Proportional and Excess of Loss Reinsurance

A small property insurer writes a $20 million policy. A single total loss would wipe out surplus. It could cede 60% of every policy to a reinsurer (simple but expensive. Or it could retain the first $5 million per claim and let the reinsurer handle the excess) cheaper on average, but the insurer keeps all small claims.

Under quota share, the cedant cedes a fixed percentage $c$ of every policy and every loss. The cedant retains $1 - c$:

$$\text{Cedant's retained loss} = (1 - c) \cdot X$$

HIGH-FREQUENCY: In quota share, every loss is split in the same ratio. The retained distribution has the same shape, just scaled down by $(1-c)$.

KEY: Variance scales by $(1-c)^2$, CV is unchanged, loss ratio is unchanged. Quota share does not improve the loss ratio, it reduces absolute dollar impact.

Under surplus share, the cession varies by policy. If the insured value is $V$ and the cedant's retention line is $R$:

Common mistakes

• Confusing quota share with surplus share. A 40% quota share cedes 40% of every loss regardless of size. Surplus share varies by policy. Trap: computing a variable cession when the treaty is quota share.
• Treating XOL retention as a policyholder deductible. The XOL retention is what the insurer retains, not what the policyholder pays. These are different layers. Trap: double-subtracting the policyholder's deductible.
• Using the wrong layer bounds. "$500K xs $200K" means the layer runs from $200K to $700K, not to $500K. Trap: $E[X \wedge 500{,}000] - E[X \wedge 200{,}000]$ instead of $E[X \wedge 700{,}000] - E[X \wedge 200{,}000]$.

Bottom line

• Quota share: cedes a fixed fraction $c$ of every loss; the cedant's retained distribution keeps the same shape, scaled by $(1-c)$
• Surplus share: cession varies by policy size. Small policies are fully retained, larger policies are partially ceded
• Per-risk XOL: reinsurer pays $\min((X-M)_+, L)$; expected payment $= E[X \wedge (M+L)] - E[X \wedge M]$
• Per-occurrence XOL triggers on the aggregate loss from one event; stop-loss triggers on the annual aggregate

Exam shortcut

When you see "xs" notation, immediately translate: lower = retention, upper = retention + limit. Write both bounds before calculating. For proportional problems, check whether the question asks for cedant's or reinsurer's share, they must sum to the total. "QS = Same Shape, Smaller Scale." Loss ratio unchanged. "L xs M" = layer from M to M+L. The word "excess" tells you M is the bottom.

The full lesson (about 1,955 words, 13 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 1d

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