumptions, increase $D$ from 1.10 to 1.13. Calculate the new rate and its increase over baseline. (Answer: \$1,124.49 per exposure; increase of \$16.20)
Using the same baseline, change only $Z$ from 0.60 to 0.50. Explain the direction and calculate the new rate. (Answer: \$1,109.29 per exposure; it increases because experience is below the complement)
Using the same baseline, change only $v$ from 0.25 to 0.26. Recalculate the indication exactly rather than using a derivative. (Answer: \$1,124.35 per exposure)
A different hypothetical portfolio has $E=\$500$, $D=1.20$, $T=1.10$, $Z=0.50$, $C=\$680$, $F=\$70$, $v=0.20$, and $p=0.04$. Calculate its indicated rate and state the prospective loss cost before fixed expense. (Answer: \$973.68 per exposure; prospective loss cost is \$670.00)

What You'll Learn