The Kelly Criterion and Learning in Competitive Insurance Underwriting
What an insurer learns about its accounts, how it learns it, and what the learning adds to the mean geometric rate at which its surplus grows: three questions of one market.
Three findings on a record of 200 seeds
Each a difference in terminal surplus between two fillings of one seat, the same market around both.
Learning is what grows surplus beyond a conventional insurer's. A second Loss-ratio insurer in the fourth seat grows at 1.93 percent a year; the plan route, the learned estimate priced through the rating plan, grows at 3.39 percent a year.
Learning pays on new business, and the rating plan does better on renewals. Learning on new business, priced through the rating plan, grows at 3.71 percent a year, the highest growth of any filling of the fourth seat in the record's market.
What the Kelly decision adds depends on the estimate it prices: it adds growth on the rating plan's estimate, matches the plan's pricing on the learned estimate, and falls short of it with learning on new business.
Growth of surplus as the objective
A rating plan gives a price for an account and no statement of what the price is for. The Kelly criterion gives the objective: the expected mean geometric rate at which surplus grows.
For one account the objective reduces to a rule: the margin as a share of surplus, the return the premium earns before the loss is paid, and the value of keeping the account, less half the variance of the loss over the square of the surplus. The weight on the variance is set by the objective, not by the insurer. The Kelly decision is the premium that maximizes the chance of winning the account times this increment, so it answers both questions an insurer faces at a quote: the price, and whether it wants the account at that price.
The expected loss
A rating plan supplies a version of it: the class loss cost, the experience and schedule modifications, the rate level.
The variability of the loss against surplus
Enters through the variance term: of two accounts with one expected loss, the more variable is charged no less.
The value of keeping the account
The margin of each later renewal times the chance of reaching it, discounted: the renewal credit.
The chance that a quote wins
The insurer's own hold curve, fitted to the quotes it won and lost.
How it is learned
From what an insurer already holds: each part of the learning reads data the insurer holds.
The loss distribution
From its claim files and the loss runs it reads: each account's claim frequency relative to its class and the scale of its claim sizes, tracked by credibility weighting; the persistence of an account's departure from its class is estimated from the insurer's own files.
The chance of winning
From the broker's figure: the hold curve, fitted to the quotes the insurer lost, with the broker's figure, and the quotes that bound.
The value of keeping an account
From its own book: the retention and the margin at a first renewal and at later renewals, over a window of years, discounted.
The paper crosses three estimates with two ways to price each. Learning on new business is the learned estimate on new business and the rating plan's estimate on renewals. The Kelly decision prices an estimate as a deviation from the price the rating plan gives it, inside the filed band of 25 percent either way; on a learned estimate the plan's pricing is the plan route and the Kelly decision the Kelly route.
| The estimate | The plan's pricing | The Kelly decision |
|---|---|---|
| The rating plan's estimate | A second Loss-ratio insurer | The Kelly decision on the plan's estimate |
| The learned estimate | The plan route | The Kelly route |
| Learning on new business | The plan route | The Kelly route |
A stylized market
Middle-market general liability: 4,000 accounts, four seats, 28 record years.
Bureau insurer
Refiles its rate level when the indication on its own experience departs from the plan by a stated threshold.
Loss-ratio insurer
Moves its rate level every year by the loss-ratio method, with competitors' winning prices as the complement of credibility.
GLM insurer
Adds relativities by class and size band from a generalized linear model fitted to its own book.
The fourth seat
Filled in turn by the insurers under study, the three conventional insurers the same in each filling.
Growth is the logarithm of terminal surplus over the endowment, net of the return on funds: a surplus that earns that return and nothing else has a growth of zero. Every filling of the fourth seat runs on the same 200 seeds, so a comparison between two fillings is a comparison of the same market under two insurers, paired by seed. The market is stylized, and no observed market has an insurer that prices by the Kelly decision.
What learning is worth
At the plan's pricing only the estimate changes.
