PDLD Ratio

PDLD Ratio (premium development to loss development ratio) is Teng and Perkins's measure of how retrospective premium responds to loss on a book of retro rated policies. For each retro adjustment it is the premium developed in that adjustment divided by the loss developed over the matching loss valuations. Accumulated into CPDLD ratios and applied to expected future loss emergence, it gives the future premium whose sum is the premium asset (Retrospective Premium Reserve).

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

$\text{PDLD}_n = \frac{P_n - P_{n-1}}{L_n - L_{n-1}}$

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

$\text{CPDLD}_k = \frac{\sum_{n \ge k} \text{PDLD}_n \, \Delta\%L_n}{\sum_{n \ge k} \Delta\%L_n}$

Symbols. $P_n$ is the retro premium at the $n$ th adjustment, and $L_n$ the loss it is computed on. $CL_n$ is that loss capped by the retro maximum and minimum and any per-accident limit. BP is the basic premium, LCF the loss conversion factor, TM the tax multiplier, and $\Delta\%L_n$ the share of ultimate loss emerging in period $n$ . $\text{BP}/L_1$ is approximated by $\text{BPF}/(\text{ELR} \times \%L_1)$ , the basic premium factor over the expected loss ratio emerged at the first adjustment.
First versus subsequent.
- $\text{PDLD}_1$ is usually above 1. It carries the whole basic premium, and little loss is capped by the first adjustment.
- Later ratios are the incremental loss capping ratio times $\text{LCF} \times \text{TM}$ . They are usually well below 1, as more new loss lands above the caps.
- An individual policy quarter's historical ratio can even be negative. That happens when upward development above the limits coincides with downward development within them.
Formula versus empirical.
- The formula ratios respond to changes in the plan parameters sold. They should be tested against actual emergence for bias from averaged parameters.
- Empirical ratios divide booked premium by reported loss with the booking lag: premium to 27 months over loss to 18, then 39 over 30, then 51 over 42.
- Teng and Perkins selected the first ratio at $1.750$ , against a historical $1.460$ and a formula $1.426$ . Recent quarters had trended up, which they attribute more likely to improved loss experience.
Feldblum's refinements (his Sections 1–2).
- Plotted against the reported loss ratio, retro premium is a chain of line segments of decreasing slope, one per adjustment. Responsiveness falls as a book matures and as its loss ratio rises.
- Each segment starts from the actual point reached (his Assumptions A and B). That is what lets the method track emerging experience where Fitzgibbon's single regression line cannot.
- The first segment should not pass through the origin: $\text{PDLD}_1$ blends the basic premium ratio with the true first slope. To separate them, subtract the average basic premium charge divided by the expected loss ratio from $\text{CPDLD}_1$ .
- A premium chain ladder would wait until at least nine months after expiry and update only annually. The loss-based method updates every quarter.
See Teng and Perkins (1996) and Loss Sensitive Rating.

Empirical PDLD Ratios from Booked Premium and Reported Loss Two policy quarters of retro rated workers compensation ($000). Losses are valued for the first three adjustments at 18, 30 and 42 months. The premium they produce is booked by 27, 39 and 51 months.

