Implied Development Method

Implied Development Method is Siewert's approach to the excess layer of a high deductible program: develop full coverage (unlimited) losses to ultimate, develop deductible (limited) losses to ultimate with factors for inflation-indexed limits, and take the ultimate excess losses as the difference. Excess development is implied by the two developments rather than measured on the thin excess data itself.

$\text{Ult}^{XS} = C_t \cdot \text{LDF}_t - C^L_t \cdot \text{LDF}^L_t$

$\text{IBNR}^{XS} = \text{Ult}^{XS} - (C_t - C^L_t)$

Symbols. $C_t$ is unlimited reported loss at age $t$ and $C^L_t$ the loss limited to the deductible $L$ . $\text{LDF}_t$ and $\text{LDF}^L_t$ are their age-to-ultimate factors. Siewert applies the factors account by account, each at its own deductible, and sums the account ultimates. That reflects the program's deductible mix.
Consistency and indexing.
- The full coverage tail must agree with the limited tails. Limited losses must never be developed beyond unlimited losses, nor a lower limit beyond a higher one.
- Limits are indexed for inflation so experience years can be combined. At $10\%$ cost inflation, losses over a $100,000 deductible one year match those over $110,000 the next. Siewert's index of $1.095$ makes a current $250,000 limit equivalent to $228,311 a year earlier.
Strengths and weakness. It gives an excess estimate at early maturities even when no excess losses have emerged. Limited factors are more stable than excess ones. Its by-product, ultimate deductible losses, feeds the service revenue asset: a loss multiplier on deductible losses, less known recoveries (see Deductible Recovery). Its weakness, in Siewert's words, is that its focus is misplaced: one would like to recognize excess development explicitly.
The alternatives Siewert compares it with. P is premium, E the expected loss ratio, \chi the per-occurrence charge and O_t the observed loss.
- The loss ratio approach sets excess losses at $P \cdot E \cdot \chi$ from pricing. It is practical for immature years but ignores emerging experience.
- Direct development applies an excess loss development factor to reported excess losses. It is leveraged and volatile, and it gives nothing when nothing has emerged.
- Credibility weighting takes $Z = 1/\text{LDF}_t$ , giving the Bornhuetter-Ferguson result $O_t + E(\text{LDF}_t - 1)/\text{LDF}_t$ . It is stable and tied to pricing, but it ignores actual experience to the extent of the complement of credibility.
- The implicit and explicit methods need not give the same reserve. Their underlying factors should balance, though, to the unlimited factor.
See Siewert (1996), and Sahasrabuddhe (2010) for moving a pattern between layers.

Excess IBNR for Two Accounts with Different Deductibles At 36 months, one accident year of a workers compensation program has two accounts ($000):

Account Deductible Unlimited reported Limited reported

A $100,000 $1{,}200$ $900$

B $250,000 $2{,}000$ $1{,}750$

The unlimited 36-to-ultimate factor is $1.300$ . The limited factors are $1.080$ at $100,000 and $1.150$ at $250,000. Estimate each account's excess IBNR by implied development. Then find the service revenue asset for a loss multiplier of $8\%$ , collected so far on reported deductible losses.
Answer Account A:
 $\begin{align*} \text{Ult unlimited} &= 1{,}200 \times 1.300 \\ &= 1{,}560 \\ \text{Ult limited} &= 900 \times 1.080 \\ &= 972 \\ \text{Ult excess} &= 1{,}560 - 972 \\ &= 588 \\ \text{IBNR}^{XS} &= 588 - 300 \\ &= 288 \end{align*}$ 
Account B:
 $\begin{align*} \text{Ult unlimited} &= 2{,}000 \times 1.300 \\ &= 2{,}600 \\ \text{Ult limited} &= 1{,}750 \times 1.150 \\ &= 2{,}012.5 \\ \text{Ult excess} &= 2{,}600 - 2{,}012.5 \\ &= 587.5 \\ \text{IBNR}^{XS} &= 587.5 - 250 \\ &= 337.5 \end{align*}$ 
Total excess IBNR is $625.5$ , or $625,500. The factors pass Siewert's consistency check, since $1.080 < 1.150 < 1.300$ .

The implied excess factors are $588/300 = 1.96$ for A and $587.5/250 = 2.35$ for B. The higher retention's excess layer develops more.

Service revenue. The multiplier applies to the deductible losses still to emerge:
 $\begin{align*} \text{Asset} &= 0.08 \times \big[(972 - 900) + (2{,}012.5 - 1{,}750)\big] \\ &= 0.08 \times 334.5 \\ &= 26.76 \end{align*}$ 
That is an asset of $26,760, built on the same ultimate deductible losses the excess estimate used.

Account	Deductible	Unlimited reported	Limited reported
A	$100,000	$1{,}200$	$900$
B	$250,000	$2{,}000$	$1{,}750$

Implied, Direct and Bornhuetter-Ferguson Compared At 24 months an accident year with a $250,000 deductible has unlimited reported losses of $4{,}000$ and limited reported losses of $3{,}800$ ($000). The factors to ultimate are $1.400$ unlimited and $1.250$ limited. They were derived with a 24-month relativity $R^L_{24} = 0.96$ , and the expected excess ultimate from pricing is $800$ .

Estimate the excess IBNR by implied development, by direct development and by Bornhuetter-Ferguson. Explain why they differ.
Answer Implied development:
 $\begin{align*} \text{Ult}^{XS} &= 4{,}000(1.400) - 3{,}800(1.250) \\ &= 5{,}600 - 4{,}750 \\ &= 850 \\ \text{IBNR}^{XS} &= 850 - 200 \\ &= 650 \end{align*}$ 
Direct development. The excess factor that balances with these factors is:
 $\begin{align*} \text{XSLDF}_{24} &= \frac{1.400 - 0.96(1.250)}{0.04} \\ &= 5.000 \\ \text{IBNR}^{XS} &= 200(5.000) - 200 \\ &= 800 \end{align*}$ 
Bornhuetter-Ferguson with the pricing expectation:
 $\begin{align*} \text{IBNR}^{XS} &= 800 \times (1 - 1/5.000) \\ &= 640 \end{align*}$ 
The factors expect $4\%$ of reported losses above the deductible at 24 months, which is $160$ . Actual excess emergence is $200$ , $40$ more.

Direct development multiplies the extra $40$ by $5.0$ , adding $200$ to the ultimate.

Implied development sees it only as $40$ less limited loss, developed at $1.25$ , adding $50$ .

Bornhuetter-Ferguson passes it through one for one, adding $40$ .

All three rest on the same balanced factors, as Siewert requires. They differ in how far they let an early, volatile emergence move the reserve.