Reinsurance

Reinsurance transfers part of an insurer’s claim liability to another insurer. If aggregate claims are $S$, a retained payout $g(S)$ with $0\le g(S)\le S$ leaves the reinsurer with $S-g(S)$. Quota share reinsurance retains a fixed fraction, whereas aggregate stop loss reinsurance retains losses only up to a fixed aggregate threshold.

A threshold limiting the insurer's retained payment under a reinsurance contract. Under excess of loss reinsurance it applies to each claim size; under aggregate stop loss reinsurance it applies to the total loss over the stated period. These contracts therefore cap different quantities even when their thresholds have the same numerical value.

For independent exponential claims of mean $\mu$ and a zero-based geometric count of parameter $p$, excess of loss reinsurance at per-claim reinsurance retention $M$ has expected recovery $(1-p)\mu e^{-M/\mu }/p$. The zero-based geometric sum of exponential variables has tail $(1-p)e^{-px/\mu }$, so aggregate stop loss reinsurance at reinsurance retention $\tilde{M}$ has expected recovery $(1-p)\mu e^{-p\tilde{M}/\mu }/p$. Equating them gives the displayed reinsurance retention relation. This comparison concerns claim payouts, before any reinsurance premium is charged.

Per-claim reinsurance under which the insurer retains each claim up to a level $M$ and the reinsurer pays its positive part $(x-M{)}_{+}$. The annual retained amount is a sum of capped claims, so it is generally different from the cap of the annual sum in aggregate stop loss reinsurance.

A reinsurance layer with attachment $M$ and payment limit $A$ pays nothing below $M$, the excess between $M$ and $M+A$, and $A$ above that. Its expected recovery is ${\int }_M^{M+A}\mathbb{P}(X>x)dx$, by the tail integral formula for moments. For continuous claims, conditioning on penetration of the layer divides this expectation by $\mathbb{P}(X>M)$.

For a lognormal distribution, let $d_j(t)=(\log t-\mu -j{\sigma }^2)/\sigma$ for $j=0,1$. The truncated lognormal moment formula gives the recovery numeratorDivide by $1-\Phi (d_0(M))$ to condition on $X>M$. This handles both the partially penetrating claims and the capped payments beyond the upper attachment.

For positive inflation factor $h$, $L_{M,A}(hX)=h{L}_{M/h,A/h}(X)$. The conditioning event also changes to $X>M/h$. If claims have a lognormal distribution, the same result follows by replacing their log-mean $\mu$ by $\mu +\log h$ and keeping the contractual attachment and limit fixed. Merely multiplying the original conditional mean by $h$ would hold those effective thresholds incorrectly fixed.

For positive claim sizes with finite second moment in a compound Poisson distribution aggregate of parameter $\lambda$, the two excess of loss reinsurance payouts are $\min (X,M)$ and $(X-M{)}_{+}$. The derivative of their total aggregate variance is $2\lambda (M\overline{F}(M)-\mathbb{E}[(X-M{)}_{+}])$. Thus the displayed condition characterizes stationarity. When the tail probability is positive it says that the retention equals the mean residual life. Global minimality requires an additional sign or comparison argument.

For a Pareto distribution of shape three and minimum $d>0$ with count rate $\lambda >0$, the sum of insurer and reinsurer aggregate variances is $V(M)=\lambda (3d^2-3dM+2M^2)$ for $0\le M\le d$, and $\lambda (3d^2-d^3/M)$ for $M\ge d$. The first branch has its unique minimum at $3d/4$; the second is increasing. The branches agree in value and slope at $d$, giving the displayed global minimum. This criterion minimizes the sum of party variances; the total portfolio variance includes their covariance and remains $3\lambda d^2$.

For exponential distribution claims of mean $\mu$ and compound Poisson distribution count parameter $\lambda$, the total party variance under excess of loss reinsurance is $g(M)=2\lambda {\mu }^2(1-(M/\mu )e^{-M/\mu })$. Its derivative is $2\lambda e^{-M/\mu }(M-\mu )$, so $M=\mu$ is the unique global minimum, with value $2\lambda {\mu }^2(1-e^{-1})$.

