Credibility Blends Of Chain Ladder And Bornhuetter-Ferguson

Why Blend Reserving Methods?

Chain ladder and Bornhuetter-Ferguson use different information to estimate ultimate losses. Chain ladder projects observed experience using selected development factors. Bornhuetter-Ferguson (BF) combines observed losses with an a priori expected ultimate, applying that prior to the portion of losses expected to remain undeveloped. A credibility blend can make the final estimate responsive to both components: more responsive to observed development when experience receives greater weight, and more responsive to the prior when the prior receives greater weight.

The blend does not, by itself, determine how much credibility to assign. Treat the credibility weight as an input to this calculation, and make sure that the chain ladder estimate, BF estimate, and prior all refer to the same loss basis, period, and valuation date. This tutorial is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: For an accident period, let \(C\) be the observed cumulative loss at the valuation date, \(L\) the selected cumulative development factor from that valuation age to ultimate, and \(P\) the a priori expected ultimate loss on the same basis. Let \(Z\) be the credibility weight assigned to chain ladder, with \(0 \leq Z \leq 1\). The chain ladder ultimate is \(CL=CL_{\text{ult}}=CL\) in the shorthand below, or, stated without ambiguity, \(C_{\text{CL}}=CL\). The BF ultimate is \(C_{\text{BF}}=C+P(1-1/L)\). The credibility-blended ultimate is \(C_{\text{blend}}=ZC_{\text{CL}}+(1-Z)C_{\text{BF}}\).

Here \(CL\) in the product \(C L\) means observed cumulative loss multiplied by the development factor; \(C_{\text{CL}}\) denotes the resulting chain ladder estimate. The notation \(C_{\text{CL}}\), \(C_{\text{BF}}\), and \(C_{\text{blend}}\) will be used below to keep the three estimates distinct. The factor \(L\) must be a cumulative factor from the current development age to ultimate, not merely the next age-to-age factor. Ordinarily \(L \geq 1\) for cumulative losses that develop upward.

How the Two Components Are Constructed

The chain ladder ultimate estimate applies the selected factor directly to observed cumulative losses:

C_{\text{CL}}=CL.

The BF estimate instead adds an expected amount of future development to the observed cumulative amount. Under the BF assumption that the selected development pattern applies to the expected ultimate, the estimated proportion emerged is \(1/L\), so the estimated proportion not yet emerged is \(1-1/L\). Applying that remaining proportion to the a priori expected ultimate gives the BF unpaid amount \(P(1-1/L)\).

C_{\text{BF}}=C+P\left(1-\frac{1}{L}\right).

The credibility blend weights ultimate estimates, not development factors. Substituting the two formulas gives a useful equivalent form:

\begin{aligned} C_{\text{blend}} &=Z(CL)+(1-Z)\left[C+P\left(1-\frac{1}{L}\right)\right]\\ &=C+ZC(L-1)+(1-Z)P\left(1-\frac{1}{L}\right). \end{aligned}

This rewrite shows that the blended estimate begins with the observed cumulative amount \(C\). It then adds a credibility-weighted chain ladder development amount and a complementary-weighted prior-based development amount. It also provides an arithmetic check: the result must lie between the chain ladder and BF estimates whenever \(0 \leq Z \leq 1\).

Worked Triangle: Calculate Each Period Separately

Consider the following hypothetical cumulative loss triangle, with amounts in thousands of dollars. The selected age-to-age factors are 1.45 from 12 to 24 months and 1.20 from 24 to 36 months. These are assumed selections for this example; they need not equal every individual ratio visible in this small triangle. Thus the cumulative factors to ultimate are 1.74 at 12 months, 1.20 at 24 months, and 1.00 at 36 months.

Accident period	12 months	24 months	36 months
2022	600	900	1,080
2023	700	980	—
2024	800	—	—

Assume the following additional inputs, all in thousands: for 2023, \(P=1{,}200\) and \(Z=0.60\); for 2024, \(P=1{,}500\) and \(Z=0.40\). The 2022 period is fully developed. Calculate the three estimates for each period and aggregate them.

