An Implementation Guide to Menu-of-Contracts Reinsurance Pricing

1 Purpose and Audience

This guide is written for heads of reinsurance business units, chief underwriting officers (CUOs), and the pricing actuaries and portfolio managers who support them. It takes the model of Optimal Menu-of-Contracts Design in Reinsurance (Schmid, 4 November 2026; cited below as the paper) and states what a reinsurer does with it: which contract to offer beside the market’s single price, how to price it, how to size the book, what to expect of the rivals, and what to expect of time. The paper’s model is a screening model in the line of Rothschild and Stiglitz,¹ ranked by the growth criterion of Kelly² under the market-selection argument of Blume and Easley;³ the guide uses the paper’s results and not its proofs.

The paper’s market has two kinds of reinsurer. The rivals quote one price for full coverage, the pooled price, set at the population’s mean loss ratio or at a loading on their experience. The menu reinsurer offers, beside that contract, one partial-cover contract priced so that a low-risk cedant is indifferent between it and the pooled contract. A low-risk cedant takes the partial contract and a high-risk cedant takes the pooled one; the menu reinsurer writes the better risks up to its capacity, at a positive margin at every coverage above the one at which that margin is zero, and the rivals keep the worse risks and the better ones the menu reinsurer cannot absorb. Both reinsurers maximize the expected logarithm of their surplus, the Kelly criterion, and under the fixed pooled price, where the menu reinsurer’s expected growth rate exceeds the rivals’ by a margin bounded below by a positive constant at every division of the market’s surplus, the menu reinsurer comes to hold the market’s capacity. The central findings it draws on are three, at the paper’s calibration: the cream-skimming contract earns a positive expected margin at every coverage above 0.2929 of the loss; the rivals’ expected growth rate lies below the risk-free rate at every share of the market the paper computes, on its coarse grid of shares and its grid of ten-thousandths; and the menu reinsurer’s advantage shows in its share over decades rather than years.

2 The Market the Guide Assumes

The values in this guide are at the paper’s calibration. The paper prints them, apart from the worked example of Section 4 and the doubling times of Section 6, which are computed from the paper’s primitives and its Table 2, and the figures, which draw the paper’s model at points its tables do not print. The procedure applies at any other calibration with the same steps. A practitioner supplies two cedant types, low-risk and high-risk, with Normal loss ratios of means $\mu_L$ and $\mu_H$ and volatilities $\sigma_L$ and $\sigma_H$, floored at zero and capped at $\bar X$, whose capped means $m_L, m_H$ and variances $v_L, v_H$ are what the pricing uses; the population share $\lambda$ of low-risk cedants; and the cedants’ risk aversion $\rho$, which sets how much a cedant pays for variance removed: a cedant’s gain from a contract of coverage $\alpha$ at a price $p$ per unit of subject premium is $$u_\theta(\alpha, p) = \alpha m_\theta + \tfrac{\rho}{2}\bigl[1 - (1-\alpha)^2\bigr] v_\theta - p ,$$(1) the expected loss transferred plus the value of the variance removed, less the premium. The practitioner also supplies the market’s size, $N$ submissions a period, a period being a year, each of $B$ units of subject premium in units of industry surplus, and a capacity multiple $\kappa$, the reinsurance premium a reinsurer may write per unit of its surplus; the risk-free rate $r_f$; and the menu reinsurer’s starting share $s_0$ of industry surplus. Table 1 states each at the calibration with the rule that set it.

The population mean loss ratio $\bar m = \lambda m_L + (1-\lambda) m_H$ is 0.7793; the pooled price of a competitive market, the price at which the rivals’ expected profit on the population is zero, is $p_P = \bar m = 0.7793$ per unit of subject premium. At that price a low-risk cedant’s gain from the pooled contract $C_P = (1, p_P)$ is $u_L(C_P) = 0.01384$ and a high-risk cedant’s $u_H(C_P) = 0.1408$: the pooled contract is attractive to both types, a high-risk cedant paying less than its expected loss and a low-risk cedant paying more but valuing the variance removed, its full-coverage risk premium $\tfrac{\rho}{2} v_L = 0.02769$. The calibration sets $\mu_L$ so that the low-risk gain is one half of that risk premium, which is the half-risk-premium rule of Table 1; the rule keeps the better risks in the pooled market, which a market under voluntary purchase requires.

