Recoveries — Qualified

Salvage & Subrogation (S&S)

Salvage and subrogation are the primary sources of post-claim recovery for insurers:

Salvage: The sale of damaged property (e.g., a totaled vehicle) acquired by the insurer during claim settlement.
Subrogation: The legal right to recover payments from a negligent third party responsible for the loss.
Recoverable S&S (Unpaid S&S): $\text{Recoverable S\&S} = \text{Ultimate S\&S} - \text{Received (Paid) S\&S}$

S&S Estimation Approaches

Insurers can estimate ultimate salvage and subrogation recoveries using two main methods:

1. Direct Development Approach

Uses traditional loss development techniques directly on historical S&S triangles (either paid/received S&S or reported S&S).
Evaluation:
- Works well for Salvage (property coverages) because it develops rapidly and is settled soon after the claim is reported.
- Less effective for Subrogation (liability coverages) because legal proceedings can cause significant reporting and settlement lags.

2. S&S-to-Loss Ratio Approach

Methodology:
1. Construct a triangle of historical S&S-to-loss ratios: $\frac{\text{S\&S}}{\text{Gross Claims}}$ (paid or reported).
2. Develop these ratios to ultimate using ratio development factors.
3. Apply the ultimate S&S-to-loss ratio to the estimated ultimate gross losses to find ultimate S&S.
Advantages of the Ratio Approach:
- Less Leveraged: Ratio triangles are less volatile and less leveraged than S&S dollar triangles, which can have small numbers and high percentage fluctuations.
- Diagnostic Value: The resulting ultimate ratio ( $\frac{\text{Ultimate S\&S}}{\text{Ultimate Gross Loss}}$ ) acts as a diagnostic test. If a year’s ratio is anomalous, the actuary can manually adjust or select a normalized ratio.

Example: S&S Ratio Approach Calculation

AY	Dev S&S %	Sel S&S %	Est Ult Gross Loss	Ult S&S	Rec S&S	Recoverable S&S
2006	36.10%	36.10%	$17,000	$6,137.00	$5,600.19	$536.81
2007	38.17%	38.17%	$17,248	$6,583.51	$5,900.27	$683.24
2008	32.18%	37.18%	$16,500	$6,134.70	$2,700.13	$3,434.57
Total				$18,855.21	$14,200.59	$4,654.62

[!TIP] Selection Rationale: For mature accident years (e.g., 2006 and 2007), the developed ratio is highly reliable and should be selected directly. For immature years, select ratios based on a historical average or trend. If a downward trend exists, select lower developed ratios to reflect deteriorating recoveries.

Reinsurance Net-of-Reinsurance Estimation

Insurers must often project unpaid losses net of reinsurance. The two standard approaches are:

Direct Net Development: Develop the net loss triangle directly. Preferred if ceded loss data is thin or volatile.
Indirect Net Development: Develop gross and ceded loss triangles separately, then subtract ultimate ceded from ultimate gross: $\text{Net Ultimate Losses} = \text{Gross Ultimate Losses} - \text{Ceded Ultimate Losses}$

Reinsurance Structures & Tail Development

The impact of reinsurance on loss development and tail factors depends heavily on the contract structure:

The reinsurer shares a fixed percentage of all premiums and losses (e.g., 70% Quota Share).
Tail Factor Impact: The tail factors for gross, net, and ceded triangles are identical because net and ceded losses are constant proportions of gross losses.

2. Excess of Loss (XOL) / Stop Loss (Non-Proportional)

Per-Risk/Per-Occurrence XOL: Cedes individual claim amounts above a specific retention level $R$ up to a limit.
Stop Loss (Aggregate XOL): Cedes aggregate annual losses above a retention up to a limit.
Tail Factor Impact:
- Ceded Tail Factor > Gross Tail Factor: As claims mature, larger claims pierce the retention limit. Future development on these large claims occurs entirely in the ceded layer.
- Net Tail Factor < Gross Tail Factor: Because net losses are capped at retention $R$ , net development slows down or halts once the retention is breached.

Common Calculation Pitfalls

Incorrect Development Application: Do not apply gross loss development factors (LDFs) directly to net or ceded loss triangles, as non-proportional reinsurance alters development speeds.
XOL Transaction Mechanics: To find net payments for a calendar year under an XOL treaty:
1. Determine cumulative gross paid losses at the beginning ( $G_{t-1}$ ) and end ( $G_t$ ) of the period.
2. Compute cumulative net paid losses at both points: $N_t = \min(G_t, R) \quad \text{and} \quad N_{t-1} = \min(G_{t-1}, R)$
3. Calculate the incremental net paid loss for the period: $\Delta N_t = N_t - N_{t-1}$
4. The incremental ceded paid loss is the remainder: $\Delta C_t = \Delta G_t - \Delta N_t$