Why Test the Selections?
Berquist-Sherman adjustments respond to assumptions about how case reserves and claim closures would look on a consistent basis. Those assumptions are selections, not facts observed directly in the triangle. A reserve estimate can therefore appear precise while depending materially on a judgment about reserve adequacy, a target closure rate, or both. Sensitivity analysis makes that dependence visible.
This tutorial focuses on changing those selections and tracing the resulting changes in adjusted amounts and ultimate estimates. To isolate the effects, examples initially hold development factors fixed. In an actual analysis, an actuary may also recalculate the factors from each adjusted triangle; that additional effect should be distinguished from the direct effect of changing the selection.
For a paid-loss settlement adjustment, let \(q_j\) be the observed proportion of claims closed by development age \(j\), and let \(q_j^*\) be the selected target closure proportion. Under the simplifying assumption that paid losses are proportional to the closure proportion, the standardized paid amount is \(P_j^{S}=P_j(q_j^*/q_j)\). This is a selected-basis adjustment, not a claim that every closed claim has the same payment or that the adjusted amount is a separately observed payment total.
The assumptions matter. The adequacy multiplier should represent the relationship between reported case reserves and the chosen standard basis, and be appropriate for the relevant development age and claim population. The closure adjustment assumes that the proportional change in closure corresponds to a proportional change in cumulative paid losses, with claim mix and payment severity sufficiently comparable. If mix, severity, or closure behavior differs materially, a simple multiplier may not be credible.
How a Selection Change Reaches an Ultimate
Suppose a selected age-to-ultimate factor \(f_j\) is applied to the adjusted amount at age \(j\). For incurred losses, the projected ultimate is \(U_j^{A}=f_j(P_j+a_jC_j)\). Comparing two adequacy selections \(a_1\) and \(a_2\), while holding the data and factor fixed, gives:
This follows by subtracting the two projections: \(f_j(P_j+a_2C_j)-f_j(P_j+a_1C_j)\). The paid term \(f_jP_j\) cancels, leaving \(f_jC_j(a_2-a_1)\). Thus, the direct ultimate impact is larger when case outstanding is large, the factor is high, or the adequacy selections are far apart. The result holds only when the factor and all other inputs are held fixed.
For the closure adjustment, the corresponding direct change between two target closure selections \(q_1^*\) and \(q_2^*\), at fixed observed closure proportion \(q_j\), is:
Here the projected paid ultimate is \(f_jP_j(q_j^*/q_j)\). Subtraction cancels the common observed-basis terms and leaves the expression shown. This comparison requires \(q_j>0\); if no claims have closed, the ratio is undefined and this simple adjustment cannot be used.
Worked Example: Comparing Case-Reserve Adequacy Patterns
A hypothetical liability portfolio has the following cumulative amounts, all in thousands of dollars. Compare two adequacy patterns: Pattern L uses \(a_{12}=0.90\) and \(a_{24}=0.90\); Pattern H uses \(a_{12}=1.05\) and \(a_{24}=1.05\). Selected incurred age-to-ultimate factors are 2.00 at 12 months and 1.20 at 24 months. Hold these factors fixed to isolate the effect of the adequacy pattern.
| Age | Paid \(P_j\) | Case outstanding \(C_j\) | Incurred \(I_j\) |
|---|---|---|---|
| 12 months | 400 | 600 | 1,000 |
| 24 months | 850 | 150 | 1,000 |
At 12 months, Pattern L. Adjusted incurred is \(400+0.90(600)=400+540=940\). The ultimate estimate is \(940(2.00)=1,880\) thousand.
At 12 months, Pattern H. Adjusted incurred is \(400+1.05(600)=400+630=1,030\). The ultimate estimate is \(1,030(2.00)=2,060\) thousand. The Pattern H estimate exceeds Pattern L by \(2,060-1,880=180\) thousand.
At 24 months, Pattern L. Adjusted incurred is \(850+0.90(150)=850+135=985\). The ultimate estimate is \(985(1.20)=1,182\) thousand.
At 24 months, Pattern H. Adjusted incurred is \(850+1.05(150)=850+157.5=1,007.5\). The ultimate estimate is \(1,007.5(1.20)=1,209\) thousand. The difference is \(1,209-1,182=27\) thousand.
