Coinsurance¶

Definition¶

Under coinsurance, the insurer pays a specified proportion $a\in[0,1]$ of a covered loss and the insured or another insurer pays the rest. Without other modifications, insurer payment is $Y=aX$ $a X$ .

Interpretation¶

Both parties share each covered monetary unit. Combining coinsurance with an ordinary deductible or policy limit requires specifying the order of those operations.

Two bars splitting a loss X = 1000 with a = 0.8 and a deductible of 100: the insurer pays 720 when the deductible is applied first and 700 when the loss is shared first.

The figure splits the Example's loss $X=1000$ $1000$ , with $a=0.8$ and deductible $d=100$ $100$ , between the insured (blue) and the insurer (red). Applying the deductible first leaves an excess of 900, of which the insurer pays $0.8(900)=720$ and the insured keeps $100+180=280$ $280$ . Sharing first gives the insurer $aX=800$ before the deductible, so it pays $700$ and the insured keeps $100+200=300$ $300$ . The gap of 20 is $(1-a)d$ $a) d$ : sharing after the deductible lowers the insurer's payment below $aX$ by only $ad=80$ $80$ , while deducting after sharing lowers it by the full $d=100$ $100$ .

Change the insurer share $a$ , the deductible $d$ and the loss $X$ : deducting first pays $a(X-d)_+$ $d)_{+}$ , sharing first pays $(aX-d)_+$ $d)_{+}$ , and while both are positive they differ by $(1-a)d$ $a) d$ . The default $X=1000$ $1000$ , $a=0.8$ $0.8$ , $d=100$ gives 720 against 700.

Variance and standard deviation explains why proportional sharing scales the variance quadratically: multiplying each realized loss by $a$ multiplies its deviation from the mean by $a$ , and its squared deviation by $a^2$ .

Key Results¶

$\mathbb{E}[aX]=a\mathbb{E}[X]$ and $\operatorname{Var}(aX)=a^2\operatorname{Var}(X)$ when finite.
For $a>0$ $0$ , $F_{aX}(y)=F_X(y/a)$ $F_{X} (y / a)$ .
$a\min((X-d)_+,u)$ means sharing an already capped excess. It differs from $\min((aX-d)_+,u)$ $d)_{+}, u)$ .
In reinsurance, a quota-share treaty applies the same proportional split to the insurer's portfolio losses: the reinsurer pays a fixed share of every loss.

Examples¶

With $a=0.8$ $0.8$ , $X=1000$ and deductible 100, sharing after the deductible gives $0.8(900)=720$ $720$ . Applying the deductible after sharing gives $800-100=700$ $700$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 3

Aliases: proportional insurance, quota-share payment, co-insurance, proportional coverage Keywords: Y = aX, insurer share a, order of deductible and coinsurance, Var(aX) = a^2 Var(X), quota-share treaty Symbols: element of

Last updated: 14 October 2026 08:21