Actuarial Notes

Reinsurance

Study guides: Exam FAM, Exam DISC-RM, Exam 5, Exam 6U, Exam 6I, Exam 7, Exam 9, Exam CFE-101, Exam CFE-201, Exam CP-341, Exam GH-201U, Exam GI-201, Exam GI-301

Reinsurance is insurance bought by an insurer. The cedant transfers part of its exposure to a reinsurer for a ceded premium, reducing net retained loss, buying protection against large or accumulated losses, and freeing surplus to support more writings.

\text{Net} = \text{Gross} - \text{Ceded}

\text{Ceded}_{\text{XOL}} = \min\!\left(\max(X - R,\, 0),\, L\right)

Proportional — the reinsurer takes a share of premium and losses alike:

  • Quota share — a flat percentage of every policy. Ceded losses are a scalar multiple of gross, so the ceded triangle has the same shape as the gross one. Provides capacity and surplus relief, and protects against frequency and severity alike — but only in proportion: the cedant keeps its share of every large loss.
  • Surplus share — cedes the portion of each policy's limit above the cedant's retention, so the cession percentage varies by policy. Retains small policies fully and shares large ones.

Non-proportional (excess of loss) — the reinsurer pays only above an attachment point:

  • Per-risk XOL — responds to a single large claim above retention R, up to limit L.
  • Per-occurrence / catastrophe XOL — responds to the accumulation from one event, which is how catastrophe exposure is managed.
  • Aggregate stop loss — responds when the whole year's losses exceed a threshold.

Further points:

  • Functions (DISC-RM). Reinsurance gives the cedant capacity, both to write a single large risk and to write more premium than its surplus alone supports; stability, by taking out the peaks in its loss results; catastrophe protection against the accumulation of losses from one event; surplus relief while premium and unearned premium grow; services such as underwriting, claims and actuarial reviews and help with new products; and a way to withdraw from a territory or line, through a 100% quota share or portfolio reinsurance that carries the policies until they run off. See Pro Rata Reinsurance and Excess of Loss for how each form delivers them.
  • Sources (DISC-RM). Reinsurance is bought from professional reinsurers, from the reinsurance departments or subsidiaries of primary insurers, from Lloyd's syndicates and the Bermuda and international market, and from pools, which may be intercompany, voluntary (nuclear, aviation) or involuntary (FAIR plans, coastal pools). It is placed either through a reinsurance broker or intermediary, paid by the reinsurer, or directly with a direct-writing reinsurer whose own employees produce the business.
  • Gross and ceded develop differently. Excess cessions attach only to large claims, which report late and settle slowly, so the ceded pattern is longer-tailed and far more volatile than gross. Friedland's basic text prescribes no single route: some actuaries analyze gross and ceded and derive net, others gross and net, depending on data availability, the reinsurance programme and preference, with gross, ceded and net kept consistent throughout (net IBNR generally no greater than gross). Her reinsurance study note adds that ceded data often have limited credibility — low volume, large-loss and catastrophe volatility, frequent changes in terms — so actuaries typically select patterns and expected loss ratios gross and net, and take ceded as the difference.
  • Net experience is smoother than gross, because reinsurance caps exactly the large losses that drive volatility; net development patterns are frequently at or below gross ones.
  • In ratemaking, Werner reduces projected losses by the expected non-proportional recoveries and premium by the amount ceded — or, alternatively, includes the net cost of reinsurance (the cost of the reinsurance minus the expected recoveries) as an expense item in the indication. Proportional reinsurance may need no explicit treatment. See Net of Reinsurance.
  • Collectability is a real exposure: reinsurance recoverables are an asset subject to dispute, commutation and reinsurer insolvency, and the cedant remains liable to its policyholders regardless.

Net Losses Under Quota Share and Excess of Loss A cedant has gross ultimate losses of \$5{,}000{,}000 across five claims: \$2{,}600{,}000, \$900{,}000, \$800{,}000, \$500{,}000 and \$200{,}000.

Compute net losses under (a) a 40\% quota share and (b) a \$1{,}000{,}000 excess of \$500{,}000 per-risk treaty.

Answer (a) Quota share 40\%:

\begin{align*}
\text{Ceded} &= 0.40 \times \$5{,}000{,}000 = \$2{,}000{,}000 \\
\text{Net} &= \$3{,}000{,}000
\end{align*}

(b) \$1M xs \$500K per risk — cede \min(\max(X - 500\text{K},0),\, 1\text{M}) on each claim:

Claim Gross Ceded Net
1 \$2{,}600K \$1{,}000K \$1{,}600K
2 \$900K \$400K \$500K
3 \$800K \$300K \$500K
4 \$500K \$0 \$500K
5 \$200K \$0 \$200K
Total \$5{,}000K \$1{,}700K \$3{,}300K

The two structures cede similar amounts on this year's experience but do completely different things. Quota share cedes 40\% of every claim including the \$200K one; the excess treaty cedes nothing below \$500K and holds the cedant to \$500K on any claim up to \$1.5M — claim 1 pierces the top of the layer, so the cedant retains \$1.6M of it.

Note also what the excess treaty does not cover: a second \$2.6M claim would again leave \$1.6M net. Only an aggregate cover limits the total.

Why the Ceded Triangle Cannot Be Scaled from Gross Under the same \$1M xs \$500K treaty, gross and ceded reported losses for one accident year:

Maturity Gross Ceded Ceded ÷ gross
12 mo \$3{,}000K \$150K 5.0\%
24 mo \$4{,}200K \$700K 16.7\%
36 mo \$4{,}800K \$1{,}400K 29.2\%
48 mo \$5{,}000K \$1{,}700K 34.0\%

What do the patterns show?

Answer

\begin{align*}
\text{Gross CDF}_{12 \to 48} &= \frac{5{,}000}{3{,}000} = 1.667 \\[4pt]
\text{Ceded CDF}_{12 \to 48} &= \frac{1{,}700}{150} = 11.333
\end{align*}

The ceded layer develops nearly seven times as much as gross. The reason is structural: a claim only enters the layer once its reserve passes \$500{,}000, so at 12 months almost nothing has attached, while by 48 months the large claims have matured into the layer.

Two consequences:

  • Applying the gross CDF to ceded losses would project a ceded ultimate of \$150\text{K} \times 1.667 = \$250K against an actual \$1{,}700K — understating ceded recoveries by 85\% and correspondingly overstating net reserves.
  • A change in the attachment point changes both the ceded and the net triangle: every prior year then reflects different terms. Friedland's reinsurance study note counts frequent changes in attachment points, limits and participation among the reasons ceded data are not homogeneous.