Optimal Menu-of-Contracts Design in Reinsurance

A cedant knows more about the risk it cedes than the reinsurer that takes it; a reinsurer meets the asymmetry with a single price or with a menu. Where the two designs compete for the same cedants under a fixed pooled price and the menu reinsurer's expected growth rate of surplus exceeds the rivals' by a margin bounded away from zero, the menu reinsurer's share of industry surplus converges to one almost surely.

Competitive screening settles which contracts survive deviation when firm size is fixed; market selection settles which agents come to hold a market when wealth compounds. The paper joins the two where a reinsurer's capacity is a multiple of its surplus.

Two cedant types with mean–variance valuations; a menu reinsurer and single-price rivals whose pooled contract is priced at the population mean (the fixed pooled price) or at their experienced mean (experience rating); submissions placed in arrival order up to each reinsurer's capacity, with one partial line; and each reinsurer's objective over the period, its expected logarithmic growth, the Kelly criterion. At the calibration the pooled price is 0.7793 and the low-risk type's gain from the pooled contract 0.01384.

Theorem 1: where the pooled price exceeds the low-risk type's expected loss and leaves that type a non-negative gain, as at the calibration, the contract that leaves the low-risk type indifferent to the pooled contract is taken by that type and left by the high-risk type at every coverage below one, and its expected margin on a low-risk line is positive above the coverage 0.2929. Where the low-risk type would lose from the pooled contract, the margin is positive at every positive coverage, and some contract of the family separates the types if and only if the high-risk type gains from the pooled contract. At a share of 0.25 the menu reinsurer's growth-optimal coverage is 0.7045 at a price of 0.5507.

At the pooled price of a competitive market, wherever one of the two reinsurers can write all the business that comes to it, the rivals' expected underwriting result is negative (Proposition 1) and their expected growth rate lies below the risk-free rate (Proposition 2). At the calibration the rivals can write every submission at every share at or below 0.107, and their growth rate lies below ln(1 + r_f) = 0.04402 at every share of the coarse grid and the grid of ten-thousandths of Section 7.

Equations (9) and (10): the log-odds of the menu reinsurer's share move as a random walk whose drift, the growth gap, and conditional variance depend on the state. Theorem 2: under the fixed pooled price, where the gap is bounded away from zero the share converges to one almost surely; at a finite horizon the dispersion is that of a random walk from a small base.

The gap at the menu reinsurer's best contract is positive at every share of the coarse grid, from +0.00959 at a share of 0.01 to +0.10427 at 0.99, and on the grid of ten-thousandths its smallest value is +0.00935 with 0 non-positive points; the rivals' growth rate lies below the risk-free rate at every share of both grids. A grid does not establish the uniform bound that Theorem 2 assumes, as Section 7.2 states.

Under the fixed pooled price the share's cross-path mean reaches 0.992 after 300 periods from 0.05, with 998 of 1,000 paths above one half. Under experience rating at a zero loading the rivals' price rises toward the high-risk type's cost, and the noise of a small book lifts it above that type's willingness to pay for full coverage, 0.9201, on 689 of the 1,000 paths by the final period, a state in which no reinsurance is written. At a loading of 0.01 on the experienced mean the rivals' realized logarithmic return, 0.04917, stands above ln(1 + r_f) = 0.04402 while they are diluted; at a share of one half, above a loading of 0.045, they outgrow the menu reinsurer in the separated configuration, in which the menu reinsurer offers the high-risk type no contract. The menu reinsurer has nothing to learn from a cedant's losses: the choice reveals the type, and experience rating is the rivals' learning alone. As a monopolist the menu reinsurer's growth-optimal menu serves the high-risk type alone, at a coverage of 0.99 and a price of 0.9120.

Where the industry's capitalization, its capacity over the premium of writing every submission at the population mean loss ratio, stands on the paper's grid from 0.5 through the calibration's level of 1.120, a common loss factor widens the gap at a share of 0.25, from +0.0278 to +0.0312 at the calibration's capitalization as the common share of the variance rises to one half; at a share of 0.25 and the grid's capitalizations from 1.5 through 3, the common loss factor narrows the gap. At a share of 0.25 the menu reinsurer's growth-optimal exposure, the share of its capacity up to which it writes, leaves full capacity at a common share of 0.897 at a capitalization of 1 and of 0.877 at a capitalization of 3, found by a bisection in twelve steps over common shares up to 0.98, on the presumption that the exposure leaves full capacity once on that interval, so that at those capitalizations the Kelly criterion sizes the book below capacity only where losses are almost entirely common. On 500 simulated paths of 60 periods from a share of 0.05, with demand growing at the risk-free rate, the Kelly criterion sizes the book below capacity on some paths at a common share of 0.3, to a smallest exposure of 0.093 of capacity, and at that common share the rivals' capacity binds in 42.0 percent of the path-periods.

On a continuum of types a menu of contracts partitions the types by thresholds, each price set by the marginal type's indifference. One contract earns a gap of 0.03827 at a share of 0.05, above the two-type core's 0.01898, since every type below the threshold pays the marginal type's price on a lower expected loss. Where the first contract skims the better risks, at shares of 0.05, 0.25, and 0.50, the second contract adds 0.00011, 0.00027, and 0.00006 to the gap and the third less than the second; at 0.75, where the rivals' capacity binds in every period, the menu reinsurer is nearly indifferent between skimming the better risks and pooling the rationed ones, and the third contract lifts the gap from 0.01049 with two contracts to 0.03395.

With the mean and the volatility of the loss ratio private, one contract screens on the mean and a second separates on the volatility, and the second adds at most 4.6 percent to the one-contract gap. The type a second contract draws, low in mean and high in volatility, gains 0.1061 from the pooled contract against the low-mean, low-volatility type's 0.0138; a contract that draws it must leave it that gain, and at a share of 0.25 the contract it buys earns the menu reinsurer 0.0136 per unit of subject premium on its line against 0.0118 on the low-volatility type's.

At the Normal's mean and variance, a heavier tail changes the gap only through the moments after the floor and the cap: Student's t lifts the high-risk type's capped mean by 0.0015 and widens the gap at a share of 0.50 by 0.00116, the log-normal narrows it by 0.00008, and the quadrature at each family's capped moments accounts for the measured differences; the single-price rivals, who price on the Normal's moments, carry the shift of the high-risk type's capped mean in their margin.

A menu reinsurer and single-price rivals compete through the growth of their surplus, and the growth gap decides the competition. The cream-skimming contract earns a positive expected margin on every low-risk line whenever the pooled price exceeds the low-risk type's expected loss and the high-risk type gains from the pooled contract. At the pooled price of a competitive market, wherever one of the two reinsurers can write all the business that comes to it, the rivals' expected underwriting result is negative and their growth rate lies below the risk-free rate; where the gap is bounded away from zero the menu reinsurer's share converges to one almost surely. The mechanism carries onto a continuum of types, two dimensions of private information, and heavy-tailed losses, and the reinsurer that won the market by serving the better risks would, as a monopolist, serve the worse ones alone.