Range of Indications

Study guide: Exam 7

A range of indications is the spread of unpaid-claim estimates obtained when several reasonable methods, models or assumption sets are applied to the same data. It is the evidence from which the actuary selects a point estimate and states a range of reasonable estimates. It measures uncertainty about the estimate of the mean. That is a different, and usually much narrower, thing than the range of possible outcomes described by an unpaid claim distribution.

$\text{Range} = \Big[\min_{m \in M} \hat R_m,\; \max_{m \in M} \hat R_m\Big]$

$\text{Range of reasonable estimates} \subset \text{Range of possible outcomes}$

$M$ is the set of methods judged appropriate for the data, and $\hat R_m$ is method $m$ 's estimate of unpaid claims or IBNR. Build it by accident year: paid and reported development, expected claims, BF, Cape Cod, Benktander and frequency-severity. First drop any method whose assumptions a diagnostic has shown to fail.
A method the data contradicts does not widen the range. If settlement has sped up, the paid chain ladder's high answer is a known bias, not evidence of uncertainty. Leaving it in overstates the range; leaving out a sound method that disagrees understates it.
Stochastic version (Shapland). Run several bootstrap models (paid and incurred chain ladder, BF, Cape Cod, GLM) and weight them by year to get a best-estimate distribution. The modeled range spans the means of every model used anywhere. The weighted range uses, for each year, only the models given weight in that year. It is the more representative view of uncertainty in the actuarial central estimate.
Reinsurance ranges are wider. In Friedland's reinsurance examples, the range of indicated IBNR (maximum less minimum across methods) was wider for reinsurance than for primary business, for non-proportional than for proportional covers, and for catastrophe than for non-catastrophe property. See Reinsurance Reserving.
Professional basis. The CAS Statement of Principles on unpaid claims estimates (Principle 2, see Actuarial Principles) says uncertainty "implies that a range of estimates can be reasonable". ASOP 43 allows a point estimate, a range, or both. In the U.S., ASOP 36 treats carried reserves as a reasonable provision when they fall within a range of estimates the actuary considers reasonable.

Screening Methods Before Stating a Range Total IBNR indications ($000s) for a commercial auto book: reported development $4{,}200$ ; paid development $6{,}100$ ; expected claims $5{,}000$ ; BF (reported) $4{,}700$ ; BF (paid) $5{,}300$ . Closure rates at every maturity have risen sharply over the last two calendar years, and paid-to-reported ratios with them.

State a range of reasonable estimates.
Answer The raw spread is $4{,}200$ to $6{,}100$ , which is $1{,}900$ wide.

Rising closure rates mean payments are running ahead of the historical pattern. The paid development method applies old, slower factors to a faster diagonal, so it is biased high. The same is true, to a lesser degree, of paid BF, whose "percentage unpaid" comes from the same stale pattern.

Dropping paid development and noting the upward lean of paid BF:
 $\begin{align*} \text{Range} &= [4{,}200,\ 5{,}300] \\ \text{Width} &= 1{,}100 \end{align*}$ 
A selection near $4{,}800$ , between reported BF and expected claims, is defensible. If the paid triangle were restated to current disposal rates (Berquist-Sherman), its result could come back into the range.

A bootstrap on the same data might put the 25th to 75th percentiles of outcomes at $4{,}300$ to $5{,}600$ . That band is not a competing range of reasonable estimates. It says where the actual payments may land, not where a reasonable estimate of their mean may lie.

Modeled Versus Weighted Range Three bootstrap models give mean unpaid ($000s) for two accident years:

AY A (mature): paid CL $100$ , incurred CL $110$ , BF $105$ . Weights $50\% / 50\% / 0\%$ .

AY B (latest): paid CL $900$ , incurred CL $700$ , BF $760$ . Weights $0\% / 25\% / 75\%$ .

Find the best estimate and both ranges.
Answer
 $\begin{align*} \text{Best}_A &= 0.5(100) + 0.5(110) \\ &= 105 \\ \text{Best}_B &= 0.25(700) + 0.75(760) \\ &= 745 \\ \text{Best} &= 850 \end{align*}$ 
Weighted range, using only the models given weight in each year: A is $[100, 110]$ and B is $[700, 760]$ , so the total is $[800, 870]$ .

Modeled range, using every model used anywhere: A is $[100, 110]$ and B is $[700, 900]$ , so the total is $[800, 1{,}010]$ .

The modeled range's top end is set by the paid chain ladder on AY B, a model the actuary gave no weight for that year. The weighted range reflects the actual judgment and is the better statement of uncertainty in the central estimate.