Frank Schmid and Can Wang · 26 October 2026 · The flow of the paper, section by section
The Kelly Criterion and Learning in Competitive Insurance Underwriting
Learning each account's loss distribution from the insurer's own records is what grows surplus beyond a conventional insurer's. The Kelly criterion supplies the objective and one way to put the learning to work; the insurer's own rating plan is another, and the two reach equivalent growth by different books and different paths.
The Kelly criterion
The objective is the long-run growth of surplus. For each account the criterion names four things: its expected loss, the variability of that loss against the insurer's surplus, what keeping the account is worth, and the chance that a quote wins it.
Learning
A rating plan supplies a version of the expected loss and none of the other three. The insurer can learn all four from what it already holds: its claim files, the loss runs of the accounts it quotes, its own retention, and the broker's figure, by how much a quote lost where the account went to market.
The line of sight
the section belongs to the Kelly criterionthe section belongs to learningthe theme passes through
1
Introduction
Kelly criterionLearning
Establishes
The question: a rating plan gives an insurer a price for an account and no statement of what the price is for. What must an insurer know to grow its surplus, and what of that can it learn from its own operations? The three findings, the markets in which the two routes part, what follows for an insurer, the limits, and three literatures: the Kelly criterion, pricing under asymmetric learning and its lowballing, and pricing algorithms in competition.
Leads to
What does the objective ask of a price, and where does an insurer find what it asks?
2
Growth as the Objective, and Learning as Its Source
Kelly criterionLearning
Establishes
The objective is the long-run growth of surplus, which the Kelly criterion states as the maximization of the expected mean geometric rate of growth. For one account the criterion names four things: the expected loss, the variability of the loss against the insurer's surplus, the value of keeping the account, and the chance that a quote wins it; the price it chooses from the four is the Kelly decision. In the paper's construction the value of keeping the account enters at the price the decision deviates from, the anchored credit; the other form, the quote-anchored credit, lets it rise with the price quoted, and the record measures both. An insurer can learn all four from what it already holds, its claim files, the loss runs of the accounts it quotes, its own retention, and the broker's figure, by how much a quote lost where the account went to market, and put the learning to work in two ways: through its rating plan (the plan route) or through the Kelly decision as a deviation from the plan route's price (the Kelly route).
Exhibits
Equation 1, what one account adds to the growth of surplus
Leads to
In what market is this measured?
3
A Stylized Market
Establishes
Middle-market general liability: 4,000 accounts, 28 record years, three conventional insurers that price by rating plans, the Bureau insurer, the Loss-ratio insurer, and the GLM insurer, and a fourth seat filled in turn by the insurers under study. A difference between two fillings of the fourth seat is taken within each of the 200 seeds of the record, the same draws for every filling, and averaged. Growth is measured on the surplus at the close of the record years, the terminal surplus, against the endowment grown at the return on funds.
Exhibits
Figure 1, one year of the market
Leads to
What is learning worth?
4
What Learning Is Worth
Kelly criterionLearning
Establishes
Learning is the largest effect in the record's market. A fourth conventional insurer grows at 1.93 percent a year; an insurer that learns each account's loss distribution and prices it through its rating plan grows at 3.39 percent, 49.2 percent more terminal surplus, and it does not win its new business mainly where its estimate is low. Learning adds to growth at four other settings of the market as well. The form in which the value of keeping an account enters the decision moves terminal surplus by more than the decision itself moves it.
1.93 percentgrowth a year without learning
3.39 percentgrowth a year, learning through the plan
+49.2 percentterminal surplus from learning alone
Exhibits
Table 1, the fillings of the fourth seat · Figure 2, growth by insurer and seed
Leads to
Does it take the Kelly decision to realize the gain from learning?
5
Two Routes to Equivalent Growth
Kelly criterionLearning
Establishes
The plan route grows at 3.39 percent a year and the Kelly route at 3.43 percent, the highest growths of any filling of the fourth seat; their difference, 1.1 percent of terminal surplus, lies within an equivalence margin of 0.10 percentage points of growth a year, set with the difference already on the record. It does not take the Kelly decision to realize the gain from learning. The routes differ in what a reader of quotes and of the yearly accounts sees: the Kelly route quotes renewals above the plan route's price and new business below it, in the shape of the lowballing the theory of pricing under asymmetric learning predicts, arrived at without a model of that equilibrium, holds a smaller book in a market that turns over more, and trails the plan route through record year 23, the two meeting only over the final years, its growth steadier from year to year and its terminal surplus more dispersed across seeds. Against the true expected loss of the business each writes, the Kelly route's premium on new business is 1.42 times that loss and the plan route's 1.57; net of expenses of 0.30 of premium, the Kelly route's new business is written below its true expected loss, by a difference the record resolves, and above that loss discounted at the return on funds over the mean time to payment; the plan route's is written above both.
+12.6 percentrenewals, against the plan route's price
−8.4 percentnew business, against the plan route's price
49 against 67 percentshare of accounts at the close, Kelly route against plan route
year 23the record year through which the Kelly route trails
Exhibits
Table 2, the two routes side by side · Figure 3, the difference at the close and year by year
Leads to
Is the equivalence a property of the market?
6
When the Routes Part
Kelly criterion
Establishes
The equivalence is a property of the market the record runs. Where the filed band is 40 percent either way in place of 25, the Kelly route trails the plan route. A buyer more responsive to price moves it behind and a buyer less responsive ahead, and where the insurers are smaller it leads: three differences the record resolves, the interval of each crossing the margin. In a market with a tenth of the accounts or under a rising severity trend that no insurer projects it leads beyond the margin. Beside a second insurer that learns, the two stay within the margin of each other. The gain from learning stands in every market in which the two were run, its size moving with the market.
−14.2 percentthe Kelly route at the wider filed band
+2.3 percentwhere the insurers are smaller
+12.3 percentwith a tenth of the accounts
+7.2 percentunder a rising severity trend no insurer projects
Exhibits
Table 3, the Kelly route less the plan route, market by market, in percent of terminal surplus
Leads to
What follows for an insurer?
7
What Follows for an Insurer, and Conclusion
Kelly criterionLearning
Establishes
Learning is what grows an insurer's surplus beyond a conventional insurer's. An insurer that estimates each account's loss distribution from its own files and prices it through the rating plan it already has keeps its rate level, its schedule credit, and its provision, the learned estimate taking the place of the class loss cost and the experience modification, and it reaches the highest growth in the record's market, equivalent to that of the Kelly route. Pricing the same estimate by the Kelly decision reaches that growth by a different book and a different path. The learning of the paper is econometric; the same data admit a deep learner, whose estimates either route prices as it prices the paper's. Section 7.2 states the limits of the measurement.
Appendices
A, the market: its assumptions, its insurers, and its loss process · B, the rating plans · C, the capital standard · D, the Kelly decision and the estimators · E, the Kelly route's computation, step by step · F, the registry of predictions and outcomes · G, the quote data and the further markets