Why Construct a One-Way Sensitivity Table?
A rate indication is calculated from assumptions about losses, expenses, profit, and other provisions. Even when the calculation is correct, some inputs are uncertain or subject to judgment. A one-way sensitivity table shows how the indicated result changes when one selected assumption is varied across a range of plausible values while all other inputs remain fixed.
The table is useful for answering a focused question: for example, how much does the indicated rate change move if the variable expense ratio is higher than the baseline assumption? It does not assign probabilities to the alternative assumptions, and it does not show the combined effect of changing multiple assumptions. Each row is a separate, controlled calculation.
Set Up the Indication Before Varying an Assumption
For the calculations in this tutorial, use a deterministic premium-requirement model. Let \(L\) be the projected loss and loss adjustment expense amount, \(F\) the fixed expense amount, \(v\) the variable expense ratio, \(q\) the target profit and contingency provision as a percentage of premium, and \(P\) the current premium. Let \(R\) denote the required premium under a scenario. The model assumes that variable expenses and the target provision are each calculated as a proportion of \(R\).
Solving for the required premium gives the formula used in each row of the table:
The condition \(v+q<1\) is necessary for a positive finite denominator. This model applies when both \(v\) and \(q\) are premium-based proportions as defined above. If a provision is defined on another base, such as a percentage of losses, it must be represented on that base instead; it cannot simply be inserted as \(q\) in this formula.
The indicated rate change is measured relative to the current premium. If the exposure base and mix are held constant, the required premium scales with the rate level, so the percentage change is:
A positive result indicates an increase relative to current premium; a negative result indicates a decrease. The formula does not measure a policyholder’s premium change when exposure, coverage, or mix also changes.
Why the Required-Premium Formula Works
The model follows from a premium balance. Required premium must cover projected losses, fixed expenses, variable expenses, and the target profit and contingency provision. Since variable expenses equal \(vR\) and the target provision equals \(qR\), both depend on the unknown required premium. Moving those two terms to the left gives:
Factoring \(R\) yields \(R(1-v-q)=L+F\). When \(1-v-q>0\), division by that denominator gives \(R=(L+F)/(1-v-q)\). This derivation also explains why it is incorrect to add \(v\) and \(q\) to the numerator under this model: they are proportions of the required premium, not fixed dollar amounts.
A Reliable Construction Process
A useful table makes the comparison auditable. State the baseline, identify the one assumption being changed, label its units, and calculate every row from the same formula. Include the baseline row so the reader can see the reference point. Retain full precision through the calculations and round only the reported outputs.
List \(P\), \(L\), \(F\), \(v\), and \(q\), including units and whether ratios are expressed as decimals or percentages.
Choose values that represent a meaningful low-to-high sensitivity, and include the baseline value.
Change only the selected assumption from row to row. Do not quietly revise losses, expenses, or current premium along with it.
Find \(R\) using the same premium requirement formula, then find \(R/P-1\) using the same current premium.
Verify the baseline and the direction of movement, then state what the table does and does not establish.
Worked Example: Varying the Variable Expense Ratio
A hypothetical book has current premium \(P=\$1{,}000\), projected losses \(L=\$600\), fixed expenses \(F=\$100\), and a target profit and contingency provision \(q=5\%\) of required premium. Construct a one-way table for variable expense ratios \(v=10\%, 15\%, 20\%\). Report required premium and indicated rate change to two decimals. Use the premium-based provision model; no investment income or stochastic loss distribution is included in this deterministic calculation.
Method. Vary \(v\) only. The numerator is \(L+F=\$600+\$100=\$700\), and the rate change in every row is \(R/\$1{,}000-1\).
At \(v=10\%\), the denominator is \(1-0.10-0.05=0.85\). Thus \(R=\$700/0.85=\$823.5294\), and the indicated change is \(\$823.5294/\$1{,}000-1=-0.1764706\), or \(-17.65\%\).
At \(v=15\%\), the denominator is \(1-0.15-0.05=0.80\). Thus \(R=\$700/0.80=\$875.00\), and the indicated change is \(\$875.00/\$1{,}000-1=-0.125\), or \(-12.50\%\).
