Scenario Analysis for Rate Selections

Why Use Scenario Analysis?

A rate indication is calculated from assumptions about losses, expenses, and other provisions. Those assumptions are uncertain, and uncertainty often affects more than one input at a time. For example, a smaller-than-expected book may raise fixed expense per exposure, while adverse claim experience may increase projected loss costs. Looking at each input separately can miss how such assumptions might occur together.

Scenario analysis addresses this by defining a small number of internally consistent sets of assumptions and calculating the indication under each set. A best-estimate scenario represents the selected central assumptions. A high-cost scenario combines plausible assumptions that increase the required rate; a low-cost scenario combines plausible assumptions that reduce it. These are structured alternatives, not claims that any one outcome is certain.

Definition: A scenario is a specified, coherent set of assumptions used to calculate a rate outcome. A best-estimate, high-cost, or low-cost label describes the direction of the resulting rate relative to the central scenario; it does not by itself assign a probability to that outcome.

For the examples below, let $R$ be the indicated rate per exposure, $L$ the projected loss cost per exposure, and $F$ the fixed expense per exposure. Let $v$ be the variable expense ratio and $q$ the profit and contingency provision, each expressed as a proportion of premium. The rate is calculated under the indicated model below. A different expense or profit basis may require a different equation; do not use this formula when a provision is not a percentage of premium.

R=L+F+vR+qR=\frac{L+F}{1-v-q}, \qquad 1-v-q>0.

The assumption $1-v-q>0$ is required for a positive finite solution when $L+F>0$. For each scenario, apply the same model and definitions, changing only the assumptions specified for that scenario. This makes the resulting rates comparable.

Building and Calculating Scenarios

Set the central assumptions.
Identify the selected best-estimate values for losses, expenses, exposure, and provisions, using consistent units.

Specify plausible alternatives together.
Choose combinations that describe a coherent outcome. For example, lower enrollment may coincide with higher fixed expense per exposure, not lower fixed expense per exposure.

Calculate each indication.
Use the same rate formula, denominator convention, and unit of exposure in every scenario.

Compare and select.
Review the range and drivers of the indications. If scenario weights are used, state their purpose and calculate the weighted result explicitly.

The formula follows by balancing the indicated rate against the amounts it must fund. Per exposure, the required premium $R$ covers losses $L$, fixed expenses $F$, variable expenses $vR$, and the profit and contingency provision $qR$. Thus $R=L+F+vR+qR$. Moving the premium-based provisions to the left gives $R(1-v-q)=L+F$; dividing by the positive denominator gives the stated expression.

Worked Example: Calculating a Three-Scenario Rate Range

A hypothetical program is pricing coverage per enrolled unit. Its selected central assumptions are $L=\$600$, $F=\$60$, $v=0.20$, and $q=0.05$. For a high-cost scenario, the actuary selects $L=\$690$, $F=\$66$, $v=0.22$, and $q=0.05$. For a low-cost scenario, the assumptions are $L=\$540$, $F=\$55$, $v=0.18$, and $q=0.04$. Calculate all three indications and check whether their ordering matches the scenario labels.

Method. Apply $R=(L+F)/(1-v-q)$ separately to each complete set of assumptions. The expense and profit ratios are fractions of premium, so each belongs in the denominator.

Best estimate. The numerator is $600+60=\$660$. The denominator is $1-0.20-0.05=0.75$. Therefore,

R_{\text{best}}=\frac{660}{0.75}=\$880.00.

High cost. The numerator is $690+66=\$756$. The denominator is $1-0.22-0.05=0.73$. Therefore,

R_{\text{high}}=\frac{756}{0.73}=\$1{,}035.6164\ldots \approx \$1{,}035.62.

Low cost. The numerator is $540+55=\$595$. The denominator is $1-0.18-0.04=0.78$. Therefore,

R_{\text{low}}=\frac{595}{0.78}=\$762.8205\ldots \approx \$762.82.

Rounded to the nearest cent, the ordering is $\$762.82 < \$880.00 < \$1{,}035.62$, as expected. The high-cost scenario raises both the numerator and the premium-based expense share; the latter also reduces the denominator. The common mistake this method avoids is changing the loss cost but leaving the other explicitly changed assumptions at their central values.

Exposure and Fixed Expense Must Be Consistent

Fixed expense is often estimated as a total amount for a book of business, not as a cost per exposure. If the scenario changes the number of exposures, convert the total fixed expense to a per-exposure amount within that scenario. Otherwise, a scenario can appear to have an implausibly low or high indication simply because its expense allocation was not updated.

Worked Example: Changing Enrollment and Fixed Expense per Exposure

A hypothetical program has a current rate of $\$590$ per exposure. Its best-estimate scenario has 8,000 exposures, projected loss cost of $\$420$ per exposure, total fixed expenses of $\$480{,}000$, $v=0.18$, and $q=0.02$. A high-cost scenario has 7,500 exposures, loss cost $\$460$, total fixed expenses $\$510{,}000$, $v=0.20$, and $q=0.02$. A low-cost scenario has 8,500 exposures, loss cost $\$395$, total fixed expenses $\$467{,}500$, $v=0.16$, and $q=0.02$. Calculate each rate indication and its percentage change from the current rate.

Method. For each scenario, first divide total fixed expenses by that scenario's exposure count to obtain $F$. Then calculate $R$, followed by $100(R/590-1)\%$. Do not divide every scenario's fixed expenses by the central exposure count.

Best estimate. Fixed expense per exposure is $\$480{,}000/8{,}000=\$60$. The numerator is $420+60=\$480$, and the denominator is $1-0.18-0.02=0.80$. Thus $R=480/0.80=\$600.00$. The change from the current rate is $100(600/590-1)=1.6949\%$, or $+1.69\%$.

