How the study works
A simulation of one insurance market, in which one insurer at a time is tested against the same competitors and the same accounts.
- The study is a computer simulation of one insurance market, in which four insurers compete for the same 4,000 accounts of middle-market general liability. Three of them price by a rating plan: a rating bureau publishes a loss cost for each class of account, the class loss cost, and the plan multiplies it by the experience modification, an adjustment for the account's own claims, by the schedule modification, the underwriter's adjustment, and by the insurer's own rate level.
- Three seats hold the same three conventional insurers in every run, each pricing by the rating plan and setting its rate level in its own way (the cards below). The fourth place in the market, the “fourth seat,” holds the insurer the paper is testing. The same market is run for each insurer tested, with the same competitors, the same accounts, and the same random claims, so a difference between two insurers in the fourth seat comes from the insurer in the fourth seat and from how the rest of the market responds to it: the other three insurers' books, rate levels, and capital.
Bureau insurer
Like the other two, prices by the rating plan. It refiles its rate level, the multiplier it applies to the bureau's loss costs, only when its own experience departs from the plan by more than a set threshold.
Loss-ratio insurer
Moves its rate level every year by its own loss ratio. Where its own experience is thin, it gives the remaining weight to competitors' winning prices.
GLM insurer
Adds price adjustments by class and account size from a statistical model, a generalized linear model (GLM), fitted to its own book.
- The insurers tested differ in two choices. The first is the estimate of an account's loss they price on: the rating plan's estimate; the learned estimate, which the insurer builds from the account's own claims, how often the account has claims compared with its class and how large they are, on every account; or learning on new business, the learned estimate on new business and the rating plan's estimate on renewals. The second is how they turn the estimate into a price: through the rating plan, the plan route, or by the Kelly decision, the Kelly route. The Kelly decision moves the undeviated price, the price the rating plan gives the estimate, to the price that adds most to the expected growth of surplus, within the filed band, the range within which the plan lets a price move: 25 percent either way in the main market.
- The paper's learning has two parts. Learning the loss is the learned estimate of each account's loss. Learning the hold curve is fitting the chance that a quote wins the account to the insurer's own quotes won and lost. On the learned estimate and with learning on new business, the plan route learns the loss alone; the Kelly route learns the hold curve on any estimate it prices and reads the value of keeping an account from its own book.
- The benchmark is the plan route on the rating plan's estimate: a second Loss-ratio insurer in the fourth seat, which prices by the rating plan, moves its rate level every year by its own loss ratio, and learns neither the loss nor the hold curve.
| The estimate it prices on | Priced through the rating plan: the plan route | Priced by the Kelly decision: the Kelly route |
|---|---|---|
| The rating plan's estimate | The benchmark, a second Loss-ratio insurer | The Kelly route on the rating plan's estimate |
| The learned estimate, on every account | The plan route on the learned estimate | The Kelly route on the learned estimate |
| Learning on new business | The plan route with learning on new business | The Kelly route with learning on new business |
- Learning on new business uses the learned estimate on new business and the rating plan's estimate on renewals. The paper also tests learning on renewals alone and learning one trait of an account alone.
- Each run builds up the books, then runs 10 practice years, which are not scored, and 28 measured years, which are. The market is run 200 times, the paper's seeds, each with its own draws of chance, and every insurer in the fourth seat meets the same 200 runs. The market at the paper's main values is the main market; the paper also runs others. This page reports the main market, two of the further markets in one section, and the market of one tenth of the accounts among the limits.
- Growth is measured beyond what the starting capital would earn invested at the return on funds, the industry's investment yield: an insurer that earns only that return grows by zero. A rate of growth is stated in percent a year above the return on funds. A difference between two insurers is stated in percent of terminal surplus, the surplus at the close of the 28 measured years: how much more one holds than the other, as a percent of the other's.
Three findings in the main market, over 200 runs
Each finding compares two insurers that took the fourth seat in turn, in the same market: how much more surplus one holds than the other at the close of the 28 measured years.
First, in the main market learning the loss adds more growth than anything else the paper measures. With the benchmark in the fourth seat, its surplus grows 1.93 percent a year above the return on funds; with the plan route on the learned estimate in its place, the plan route's grows 3.39 percent a year.
Second, in the main market, on the plan route, learning the loss pays on new business, and on renewals the rating plan's estimate grows surplus more. The plan route with learning on new business grows the most of any insurer the paper reports in the fourth seat in the main market.