At \(v=20\%\), the denominator is \(1-0.20-0.05=0.75\). Thus \(R=\$700/0.75=\$933.3333\), and the indicated change is \(\$933.3333/\$1{,}000-1=-0.0666667\), or \(-6.67\%\).
| Variable expense ratio \(v\) | Required premium \(R\) | Indicated rate change |
|---|---|---|
| 10% | $823.53 | -17.65% |
| 15% (baseline) | $875.00 | -12.50% |
| 20% | $933.33 | -6.67% |
Check. The denominator decreases as \(v\) increases, so required premium rises. Each premium multiplied by its denominator returns the same numerator of \(\$700\), before rounding. The common mistake avoided here is using \(L\) alone in the numerator and omitting fixed expenses.
Changing a Fixed Expense Assumption
Not every input enters the formula in the same way. A fixed expense \(F\) is part of the numerator, while a premium-based variable expense ratio changes the denominator. The table should make the selected input explicit so that a reader can see which part of the indication is being tested.
Worked Example: Varying Fixed Expenses
For a hypothetical program, current premium is \(P=\$900\), projected losses are \(L=\$640\), the variable expense ratio is \(v=12\%\), and the target provision is \(q=6\%\) of required premium. Compare fixed expense assumptions \(F=\$80,\$100,\$120\). Calculate required premium and indicated rate change to two decimals. All other inputs remain fixed; the example is deterministic and assumes no investment income.
Method. The denominator is \(1-0.12-0.06=0.82\) in all three scenarios. Only \(F\), and therefore the numerator \(L+F\), changes.
For \(F=\$80\), the numerator is \(\$640+\$80=\$720\). Required premium is \(\$720/0.82=\$878.0488\). The rate change is \(\$878.0488/\$900-1=-0.0243902\), or \(-2.44\%\).
For \(F=\$100\), the numerator is \(\$640+\$100=\$740\). Required premium is \(\$740/0.82=\$902.4390\). The rate change is \(\$902.4390/\$900-1=0.0027100\), or \(0.27\%\).
For \(F=\$120\), the numerator is \(\$640+\$120=\$760\). Required premium is \(\$760/0.82=\$926.8293\). The rate change is \(\$926.8293/\$900-1=0.0298103\), or \(2.98\%\).
| Fixed expenses \(F\) | Numerator \(L+F\) | Required premium \(R\) | Indicated rate change |
|---|---|---|---|
| $80 | $720 | $878.05 | -2.44% |
| $100 (baseline) | $740 | $902.44 | 0.27% |
| $120 | $760 | $926.83 | 2.98% |
Check. The required premiums increase with fixed expenses, while the denominator remains \(0.82\). For example, \(\$926.8293(0.82)=\$760\), matching the numerator in the highest-expense row. The table also shows why an indication can move from a small decrease to a small increase as one assumption changes.
Changing the Profit and Contingency Provision
A sensitivity table can also test a target provision, provided its definition is kept consistent. In the model here, \(q\) is a proportion of required premium. A higher \(q\) reduces the denominator and increases the required premium. This interpretation would be different if the provision were defined on another base.
Worked Example: Varying the Target Provision
A hypothetical book has \(P=\$800\), \(L=\$620\), \(F=\$20\), and \(v=8\%\). Compare target provisions \(q=4\%,6\%,8\%\), each defined as a percentage of required premium. Report required premium and rate change to two decimals. Keep the loss and expense amounts fixed; no probability distribution or investment income is assumed.
Method. The numerator is \(L+F=\$620+\$20=\$640\), and the rate change uses the same current premium of \(\$800\) in every row.
At \(q=4\%\), the denominator is \(1-0.08-0.04=0.88\). Then \(R=\$640/0.88=\$727.2727\), and the change is \(\$727.2727/\$800-1=-0.0909091\), or \(-9.09\%\).
At \(q=6\%\), the denominator is \(1-0.08-0.06=0.86\). Then \(R=\$640/0.86=\$744.1860\), and the change is \(\$744.1860/\$800-1=-0.0697674\), or \(-6.98\%\).
