Why Use a One-Way Sensitivity Table?
A reserve estimate depends on selected assumptions as well as observed claim data. A one-way sensitivity table makes the effect of one selected assumption visible: it recalculates the estimate at several values of that assumption while holding the data, methods, and all other selections constant. This is useful when explaining how much a reserve estimate changes if, for example, a selected development factor or trend rate is adjusted.
A table is only interpretable when its baseline and varied assumption are explicit. The baseline is the estimate under the central or current selection. Each alternative row changes one assumption from that baseline. The results are conditional calculations, not a range of equally likely outcomes and not a statement that the reserve will fall within the displayed values.
For a reserve estimate \(R\), let \(a_0\) be the baseline assumption and \(R(a_0)\) its resulting estimate. For an alternative assumption \(a\), the absolute change and percentage change from baseline are
A positive change means the estimate is higher than baseline; a negative change means it is lower. The denominator for the percentage comparison is the baseline estimate. If the baseline estimate is zero, this percentage is undefined, so report an absolute change or another suitable measure instead.
Build the Table From a Defined Calculation
Before calculating alternatives, specify the reserve measure and the assumption being varied. For a paid-loss development calculation, suppose \(P\) is cumulative paid loss at the valuation date and \(f\) is a selected paid loss development factor to ultimate. Under the deterministic chain-ladder projection, the projected ultimate is \(P f\), and unpaid claims are projected ultimate less paid to date:
This calculation assumes that the selected factor applies to the stated paid amount, that the factor is on a consistent basis, and that the reserve measure sought is unpaid loss excluding any separately modeled expenses or other liability components. It does not, by itself, estimate uncertainty or account for changes in claim handling, payment patterns, or portfolio composition.
Record the current estimate and the assumptions that produce it. State the valuation basis and the unit of measure.
Use a meaningful range around the baseline, and identify the assumption precisely—for example, a paid development factor for a specified maturity.
For each alternative value, use the same data and method and change no other assumption. Recalculate the reserve, rather than merely adjusting the displayed result by an unexplained amount.
Calculate absolute changes and, if appropriate, percentage changes using the baseline reserve as denominator. Label units and rounding.
Worked Example: Sensitivity to a Paid Development Factor
A hypothetical liability portfolio has cumulative paid losses of $1,200,000 at the valuation date. The selected paid development factor to ultimate is 1.50. To assess sensitivity, an analyst tests factors of 1.45 and 1.55, holding the paid amount, valuation date, and all other assumptions fixed. Calculate the reserve at each factor and compare it with baseline.
Method. Use the paid chain-ladder projection \(R(f)=P(f-1)\). The baseline factor is \(f_0=1.50\).
Baseline. Projected ultimate is \(1{,}200{,}000\times1.50=\$1{,}800{,}000\). Unpaid claims are \(\$1{,}800{,}000-\$1{,}200{,}000=\$600{,}000\). Equivalently, \(1{,}200{,}000\times(1.50-1)=\$600{,}000\).
Lower factor, 1.45. Projected ultimate is \(1{,}200{,}000\times1.45=\$1{,}740{,}000\). Reserve is \(\$1{,}740{,}000-\$1{,}200{,}000=\$540{,}000\). The absolute change is \(\$540{,}000-\$600{,}000=-\$60{,}000\), and the percentage change is \(-60{,}000/600{,}000\times100\%=-10\%\).
Higher factor, 1.55. Projected ultimate is \(1{,}200{,}000\times1.55=\$1{,}860{,}000\). Reserve is \(\$1{,}860{,}000-\$1{,}200{,}000=\$660{,}000\). The absolute change is \(\$660{,}000-\$600{,}000=\$60{,}000\), and the percentage change is \(60{,}000/600{,}000\times100\%=10\%\).
| Paid factor | Unpaid claims | Change from baseline | Percentage change |
|---|---|---|---|
| 1.45 | $540,000 | -$60,000 | -10% |
| 1.50 (baseline) | $600,000 | $0 | 0% |
| 1.55 | $660,000 | $60,000 | 10% |
The table is symmetric because reserve is linear in the factor and the tested factor values are equally spaced around 1.50. The common mistake is to report \(P f\), such as $1,800,000 at the baseline, as unpaid claims. That amount is projected ultimate; subtracting the $1,200,000 already paid gives the $600,000 unpaid estimate.
