Frank Schmid and Can Wang · 23 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. 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 the same 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 learns 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 report of the quotes it lost.
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 should an insurer know in order to grow its surplus, and how much 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 value of keeping the account enters at the price the plan would charge. 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 report of the quotes it lost, 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'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. The market is run 200 times from 200 draws of its random streams, the same draws for every filling, so that a difference between two insurers is a difference in the same market. Growth is the surplus at the close of the record years over the endowment, net of the return on funds.
Exhibits
Figure 1, one year of the market
Leads to
What is learning worth?
4
What Learning Is Worth
Learning
Establishes
Learning is what grows surplus. 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 wrong. The Kelly decision applied to the plan's own estimate adds nothing: a pricing rule is worth what the information behind it is worth. And the form in which the value of keeping an account enters the decision is worth more than the decision itself.
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 the Same Growth
Kelly criterionLearning
Establishes
The plan route grows at 3.39 percent a year and the Kelly route at 3.43 percent, the two highest growths on the record; their difference, 1.1 percent of terminal surplus, lies within a margin of equivalence of 0.10 percentage points of growth a year, set before the test. 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's price and new business below it, the lowballing the theory predicts, holds a smaller book in a market that turns over more, and trails the plan route through the first two decades before the two converge.
+12.6 percentrenewals, against the plan's price
−8.4 percentnew business, against the plan'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 equality a property of the market?
6
When the Routes Part
Kelly criterion
Establishes
The equality holds at the record's market. Where the filed rate band is 40 percent either way in place of 25, the Kelly route trails the plan route, and a buyer more responsive to price moves it behind; in a smaller insurer, a market with a tenth of the accounts, or under a rising severity trend, it leads. Beside a second insurer that learns, the two stay together.
−14.2 percentthe Kelly route at the wider rate band
+2.3 percentin a smaller insurer
+12.3 percentwith a tenth of the accounts
+7.2 percentunder a rising severity trend
Exhibits
Table 3, the Kelly route less the plan route, market by market, in percent of terminal surplus
Leads to
What does an insurer gain by building the learning and the decision into one architecture?
7
The Kelly Underwriter
Kelly criterionLearning
Establishes
The Kelly underwriter builds the learning and the decision into one architecture from the first year. It outgrows the three conventional insurers across 79 variations of the market but one, and trails both routes; under a rising severity trend it trails the conventional insurers too. The one part it cannot do without is the value of keeping an account: without it, it falls behind all three. Around it the market changes: new business priced below renewals, more turnover, and conventional insurers at higher rate levels and lower growth.
2.50 percentthe Kelly underwriter as first built, growth a year
−17.3 percentwith the anchored credit, against the plan route, in terminal surplus
−20.1 percentwithout the value of keeping an account
Exhibits
Figure 4, shares of the accounts and rate levels over the record years
Leads to
What follows for an insurer?
8
What Follows for an Insurer, and Conclusion
Kelly criterionLearning
Establishes
The return is in the learning, and the rating plan can carry it: an insurer that estimates each account's loss distribution from its own files and prices through the plan it already has reaches the highest growth on the record without changing its pricing architecture. The Kelly decision is a choice about the book the insurer builds and the path it travels; where the market departs from the record's it adds to growth or costs it. The limits: one line of business, no cycle and no reinsurance, conventional insurers that do not respond to a learner, the criterion's weight on risk small at this market's surplus, a horizon the insurer knows, one model. The learning extends to deep learning: a neural network identifies nonlinearities and dependencies that estimators of a stated form do not, and the two routes carry over to it unchanged.
Appendices
A, the market in full · B, the rating plans · C, the capital standard · D, the Kelly decision and the estimators · E, the Kelly underwriter's computation, step by step · F, the registry of predictions and outcomes · G, the Kelly underwriter's parts and the market around it · H, the sensitivity program · I, the quote data and the further markets