Optimal Menu-of-Contracts Design in Reinsurance

The argument

A reinsurer that screens cedants with a menu of contracts and one that quotes a single price compete through the growth of their surplus: at the calibration the menu reinsurer's cream-skimming contract earns a positive expected margin on every low-risk line above the coverage 0.2929, the growth gap is positive, and the single-price rivals' growth rate lies below the risk-free rate at every share of the coarse grid, 0.01 through 0.99, and of the grid of ten-thousandths; under the fixed pooled price, where the gap is bounded below by a positive constant, the menu reinsurer's share of industry surplus converges to one almost surely, a bound that a grid does not establish over the continuum of shares (Section 7.2).

The growth gap is the difference between the menu reinsurer's and the rivals' expected logarithmic growth rates per period at the menu reinsurer's best contract, a period being a year; the share is the menu reinsurer's share of industry surplus. The paths start from a share of 0.05.

The gap across the paper's settings

At a share of 0.25 the gap is positive in each of the six computations charted, the two-type core's by quadrature and under Normal losses by Monte Carlo among them: largest on the continuum of types and smallest under two dimensions of private information.

The core's value is the quadrature's of Section 7; the continuum's is the count Monte Carlo of Section 10, over 20,000 count vectors; the two-dimensional value is the quadrature of Section 11 over the 23,426 count vectors of the four types, enumerated; the three families' values are the Monte Carlo of 100,000 periods of Section 12, in which the Normal's gap stands beside the quadrature's.

The panels

The panels take the paper's parts in its order; the sources panel names the paper, the flow map, and the computations behind the values.

• The market: Section 3, the primitives and the calibration.
• Cream skimming: Sections 4 and 5, the contract and the rivals' book.
• Selection and the gap: Sections 6 and 7, the theorem and the quadrature.
• Share paths: Section 8, three pricing regimes and the monopolist's menu.
• Correlation and the cycle: Section 9, the common factor and demand tied to the economy.
• Extensions: Sections 10 through 12, a continuum of types, two dimensions, heavy tails.

Cedants and contracts

The market has two cedant types with mean–variance valuations of a contract, the high-risk type's marginal willingness to pay above the low-risk type's at every coverage; the rivals offer full coverage at a pooled price, and the menu reinsurer one partial-cover contract priced at the low-risk type's indifference to its best alternative.

The first equation is a cedant's gain from a contract that cedes a share of its loss at a price per unit of subject premium: the expected loss transferred plus the value of the variance removed, less the premium. The second is the family the menu reinsurer's contract is drawn from.

Returns and the objective

A reinsurer's gross return is the risk-free return on its whole surplus plus its underwriting result, and its objective over the period is its expected logarithmic growth under a ruin floor, the Kelly criterion.

The underwriting result sums, over the lines the reinsurer wrote, the premium less the ceded loss, each line weighted by the fraction of the submission it carries and by the subject premium of a submission in units of surplus.

The primitives

The table states every primitive at the calibration, its source, and whether the runs of Sections 7 through 9 vary it; the low-risk mean follows the half-risk-premium rule, the risk aversion the willingness-to-pay rule, and the book size the capacity rule.

The capped means and variances are computed from the Normal parameters of the loss ratio, floored at zero and capped at the cap.

The calibration

At the calibration the population mean loss ratio is 0.7793, the fixed pooled price 0.7793, and the pooled contract leaves the low-risk type a gain of 0.01384 against a full-coverage risk premium of 0.02769, and the high-risk type a gain of 0.1408.

Gains and premiums are per unit of subject premium.

The cream-skimming contract

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 the menu reinsurer's expected margin on a low-risk line is positive above the coverage 0.2929; at a share of 0.25 the growth-optimal coverage is 0.7045 at a price of 0.5507.

The margin per unit of subject premium on a low-risk line is the low-risk type's value of the variance removed less its gain from the pooled contract, continuous and increasing in the coverage, with one zero. Where the low-risk type would lose from the pooled contract, a case Section 4 treats after the theorem, 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.

The rivals' book

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, by a tangent bound under the ruin floor); at the calibration their growth rate lies below the logarithm of one plus the risk-free rate, 0.04402 per period, at every share of the coarse grid and the grid of ten-thousandths, its largest value 0.04361 at a share of 0.0001.

The equation is the rivals' expected underwriting result given the low-risk count where at most one of the two reinsurers rations: the high-risk lines and the overflow of low-risk lines that come to them, of which they accept a fraction. The chart's values are the quadrature's of Section 7 at the menu reinsurer's best contract, at the nine shares of the paper's Table 2 joined by straight segments; the paper's Figure 2 joins the values of the grid of ten-thousandths at every hundredth of the share.