For a positive claim and $r>0$, the tail integral formula for moments gives the displayed expression. In particular the first moment is ${\int }_0^M\overline{F}(x)dx$ and the raw second moment is $2{\int }_0^{M}x\overline{F}(x)dx$. If the claim law has a density, capping adds an atom of a measure at $M$ with mass $\mathbb{P}(X\ge M)$.

With aggregate retention $M\ge 0$, the direct insurer pays $\min (S,M)$ and the reinsurer pays $(S-M{)}_{+}$, where $+$ denotes the positive part. The threshold applies to the whole annual loss; applying a threshold to individual claims is a different contract.

If $S$ has exponential distribution with expected value $\mu$, the tail integral formula for moments gives $\mathbb{E}\min (S,M)=\mu (1-e^{-m})$ and $\mathrm{Var}(\min (S,M))={\mu }^2(1-2m{e}^{-m}-e^{-2m})$, where $m=M/\mu$. At matching retained expected value, the excess variance under quota share reinsurance is $2{\mu }^2{e}^{-m}(m-1+e^{-m})\ge 0$.

Among retained payouts $g(S)$ with $0\le g(x)\le x$ and the same expected value as $\min (S,M)$, aggregate stop loss reinsurance minimizes the variance. Pointwise $(g(x)-M{)}^2\ge (\min (x,M)-M{)}^2$, and subtracting the identical squared distance of their common expected value from $M$ proves the claim. Equality requires equal payouts almost surely.

Quota share reinsurance retains a fixed proportion $\alpha \in [0,1]$ of each claim. The annual retained aggregate is $\alpha S$, whose expected value and variance are $\alpha \mathbb{E}S$ and ${\alpha }^2\mathrm{Var}(S)$.

If insurer and reinsurer use the same positive relative safety loading $\theta$, retained claim sizes and net premiums both scale by the retention $\alpha$: $X_{\alpha }=\alpha X$ and $c_{\alpha }=\alpha c$. Substitution into the adjustment coefficient equation gives $R_{\alpha }=R_1/\alpha$. Thus there is no maximizing positive retention on $(0,1]$; the supremum is approached as $\alpha \downarrow 0$. At full cession, if allowed, both retained claims and net premiums vanish, and the insurer's nonnegative capital remains constant. The coefficient is then not uniquely defined by the degenerate equation, although ultimate ruin probability is zero in this idealized model.

For independent Poisson process arrivals and exponential distribution claim sizes of mean $\mu$, minimizing the logarithm of the negative expected exponential utility gives $H^{\prime }(\alpha )=\lambda \beta \mu [(1-\beta \mu \alpha {)}^{-2}-(1+{\theta }_R)]$. Its strictly positive second derivative makes the displayed stationary point unique. On $0\le \alpha \le 1$, the optimum is $\min (1,{\alpha }_{\ast })$. On the strictly open interval $0<\alpha <1$, there is no maximizer when ${\alpha }_{\ast }\ge 1$, only a supremum at full retention. The stationary retention increases with reinsurer loading and is independent of the direct insurer's loading.

Suppose $S=S_I+S_R$ has finite positive variance $v$, and a matching quota share reinsurance contract exists with fraction ${\alpha }^{\ast }=\sqrt{\mathrm{Var}(S_I)/v}\in [0,1]$. The Cauchy-Schwarz inequality bounds $\mathrm{Cov}(S,S_I)\le {\alpha }^{\ast }v$. Expanding $\mathrm{Var}(S-S_I)$ then proves that the matching quota share minimizes the other party's variance, and hence the sum of party variances. Claimwise retentions $0\le h(x)\le x$ in a compound Poisson distribution automatically satisfy the required variance range, because $\mathrm{Var}(S_I)=\lambda \mathbb{E}[h(X{)}^2]\le \lambda \mathbb{E}[X^2]$.

For any aggregate with positive finite variance $v$, quota share reinsurance with retained fraction $\alpha$ has total party variance $v({\alpha }^2+(1-\alpha {)}^2)=v/2+2v(\alpha -1/2{)}^2$. Equal sharing uniquely minimizes this objective. The two party totals remain dependent; adding their individual variances is a different objective from the fixed variance of their sum.