Retrospective Premium Reserve

Study guides: Exam 7, Exam GI-301

The retrospective premium reserve is the provision for premium adjustments still to come on retrospectively rated policies: the expected ultimate retrospective premium, given the losses still to emerge, less the premium already booked. When positive it is an asset, the premium asset (earned but not reported premium, shown as accrued retrospective premiums). When negative it is a liability for return premium. The "retro reserve" of the older literature (Berry, Fitzgibbon) is the same quantity with the opposite sign, the premium deviation to date less the ultimate premium deviation (Teng and Perkins).

$\text{Premium asset} = \text{Expected ultimate premium} - \text{Premium booked}$

$\text{Ult. premium} = P_{\text{last adj.}} + \text{CPDLD}_k \times \text{Future loss emergence}$

Teng and Perkins estimate how premium develops as losses develop. The retro formula at adjustment $n$ is $P_n = (\text{BP} + CL_n \times \text{LCF}) \times \text{TM}$ , with capped losses $CL_n$ . The ratio of premium development to loss development (PDLD) is then:

$\text{PDLD}_1 = \frac{\text{BPF} \times \text{TM}}{\text{ELR} \times \%L_1} + \frac{CL_1}{L_1}\,\text{LCF} \times \text{TM}$

$\text{PDLD}_n = \frac{CL_n - CL_{n-1}}{L_n - L_{n-1}}\,\text{LCF} \times \text{TM}$

The symbols:
- BPF is the basic premium factor (basic premium divided by standard premium), TM the tax multiplier and LCF the loss conversion factor;
- ELR is the expected loss ratio to standard premium, and $\%L_1$ the share of losses emerged at the first adjustment;
- $CL/L$ is the loss capping ratio, the share of loss development that the per-accident limit and the retro maximum and minimum let through.
$\text{PDLD}_1$ is usually above 1. The whole basic premium is collected at the first adjustment and few losses are capped yet. Later PDLDs fall well below 1 as more of each new dollar of loss lands above a cap. Assuming premium tracks losses one-for-one overstates the asset.
Cumulative ratios. $\text{CPDLD}_k$ averages the PDLDs from adjustment $k$ onward, weighted by the expected loss emerging in each: $\sum_{j \ge k} \text{PDLD}_j\, \Delta\%L_j \,/\, \sum_{j \ge k} \Delta\%L_j$ . It is the premium expected per dollar of loss still to emerge.
Formula or empirical. PDLDs can come from the plan parameters sold, which respond to changes in plan design, or from history, dividing booked premium by reported loss. Premium is booked with a lag: in Teng and Perkins' example, premium at 27 months reflects losses valued at 18 months, the first adjustment. Capping ratios are estimated with Table M charges and a loss elimination ratio for the per-accident limit.
Tie it to the loss reserve. Future loss emergence should come from the same analysis that sets IBNR, so a reserve increase raises the premium asset consistently. The asset is collected from insureds after the fact, so collectability and collateral matter. See Premium Reserve for the other premium-sensitive reserves.

PDLD and CPDLD Ratios from Plan Parameters A book of retro plans has BPF $0.25$ , TM $1.04$ , LCF $1.10$ and ELR $0.65$ . Losses emerge $80\%$ by the first adjustment, $12\%$ by the second and $8\%$ by the third (the last). The loss capping ratios are $0.90$ , $0.60$ and $0.40$ .

Compute the PDLD and CPDLD ratios.
Answer
 $\begin{align*} \text{PDLD}_1 &= \frac{0.25(1.04)}{0.65(0.80)} + 0.90(1.10)(1.04) \\ &= 0.5000 + 1.0296 \\ &= 1.5296 \\ \text{PDLD}_2 &= 0.60(1.10)(1.04) \\ &= 0.6864 \\ \text{PDLD}_3 &= 0.40(1.10)(1.04) \\ &= 0.4576 \end{align*}$ 
 $\begin{align*} \text{CPDLD}_1 &= 1.5296(0.80) + 0.6864(0.12) + 0.4576(0.08) \\ &= 1.3427 \\ \text{CPDLD}_2 &= \frac{0.6864(0.12) + 0.4576(0.08)}{0.20} \\ &= 0.5949 \\ \text{CPDLD}_3 &= 0.4576 \end{align*}$ 
Before the first adjustment each dollar of expected loss brings $\$1.34$ of premium. After it, each further dollar brings only $\$0.59$ , because the basic premium has been collected and caps increasingly bind.

Premium Asset at a Valuation Date Using the CPDLDs above:

Policy year A has had no retro adjustment. Its expected losses through all adjustments are $\$10.0$ M, and $\$12.0$ M of standard premium has been booked.

Policy year B has had its first adjustment, which produced $\$11.5$ M of premium. Losses still expected to emerge are $\$1.8$ M, and booked premium is now $\$11.6$ M.

Compute the premium asset.
Answer
 $\begin{align*} \text{Ult}_A &= 1.3427 \times 10.0 \\ &= 13.43 \\ \text{Asset}_A &= 13.43 - 12.00 \\ &= 1.43 \\ \text{Ult}_B &= 11.50 + 0.5949 \times 1.8 \\ &= 12.57 \\ \text{Asset}_B &= 12.57 - 11.60 \\ &= 0.97 \end{align*}$ 
The total premium asset is $\$2.40$ M. Note how year B uses booked premium from its last adjustment as the base and applies only the later, smaller CPDLD to the losses still to come. Had the actuary applied $1.34$ to B's remaining $\$1.8$ M, the asset would be overstated by $\$1.3$ M.