3 The Cream-Skimming Contract

The menu reinsurer offers one contract $C_L = (\alpha, p(\alpha))$: a share $\alpha$ of the loss ceded at a price that leaves the low-risk cedant exactly its gain from its best alternative, the pooled contract where that contract leaves it a non-negative gain and no reinsurance otherwise, $$p(\alpha) = W_L(\alpha) - \max\{u_L(C_P), 0\}, \qquad W_L(\alpha) = \alpha m_L + \tfrac{\rho}{2}\bigl[1 - (1-\alpha)^2\bigr] v_L ,$$(2) where $W_L(\alpha)$ is the low-risk type’s willingness to pay for the coverage. Where the pooled price exceeds the low-risk type’s expected loss and leaves that type a non-negative gain, $u_L(C_P) \ge 0$, as at the calibration, the paper’s Theorem 1 states what follows for every coverage below one: the low-risk cedant is indifferent between the two contracts and, by the paper’s rule that an indifferent cedant takes the contract of the lower premium, takes the partial one; the high-risk cedant prefers the pooled contract, since its willingness to pay for the coverage it gives up exceeds the low-risk cedant’s at every coverage; and the menu reinsurer’s expected margin on a low-risk line, $$\pi(\alpha) = p(\alpha) - \alpha m_L = \tfrac{\rho}{2}\bigl[1 - (1-\alpha)^2\bigr] v_L - u_L(C_P) ,$$(3) rises with the coverage from $-u_L(C_P)$ at no coverage to $p_P - m_L$ at full coverage, and is positive above one coverage $\alpha^\ast$, at which the low-risk cedant’s value of the variance removed equals its gain from the pooled contract. At the calibration $\alpha^\ast = 0.2929$: a contract that cedes less than 0.2929 of the loss is sold below the expected loss it takes on, and a contract that cedes more earns a positive margin on every low-risk line it writes. Where the low-risk cedant would lose from the pooled contract, the price is its full willingness to pay, the margin is positive at every positive coverage, and some contract of the family separates the types if and only if the high-risk cedant gains from the pooled contract (the paper’s Section 4).

3.1 The Steps

The procedure has five steps, each a quantity of Section 2 or a line of Table 2.

Estimate the two types. Set the capped means $m_L, m_H$ and variances $v_L, v_H$ from the loss experience of the two classes the menu is to separate, and the population share $\lambda$ from the submission flow. The classes are the reinsurer’s own reading of the market, not a cedant’s declaration: the mechanism needs the two types to exist and to differ in mean and in variance, not to be known cedant by cedant.

Read the market’s price. Observe the pooled price $p_P$ the single-price rivals quote for full coverage. In a competitive market at a zero loading it is the population mean loss ratio $\bar m$; where the rivals load their price, it is $(1 + \delta)\bar m$, and Section 7 states what the loading changes.

Compute the low-risk gain and the margin’s zero. From (1), the low-risk gain from the pooled contract is $u_L(C_P) = m_L + \tfrac{\rho}{2} v_L - p_P$, and the margin’s zero is the coverage at which $\tfrac{\rho}{2}[1 - (1 - \alpha)^2] v_L = u_L(C_P)$. Every coverage above it and below one is admissible; a coverage below it at which the price is positive is admissible as well, and there the contract is sold below the expected loss it takes on.