Third, in the main market what the Kelly decision adds depends on the estimate it prices: the Kelly route grows more than the plan route on the rating plan's estimate; on the learned estimate it trails the plan route for most of the measured years and ends within an equivalence margin of it; and it falls short of the plan route with learning on new business.
Growth of surplus as the objective
A rating plan gives a price for an account; it does not say what the price is meant to achieve. The Kelly criterion supplies the objective: the long-run growth of surplus.
The Kelly criterion is a rule for growing wealth through repeated bets: in repeated independent bets on known odds, no other rule grows wealth faster in the long run. An insurer that follows it chooses its prices so as to maximize the expected logarithm of its surplus at the horizon, which is the same as maximizing the expected compound rate at which its surplus grows from year to year. An insurer does not know the odds; the study measures what the objective is worth to an insurer that has to estimate them.
For one account small against surplus the objective reduces, approximately, to a rule. What writing the account adds to the logarithm of surplus, in expectation, is the margin of the premium over expenses and the expected loss, plus the investment income the reserve held against it earns until the loss is paid, plus the value of keeping the account at later renewals, all as a share of surplus, less half the variance of the loss over the square of surplus: the variance term, the criterion's weight on risk. The weight on that term is set by the objective, not chosen by the insurer.
G(P), what writing the account at premium P adds to the logarithm of surplus, in expectation; n, the premium net of expenses, and , the same at the undeviated price, from which the decision moves; and , the account's estimated expected loss and its variance; , the return the reserve held against the loss earns before the loss is paid; V, the value of keeping the account per unit of ; S, surplus.
The Kelly decision is the premium that maximizes the chance of winning the account at that premium times what the account would add. It gives both answers an insurer needs at a quote: what price to offer, and, by whether the account adds anything at that price, whether to want the account at all. The main construction of the decision makes three choices: it values later renewals at the undeviated price, the anchored lifetime value; it discounts them at the return on funds; and it lowers the center of its fitted hold curve by one posterior standard deviation, so that the chance of winning it assumes is lower at every price, the pessimism variant. It needs four things of an account:
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
It enters through the variance term: of two accounts with one expected loss, the more variable one is never charged less.
The value of keeping the account
The margin of each later renewal times the chance of reaching it, discounted and summed to the horizon. The paper calls it the account's lifetime value: what its later renewals are worth to the insurer, counted in the insurer's objective; it is not a discount granted to the buyer, on renewal or on new business.
The chance that a quote wins
The insurer's own hold curve: the chance that a quote at each price wins the account, fitted to the quotes it won and lost.
How it is learned
From data the insurer already has: its claim files, the loss runs brokers send with the accounts it is asked to quote, its won and lost quotes, and its own book.
The loss distribution
From its own claim files and the loss runs of the accounts it is asked to quote. For each account it tracks two traits: how often the account has claims compared with its class, and how large its claims are. Credibility weighting sets how much each year of the file counts, and how long an account that runs better or worse than its class stays that way is estimated from the insurer's own files, not assumed.
The chance of winning
From its won and lost quotes. Where an account went to market and the insurer's quote lost, the broker tells it by how much: the broker's figure. From these it fits its hold curve, allowing for the quotes that won, whose competing prices it never sees.
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.
and , the learned expected loss of one account and its variance, which the schedule modification and the rate level scale into the and of the equation above; , the tracked claim rate; , the expectation; Y, the claim size at the tracked scale; u, the occurrence limit; a, the loss adjustment expense's share, the cost of settling claims as a share of the loss.
Learning the loss gives the expected loss of the first card in place of the rating plan's estimate; the plan route uses it alone. Every insurer that prices by the Kelly decision learns the hold curve and reads the value of keeping an account from its own book. On the learned estimate it uses the learned variance as well, and with learning on new business it uses the learned variance on new business and on renewals alike.
The two routes, step by step
On the learned estimate both routes learn the loss; the Kelly route also learns the hold curve, reads the lifetime value from its own book, and prices by the Kelly decision.
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A stylized market
A simplified market of middle-market general liability insurance: 4,000 accounts, four insurers, 28 measured years.
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g, growth; , surplus at the close of the measured years; , the endowment, the surplus each insurer starts with; T, the number of measured years; r, the return on funds, the industry's investment yield, which surplus and reserves earn.
An insurer that earns only the return on funds has a growth of zero. The rate a year the page states is the exponential of growth over the number of measured years, less one, in percent above the return on funds; surplus at the close as a multiple of what the starting capital would earn at the return on funds restates the same growth as a factor. The market is stylized, and the Kelly decision is the paper's construction, not an observed insurer's practice.