Selection

Theorem 2: under the fixed pooled price, where the gap is bounded below by a positive constant at every share, the menu reinsurer's share converges to one almost surely, by the martingale strong law; the log-odds of the share move as a random walk whose drift, the growth gap at the state, and conditional variance depend on the state.

The first equation is the exact law of the log-odds of the share; the second is its drift given the history, the growth gap at the contract the menu reinsurer chooses at that state.

The gap over the share

At the menu reinsurer's best contract the gap 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, and 0 of its 9,999 values are non-positive; a grid does not establish the bound over the continuum of shares that Theorem 2 assumes (Section 7.2).

The menu reinsurer's best coverage and price at each share, the two expected growth rates per period, their gap, and the expected number of lines each reinsurer writes, by the quadrature under the fixed pooled price.

The chart joins the table's nine shares by straight segments and prints each series' value at its two ends; the table carries every value. The quadrature sums over the period's type counts and integrates the floored logarithm of each book's gross return (Section 7).

The quadrature against a Monte Carlo

At four shares a Monte Carlo of 200,000 periods that draws every loss and places every submission in its arrival order is set against the quadrature: the largest deviation in absolute value of the Monte Carlo from the quadrature is 1.5 standard errors, the two ends of the grid included.

Monte Carlo standard errors in parentheses and the deviation, the Monte Carlo less the quadrature, in standard errors in brackets; 200,000 periods at each share at the menu reinsurer's best contract.

Three pricing regimes

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 on 689 paths the noise of a small book lifts it above that type's willingness to pay, 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 the logarithm of one plus the risk-free rate while they are diluted; and 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 cross-path mean of the menu reinsurer's share at four periods under the three pricing regimes, period 59 the one at which the paper compares experience rating with the fixed price, and at the end of the paths the median, the number of paths at or below one half, and the number of paths on which the rivals' price stands above the high-risk type's willingness to pay for full coverage, the paths on which the market has closed; a closed path enters the means at the share at which it stopped; 1,000 paths of 300 periods from a share of 0.05.

The menu reinsurer has nothing to learn from a cedant's losses, since the choice reveals the type; experience rating is the rivals' learning alone, and at a zero loading it raises their price toward the high-risk type's cost.

The monopolist's menu

If the menu reinsurer re-optimizes as a monopolist once the rivals have left the market on a path, its growth-optimal menu at the calibration, computed at a share of one on a grid of hundredths of each type's coverage, serves the high-risk type alone, at a coverage of 0.99, the grid's largest, and a price of 0.9120, with an expected growth rate of 0.09500 per period; the best menu that serves both types earns 0.09444.

The exclusion of the better risks is that of the monopoly screening problem; against a posted price, the menu reinsurer distorts their coverage instead.

The common factor and the exposure

With a common loss factor carrying a common share of each loss ratio's variance, the variance of a book's result no longer vanishes with diversification, and the growth-optimal choice has a second dimension beside the contract: the exposure, the share of its capacity up to which the menu reinsurer writes. At a share of 0.25 the exposure is the full capacity at every point of the grid over the common share, up to 0.5, and the industry's capitalization, from 0.5 through 3, the capitalization being its capacity over the premium of writing every submission at the population mean loss ratio; the exposure 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.

The first equation adds a factor common to every submission of the period to each loss ratio, leaving its marginal distribution the baseline's; the second is the variance of a book's result, whose common part scales with the square of the book. Against a Monte Carlo of 100,000 periods at a common share of 0.3, a share of 0.25, and a capitalization of 1, the Monte Carlo less the quadrature is −0.68 standard errors for the menu reinsurer's growth rate and −0.23 for the rivals'.

The gap over correlation and capitalization

At a share of 0.25, the common factor widens the gap at the grid's capitalizations from 0.5 through the calibration's level, from +0.0278 to +0.0312 at the calibration's capitalization as the common share rises to one half, and narrows it at the grid's capitalizations from 1.5 through 3.

The growth gap per period at a share of 0.25 under the fixed pooled price, over the common share r of the loss-ratio variance (rows) and the industry's capitalization C (columns; the column at 1.120 is the calibration's).

The chart prints no values; its vertical scale reads the lines, and the table carries every value.

