Why Vary Two Assumptions at Once?
A rate indication depends on multiple assumptions. A one-way sensitivity table shows how the result changes when one input changes and the others are held fixed. A two-way sensitivity table extends that idea: it displays the indicated result for each combination of values of two assumptions. This makes it possible to see both the effect of each input and whether conclusions depend on the value chosen for the other input.
For example, an actuary might want to see how an indicated rate change responds to both projected loss trend and the variable expense provision. The table is not a forecast of which combination will occur, and it does not assign probabilities to the combinations. It is a structured comparison of calculated results under stated assumptions.
Set Up the Rate Indication
For the examples, use a simplified aggregate rate-level model. Let \(L\) be projected losses and loss adjustment expenses, \(F\) be fixed expenses, \(P\) be current premium at the current rate level, \(v\) be the variable expense ratio, and \(q\) be the profit and contingency provision as a proportion of premium. Let \(t\) be a multiplier applied to the projected losses. The required premium \(Q\) satisfies the model equation below.
This model assumes that losses and fixed expenses are dollar amounts for the portfolio being analyzed; variable expenses and the profit and contingency provision are each calculated as a proportion of required premium; and the proportions \(v\) and \(q\) apply to the same premium base. It also assumes \(1-v-q>0\), so the required premium is positive and finite when the numerator is positive. The indicated rate change is the required premium divided by current premium, less one:
Here, \(r=0.08\) means an indicated increase of 8%, while \(r=-0.08\) means an indicated decrease of 8%. This aggregate model is useful for sensitivity comparisons; it should not be applied without adjustment if the indicated premium changes exposure or coverage, or if the expense or profit assumptions are not proportional to premium as specified.
Constructing the Table
State whether each cell will show an indicated rate change, a required premium, or another defined output. Do not mix outputs in one grid.
Assign one assumption to rows and the other to columns. Choose values that span a useful range and label them with units where relevant.
Record the fixed inputs and use them identically in every cell. Only the row and column inputs should change.
Substitute both labels into the model, calculate the result, and check that direction and magnitude make sense before rounding consistently.
A table is easiest to audit when the row and column labels are explicit and the cells use a consistent format. If the cells show rate changes, report them as percentages, not as premium dollars. State whether a displayed negative percentage means a rate decrease.
Worked Example: Loss Trend and Variable Expense
A hypothetical portfolio has current premium \(P=\$1{,}000{,}000\), projected losses \(L=\$600{,}000\), fixed expenses \(F=\$100{,}000\), and profit and contingency provision \(q=5\%\). Build a two-way table of indicated rate changes for loss multipliers \(t=0.98, 1.00, 1.02\) and variable expense ratios \(v=14\%, 15\%, 16\%\). Round table entries to two decimal places.
Method. Use the required-premium model and calculate \(r=(tL+F)/(P(1-v-q))-1\). The fixed inputs are \(P=\$1{,}000{,}000\), \(L=\$600{,}000\), \(F=\$100{,}000\), and \(q=0.05\). The loss amounts after applying each multiplier are \(0.98(\$600{,}000)=\$588{,}000\), \(1.00(\$600{,}000)=\$600{,}000\), and \(1.02(\$600{,}000)=\$612{,}000\). Thus the numerators \(tL+F\) are respectively \(\$688{,}000\), \(\$700{,}000\), and \(\$712{,}000\).
For \(v=14\%\), the premium portions available for losses and fixed expenses are \(1-0.14-0.05=0.81\), so the denominator is \(\$1{,}000{,}000(0.81)=\$810{,}000\). The three results are \(\$688{,}000/\$810{,}000-1=-15.06\%\), \(\$700{,}000/\$810{,}000-1=-13.58\%\), and \(\$712{,}000/\$810{,}000-1=-12.10\%\).
For \(v=15\%\), the denominator is \(\$1{,}000{,}000(1-0.15-0.05)=\$800{,}000\). The results are \(\$688{,}000/\$800{,}000-1=-14.00\%\), \(\$700{,}000/\$800{,}000-1=-12.50\%\), and \(\$712{,}000/\$800{,}000-1=-11.00\%\).
