Why Combine Paid and Incurred Evidence?
Paid and incurred loss development provide two views of the same underlying claims. Paid amounts reflect cash already disbursed; incurred amounts add case reserves to those payments. A paid projection may respond slowly when claims take time to settle, while an incurred projection can respond to changes in case estimates before any additional cash is paid. Both can be useful, but neither should be treated as independent evidence: they overlap because incurred losses include paid losses.
A credibility-weighted estimate can blend a paid-based ultimate estimate, an incurred-based ultimate estimate, and an expected-loss benchmark. The resulting estimate must then be converted carefully from ultimate losses to unpaid losses. In particular, a case-reserve change is not a payment. It changes incurred evidence, but it does not change cumulative paid losses.
For the methods below, a development factor is a factor from the current valuation age to ultimate. If \(L_P\) is the selected paid development factor and \(L_I\) is the selected incurred development factor, the corresponding ultimate estimates are \(U_P=P_tL_P\) and \(U_I=I_tL_I\). These are projections, not final observed outcomes.
A Joint Credibility Estimate
Let \(U_0\) be an expected ultimate loss benchmark, such as one based on expected losses for the period. A simple credibility-weighted estimate is
The weights must satisfy \(w_P\geq 0\), \(w_I\geq 0\), and \(w_P+w_I\leq 1\), so that \(w_0\geq 0\). The weights represent the influence assigned to each source in this combined estimate. Because paid and incurred projections share information, their weights should be calibrated or selected jointly for the combined method. Applying separate credibility weights as if the two projections were independent can give the overlapping evidence too much influence.
The identity behind this result is direct. For a particular outcome, total ultimate losses equal paid-to-date losses plus payments still to be made. Thus, remaining payments equal ultimate losses minus paid-to-date losses. At the valuation date, \(P_t\) is already known, so estimating remaining payments amounts to estimating ultimate losses and subtracting \(P_t\). Case reserves are not subtracted in this identity; they are estimates of some of the remaining payments.
Calculation Sequence and Interpretation
A reliable calculation keeps the development projections, credibility blend, and reserve conversion distinct. This makes it easier to see whether a change is due to cash paid, incurred estimates, development assumptions, or the credibility weights.
Record cumulative paid losses \(P_t\) and cumulative incurred losses \(I_t\). Compute case reserves as \(I_t-P_t\) when useful as a check.
Apply the selected paid development factor to paid losses and the selected incurred development factor to incurred losses. This gives \(U_P\) and \(U_I\).
Blend \(U_P\), \(U_I\), and the expected ultimate \(U_0\), confirming that the weights are nonnegative and sum to one when the expected-loss weight is included.
Subtract cumulative paid losses once: \(R^*=U^*-P_t\). Interpret this as estimated remaining payments, which include amounts represented by case reserves as well as amounts not yet case-reserved.
This method assumes the selected development factors are appropriate for the relevant experience and that the expected-loss benchmark uses a compatible loss and exposure basis. It does not require a specified probability distribution for the worked calculations below, and no discounting is applied. In practice, the benchmark and the development projections must also be on a consistent basis, such as the same treatment of loss adjustment expenses and the same valuation date.
Worked Example: Blending Two Development Projections
A hypothetical liability portfolio is evaluated at year-end. Amounts are in thousands of dollars. Cumulative paid losses are \(P_t=420\), cumulative incurred losses are \(I_t=700\), the paid development factor is \(L_P=2.00\), and the incurred development factor is \(L_I=1.25\). The expected ultimate loss benchmark is \(U_0=1{,}000\). Use weights \(w_P=0.30\) and \(w_I=0.40\). Assume no discounting and no explicit probability distribution; the factors and weights are supplied selections.
Method. Project each basis to ultimate, combine the projections with the benchmark, then subtract paid losses to obtain the unpaid reserve.
The case reserve is \(700-420=280\). The paid-based ultimate estimate is
The incurred-based ultimate estimate is
The expected-loss weight is \(w_0=1-0.30-0.40=0.30\). Therefore,
The unpaid reserve estimate is \(902-420=482\) thousand dollars. As a check, it exceeds the current case reserve of 280 by 202 thousand dollars; that difference is the amount of the estimated unpaid reserve not represented by the current case reserve. The common mistake avoided here is subtracting incurred losses instead of paid losses: \(902-700=202\) is not the total unpaid reserve.
How Case-Reserve Changes Affect the Estimate
Suppose case reserves are revised while paid losses and development factors stay fixed. Since \(I_t=P_t+C_t\), a change \(\Delta C_t\) changes incurred losses by the same amount: \(\Delta I_t=\Delta C_t\). Under a fixed incurred development factor and fixed joint weights, the resulting change in the blended ultimate estimate is
This follows by substituting \(U_I=I_tL_I\) in the blend. The paid projection and expected-loss benchmark do not change under these assumptions. The change in unpaid reserve is also \(\Delta R^*=\Delta U^*\), because paid losses are fixed. This relationship is conditional on holding the development factor, benchmark, and weights fixed; if any of those are reselected, the total effect can differ.