Surplus paths with demand tied to the economy

With demand growing at the risk-free rate the industry's capitalization drifts down without a common factor, from 1.120 to a cross-path mean of 1.088 at the end of 60 periods, and the rivals' capacity binds in 18.0 percent of the path-periods; at a common share of 0.3 an adverse year reduces both surpluses together, and the capacity binds in 42.0 percent of them; at that common share the menu reinsurer's exposure leaves full capacity on some paths, its smallest value 0.093 of capacity, and on none without a common factor.

500 paths per setting, each of 60 periods, from a share of 0.05 at the calibration's capitalization.

The continuum of types

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 third contract lifts the gap from 0.01049 with two contracts to 0.03395.

The growth gap per period at each share for menus of one to 5 contracts on the continuum of types, and the two-type core's gap at the same share.

The equation prices the chain of contracts from the top down: the price of the top contract leaves the highest type its gain from the pooled contract, and each lower price leaves the type at the threshold indifferent between adjacent contracts. The types, expected loss ratios before the cap, are uniform from 0.7499 through 0.8086, and the gap is computed over 20,000 draws of the period's class counts. Against a Monte Carlo of 100,000 periods that draws every type and every loss, at a share of 0.50, the Monte Carlo less the count Monte Carlo is −0.34 and −0.47 standard errors for the menu reinsurer's growth rate with one and two contracts and −2.24 and −2.14 for the rivals'; the cause of the rivals' deviation is not established.

Two dimensions of private information

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 4.6 percent to the one-contract gap at a share of 0.25, the most at any share computed. 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 a share of 0.05 any second contract that some type buys lowers the gap, from 0.01906 to 0.01079, because the first contract's low-volatility business fills the menu reinsurer's capacity and a contract the high-volatility type takes displaces that business in arrival order with lines of more than four times the variance.

The four types: the Normal parameters, the capped mean and variance, the population share, and the gain from the pooled contract at a pooled price of 0.7793; the first letter is the mean (L or H), the second the volatility.

The one-contract menu (the first four columns after the share) and the two-contract menu: coverages, prices, the gap per period, and the types whose first choice each contract is (a comma joins the buyers of one contract; for two contracts the first contract's buyers stand before the semicolon and the second's after it).

The chart prints no values; its vertical scale reads the two lines, and the second table carries every value. The gaps are the quadrature over the 23,426 count vectors of the four types, enumerated. Against a Monte Carlo of 100,000 periods at a share of 0.50, the Monte Carlo less the quadrature is −0.31 and −0.29 standard errors for the menu reinsurer's growth rate with one and two contracts and +0.00 and +0.05 for the rivals'.

Heavy-tailed losses

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 two expected growth rates per period and their gap under each family at three shares, from a Monte Carlo of 100,000 periods on common draws; standard errors in parentheses, the difference from the Normal's gap with the standard error of the paired difference.

The capped mean and variance of each type's loss ratio under each family, over the draws of the Monte Carlo; the Normal's exact values, those of the market's primitives, are 0.7654 and 0.0144 for the low-risk type and 0.8000 and 0.0624 for the high-risk type.

Student's t has 4 degrees of freedom; 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. The chart prints no values: the Normal's and the log-normal's gaps differ by 0.00006, 0.00007, and 0.00008 at the three shares, and their lines are drawn one over the other; the vertical scale reads the lines, and the first table carries every value with its standard error.

The paper

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; under the fixed pooled price, where the gap is bounded away from zero, the menu reinsurer's share of industry surplus converges to one almost surely; and the mechanism carries onto a continuum of types, two dimensions of private information, and heavy-tailed losses.

Optimal Menu-of-Contracts Design in Reinsurance, 4 November 2026. Flow map.

The paper is the source of the values in the statements, tiles, and tables of this page.

The computations

The growth rates are a quadrature over the period's type counts (Section 7), set against Monte Carlo runs that draw every loss; the share paths, the surplus paths, and the heavy-tailed losses are Monte Carlo runs, the continuum of types a quadrature over count vectors drawn by Monte Carlo, and the two dimensions of private information a quadrature over every count vector, of the sizes below.

A value in parentheses on this page is a Monte Carlo standard error; a value in brackets is the Monte Carlo less the quadrature, in standard errors.

Reading the values

Growth rates and gaps are per period; prices, premiums, gains, and margins are per unit of subject premium; shares are the menu reinsurer's share of industry surplus; coverages are fractions of a line.

The loss ratio is a Normal floored at zero and capped at the cap, with the two heavier-tailed families of Section 12 matched to its mean and variance before the floor and the cap.