For \(v=16\%\), the denominator is \(\$1{,}000{,}000(1-0.16-0.05)=\$790{,}000\). The results are \(\$688{,}000/\$790{,}000-1=-12.91\%\), \(\$700{,}000/\$790{,}000-1=-11.39\%\), and \(\$712{,}000/\$790{,}000-1=-9.87\%\).
| Loss multiplier \(t\) | \(v=14\%\) | \(v=15\%\) | \(v=16\%\) |
|---|---|---|---|
| 0.98 | -15.06% | -14.00% | -12.91% |
| 1.00 | -13.58% | -12.50% | -11.39% |
| 1.02 | -12.10% | -11.00% | -9.87% |
Interpretation and check. Within a row, a higher variable expense ratio leaves a smaller share of premium for losses and fixed expenses, so the indicated rate change rises. Within a column, higher projected losses also raise the indicated rate change. All cells are negative, so this model indicates decreases throughout the displayed range. A common mistake is to calculate the denominator as \(P(1-v)\), omitting the profit provision; for the middle cell that would give \(\$700{,}000/(\$1{,}000{,}000(0.85))-1=-17.65\%\), rather than the correct \(-12.50\%\). Include both proportions in the premium share, as required by the model.
Why the Formula Works
The formula follows from balancing the premium available for losses and fixed expenses against the amount required for them. Under the stated assumptions, each dollar of required premium has \(1-v-q\) dollars left after variable expenses and the profit and contingency provision. Therefore required premium \(Q\) provides \(Q(1-v-q)\) dollars for losses and fixed expenses. Equating that amount to \(tL+F\) gives:
Dividing \(Q\) by current premium \(P\) gives the indicated premium factor. Subtracting one converts that factor to a percentage rate change. In a two-way table, each cell uses this same relationship; only the selected row and column values differ.
Worked Example: Loss Trend and Profit Provision
A hypothetical book has current premium \(P=\$500{,}000\), projected losses \(L=\$300{,}000\), fixed expenses \(F=\$50{,}000\), and variable expenses \(v=20\%\). Calculate indicated rate changes for loss multipliers \(1.00,1.05,1.10\) and profit and contingency provisions \(q=4\%,5\%,6\%\). Round to two decimal places.
Method. The numerator \(tL+F\) is \(\$350{,}000\) at \(t=1.00\), \(\$365{,}000\) at \(t=1.05\), and \(\$380{,}000\) at \(t=1.10\). The denominator is \(P(1-v-q)\). For \(q=4\%\), it is \(\$500{,}000(1-0.20-0.04)=\$380{,}000\). The three rate changes are \(\$350{,}000/\$380{,}000-1=-7.89\%\), \(\$365{,}000/\$380{,}000-1=-3.95\%\), and \(\$380{,}000/\$380{,}000-1=0.00\%\).
For \(q=5\%\), the denominator is \(\$500{,}000(0.75)=\$375{,}000\). The results are \(\$350{,}000/\$375{,}000-1=-6.67\%\), \(\$365{,}000/\$375{,}000-1=-2.67\%\), and \(\$380{,}000/\$375{,}000-1=1.33\%\). For \(q=6\%\), the denominator is \(\$500{,}000(0.74)=\$370{,}000\). The results are \(\$350{,}000/\$370{,}000-1=-5.41\%\), \(\$365{,}000/\$370{,}000-1=-1.35\%\), and \(\$380{,}000/\$370{,}000-1=2.70\%\).
| Loss multiplier \(t\) | \(q=4\%\) | \(q=5\%\) | \(q=6\%\) |
|---|---|---|---|
| 1.00 | -7.89% | -6.67% | -5.41% |
| 1.05 | -3.95% | -2.67% | -1.35% |
| 1.10 | 0.00% | 1.33% | 2.70% |
Interpretation and check. Increasing either the loss multiplier or the provision \(q\) increases the indicated rate change. The table also shows that the sign of the indication depends on the combination: the \(t=1.10,\ q=4\%\) cell is exactly break-even, while the \(t=1.10,\ q=6\%\) cell indicates a 2.70% increase. The common mistake to avoid is treating a higher profit provision as reducing required premium. In this model, a higher \(q\) reduces the share \(1-v-q\) available for losses and fixed expenses, so required premium rises.
Reading the Combined Effects
A two-way table is more useful than a list of individual outputs when it makes the joint effect visible. Compare cells along one row to see how the column assumption changes the result at a fixed row assumption. Compare cells down one column to see how the row assumption changes the result at a fixed column assumption. The difference between two cells is a change in the indicated result, usually expressed in percentage points. For example, a movement from \(-5\%\) to \(2\%\) is a 7 percentage-point change, not a 7% relative increase.
The table can also reveal that a conclusion is conditional on the other assumption. A result may remain a rate decrease for one column value but become an increase for another. That is important information to report: a single central indication can conceal how much the conclusion changes across plausible input values. A two-way table does not establish which input values are most likely; selection of assumptions and interpretation require actuarial judgment.
Worked Example: Loss Trend and Fixed Expenses
A hypothetical portfolio has \(P=\$1{,}000{,}000\), projected losses \(L=\$500{,}000\), variable expenses \(v=18\%\), and profit and contingency provision \(q=4\%\). Build a table for loss multipliers \(t=1.00,1.05,1.10\) and fixed expenses \(F=\$100{,}000,\ \$120{,}000,\ \$140{,}000\). Show rate changes to two decimal places.