Quarter Loss 18 Loss 30 Loss 42 Prem 27 Prem 39 Prem 51

Q1 $4{,}000$ $4{,}500$ $4{,}700$ $5{,}800$ $6{,}150$ $6{,}110$

Q2 $3{,}000$ $3{,}400$ $4{,}410$ $4{,}650$

Compute each quarter's PDLD ratios and the pooled first and second ratios. Compare them with formula ratios for BPF $0.20$ , TM $1.03$ , ELR $0.70$ , $78.4\%$ of loss emerged at the first adjustment, LCF $1.20$ , and loss capping ratios of $0.85$ (first) and $0.55$ (second, incremental).
Answer By quarter:
 $\begin{align*} \text{Q1: } \text{PDLD}_1 &= 5{,}800 / 4{,}000 \\ &= 1.450 \\ \text{PDLD}_2 &= 350 / 500 \\ &= 0.700 \\ \text{PDLD}_3 &= -40 / 200 \\ &= -0.200 \\ \text{Q2: } \text{PDLD}_1 &= 4{,}410 / 3{,}000 \\ &= 1.470 \\ \text{PDLD}_2 &= 240 / 400 \\ &= 0.600 \end{align*}$ 
Pooled:
 $\begin{align*} \text{PDLD}_1 &= \frac{5{,}800 + 4{,}410}{4{,}000 + 3{,}000} \\ &= 1.459 \\ \text{PDLD}_2 &= \frac{350 + 240}{500 + 400} \\ &= 0.656 \end{align*}$ 
Formula:
 $\begin{align*} \text{PDLD}_1 &= \frac{0.20(1.03)}{0.70(0.784)} + 0.85(1.20)(1.03) \\ &= 0.375 + 1.051 \\ &= 1.426 \\ \text{PDLD}_2 &= 0.55(1.20)(1.03) \\ &= 0.680 \end{align*}$ 
The first ratios agree well. The third-period ratio for Q1 is negative: its loss rose by $200$ while its premium fell by $40$ . That happens when losses develop upward above the loss limits, earning no premium, while developing downward within them, returning premium.

A single negative point is noise to be averaged over as many quarters as possible, or replaced by the formula ratio. It is not a sign that premium falls as losses rise.

Quarter	Loss 18	Loss 30	Loss 42	Prem 27	Prem 39	Prem 51
Q1	$4{,}000$	$4{,}500$	$4{,}700$	$5{,}800$	$6{,}150$	$6{,}110$
Q2	$3{,}000$	$3{,}400$		$4{,}410$	$4{,}650$

Separating the Basic Premium from the Slope A book's retro plans have a basic premium ratio including tax of $0.20$ of standard premium and an expected loss ratio of $0.75$ . The slopes of premium against loss are $1.10$ to the first adjustment, $0.60$ in the second period and $0.40$ in the third (the last). Losses emerge $80\%$ , $15\%$ and $5\%$ .

(a) Compute $\text{PDLD}_1$ and $\text{CPDLD}_1$ as Teng and Perkins would, and split $\text{CPDLD}_1$ as Feldblum suggests.

(b) Reporting slows to $70\%$ , $25\%$ and $5\%$ , with nothing else changed. Recompute.
Answer (a) The first ratio adds the basic premium, spread over the loss emerged at the first adjustment, to the first slope:
 $\begin{align*} \text{PDLD}_1 &= \frac{0.20}{0.75(0.80)} + 1.10 \\ &= 0.3333 + 1.10 \\ &= 1.4333 \end{align*}$ 
 $\begin{align*} \text{CPDLD}_1 &= 1.4333(0.80) + 0.60(0.15) + 0.40(0.05) \\ &= 1.1467 + 0.0900 + 0.0200 \\ &= 1.2567 \end{align*}$ 
Feldblum's split takes the basic premium over the expected loss ratio, $0.20/0.75 = 0.2667$ . The rest, $1.2567 - 0.2667 = 0.9900$ , is the loss-weighted premium responsiveness.

(b) With $70\%$ emerged by the first adjustment:
 $\begin{align*} \text{PDLD}_1 &= \frac{0.20}{0.75(0.70)} + 1.10 \\ &= 1.4810 \\ \text{CPDLD}_1 &= 1.4810(0.70) + 0.60(0.25) + 0.40(0.05) \\ &= 1.2067 \end{align*}$ 
The first PDLD rose from $1.433$ to $1.481$ although no plan parameter and no slope changed. The same basic premium is simply spread over less emerged loss.

Split Feldblum's way, the basic premium component stays $0.2667$ and responsiveness falls to $0.9400$ . More loss now emerges in the flatter later periods.

Blended into one first ratio, this looks like more responsive plans. Separated, it is visibly a slower reporting pattern. That is the ambiguity Feldblum's enhancement removes.