Claims Development and Reserving

Bornhuetter–Ferguson Method

Blend an expected-loss prior with observed maturity without letting early random payments dominate.

Bornhuetter–Ferguson Method

The Bornhuetter–Ferguson (BF) method is designed for immature origin years. It uses observed loss for the developed portion and an independent prior for the unobserved portion.

1. Build the prior ultimate

Let $P_i$ be earned premium on a consistent on-level basis and $q_i$ the selected expected loss ratio (ELR):

U_i^{\text{prior}}=P_iq_i.

For Harbour Mutual, use an 86% ELR:

AY	Earned premium (£m)	ELR	Prior ultimate (£m)
2020	324	86%	278.64
2021	365	86%	313.90
2022	380	86%	326.80
2023	580	86%	498.80

An ELR may come from pricing, plan, exposure × frequency × severity, or comparable mature periods. “Independent” means it should not simply repackage the same immature paid losses.

2. Estimate the unobserved proportion

From the chain-ladder CDF, the expected paid proportion at age $j$ is $p_j=1/F_j$ . BF assigns the prior only to the unpaid proportion:

\widehat R_i^{BF}=U_i^{\text{prior}}(1-p_{j_i}),

\widehat U_i^{BF}=C_{i,j_i}+\widehat R_i^{BF}.

Notice what does not happen: BF does not multiply current paid loss by a large CDF. Early random payments therefore have less influence on the youngest years.

3. Work the portfolio

AY	Latest paid	Paid proportion	Prior ultimate	BF reserve	BF ultimate
2020	280.00	100.00%	278.64	0.00	280.00
2021	300.00	85.71%	313.90	44.84	344.84
2022	260.00	63.49%	326.80	119.31	379.31
2023	160.00	34.63%	498.80	326.06	486.06
Total	1,000.00			490.21	1,490.21

For AY 2023:

\widehat R_{2023}^{BF} =498.8\left(1-\frac{1}{2.8875}\right) =326.06.

The chain ladder gives £302.0m for the same year. BF is higher because pricing's prior ultimate (£498.8m) exceeds the chain-ladder ultimate (£462.0m).

4. Understand the credibility mechanism

Rewrite BF as:

\widehat U_i^{BF} =p_i\underbrace{\left(\frac{C_i}{p_i}\right)}_{\text{chain-ladder ultimate}} +(1-p_i)\underbrace{U_i^{\text{prior}}}_{\text{prior ultimate}}.

Thus $p_i$ acts like credibility given to the chain-ladder indication:

at 35% paid, most weight remains on the prior;
at 86% paid, observed experience dominates;
at 100% paid, BF ultimate equals observed paid loss.

This algebra explains BF more clearly than memorising a separate reserve formula.

5. Sensitivity belongs in the answer

For a change $\Delta q$ in ELR,

\Delta R_i^{BF}=P_i(1-p_i)\Delta q.

AY 2023 has premium £580m and is 34.63% paid. A five-percentage-point ELR increase changes its reserve by

580\times(1-0.3463)\times0.05\approx£18.96\text{m}.

Report this sensitivity rather than presenting £326.06m as if it were exact.

6. When the method helps—and when it hides problems

Situation	BF response	Required challenge
very immature recent year	stabilises against early random payments	is the ELR current and on-level?
exposure or rate changed	prior can reflect new portfolio	are premium and expected loss both adjusted consistently?
large early catastrophe payment	limits overreaction	should the catastrophe be modelled separately?
persistent adverse emergence	gradually responds as maturity increases	is slow response delaying recognition?
weak or politically selected plan	gives false stability	use independent benchmarks and scenarios

It is higher than chain ladder only when its prior for the unobserved portion is higher. A low prior can suppress genuine adverse emergence.

7. Cape Cod as the next idea

BF takes the ELR as external. Cape Cod estimates an ELR from the triangle and exposure, broadly:

\widehat q =\frac{\sum_i C_i} {\sum_i P_i p_i},

then uses this estimated ratio in a BF-style projection. This can improve internal consistency, but it makes the prior less independent from the observed data.

Practice

A year has £200m earned premium, 75% ELR, and is 40% paid. Calculate the BF reserve.
If £100m has already been paid, calculate BF ultimate and chain-ladder ultimate under the same 40% paid pattern.
What happens if the early paid amount doubles but premium, ELR, and paid proportion remain fixed?

Answers

Prior ultimate $=200\times0.75=150$ ; reserve $=150\times0.60=£90$ m.
BF ultimate $=100+90=£190$ m; chain-ladder ultimate $=100/0.40=£250$ m.
BF reserve remains £90m, so BF ultimate rises only by the additional paid amount. Chain-ladder ultimate doubles. This is the intended stability—and a reason to monitor whether the prior has become stale.

Next, quantify and test what the point estimates conceal in Uncertainty and Diagnostics.

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Frequency–Severity Reserving

Separate claim emergence from average cost to explain why a triangle develops.

Uncertainty and Diagnostics

Distinguish process, parameter, and model uncertainty; backtest reserving assumptions.