Choose the coverage by the share. Among the coverages above $\alpha^\ast$ the menu reinsurer chooses the one that maximizes its expected logarithmic growth over the period, the expected logarithm of its gross return floored at the ruin floor, the smallest fraction of its surplus a reinsurer carries into the next period, given its share $s$ of industry surplus and the placement of submissions in arrival order up to each reinsurer’s capacity, $\kappa s/(B p_L)$ lines for the menu reinsurer at the price $p_L$ of its contract (the paper’s Section 3); the paper computes it by a quadrature over the period’s type counts (its Section 7). The choice is not the coverage of the largest margin per line: a larger coverage earns more per line but carries more variance per line, and the Kelly criterion weighs the two at the reinsurer’s own scale. Table 2 reports the paper’s choice at nine shares, with the price it implies and the two reinsurers’ growth rates.

Price it. The price is (2) at the chosen coverage: the low-risk willingness to pay less the low-risk gain from the pooled contract, where that gain is non-negative. It is quoted per unit of subject premium and scales with the treaty, as Section 4 shows; the coverage of Table 2 is chosen for submissions of the paper’s size $B$ in units of industry surplus, and for submissions of another size the step above chooses it again.

First, the chosen coverage moves with the share, and not in one direction: on the paper’s fine grid it lies between 0.5210 and 0.9648, and between grid points the best contract can jump between two maximizers. At small shares the menu reinsurer’s capacity binds, it writes a fraction of the low-risk submissions, and its growth is the most variable; as the share approaches one the rivals’ capacity vanishes, and the high-risk submissions they turn away spill over to the menu reinsurer’s contract at the low-risk price. The paper’s cycle extension reads the coverage distortion that separates the types as, in part, a response to the menu reinsurer’s own capacity constraint: with slack capacity it writes every low-risk submission at a coverage near one (Section 8). Second, the price moves with the coverage along the family (2) and not with the share directly: a reinsurer that holds a different share of the market offers a different coverage, and the price follows from it.

Figure 1 draws the margin over the coverage, with its zero and the coverages chosen at four shares, and beside it the two cedants’ gains at the contract’s price: the low-risk cedant’s, held at its gain from the pooled contract, and the high-risk cedant’s, below its gain from the pooled contract at every coverage below one.

4 A Worked Example at the Calibration

Take a treaty of 100 million in subject premium, in the currency the treaty is written in, at the calibration, the menu reinsurer holding one quarter of industry surplus. The rivals quote full coverage at the pooled price, 77.93 million. The menu reinsurer offers coverage of 0.7045 of the loss at 55.07 million, the price of Table 2 at $s = 0.25$ scaled to the treaty. Table 3 sets the example out.

The menu reinsurer’s contract runs at an expected loss ratio of 97.9 percent on its own premium, a margin of 2.1 percent: the cream-skimming contract is priced at the low-risk cedant’s indifference and not at a loading on its expected loss, and its advantage is that the lines it writes are low-risk lines wherever the rivals’ capacity does not bind; where it binds, the high-risk submissions the rivals turn away spill to its contract at the low-risk price (Section 8). The rivals’ pooled contract, priced at the population mean, earns zero in expectation on the population and a non-zero margin on each line: its expected margin is −2.08 million on a high-risk treaty of this size and 1.38 million on a low-risk one, and the menu takes the low-risk ones away. The low-risk cedant gains the same 1.38 million from either contract; a high-risk cedant would gain 12.25 million from the menu contract against 14.08 million from the pooled one, and so does not take it. At the calibration the low-risk cedant’s gain from the pooled contract and the rivals’ margin on a low-risk line coincide, both being one half of the low-risk risk premium by the calibration’s rule.

5 What the Rivals Are Left With

A single price for full coverage carries the two underwriting errors the trade names: business written below its cost (the high-risk lines, an error of commission) and profitable business lost to a competitor (the low-risk lines, an error of omission). The paper measures both. At the pooled price of a competitive market the rivals’ expected margin is $-\lambda(m_H - m_L)$ on a high-risk line and $(1-\lambda)(m_H - m_L)$ on a low-risk one, so their book earns a non-positive expected result whenever its low-risk share is at most $\lambda$ (the paper’s Corollary 1); once the menu reinsurer takes the low-risk submissions, 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 bound on the logarithm under the ruin floor); there Treasury bills, at the risk-free rate, grow a reinsurer’s surplus faster than the book the menu leaves the rivals.