The check using the direct-change formula gives \(2.00(600)(1.05-0.90)=180\) thousand at 12 months and \(1.20(150)(1.05-0.90)=27\) thousand at 24 months. Both match the full calculations. The younger age is more sensitive here because it has substantially more case outstanding and a larger factor. The method avoids applying the multiplier to total incurred rather than only to case outstanding.
Testing a Settlement-Rate Basis
A closure selection changes standardized paid amounts through the ratio of the selected target closure proportion to the observed closure proportion. If the target is higher than the observed rate, the standardized paid amount increases under the proportionality assumption. The amount of that increase is not automatically evidence of additional ultimate loss: it is the consequence of expressing the observed data on the selected settlement-rate basis.
Worked Example: Two Target Closure Selections
A hypothetical portfolio has cumulative paid losses of \(480\) thousand at an age when 40% of claims are closed. Compare target closure selections of 45% and 50%. For this illustration, use a fixed paid age-to-ultimate factor of 2.40. Assume the proportional paid-to-closure relationship used by the adjustment is reasonable for this comparison.
Target closure rate of 45%. The scaling ratio is \(0.45/0.40=1.125\). Standardized paid losses are \(480(1.125)=540\) thousand. The projected ultimate is \(540(2.40)=1,296\) thousand.
Target closure rate of 50%. The scaling ratio is \(0.50/0.40=1.25\). Standardized paid losses are \(480(1.25)=600\) thousand. The projected ultimate is \(600(2.40)=1,440\) thousand.
The higher target increases the adjusted paid amount by \(600-540=60\) thousand and the projected ultimate by \(1,440-1,296=144\) thousand. The direct-change check is \(2.40(480)(0.50-0.45)/0.40=144\) thousand. The calculation avoids reversing the ratio: multiplying by \(0.40/0.50\) would produce \(384\) thousand, moving in the wrong direction for a target closure rate above the observed rate.
Joint Sensitivity and Attribution
When both selections are uncertain, a small scenario grid is often clearer than changing one input without recording the other. Keep the transformed quantities distinct: the closure adjustment in these examples standardizes paid losses, whereas the adequacy adjustment changes the case-reserve component of incurred losses. They are not automatically added together to create a new incurred measure. Any combined measure requires a defined construction that avoids mixing bases.
Worked Example: A Two-by-Two Selection Grid
A hypothetical portfolio has \(P=500\) thousand, \(C=300\) thousand, and an observed closure proportion \(q=0.40\). Consider adequacy multipliers \(a=0.80\) and \(a=1.00\), and target closure proportions \(q^*=0.45\) and \(q^*=0.50\). Use fixed paid and incurred age-to-ultimate factors of 2.00 and 1.60, respectively. Calculate the two adjusted measures and their corresponding ultimates for each selection pair.
For the paid measure, the 45% target gives \(500(0.45/0.40)=500(1.125)=562.5\) thousand, and its ultimate is \(562.5(2.00)=1,125\) thousand. The 50% target gives \(500(0.50/0.40)=500(1.25)=625\) thousand, and its ultimate is \(625(2.00)=1,250\) thousand.
For the incurred measure with \(a=0.80\), adjusted incurred is \(500+0.80(300)=500+240=740\) thousand, and its ultimate is \(740(1.60)=1,184\) thousand. With \(a=1.00\), adjusted incurred is \(500+1.00(300)=800\) thousand, and its ultimate is \(800(1.60)=1,280\) thousand.
| Adequacy multiplier \(a\) | Target closure \(q^*\) | Standardized paid | Paid ultimate | Adjusted incurred | Incurred ultimate |
|---|---|---|---|---|---|
| 0.80 | 0.45 | 562.5 | 1,125 | 740 | 1,184 |
| 0.80 | 0.50 | 625 | 1,250 | 740 | 1,184 |
| 1.00 | 0.45 | 562.5 | 1,125 | 800 | 1,280 |
| 1.00 | 0.50 | 625 | 1,250 | 800 | 1,280 |
Across the two closure selections, paid ultimate changes by \(1,250-1,125=125\) thousand. The formula check is \(2.00(500)(0.50-0.45)/0.40=125\) thousand. Across the adequacy selections, incurred ultimate changes by \(1,280-1,184=96\) thousand. The check is \(1.60(300)(1.00-0.80)=96\) thousand. The grid also makes clear that, with these transformations and fixed factors, the closure selection does not alter adjusted incurred, and the adequacy multiplier does not alter standardized paid.
