Workers' Compensation Ratemaking and Reserving from a Real Schedule P Triangle

Background

You are the pricing/reserving actuary reviewing Allstate Ins Co Grp's countrywide workers' compensation experience, pulled from the CAS Loss Reserve Database (NAIC Schedule P, accident years 1988–1997, net of reinsurance, values in $000s). Your task mirrors a classic Exam 5 exercise: complete the paid-loss triangle to ultimate using the volume-weighted (chain-ladder) method, quantify IBNR reserves, and use the resulting loss ratios to indicate a rate level change.

Paid loss triangle ($000s)

Corresponding net earned premium by accident year ( $000s): 1988:$ 394,742; 1989: $374,252; 1990:$ 280,320; 1991: $313,982; 1992:$ 252,698; 1993: $201,055; 1994:$ 174,381; 1995: $146,366; 1996:$ 93,294; 1997: $7,651 (a partial, thin year of business).

Volume-weighted age-to-age factors

Computing each age-to-age (link ratio) factor as $f_j = \dfrac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}}$ over accident years with data at both ages $j$ and $j+1$ gives:

Age $j \to j{+}1$	1→2	2→3	3→4	4→5	5→6	6→7	7→8	8→9	9→10
$f_j$	2.2230	1.3377	1.1584	1.0927	1.0586	1.0455	1.0314	1.0361	1.0109

Chaining these forward (assuming the triangle is fully developed by age 10, i.e. the tail factor beyond age 10 is 1.000) gives cumulative development factors (CDFs) to ultimate from each age:

Age	1	2	3	4	5	6	7	8	9	10
CDF to ult	4.5011	2.0248	1.5136	1.3066	1.1957	1.1295	1.0803	1.0474	1.0109	1.0000

Projected ultimate losses and IBNR

Applying each accident year's CDF (from its latest diagonal age) to its latest cumulative paid loss produces:

AY	Latest paid ($000s)	CDF	Ultimate ($000s)	Reserve (IBNR + case, $000s)
1988	325,322	1.0000	325,322	0
1989	273,873	1.0109	276,864	2,991
1990	256,788	1.0474	268,961	12,173
1991	239,195	1.0803	258,402	19,207
1992	159,496	1.1295	180,151	20,655
1993	87,215	1.1957	104,286	17,071
1994	91,077	1.3066	119,003	27,926
1995	87,311	1.5136	132,157	44,846
1996	44,916	2.0248	90,948	46,032
1997	691	4.5011	3,110	2,419
Total	1,565,884		1,759,204	193,320

Total indicated reserves (the sum of the last column) are ** $193.3 million**, i.e. the paid-to-date total of$ 1,565.9M is projected to develop to $1,759.2M at ultimate.

Loss ratios and rate indication

Dividing each accident year's ultimate loss by its earned premium gives the ultimate loss ratio by year:

AY	1988	1989	1990	1991	1992	1993	1994	1995	1996	1997
ELR	0.824	0.740	0.959	0.823	0.713	0.519	0.682	0.903	0.975	0.407

The premium-weighted ultimate loss ratio across all ten years is $\sum \text{Ultimate}_i / \sum \text{Premium}_i = 1{,}759{,}204/2{,}239{,}101 = 0.7858$ , i.e. 78.58%. Suppose the rate filing's permissible loss ratio (i.e., $1$ minus the expense-and-profit load) is 65%, reflecting a 35% combined expense ratio and target underwriting profit provision. The indicated rate change is

$\text{Indicated change} = \frac{\text{Premium-weighted ultimate LR}}{\text{Permissible LR}} - 1 = \frac{0.7858}{0.65} - 1 = +20.9\%.$

This straightforward comparison of experience loss ratio to permissible loss ratio is the core of the loss ratio method of ratemaking, one of the two classic methods (the other being the pure premium method) tested throughout CAS Exam 5.

Your task

Treat all figures above as given (do not recompute from scratch, though you should be able to reproduce any single cell if asked). Answer the questions below about the chain-ladder mechanics, the reserve estimate, and the rate indication.