At the calibration the rivals’ capacity covers every submission at the pooled price at every share up to 0.107, one of the two cases of Proposition 1; between the two cases the sign of the rivals’ expected result is not established, and the paper’s quadrature computes their growth rate instead. It is below the logarithm of one plus the risk-free rate, $\ln(1 + r_f) = 0.04402$, at every share of the coarse grid of shares, 0.01 through 0.99, and of the grid of ten-thousandths; its largest value over both grids is 0.04361 at a share of 0.0001, where the rivals still write nearly every submission, and it falls as the menu reinsurer’s share grows, below zero toward the other end of the grid, where the rivals’ remaining book is a fraction of one high-risk line priced at $\bar m$ on a surplus that vanishes with it: at $s = 0.99$ the logarithm of their expected gross return lies 0.02583 below $\ln(1 + r_f)$, and the variance of that fraction of a line on their surplus places their growth rate a further 0.07658 below it (the paper’s Section 7). Figure 2 draws the two growth rates against the share, and beside them the lines each reinsurer writes: at the shares of Table 2 above $0.25$ the menu reinsurer’s growth rate falls toward $\ln(1 + r_f)$, and its expected lines first exceed the rivals’ between the shares of $0.25$ and $0.50$.

6 How Fast the Share Moves

The log-odds of the menu reinsurer’s share of industry surplus move as a random walk whose drift is the growth gap at the state, $g_0 - g_R$, and whose conditional variance depends on the state (the paper’s Section 6). Under the fixed pooled price, where the gap is bounded below by a positive constant at every share, the share converges to one almost surely (the paper’s Theorem 2). The gap is positive at every share of the coarse grid of shares, from +0.00959 at a share of 0.01 to +0.10427 at 0.99 (Table 2), and on the grid of ten-thousandths its smallest value is +0.00935; a grid does not establish the bound over the continuum of shares (the paper’s Section 7).

The pace is set by the gap and measured in periods, a period being a year of underwriting. The ratio of the menu reinsurer’s surplus to the rivals’ doubles every $\ln 2 / (g_0 - g_R)$ periods, which at the calibration is 36.5 years at a share of $0.05$, 24.9 years at a share of $0.25$, and 19.3 years at a share of $0.50$. The paper’s simulated paths from a share of 0.05 bear this out (Table 4, first row, and Figure 3): the cross-path mean share is 0.167 at period 59, 0.381 at period 100, 0.911 at period 200, and 0.992 at the horizon of 300 periods, with 998 of 1,000 paths above one half. The dispersion is that of a random walk from a small base: 2 paths fell early to a share at which the menu reinsurer’s book was a fraction of one line and the increment’s noise a multiple of its drift, and had not recovered at the horizon, the lowest of them standing at 0.0028. The strategy is a growth strategy over decades: on the paper’s paths the cross-path mean share stands below one half at period 100 and above it at period 200, so that a business plan that expects the share to show within a planning cycle sets a horizon the paths do not support. And a menu reinsurer that starts small is exposed to the noise of a small book. In the paper’s cycle extension, where the exposure is a choice, the growth-optimal exposure is full capacity at a share of 0.25 over the paper’s grid of common shares up to one half and capitalizations from one half through three (Section 8 below); on the paper’s 500 paths per setting, each of 60 periods, from a share of 0.05, with demand tied to the economy, the exposure stays at full capacity on every path without a common factor and leaves it on some paths at a common share of 0.3.

7 When the Rivals Reprice to Experience

The rivals of the fixed-price regime never learn. The paper’s second regime has them reprice each period to a credibility-weighted mean of the loss ratios of the business they wrote, with weight 0.3 on the period’s experience, at a loading $\delta$ on that mean.