At \(q=8\%\), the denominator is \(1-0.08-0.08=0.84\). Then \(R=\$640/0.84=\$761.9048\), and the change is \(\$761.9048/\$800-1=-0.0476190\), or \(-4.76\%\).
| Target provision \(q\) | Denominator \(1-v-q\) | Required premium \(R\) | Indicated rate change |
|---|---|---|---|
| 4% | 0.88 | $727.27 | -9.09% |
| 6% (baseline) | 0.86 | $744.19 | -6.98% |
| 8% | 0.84 | $761.90 | -4.76% |
Check. The required premium rises as the target provision rises, because a larger share of premium is reserved for that provision. A frequent error is to use the provision as a dollar amount or to omit it from the denominator. The calculation must follow the stated percentage base.
Reading the Table and Avoiding Common Traps
A one-way table is most informative when it is read as a controlled comparison rather than as a forecast of what will happen. It shows the mechanical effect of changing one assumption under the chosen model. It does not show whether a particular assumption is more likely, nor does it measure uncertainty in all inputs at once.
- Do not change more than one input. If both \(L\) and \(v\) differ between rows, the table no longer isolates the effect of \(v\).
- Do not mix percentage bases. A provision that is a percentage of premium must be treated as a premium-based proportion in the formula.
- Do not omit fixed expenses. When \(F\) is part of the indication, the numerator is \(L+F\), not just \(L\).
- Do not vary the reference premium unintentionally. Use the same \(P\) in \(R/P-1\) for all rows unless the question explicitly defines a different comparison.
- Do not overinterpret the range. A table does not make the selected low, baseline, or high value more credible; it reports the resulting indication conditional on that value.
For the premium-requirement model, the direction of a sensitivity can also be checked algebraically. Increasing \(F\) increases the numerator when \(L+F\) is positive. Increasing \(v\) or \(q\), while \(v+q<1\), decreases the denominator. In each case the required premium increases. If a table shows the opposite direction, check the arithmetic, the units, and which provision base is being used.
Check Your Understanding
For each problem, use the premium-requirement model defined above. Round required premiums to cents and rate changes to two decimals.
- A hypothetical account has \(P=\$750\), \(L=\$510\), \(F=\$60\), and \(q=4\%\). Compare \(v=12\%\) with \(v=16\%\). Find the required premium and indicated rate change for each value. (Answer: At 12%, \(R=\$678.57\) and the change is \(-9.52\%\); at 16%, \(R=\$712.50\) and the change is \(-5.00\%\).)
- A hypothetical program has \(P=\$1{,}200\), \(L=\$810\), \(v=15\%\), and \(q=5\%\). Compare fixed expenses of \(\$100\) and \(\$130\). Find both required premiums and rate changes. (Answer: At \(\$100\), \(R=\$1{,}137.50\) and the change is \(-5.21\%\); at \(\$130\), \(R=\$1{,}175.00\) and the change is \(-2.08\%\).)
- A hypothetical book has \(P=\$1{,}000\), \(L=\$700\), \(F=\$50\), and \(v=10\%\). Create a three-row sensitivity for \(q=3\%,5\%,7\%\), and state whether the indicated rate change becomes larger or smaller as \(q\) increases. (Answer: At 3%, \(R=\$862.07\) and the change is \(-13.79\%\); at 5%, \(R=\$882.35\) and the change is \(-11.76\%\); at 7%, \(R=\$903.61\) and the change is \(-9.64\%\). The change becomes larger, meaning less negative.)
- A hypothetical indication has \(P=\$950\), \(L=\$650\), \(F=\$50\), and \(q=5\%\). Compare \(v=12\%\) with \(v=17\%\). Show the denominator and calculate the rate change for each scenario. (Answer: At 12%, the denominator is 0.83, \(R=\$843.37\), and the change is \(-11.22\%\); at 17%, the denominator is 0.78, \(R=\$897.44\), and the change is \(-5.53\%\).)
- A hypothetical book has \(L=\$500\), \(F=\$50\), \(v=10\%\), \(q=5\%\), and current premium \(P=\$700\). What is the indicated rate change, and what is the effect on that change if the analyst mistakenly leaves \(F\) out of the numerator? (Answer: Correctly, \(R=\$550/0.85=\$647.06\), so the change is \(-7.56\%\). Omitting \(F\) gives \(R=\$500/0.85=\$588.24\), or \(-15.97\%\), understating the indicated change by 8.40 percentage points.)