Where an insurer wins its new business: the new business a second Loss-ratio insurer wins has a true expected loss of 1.012 times its own estimate, because an insurer wins where its estimate is low. The new business the plan route wins has a true expected loss of 0.927 times its estimate. The plan route holds 67 percent of the market's accounts at the close, a book at which the capital standard begins to act: brokers withhold 964 new-business invitations over the 28 record years and 200 seeds, where in the market with a second Loss-ratio insurer they withhold none.
What is learned: claim frequency and claim size
Percent of terminal surplus, against the class value of both traits.
The learned estimate tracks two traits of each account: its claim frequency relative to its class and the scale of its claim sizes. Learning both adds more than the two alone: each trait is worth more when the other is learned too. The claim frequency is worth more than the claim size, by 33.9 percent of terminal surplus.
Where learning is used
Percent of terminal surplus, against a second Loss-ratio insurer.
On new business the insurer quotes an account it does not hold, on the broker's loss run. On a renewal it quotes its own account, on its own claim file, which the rating plan reads through the experience modification. The renewals the plan route keeps carry a true expected loss of 1.027 times its learned estimate; with the rating plan's estimate on renewals, 0.997 times. An insurer is paid on its estimate, so an estimate that runs low on the accounts it keeps costs it margin.
Learning on new business grows the most
The learned estimate on new business, the rating plan's on renewals, the plan's pricing.
What the Kelly decision adds: the grid
Growth, logarithm; the Kelly decision less the plan's pricing, percent of terminal surplus.
| The estimate | The plan's pricing | The Kelly decision | Kelly less plan |
|---|---|---|---|
| The rating plan's estimate | 0.534 | 0.732 | +21.8 |
| The learned estimate | 0.935 | 0.945 | +1.1 |
| Learning on new business | 1.019 | 0.918 | −9.6 |
On the learned estimate the two pricings are equivalent: the 90 percent interval of their difference, from 0.2 to 2.0 percent of terminal surplus, lies inside a margin of 2.8 percent either side of zero, 0.10 percentage points of growth a year.
The better the estimate, the less the decision adds on top of it. An insurer that does not learn gains by pricing its plan's estimate by the decision. Learning on new business adds 8.8 percent to the plan route and takes 2.7 percent from the Kelly route.
With learning on new business, two books
The Kelly route's prices against the plan route's price, and the book each holds.
On the learned estimate the Kelly route quotes renewals 12.6 percent above the plan route's price and new business 8.4 percent below it: a low first price recovered on the accounts that stay, the shape of the lowballing that the theory of pricing under asymmetric learning predicts. With learning on new business it quotes both above the plan route's price.
Along any one path the Kelly route's growth is the steadier of the two; across seeds its terminal surplus is the more dispersed, the standard deviation of its growth at the close 1.272 times the plan route's.
Where to learn, and how to price
What follows for an insurer.
Where to learn
On new business, where the insurer holds no file of its own. On renewals the rating plan, through the experience modification on its own claim file, does better. The plan's architecture stands: the rate level, the schedule credit on new business, the provision for expenses and profit.
How to price
The Kelly decision is worth most to an insurer that does not learn. On the learned estimate it reaches the plan route's growth by a different book and path. With learning on new business it falls short of the plan's pricing.
An insurer that learns on new business and prices through its rating plan grows its surplus at 3.71 percent a year, against 1.93 percent for a second Loss-ratio insurer in its place.
What the record does not show
The limits the paper states, in Section 7.2.
One market, stylized
One line of business, no underwriting cycle, no reinsurance, and conventional insurers that respond to a learner only through their own rules. The grid is run in the record's market.
The criterion's rule, not its weighting of risk
The variance term is small at this market's surplus, so the record tests the rule for pricing under uncertainty and not the weighting of risk.
A known horizon
The horizon is known to the insurer, and growth is read at it.
The precision of a simulation
Every difference is measured within one model over 200 seeds; its t statistic states the precision of the simulation, not of an observed market.