At a zero loading the rivals can price themselves out of the market. As the menu reinsurer takes the low-risk business, the rivals’ experienced mean tends toward the high-risk mean $m_H$, and their price rises toward the high-risk cedant’s cost with it; at the price $m_H$ a low-risk cedant’s gain from the pooled contract is −0.00692, below zero, and the menu reinsurer prices against no reinsurance instead, which raises its margin, and the share runs ahead of the fixed-price case early (0.257 against 0.167 at period 59, and 0.507 against 0.381 at period 100) and behind it later (0.859 against 0.911 at period 200). Then, once the rivals’ book is a fraction of one line, a single adverse loss ratio lifts their price above the high-risk cedant’s willingness to pay, $W_H(1) = 0.9201$; no cedant buys the pooled contract, and without it the menu reinsurer has nothing to separate against and writes nothing either. On 689 of 1,000 paths the noise of a small book had lifted the rivals’ price above $W_H(1)$ by the horizon, and the cross-path mean of the price, the closed paths included, rises from 0.7793 to 0.8957 over the horizon. For the menu reinsurer the state is the end of the strategy as well as of the rivals’: the cream-skimming contract needs a pooled contract to skim against.

At a loading of 0.01 on the experienced mean the rivals grow while they are diluted: their price rises from 0.7871 to 0.8492 in cross-path mean, their realized logarithmic return over periods 50 through 150 is 0.04917, above the logarithm of one plus the risk-free rate, $\ln(1 + r_f) = 0.04402$, and the menu reinsurer’s share still reaches 0.899 at the horizon. The closed market is reached at that loading too, on 352 of 1,000 paths at the horizon: the price exceeds $W_H(1)$ once the estimate exceeds $W_H(1)/(1 + \delta)$, while the rivals’ book stays larger for longer. Two loadings bound the pooled market: above $\delta_L = 0.0178$ the pooled contract has no low-risk buyers at the opening and the market opens separated, and above $\delta_H = 0.1500$ the high-risk cedant leaves a pooled contract priced on a high-risk book; the simulated loading of 0.01 lies below $\delta_L$. In the separated configuration the menu reinsurer keeps the restriction of offering the high-risk cedant no contract, although the rivals’ price $(1 + \delta) m_H$ exceeds that type’s expected loss there; at a share of one half the rivals’ growth rate rises with the loading, and above $\delta^\ast = 0.045$ they outgrow the menu reinsurer, a threshold the paper computes at that share alone.

In the paper’s model, whose cedants are drawn afresh each period, the menu reinsurer learns nothing from a cedant’s losses. In a separating menu the cedant’s choice reveals its type at once, and a menu that re-priced a cedant on its loss history would screen again what the choice has already screened; experience rating is the rivals’ learning, about the composition of the book the menu leaves them. A market in which cedants persisted from period to period and a reinsurer observed a cedant’s loss history would add a second channel of information, which the model does not contain. 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 cedant alone, at a coverage of 0.99, the grid’s largest, and a price of 0.9120 per unit of subject premium, the better risks excluded, for an expected growth rate of 0.09500 per period against 0.09444 for the best menu that serves both types. The reinsurer that won the market by serving the better risks would, as a monopolist, serve the worse ones alone.

8 Capacity, Correlation, and the Cycle

The paper’s cycle extension relaxes two assumptions: losses independent across submissions, and demand that scales with industry surplus. With a common loss factor carrying a share $r$ 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. With demand growing at the risk-free rate rather than with surplus, a reinsurer’s capacity relative to the business offered to it moves with its results, and the hard market is the state in which that capacity binds.