Method. The denominator is constant across this table: \(\$1{,}000{,}000(1-0.18-0.04)=\$780{,}000\). At \(t=1.00\), losses are \(\$500{,}000\), so the three numerators \(tL+F\) are \(\$600{,}000\), \(\$620{,}000\), and \(\$640{,}000\). The resulting rate changes are \(\$600{,}000/\$780{,}000-1=-23.08\%\), \(\$620{,}000/\$780{,}000-1=-20.51\%\), and \(\$640{,}000/\$780{,}000-1=-17.95\%\).
At \(t=1.05\), losses are \(1.05(\$500{,}000)=\$525{,}000\). The numerators are \(\$625{,}000\), \(\$645{,}000\), and \(\$665{,}000\), giving \(-19.87\%\), \(-17.31\%\), and \(-14.74\%\). At \(t=1.10\), losses are \(\$550{,}000\); the numerators are \(\$650{,}000\), \(\$670{,}000\), and \(\$690{,}000\), giving \(-16.67\%\), \(-14.10\%\), and \(-11.54\%\), respectively.
| Loss multiplier \(t\) | \(F=\$100{,}000\) | \(F=\$120{,}000\) | \(F=\$140{,}000\) |
|---|---|---|---|
| 1.00 | -23.08% | -20.51% | -17.95% |
| 1.05 | -19.87% | -17.31% | -14.74% |
| 1.10 | -16.67% | -14.10% | -11.54% |
Interpretation and check. Both assumptions affect the numerator, while the premium-share denominator stays fixed. Each 5% increase in the loss multiplier adds \(\$25{,}000\) to the numerator; each \(\$20{,}000\) increase in fixed expenses adds \(\$20{,}000\). The table moves in the expected direction, and every result is negative. A common mistake is to treat a fixed expense amount as a percentage of premium and put it in the denominator. Here \(F\) is a dollar amount, so it is added to \(tL\) in the numerator.
Common Construction Checks
- Check the model definition. State precisely what each assumption means. A multiplier on losses is not the same input as a percentage-point change in an expense ratio.
- Keep units consistent. If losses, fixed expenses, and premium are in thousands, use thousands for all three. The ratio is unchanged, but mixing dollars and thousands produces an invalid result.
- Include all relevant provisions. When both variable expenses and profit are defined as percentages of premium, use \(1-v-q\) in the premium share.
- Label the axes and cell output. Show which assumption is in rows, which is in columns, and whether each cell is a rate change or a required premium.
- Do not overstate what the grid means. A table compares specified combinations; it does not assign likelihood to those combinations or select a final rate by itself.
A well-constructed two-way sensitivity table is a concise way to communicate how robust an indication is to two assumptions considered together. The method is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus: define the model, vary only the intended inputs, compute each combination consistently, and interpret the pattern in context.
Check Your Understanding
For each problem, use the model \(r=(tL+F)/(P(1-v-q))-1\), and round rate changes to two decimal places unless stated otherwise.
- A hypothetical book has \(P=\$800{,}000\), \(L=\$500{,}000\), \(F=\$100{,}000\), \(v=18\%\), and \(q=4\%\). Find the indicated rate change at \(t=1.00\). (Answer: \(-3.85\%\))
- A hypothetical portfolio has \(P=\$1{,}200{,}000\), \(L=\$700{,}000\), \(F=\$80{,}000\), \(v=16\%\), and \(q=4\%\). Find the indicated rate change at \(t=1.06\), showing the loss amount and premium-share denominator. (Answer: \(-14.38\%\))
- A hypothetical program has \(P=\$2{,}000{,}000\), \(L=\$1{,}200{,}000\), \(F=\$200{,}000\), and \(q=5\%\). For \(t=1.10\) and \(v=17\%\), calculate the indicated rate change. (Answer: \(-2.56\%\))
- A hypothetical book has \(P=\$1{,}000{,}000\), \(L=\$650{,}000\), \(F=\$100{,}000\), \(v=16\%\), and \(q=6\%\). At \(t=1.08\), determine whether the indication is an increase or decrease and calculate its size. (Answer: \(+2.82\%\), an increase)
- A hypothetical portfolio has \(P=\$1{,}500{,}000\), \(L=\$900{,}000\), \(F=\$150{,}000\), and \(q=5\%\). Across \(t=0.95,1.05\) and \(v=14\%,16\%\), identify the smallest and largest indicated rate changes and their input combinations. (Answer: smallest \(-17.28\%\) at \(t=0.95,\ v=14\%\); largest \(-7.59\%\) at \(t=1.05,\ v=16\%\))