Worked Example: Blending Estimates Across an Uneven Triangle

Method. For each accident period, use its current cumulative amount and cumulative development factor to calculate chain ladder and BF separately. Then apply that period's stated credibility weight. The common mistake to avoid is using a single factor or weight for every period just because the amounts appear in one triangle.

2022. This period is observed at 36 months, so \(C=1{,}080\) and \(L=1.00\). Chain ladder gives \(C_{\text{CL}}=1{,}080(1.00)=1{,}080\). BF adds \(P(1-1/L)=P(1-1)=0\), so \(C_{\text{BF}}=1{,}080\). Because both methods agree, any credibility weight gives \(C_{\text{blend}}=1{,}080\).

2023. The current amount is \(C=980\) at 24 months, so \(L=1.20\). Chain ladder gives \(C_{\text{CL}}=980(1.20)=1{,}176\). The BF unpaid proportion is \(1-1/1.20=1-0.833333\ldots=1/6\). Its expected unpaid amount is \(1{,}200(1/6)=200\), giving \(C_{\text{BF}}=980+200=1{,}180\). With \(Z=0.60\), the blend is \(0.60(1{,}176)+0.40(1{,}180)=705.60+472.00=1{,}177.60\).

2024. The current amount is \(C=800\) at 12 months. The cumulative factor is \(L=1.45(1.20)=1.74\), not just 1.45. Chain ladder gives \(C_{\text{CL}}=800(1.74)=1{,}392\). The BF unpaid proportion is \(1-1/1.74=1-0.5747126\ldots=0.4252874\ldots\). The expected unpaid amount is \(1{,}500(0.4252874\ldots)=637.9310\ldots\), and \(C_{\text{BF}}=800+637.9310\ldots=1{,}437.9310\ldots\). With \(Z=0.40\), the blend is \(0.40(1{,}392)+0.60(1{,}437.9310\ldots)=556.80+862.7586\ldots=1{,}419.5586\ldots\), or \(1{,}419.56\) thousand.

Aggregate and check. The blended total is \(1{,}080+1{,}177.60+1{,}419.5586\ldots=3{,}677.1586\ldots\), or \(3{,}677.16\) thousand. The 2023 result lies between 1,176 and 1,180; the 2024 result lies between 1,392 and 1,437.9310. Both are consistent with convex credibility blends. Amounts are rounded to two decimal places only at the final reporting step.

Why the Bornhuetter-Ferguson Formula Works

The BF formula follows from the assumed emergence proportion. Suppose the expected ultimate for a period is \(P\), and the selected cumulative development factor from the valuation age to ultimate is \(L\). Under the BF development-pattern assumption, the fraction of ultimate expected to have emerged by the valuation date is \(1/L\). Consequently, the fraction expected to remain is \(1-1/L\). The prior-based estimate of the remaining amount is therefore:

\text{expected amount remaining} =P\left(1-\frac{1}{L}\right).

Adding that expected remaining amount to the amount already observed yields \(C_{\text{BF}}=C+P(1-1/L)\). This argument depends on using \(P\) as an expected ultimate and \(L\) as the cumulative factor for the same loss basis and valuation age. It does not say that \(P\) is the observed cumulative amount, nor does it replace observed losses with a prior. If \(L=1\), the expected remaining proportion is zero and BF equals \(C\). For an immature period with \(L>1\), BF adds a nonnegative expected amount when \(P\geq0\).

Interpretation: \(Z=1\) gives the chain ladder estimate; \(Z=0\) gives the BF estimate. For weights between zero and one, the blend is a convex combination of the two estimates. The weight is not automatically implied by the formulas for chain ladder or BF; it must be supplied or justified by a separate credibility procedure.

Worked Example: Check the Weight and the Direction of the Blend

Worked Example: A Partially Credible Chain Ladder Estimate

For a hypothetical portfolio segment, the current cumulative losses are \(C=1{,}000\) thousand, the cumulative factor to ultimate is \(L=1.25\), the a priori expected ultimate is \(P=1{,}600\) thousand, and the chain ladder credibility weight is \(Z=0.70\). Calculate both component ultimates and the blend, then identify which component is closer to the blend.

Chain ladder. \(C_{\text{CL}}=CL=1{,}000(1.25)=1{,}250\) thousand.