First, at a share of 0.25 the growth-optimal exposure is the full capacity at every point of the grid over the common share, up to one half, and the industry’s capitalization, its capacity over the premium of writing every submission at the population mean loss ratio, from one half through three; the exposure leaves full capacity at a common share of 0.897 at a capitalization of one and 0.877 at a capitalization of three, 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 the paper’s 500 surplus paths per setting, each of 60 periods, from a share of 0.05 with demand tied to the economy, the exposure also leaves full capacity on some paths at a common share of 0.3, with a smallest value of 0.093 of capacity, and on none without a common factor; the paths do not identify the state of the market in the periods in which the exposure does so. Second, at a share of 0.25 and the grid’s capitalizations from one half through the calibration’s level of 1.120, the common factor widens the gap, from +0.0278 to +0.0312 at that level as the common share rises to one half, because the common part of a book’s variance scales with the square of its exposure and the rivals’ book carries the higher volatility; at a share of 0.25 and the grid’s capitalizations from one and a half through three, the common factor narrows the gap, because the menu reinsurer, the smaller firm, then carries the larger book relative to its own surplus. Third, on the paper’s surplus paths with demand tied to the economy the rivals’ capacity binds in 18.0 percent of the path-periods without a common factor and in 42.0 percent at a common share of 0.3. At a share of 0.25 the gap falls as capitalization rises from three quarters through three at every common share, and with slack capacity the menu reinsurer writes every low-risk submission at a coverage near one and a margin near the pooled contract’s. The high-risk submissions the rivals’ capacity turns away spill to the menu reinsurer’s contract at its own, low-risk price, where each such line loses in expectation at every coverage above $1 - \lambda$ (the paper’s Section 7). To maximize the expected logarithmic growth of the surplus at a share of 0.25, then, set the exposure at full capacity over the grid of common shares up to one half and capitalizations from one half through three and, at capitalizations of one and three, at every common share below 0.897 and 0.877, respectively, on the first point’s presumption that the exposure leaves full capacity once; the gaps of the second point are those of that exposure. To maximize each period’s expected logarithmic growth on the paper’s paths from a share of 0.05, where a common share of 0.3 takes the growth-optimal exposure below full capacity on some paths, to 0.093 of capacity at the smallest, set the exposure in each period at the growth-optimal one; the paper computes no path at that common share with the exposure held at full capacity, so the growth forgone by holding it there is not measured.

9 Richer Cedant Populations

The paper carries the mechanism onto three richer populations, each set beside the two-type market of the sections above, the core: a continuum of types, two dimensions of private information, and heavy-tailed losses.

A continuum of types. Where the cedants’ expected loss ratios run over an interval rather than two values, a menu of several contracts partitions the types by thresholds, each contract priced at the marginal type’s indifference. To earn most of the gap that menus of more contracts earn at shares of $0.05$, $0.25$, and $0.50$, where the rivals’ capacity does not bind in every period, one contract suffices, the one that skims the better risks: a second contract adds 0.00011, 0.00027, and 0.00006 to the gap at those shares, on a one-contract gap of 0.03193 at $0.25$. Against the two-type core, one contract’s gap is larger at a share of $0.05$, 0.03827 against 0.01898, since every type below the threshold pays the marginal type’s price on a lower expected loss, and smaller at a share of $0.50$, 0.03132 against 0.03596. To widen the gap at a share of $0.75$, where the rivals’ capacity binds in every period and the menu reinsurer is nearly indifferent between skimming the better risks and pooling the rationed ones, offer three contracts: they lift the gap from 0.01048 under one to 0.03395, while the menu reinsurer’s growth rate under three exceeds that under one by 0.00025, and the rivals’ falls from 0.04015 to 0.01693.

Two dimensions of private information. Where a cedant’s mean loss ratio and its volatility are both private, one contract screens on the mean and a second separates on the volatility. 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. To earn the larger gap of one contract or two, offer two at shares of $0.25$ and $0.50$ and one at $0.05$ and $0.75$: the second adds 4.6 percent to the one-contract gap at $0.25$ and 1.6 percent at $0.50$, changes the gap at $0.75$ from 0.03576 to 0.03575, and at $0.05$ 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 single contract is bought by the low-mean, low-volatility type alone at shares of $0.05$ and $0.25$, and by both low-mean types at $0.50$ and $0.75$, where it screens on the mean alone (the paper’s Table 9).

Heavy-tailed losses. With the loss ratio drawn from a log-normal or from Student’s $t$ with 4 degrees of freedom at the Normal’s mean and variance, floored and capped as before, and the contracts held at those chosen on the Normal’s capped moments, the gap changes only through the moments after the floor and the cap: Student’s $t$ lifts the high-risk 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 and write the high-risk business, take the shift of the high-risk capped mean as a shift of their margin.