Why the Size of a Sensitivity Depends on the Calculation
For the paid-loss model above, the reserve is a linear function of the factor. The change in reserve between a tested factor \(f\) and baseline \(f_0\) follows directly by subtraction:
Thus, for a fixed paid amount \(P\), an increase of 0.01 in the selected factor increases the reserve by \(0.01P\). This is a useful check on a table's arithmetic. The relationship is specific to this formula; a different reserving calculation can respond nonlinearly to an assumption, so do not assume equal input steps always produce equal reserve changes.
A sensitivity table should preserve the model's actual calculation. For example, if trend is compounded over multiple years, each alternative rate must be compounded over the same stated period. Merely adding a trend percentage to a reserve does not generally reproduce the effect of changing the trend assumption.
Worked Example: Sensitivity to a Compounded Trend Rate
A hypothetical portfolio has historical expected ultimate losses of $300,000 measured two years before the valuation date. For this simplified calculation, trend those losses to the valuation date using a constant annual compound trend rate \(t\). Paid losses at the valuation date are $210,000. Test annual rates of 3%, 5%, and 7%, keeping the historical amount, two-year period, and paid losses fixed. Find the unpaid estimates.
Method and assumption. The projected ultimate at the valuation date is \(U(t)=300{,}000(1+t)^2\). The unpaid estimate is \(R(t)=U(t)-210{,}000\). This assumes a constant annual compound rate over exactly two years and uses the same paid amount for every row.
3% trend. The trend factor is \(1.03^2=1.0609\). Projected ultimate is \(\$300{,}000\times1.0609=\$318{,}270\). Unpaid claims are \(\$318{,}270-\$210{,}000=\$108{,}270\).
5% trend (baseline). The factor is \(1.05^2=1.1025\). Projected ultimate is \(\$300{,}000\times1.1025=\$330{,}750\). Unpaid claims are \(\$330{,}750-\$210{,}000=\$120{,}750\).
7% trend. The factor is \(1.07^2=1.1449\). Projected ultimate is \(\$300{,}000\times1.1449=\$343{,}470\). Unpaid claims are \(\$343{,}470-\$210{,}000=\$133{,}470\).
| Annual trend | Two-year factor | Unpaid estimate | Change from baseline |
|---|---|---|---|
| 3% | 1.0609 | $108,270 | -$12,480 |
| 5% (baseline) | 1.1025 | $120,750 | $0 |
| 7% | 1.1449 | $133,470 | $12,720 |
The alternative changes are not equal in size: the 3% result is $12,480 below baseline, while the 7% result is $12,720 above it. Compounding makes equal percentage-point changes in annual trend produce slightly unequal changes in the two-year factor. A common mistake is to apply the annual rate only once, or to use \(1+2t\) as if it were the exact two-year compound factor. The stated model uses \((1+t)^2\).
Interpreting and Presenting the Results
The table communicates sensitivity conditional on its assumptions. It does not establish that an alternative factor or trend rate is more plausible, and it does not assign probabilities to the rows. A reader should be able to tell what changed, what did not change, which estimate is the baseline, and whether the displayed measure is a component reserve or a total reserve.
When a table concerns one component of a portfolio, distinguish that component from the total. The following example shows how to do so and how to calculate percentage changes using the correct baseline denominator.
Worked Example: Reporting Total Reserve Impact and Percentage Change
A hypothetical portfolio's baseline total reserve is made up of $148,720 for all other cohorts plus the unpaid estimate for one paid-loss cohort. That cohort has $400,000 paid to date and a baseline paid development factor of 1.5000. An analyst tests factors of 1.4586 and 1.5414, changing no other selection. Report the total reserve sensitivity and the high-factor percentage change from baseline.