This separation is deliberate. If factors are reselected from each adjusted triangle, changes in the factors can make the final results depend on both the adjustment and the development-factor selection. The fixed-factor comparison isolates the direct adjustment effect; a separate recalculation can then quantify the additional effect of factor changes.
Building an Informative Sensitivity Analysis
A sensitivity range is most useful when it represents plausible alternative assumptions rather than arbitrary extremes. An actuary should explain why the low, central, and high selections are defensible, and identify which data or judgment supports each one. For example, the selected adequacy pattern may vary by development age if case reserves are believed to be less adequate at early ages than at later ages. A single multiplier across all ages can conceal that pattern.
Specify whether the adjustment concerns case reserves, paid losses, or both as separate measures; define the development ages and the claim population being compared.
State the adequacy multipliers and target closure proportions, including how they vary by age if appropriate. Record the evidence or judgment supporting the alternatives.
Apply \(P_j+a_jC_j\) to incurred data and \(P_j(q_j^*/q_j)\) to paid data under the stated assumptions. Keep the two adjusted quantities identified by their basis.
First hold selected factors fixed to isolate direct effects. If factors are then refitted, report that result separately and explain that it combines adjustment and factor-selection effects.
State which selection drives the range, where the effect is largest, and what assumptions or data limitations prevent the sensitivity range from being treated as a probability interval.
The sensitivity range is not a confidence interval unless a statistical model and probability interpretation have been specified. It is a comparison of estimates under stated alternatives. A range can still be highly decision-useful: it reveals whether a reserve conclusion is robust to plausible selections or is driven by one uncertain assumption.
Common Traps and Interpretation
- Multiplying total incurred by an adequacy factor. If the defined selection changes case-reserve adequacy, apply it to \(C_j\), not to \(P_j+C_j\). For example, \(P=400\), \(C=600\), and \(a=0.90\) give \(400+0.90(600)=940\), not \(0.90(1,000)=900\).
- Inverting the closure ratio. The defined standardization is \(q_j^*/q_j\). A target higher than the observed closure proportion produces a multiplier above 1; the inverse produces the opposite direction.
- Changing a factor without saying so. A fixed-factor sensitivity isolates the amount adjustment. If factors are also recalculated, the change in ultimate includes both effects and should not be attributed solely to the adequacy or closure selection.
- Ignoring the assumptions behind the transformation. A closure adjustment may be misleading when claim mix, severity, or payment patterns differ across the closure populations being compared. An adequacy multiplier may also be unreliable if the case reserve data are inconsistent or the selected pattern does not fit the age and claim segment.
- Calling selected scenarios a statistical range. Low and high assumptions do not automatically have probabilities attached. Describe them as scenario estimates unless a probability model justifies a different interpretation.
Check Your Understanding
For each problem, show the adjusted amount before applying the stated factor. Money amounts are in thousands of dollars unless stated otherwise.
- A portfolio has paid losses of 700 and case outstanding of 500. Compare adequacy multipliers of 0.85 and 1.00 using a fixed incurred factor of 1.50. What are the two ultimate estimates and their difference? (Answer: 1,687.5 and 1,800; difference = 112.5 thousand.)
- Paid losses are 360, the observed closure proportion is 0.30, and the selected target is 0.45. Using a fixed factor of 2.00, calculate standardized paid and the projected ultimate. (Answer: standardized paid = 540 thousand; ultimate = 1,080 thousand.)
- At 12 months, \(P=250\), \(C=750\), and the incurred factor is 1.80. Calculate the ultimate under \(a=0.90\), then under \(a=1.10\), and find the change. (Answer: 1,665 and 1,935 thousand; increase = 270 thousand.)
- Paid losses are 640, observed closure is 0.40, and the paid factor is 1.75. Compare target closure selections of 0.45 and 0.50. What is the difference in projected ultimate? (Answer: 140 thousand.)
- A cell has \(P=300\), \(C=200\), observed closure 0.50, and fixed paid and incurred factors of 2.00 and 1.50. Compare \((a,q^*)=(0.80,0.40)\) with \((1.10,0.60)\). Report standardized paid and paid ultimate for each target, plus adjusted incurred and incurred ultimate for each adequacy selection. (Answer: paid amounts = 240 and 360 thousand, paid ultimates = 480 and 720 thousand; adjusted incurred = 460 and 520 thousand, incurred ultimates = 690 and 780 thousand.)