Reinsurance: Proportional, Non-Proportional and Optimal Retention

Fundamentals

Why reinsurance exists

Reinsurance is the mechanism by which an insurer (the ceding company) transfers part of its risk to another entity (the reinsurer). It is not a luxury: it is the tool that allows a company to write risks larger than its own capital without compromising its solvency. Its essential functions are:

Capacity: accepting policies with sums insured above the company's own retention.
Stabilization: smoothing the volatility of the underwriting result from year to year.
Catastrophe protection: limiting the loss from an event that affects many policies at once (earthquake, hurricane).
Capital relief: reducing the solvency capital requirement by ceding risk.

The fundamental split is between proportional reinsurance (a share of premiums and claims is ceded) and non-proportional reinsurance (the reinsurer pays only above a threshold).

Quota share and surplus

The ceding company and the reinsurer share premiums and claims in an agreed proportion. The reinsurer usually pays back a ceding commission.

Quota Share

Retaining a fixed share

$$X_{\text{ret}}=a\,X,\qquad X_{\text{ced}}=(1-a)\,X$$

The ceding company retains a fixed proportion $a$ of each risk (premiums and claims). Simple and effective for reducing volume, but it also cedes the small risks it could keep.

Surplus

Retention up to one line

$$a_i=\frac{\min(\text{SI}_i,\,R)}{\text{SI}_i}$$

The ceding company sets a maximum retention $R$ (the line) per policy; whatever exceeds it out of the sum insured $\text{SI}_i$ is ceded. The ceded proportion varies by risk, so the small risks are kept.

Ceding commission

Reimbursement of acquisition costs

$$C=c\cdot P_{\text{ced}}$$

The reinsurer pays the ceding company a commission on the ceded premium that recognizes its acquisition and administration costs. It may follow a sliding scale based on the loss ratio.

Excess of loss and stop-loss

The reinsurer steps in only when the loss exceeds a retention (deductible), up to a limit. It is the natural cover against large losses and catastrophes.

Excess of loss (XL)

"Limit excess of retention" layer

$$R(X)=\min\big(\max(X-d,0),\,\ell\big)$$

The reinsurer pays the part of loss $X$ that exceeds the retention $d$, up to a maximum $\ell$ (the "$\ell$ xs $d$" layer). It can be per risk (WXL) or per event (CatXL).

Stop-Loss (aggregate)

Protection on total losses

$$R(S)=\max(S-d,0)$$ $$E[R(S)]=\int_d^{\infty}\!\big(1-F_S(s)\big)\,ds$$

Covers the excess of the annual aggregate loss $S$ over a retention $d$ (often expressed as a percentage of premium). Its pure premium is the integral of the tail of $S$.

Reinstatements

Restoring the layer

$$P_{\text{total}}=P_0\Big(1+\sum_k r_k\tfrac{\text{used}_k}{\ell}\Big)$$

After a loss that exhausts the XL layer, a reinstatement restores it for an additional premium proportional to the amount used. It limits the number of events covered.

Pricing the layer

Burning cost and exposure

$$\text{Premium}=\frac{\text{Losses to the layer (present value)}}{\text{Exposure}}\times(1+\text{loading})$$

Layers are priced on experience (burning cost) and on exposure (exposure curves / catastrophe models), adjusting for inflation, trend and tail volatility.

Optimal retention and risk sharing

How much risk should be retained? Actuarial theory offers formal answers depending on the objective and the cost constraint.

Borch's result

Stop-loss minimizes variance

$$\min_{R}\;\mathrm{Var}(X-R)\;\;\text{s.t.}\;\;E[R]=p$$

Karl Borch showed that, for a given reinsurance cost, the contract that minimizes the variance of the retained loss is stop-loss. It is the "most efficient" reinsurance in the mean-variance sense.

Pareto-optimal sharing

Risk exchange between parties

$$\frac{u_c'(w_c-R_c)}{u_r'(w_r-R_r)}=\text{constant}$$

At the Pareto-optimal equilibrium, the ratio of marginal utilities is constant across states. If both parties have exponential risk aversion, the optimal sharing is linear (quota share).

Retention and solvency

Ruin probability criterion

$$\psi(u)\le e^{-Ru}$$

Retention is also chosen to keep the probability of ruin below a threshold: ceding risk raises the adjustment coefficient $R$ and reduces $\psi(u)$, at the cost of the ceded premium.

Reinsurance and regulation in Mexico

Under the Ley de Instituciones de Seguros y de Fianzas (LISF, Insurance and Surety Institutions Law), ceded reinsurance reduces the solvency capital requirement (RCS) provided the reinsurer is registered (RGRE, the registry of foreign reinsurers) and the risk transfer is effective. The actuary values the effect of reinsurance on both technical reserves and regulatory capital.