Baseline calculation. The cohort reserve is \(400{,}000(1.5000-1)=\$200{,}000\). Add the other cohorts: \(\$148{,}720+\$200{,}000=\$348{,}720\).
Lower factor. The cohort reserve is \(400{,}000(1.4586-1)=400{,}000(0.4586)=\$183{,}440\). Total reserve is \(\$148{,}720+\$183{,}440=\$332{,}160\). The change is \(\$332{,}160-\$348{,}720=-\$16{,}560\).
Higher factor. The cohort reserve is \(400{,}000(1.5414-1)=400{,}000(0.5414)=\$216{,}560\). Total reserve is \(\$148{,}720+\$216{,}560=\$365{,}280\). The change is \(\$365{,}280-\$348{,}720=\$16{,}560\). As a percentage of baseline, the increase is \(\$16{,}560/\$348{,}720\times100\%=4.7488\%\), which rounds to 4.749% to three decimal places.
The one-way table is:
| Factor for tested cohort | Total reserve | Change from baseline | Percentage change from baseline |
|---|---|---|---|
| 1.4586 | $332,160 | -$16,560 | -4.749% |
| 1.5000 (baseline) | $348,720 | $0 | 0% |
| 1.5414 | $365,280 | $16,560 | 4.749% |
The lower percentage is the negative of the higher percentage because the absolute changes are equal and both use the same baseline denominator. The common mistake is to divide the $16,560 increase by the high-factor result of $365,280, which answers a different question. For change from baseline, the denominator is $348,720.
Common Interpretation Traps
- Changing more than one assumption. If the factor and paid amount both change, the result does not isolate the sensitivity to the factor. Hold all other selections fixed.
- Leaving the baseline implicit. Label the baseline row and state the selected assumption that produces it. Without this, a “change” column has no clear reference.
- Using an inconsistent denominator. For percentage change from baseline, divide by the baseline estimate, not the alternative estimate.
- Confusing ultimate with unpaid claims. When using paid losses, projected ultimate includes amounts already paid; subtract paid to date to obtain unpaid claims.
- Reading tested values as probabilities. The table's endpoints are selected sensitivity values, not necessarily confidence limits or plausible bounds.
- Overinterpreting the result. A reserve movement under one changed assumption measures that model's response; it does not establish which assumption should be selected.
For a clear presentation, label amounts and units, retain enough precision in intermediate calculations, and round only the reported result. If a sensitivity table reports percentages, state the baseline denominator and the rounding convention. These choices make the arithmetic auditable and prevent a change in presentation from being mistaken for a change in the reserve model.
Check Your Understanding
For each problem, identify the baseline and keep every unvaried input fixed.
- A hypothetical book has $180,000 paid and a baseline paid development factor of 1.50. Calculate unpaid estimates at factors 1.40, 1.50, and 1.60. (Answer: $72,000, $90,000, and $108,000, respectively.)
- Historical expected ultimate losses are $500,000 from three years before valuation. Paid losses at valuation are $400,000. Under constant annual compound trend, compare rates of 2%, 4% (baseline), and 6%. (Answer: unpaid estimates are $130,604, $162,432, and $195,508, respectively.)
- A baseline reserve is $250,000, and a one-way alternative produces $265,000. What is the absolute and percentage change from baseline? (Answer: increase of $15,000, or 6%.)
- Other cohorts contribute $90,000 to a total reserve. A tested cohort has $120,000 paid and a baseline factor of 1.30. Calculate total reserves at factors 1.25, 1.30, and 1.35, and the total reserve change per 0.01 increase in factor. (Answer: $120,000, $126,000, and $132,000; $1,200 per 0.01 increase.)
- A baseline total reserve is $430,000. A lower assumption gives $410,000 and a higher assumption gives $462,000. Calculate the percentage changes from baseline to three decimal places. (Answer: -4.651% and 7.442%, respectively.)