Worked Example: A Downward Case-Reserve Revision
A hypothetical property portfolio has paid losses of \(P_t=500\) thousand dollars. Incurred losses are revised downward from 800 to 740 thousand dollars because case reserves are reduced; no payment is made as part of this revision. The paid factor is \(L_P=1.80\), the incurred factor is \(L_I=1.25\), and the expected ultimate benchmark is \(U_0=950\) thousand dollars. Use \(w_P=0.25\), \(w_I=0.45\), and no discounting or explicit probability distribution.
Method. Recalculate the incurred projection after the revision, retain the unchanged paid projection and benchmark, and compare the resulting unpaid estimates.
Before the revision, the paid projection is \(500(1.80)=900\), and the incurred projection is \(800(1.25)=1{,}000\). The expected-loss weight is \(1-0.25-0.45=0.30\), so the pre-revision ultimate estimate is
After the revision, the paid projection remains 900. The new incurred projection is \(740(1.25)=925\), giving
Subtracting paid losses gives unpaid estimates of \(960-500=460\) before and \(926.25-500=426.25\) thousand dollars after. The estimated unpaid reserve falls by \(33.75\) thousand dollars. This agrees with \(w_IL_I\Delta C_t=0.45(1.25)(-60)=-33.75\). The case reserve fell by 60 thousand dollars, but that is not itself a 60-thousand-dollar payment or a direct 60-thousand-dollar reduction in paid losses.
Conflicting Evidence and Common Traps
Paid and incurred projections need not agree. Differences can arise because cash payments and case estimates develop differently, or because the selected development factors differ. The credibility blend is a specified way of balancing those projections with the expected-loss benchmark; it does not make the inputs independent or eliminate the need for judgment about their basis.
A frequent error is to call \(U^*-I_t\) the unpaid reserve. That amount is the estimated amount beyond incurred losses, sometimes interpreted as an estimate of IBNR relative to the current incurred figure. It excludes the current case reserve. The total unpaid reserve, which includes both case reserves and amounts beyond them, is \(U^*-P_t\). Another error is to increase paid losses when case reserves increase. A case estimate changes incurred losses, not historical cash paid.
Worked Example: Resolving an Incurred-Projection Difference
A hypothetical specialty insurer has \(P_t=360\) thousand dollars and \(I_t=660\) thousand dollars. Its paid development factor is \(2.50\), its incurred development factor is \(1.30\), and its expected ultimate benchmark is 1,000 thousand dollars. Use weights \(w_P=0.25\) and \(w_I=0.35\), with no discounting or explicit probability distribution.
Method. Calculate both ultimate projections and retain the expected-loss weight as the residual weight.
The case reserve is \(660-360=300\). The two projections are \(U_P=360(2.50)=900\) and \(U_I=660(1.30)=858\), both in thousands of dollars. The benchmark weight is \(1-0.25-0.35=0.40\). Thus,
The unpaid reserve is \(925.3-360=565.3\) thousand dollars. The estimate beyond incurred losses is \(925.3-660=265.3\) thousand dollars; adding the 300-thousand-dollar case reserve gives the same total unpaid estimate, \(265.3+300=565.3\). The common mistake avoided is reporting 265.3 as the total unpaid reserve when it omits the existing case reserve.
These calculations are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus because they require consistent use of development evidence, expected-loss credibility, and the distinction between ultimate and unpaid losses. In a reserve review, showing each component separately also makes the result easier to audit and the effect of a case-reserve revision easier to explain.
Check Your Understanding
For each hypothetical case, assume the stated factors and weights are selected, there is no discounting, and no explicit probability distribution is required. Amounts are in thousands of dollars. Calculate the requested quantity and check whether case-reserve changes affect paid losses.
- A portfolio has \(P_t=300\), \(I_t=500\), \(L_P=2.20\), \(L_I=1.40\), and \(U_0=800\). Use \(w_P=0.20\) and \(w_I=0.50\). Find the credibility-weighted ultimate and unpaid reserve. (Answer: ultimate \(722\); unpaid reserve \(422\).)
- Paid losses are 250, incurred losses fall from 600 to 560 after a case-reserve revision, and \(L_I=1.50\). The paid projection is unchanged. With \(U_0=600\), \(w_P=0.20\), and \(w_I=0.50\), by how much does the unpaid reserve estimate change? (Answer: it decreases by \(30\).)
- A portfolio has \(P_t=400\), \(I_t=650\), \(L_P=1.50\), \(L_I=1.20\), and \(U_0=750\). Use \(w_P=0.40\) and \(w_I=0.30\). Find the unpaid reserve. (Answer: \(299\).)
- Given \(P_t=350\), \(I_t=600\), \(L_P=2.00\), \(L_I=1.40\), and \(U_0=900\), use \(w_P=0.30\) and \(w_I=0.50\). Find the residual benchmark weight and unpaid reserve. (Answer: benchmark weight \(0.20\); unpaid reserve \(460\).)
- Paid losses are 200 and incurred losses increase from 500 to 540 after a case-reserve strengthening. Use \(L_P=2.00\), \(L_I=1.25\), \(U_0=700\), \(w_P=0.25\), and \(w_I=0.45\). Find the change in the unpaid estimate, assuming all other inputs stay fixed. (Answer: it increases by \(22.5\), with paid losses unchanged.)