High cost. Fixed expense per exposure is $\$510{,}000/7{,}500=\$68$. The numerator is $460+68=\$528$, and the denominator is $1-0.20-0.02=0.78$. Thus $R=528/0.78=\$676.9231\ldots$, or $\$676.92$. The rate change is $100(676.9231/590-1)=14.7327\%$, or $+14.73\%$.

Low cost. Fixed expense per exposure is $\$467{,}500/8{,}500=\$55$. The numerator is $395+55=\$450$, and the denominator is $1-0.16-0.02=0.82$. Thus $R=450/0.82=\$548.7805\ldots$, or $\$548.78$. The rate change is $100(548.7805/590-1)=-6.9864\%$, or $-6.99\%$.

The indications are $\$600.00$, $\$676.92$, and $\$548.78$, with corresponding changes of $+1.69\%$, $+14.73\%$, and $-6.99\%$. The ordering is consistent with the assumptions. The common mistake is to treat total fixed expense as though it were already an expense per exposure, or to use the wrong scenario's exposure count.

Using Scenario Weights Carefully

A selected rate may be formed as a weighted average of scenario indications when the weights express a deliberate selection judgment or represent defensible scenario probabilities. The weights must be nonnegative and sum to one. They do not become probabilities merely because they are used in a weighted average. Explain what they mean, and do not confuse a weighted selected rate with the indication from a single “average” set of assumptions.

Worked Example: Interpreting a Weighted Scenario Selection

For a hypothetical product, the best-estimate assumptions are $L=\$330$, $F=\$25$, $v=0.18$, and $q=0.02$. The high-cost assumptions are $L=\$370$, $F=\$28$, $v=0.20$, and $q=0.02$. The low-cost assumptions are $L=\$305$, $F=\$24$, $v=0.16$, and $q=0.02$. Calculate the three indications. Then apply selection weights of 0.50 to best estimate, 0.30 to high cost, and 0.20 to low cost. Is the weighted selection above or below the central indication?

Method. Calculate each rate using the same formula, and then multiply each scenario rate by its weight. The weights sum to $0.50+0.30+0.20=1.00$.

The best-estimate indication is $(330+25)/(1-0.18-0.02)=355/0.80=\$443.75$. The high-cost indication is $(370+28)/(1-0.20-0.02)=398/0.78=\$510.2564\ldots$, or $\$510.26$. The low-cost indication is $(305+24)/(1-0.16-0.02)=329/0.82=\$401.2195\ldots$, or $\$401.22$.

Using unrounded indications for the weighted calculation gives

0.50(443.75)+0.30(510.256410\ldots)+0.20(401.219512\ldots) =455.195825\ldots \approx \$455.20.

The selection is above the central indication. Relative to $\$443.75$, the high-cost scenario contributes an upward weighted deviation of $0.30(510.2564-443.75)=\$19.95192$. The low-cost scenario contributes a downward weighted deviation of $0.20(443.75-401.2195)=\$8.50610$. The net deviation is approximately $19.95192-8.50610=\$11.44582$, which gives $443.75+11.44582=\$455.19582$, consistent with the weighted calculation. The common mistake is to conclude that the selection lies below the central indication because the low-cost scenario is present, without comparing the weighted upward and downward effects.

Checks Before Selecting

Scenario analysis matters because it shows how plausible combinations of assumptions affect a rate outcome and makes the direction of judgment visible. It does not remove uncertainty or determine which assumptions deserve the most weight. Before selecting, check that each scenario uses the same coverage basis, units, formula, and expense definitions; that each input combination is internally consistent; and that the result follows from the stated assumptions.

Do not mix scenarios accidentally. Using a high-cost loss assumption with a low-cost expense ratio may produce a mathematical result, but it is not one of the stated scenarios unless that combination is intentionally defined.
Do not treat scenario labels as evidence. A “high” scenario should produce a higher indication because of its assumptions, not merely because it is named high.
Do not imply unwarranted precision. Calculate using unrounded values where possible, then round the reported rate to the requested precision.
Do not overinterpret weights. A weighted selection is a decision tool; its meaning depends on the rationale for the weights.

Key takeaway: Define coherent assumption sets, recalculate the complete indication under each set, and explain how any scenario weights affect the final selection. Scenario analysis is only as sound as the consistency of its inputs and the interpretation of its weights.

Check Your Understanding

For each problem, use the stated rate model and round money to the nearest cent and percentage changes to two decimal places.

A scenario has loss cost of $\$320$ per exposure, fixed expense of $\$40$ per exposure, variable expense ratio $v=0.15$, and profit provision $q=0.05$. Find its indicated rate. (Answer: $\$450.00$)
A low-cost scenario has loss cost $\$280$, fixed expense $\$30$, $v=0.16$, and $q=0.04$. Calculate its indicated rate and verify that the denominator is positive. (Answer: $\$387.50$; denominator $0.80$)
A scenario indication is $\$675$ per exposure, compared with a current rate of $\$625$. Calculate the indicated percentage change. (Answer: $+8.00\%$)
Three indications are $\$500$, $\$560$, and $\$440$ for best-estimate, high-cost, and low-cost scenarios, respectively. Apply weights $0.50$, $0.30$, and $0.20$, in that order. Is the selection above or below the best-estimate indication? (Answer: $\$506.00$, above by $\$6.00$)
Two equally weighted scenarios have $(L,F,v,q)=(\$360,\$40,0.18,0.02)$ and $(\$410,\$40,0.22,0.03)$. Calculate each indication and their weighted average. Separately calculate the indication using the weighted average of the two numerators and the weighted average of their denominators. (Answer: scenario indications $\$500.00$ and $\$600.00$; weighted rate $\$550.00$; weighted-input indication $\$548.39$)

What You'll Learn