Bornhuetter-Ferguson. The emerged proportion is \(1/1.25=0.80\), so the remaining proportion is \(1-0.80=0.20\). The prior-based expected unpaid amount is \(1{,}600(0.20)=320\) thousand. Thus \(C_{\text{BF}}=1{,}000+320=1{,}320\) thousand.

Blend. The chain ladder contribution is \(0.70(1{,}250)=875\) thousand. The BF contribution is \(0.30(1{,}320)=396\) thousand. Therefore \(C_{\text{blend}}=875+396=1{,}271\) thousand.

Check. The result lies between 1,250 and 1,320. Its distance from chain ladder is \(1{,}271-1{,}250=21\) thousand; its distance from BF is \(1{,}320-1{,}271=49\) thousand. It is closer to chain ladder, as expected for a 70% chain ladder weight. Amounts are in thousands and are exact to the nearest thousand.

Common Exam Traps

A tempting but incorrect calculation in the example above is \(0.70(1{,}250)+0.30(1{,}120)=1{,}211\). The error is the BF component: 1,120 is not obtained from the stated prior and development factor. The BF unpaid amount is \(1{,}600(1-1/1.25)=1{,}600(0.20)=320\), so BF is \(1{,}000+320=1{,}320\). Correctly blending gives \(875+396=1{,}271\), not 1,211.

Using the wrong development factor. For an observation at 12 months, multiply the selected 12-to-ultimate cumulative factor, not only the next age-to-age factor. In the triangle, that factor is \(1.45(1.20)=1.74\).
Reversing the weights. Here \(Z\) weights chain ladder, so \(1-Z\) weights BF. Check the limiting cases: at \(Z=1\), the answer must be chain ladder; at \(Z=0\), it must be BF.
Blending reserves when the question asks for ultimates. If blending ultimate estimates, calculate the blend of the ultimate estimates. The corresponding unpaid reserve is the blended ultimate minus the observed cumulative losses, on the same basis.
Combining mismatched inputs. The prior \(P\), observed amount \(C\), and factor \(L\) must refer to the same period, loss definition, and valuation date. Otherwise, the BF unpaid amount and the blend do not have a consistent interpretation.

Identify the inputs by period.
Record the current cumulative amount, cumulative factor to ultimate, a priori ultimate, and chain ladder weight for each accident period.

Calculate both component ultimates.
Use \(C_{\text{CL}}=CL\) and \(C_{\text{BF}}=C+P(1-1/L)\).

Apply and verify the blend.
Calculate \(ZC_{\text{CL}}+(1-Z)C_{\text{BF}}\), then check that it lies between the component estimates.

Check Your Understanding

For each problem, calculate the requested estimate using the stated inputs. Amounts are in thousands unless otherwise stated.

A period has \(C=720\), \(L=1.50\), \(P=900\), and \(Z=0.40\). Calculate the chain ladder ultimate, BF ultimate, and blended ultimate. (Answer: chain ladder \(1{,}080\); BF \(1{,}020\); blend \(1{,}044\) thousand.)
A period has \(C=640\), \(L=1.25\), \(P=800\), and \(Z=0.25\). Calculate the three ultimate estimates and explain why the blend does not depend on \(Z\) in this case. (Answer: chain ladder \(800\); BF \(800\); blend \(800\) thousand; the components agree.)
A period has \(C=900\), \(L=1.20\), \(P=1{,}500\), and \(Z=0.60\). Calculate the blended ultimate. (Answer: \(1{,}108\) thousand.)
Two periods have these inputs: period A, \(C=400,L=1.50,P=600,Z=0.50\); period B, \(C=1{,}000,L=1.25,P=1{,}200,Z=0.20\). Calculate each blended ultimate and their total. (Answer: period A \(600\), period B \(1{,}242\), total \(1{,}842\) thousand.)
A period has \(C=800,L=1.50,P=1{,}500\). Its blended ultimate is \(1{,}240\). Determine \(Z\), the chain ladder weight. (Answer: \(Z=0.60\).)
A period has \(C=1{,}200,L=1.25,P=1{,}600,Z=0.40\). Calculate the blended ultimate and the implied unpaid reserve relative to current cumulative losses. (Answer: blended ultimate \(1{,}512\); unpaid reserve \(312\) thousand.)

What You'll Learn