10 Implementation Checklist

10.1 Before the Renewal Season

• Define the two classes the menu is to separate and estimate their capped means $m_L, m_H$ and variances $v_L, v_H$ from experience; estimate the low-risk share $\lambda$ of the submission flow.

• Read the rivals’ pooled price $p_P$ and whether it is fixed at the population mean or loaded on their experience; where it is loaded, read the loading against $\delta_L$ and $\delta_H$ of Section 7.

• To read the growth rates against what the surplus earns without underwriting, set the risk-free rate $r_f$ at the current yield of the 3-month U.S. Treasury bill, the instrument behind the paper’s 0.045: a reinsurer that writes nothing grows at $\ln(1 + r_f)$ a period, 0.04402 at the calibration, and since $r_f$ enters every growth rate, a yield other than 0.045 moves the growth rates this guide prints at the calibration, by amounts the paper does not compute. To price at the cedants’ own valuation of the variance removed, set their risk aversion $\rho$, one for both classes as the paper takes it, from the largest loading on expected loss at which the high-risk class buys full coverage: $\rho$ is twice that loading times $m_H$ over $v_H$, and the paper’s loading of 15 percent gives $\rho = 3.8451$ at the full precision of $m_H$ and $v_H$.

• Compute the low-risk gain from the pooled contract, $u_L(C_P)$, and the margin’s zero $\alpha^\ast$; confirm $u_L(C_P) \ge 0$, without which the pooled market has no low-risk buyers and the price rule changes (Section 7).

10.2 Designing the Contract

• Choose the coverage above $\alpha^\ast$ that maximizes expected logarithmic growth at the reinsurer’s share of the market, by the quadrature of the paper’s Section 7 or a Monte Carlo of the period under the placement rule of its Section 3.

• Price the contract at $p(\alpha) = W_L(\alpha) - \max\{u_L(C_P), 0\}$ per unit of subject premium and scale to the treaty.

• Confirm admissibility: the price is positive, the high-risk cedant prefers the pooled contract, and the high-risk cedant buys the pooled contract at all.

• Offer the contract beside the market’s pooled contract, not in its place: the cream-skimming contract needs the pooled contract to separate against.

10.3 Sizing and Monitoring

• To maximize the expected logarithmic growth of the surplus, set the exposure in each period at the one the Kelly criterion gives: at a share of 0.25 the paper finds it at full capacity over its grid of common shares up to one half and capitalizations from one half through three and, at capitalizations of one and three, leaving full capacity at common shares of 0.897 and 0.877, respectively, located 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; on the paper’s paths from a share of 0.05 with demand tied to the economy, the exposure falls below full capacity on some paths at a common share of 0.3; the gaps of Section 8 are those of that exposure.

• Track the realized loss ratio of the subject business in the cream-skimming book, its loss over its subject premium and not over the reinsurance premium, against $m_L$: a loss ratio above $m_L$ in expectation marks high-risk lines in the book, which under the separation reach it only where the rivals’ capacity binds and the high-risk submissions it turns away spill to the contract at the low-risk price, each such line at a loss in expectation (Section 8).

• Track the rivals’ price: under experience rating a rising price precedes the state in which no cedant buys the pooled contract and the cream-skimming contract has nothing to separate against, which the paper’s paths reach at a zero loading and at a loading of 0.01 alike; in the separated configuration at a share of one half, in which the menu reinsurer offers the high-risk cedant no contract, a loading above $\delta^\ast$ marks rivals who outgrow the menu reinsurer.

• Plan the share in decades: the ratio of surpluses doubles every $\ln 2 / (g_0 - g_R)$ periods, 24.9 years at a share of one quarter at the calibration.

11 Pitfalls

11.1 Pitfall 1: Offering a Contract to the High-Risk Cedant

At the pooled price of a competitive market no contract for the high-risk type earns more than it costs, since the pooled contract subsidizes that type. In the two-type market, while the rivals offer their pooled contract, the menu reinsurer offers one contract, which with theirs forms the menu. In the separated configuration, where the rivals load their price to $(1 + \delta) m_H$, above the high-risk type’s expected loss (Section 7 above; the paper’s Section 8), the paper keeps the restriction of offering the high-risk cedant no contract and computes the growth rates under it, so what a contract for that type would earn there is not measured; once the rivals have left the market, the paper computes the menu the menu reinsurer would offer as a monopolist, which, at the calibration, serves the high-risk cedant alone (Section 7 above).

11.2 Pitfall 2: Ceding Too Little

A contract below the margin’s zero $\alpha^\ast$ is priced below the expected loss it takes on, whatever its coverage looks like beside the pooled contract. The margin rises with the coverage; on the paper’s grid of ten-thousandths of shares the coverage the model chooses lies between 0.5210 and 0.9648, above 0.2929.

11.3 Pitfall 3: Reading the Margin as the Objective

The growth-optimal coverage is not the coverage of the largest margin per line. The Kelly criterion weighs the margin against the variance per line at the reinsurer’s scale, and the choice moves with the share and with whether capacity binds (Table 2).

11.4 Pitfall 4: Expecting the Share within a Planning Cycle

Under the fixed pooled price, where the growth gap is bounded below by a positive constant at every share, the share converges to one in the limit (the paper’s Theorem 2), and on the way it moves with the dispersion of a random walk from a small base. A small menu reinsurer that measures itself by its share after a few years reads a random walk whose conditional variance is largest at small shares.

11.5 Pitfall 5: Re-Pricing a Cedant on Its Losses

In the paper’s model, whose cedants are drawn afresh each period, the cedant’s choice has already revealed its type; experience rating adds nothing to a separating menu and is the rivals’ tool, not the menu reinsurer’s. A market in which cedants persist and a reinsurer observes a cedant’s loss history would add a second channel of information, which the model does not contain.

11.6 Pitfall 6: Ignoring the Rivals’ Loading

Under experience rating at a zero loading the rivals’ price rises toward the high-risk cedant’s cost, and the noise of a small book lifts it above that cedant’s willingness to pay on the paths that close: on 689 of 1,000 simulated paths the market had closed by the horizon, and on 352 at a loading of 0.01. At a share of one half, a loading above $\delta^\ast$ in the separated configuration, in which the menu reinsurer offers the high-risk cedant no contract, lets the rivals outgrow the menu reinsurer on the worse risks. In the paper the loading is a fixed parameter of the rivals’ pricing rule, and the outcome turns on it; the menu reinsurer’s choice each period is made against the rivals’ posted price.

12 Summary

The paper’s model gives a reinsurer one instrument against the information its cedants hold: a single partial-cover contract, offered beside the market’s pooled contract and priced at the better risks’ indifference to it. Where the pooled price exceeds the low-risk type’s expected loss and the high-risk type gains from the pooled contract, the contract earns a positive expected margin at every coverage above a zero the primitives fix; 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 who quote one price are left with the worse risks and the better ones the menu reinsurer cannot absorb, and grow below the risk-free rate; and under the fixed pooled price, where the growth gap is bounded below by a positive constant at every share, the reinsurer that offers the contract comes to hold the market’s capacity, over decades and with the dispersion of a random walk from a small base. From its loss experience and the market’s price a reinsurer obtains the contract, its price, and its size; the rivals’ price and loading mark whether the rivals are leaving or growing. On a continuum of types, one contract that skims the better risks earns most of the gap that menus of more contracts earn at shares of $0.05$, $0.25$, and $0.50$, where the rivals’ capacity does not bind in every period, and three contracts widen the one-contract gap at a share of $0.75$, where that capacity binds in every period; with two dimensions of private information, a second contract that separates on the volatility adds at most 4.6 percent to the one-contract gap; and heavy tails change the gap only through the moments after the floor and the cap, the rivals, who price on the Normal’s moments, carrying the shift of the high-risk capped mean in their margin. What the model does not supply, the reinsurer supplies: the two classes and their moments, the reading of the market’s price, and a planning horizon of decades.

Bibliography