What learning the loss is worth
What learning the loss adds to terminal surplus, both insurers priced through the rating plan; only the estimate differs.
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The three conventional insurers grow less beside the plan route (the lower panel) than beside the benchmark (the upper panel). The plan route ends with 71.8, 84.4, and 68.0 percent more terminal surplus than the Bureau insurer, the Loss-ratio insurer, and the GLM insurer beside it.
The new business the benchmark wins has a true expected loss, the one the simulation knows and the insurer does not, of 1.012 times its own estimate, consistent with the “winner's curse,” by which an insurer wins more often where its estimate is low. The new business the plan route wins has a true expected loss of 0.927 times its estimate; the paper does not measure why the ratio falls below one.
The plan route holds 67 percent of the market's accounts at the close, a book at which the capital standard begins to bind. Brokers read each insurer's capital ratio and withhold 964 invitations to quote new business over the 28 measured years and 200 runs; with the benchmark in the fourth seat they withhold none.
What is learned: claim frequency and claim size
What learning how often an account has claims, and how large they are, each adds, against learning neither.
The learned estimate tracks two traits of each account: its claim frequency relative to its class and the scale of its claim sizes. The insurer compared against learns neither and gives every account its class's values, the values the estimator gives an account with an empty file; it also leaves out the experience modification, a lower base than the benchmark's, which uses it. 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: the insurer that learns the frequency alone ends with 33.9 percent more terminal surplus than the one that learns the size alone. Such percentages compound rather than add, so 33.9 percent compares the two insurers' surpluses with each other and is not 41.7 less 5.8.
Where learning is used
What learning the loss adds on new business alone and on renewals alone.
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, from which the experience modification is computed. On the plan route, learning on new business alone adds more than learning on both, and on renewals the rating plan's estimate grows surplus more than the learned estimate.
The renewals the plan route on the learned estimate keeps carry a true expected loss of 1.027 times its estimate; for the plan route with learning on new business, with the rating plan's estimate on renewals, 0.997 times. An insurer's premium is set from its estimate, so an estimate that runs low on the accounts it keeps costs it margin.
The plan route with learning on new business grows the most in the main market
The learned estimate on new business, where the insurer holds no file of its own, and the rating plan's estimate on renewals, where it holds the claim file, priced through the rating plan.
What the Kelly decision adds
What the Kelly decision adds to the plan's pricing, on each estimate.
| The estimate | The plan route | The Kelly route |
|---|---|---|
| The rating plan's estimate | 1.93 | 2.65 |
| The learned estimate | 3.39 | 3.43 |
| Learning on new business | 3.71 | 3.33 |
In the main market, the better the estimate, measured by the growth it gives at the plan's pricing, the less the Kelly decision adds. Learning on new business, against the learned estimate, adds 8.8 percent to the plan route's terminal surplus and takes 2.7 percent from the Kelly route's. The second figure has a t of −3.52: beyond 2 in magnitude, a difference the 200 runs tell from zero, but not beyond 3.68, the level that allows for the number of tests.
On the learned estimate the Kelly route trails the plan route, by a difference the 200 runs tell from zero, in every measured year from year 1 through year 23, and ends within an equivalence margin of it: the 90 percent interval of the difference at the close, the range that allows for chance across the 200 runs, runs from 0.2 to 2.0 percent of terminal surplus, inside a margin of 2.8 percent either side of zero, 0.10 percentage points of growth a year. The paper set the margin with the difference already measured. The interval excludes zero, as it does at a t beyond 1.645, but the difference's t, 1.97, is below 2.
With learning on new business, two books
With learning on new business, the Kelly route quotes above its own undeviated price on renewals and on new business, holds fewer of the market's accounts, and ends with less surplus.
On the learned estimate, by contrast, the Kelly route quotes renewals 12.6 percent above its own undeviated price and new business 8.4 percent below it: a low first price and a higher renewal price, the shape of the “lowballing” that the theory of insurance pricing under asymmetric learning predicts of an insurer that knows its own accounts better than its competitors do. With learning on new business it quotes new business above its undeviated price.
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On average within a run the Kelly route's growth moves less from year to year than the plan route's; in 56 of the 200 runs it moves more. Across the 200 runs its terminal surplus varies more: the standard deviation of its growth at the close is 1.272 times the plan route's. The paper does not establish the cause of the shortfall; the two books come with it.
Where learning on new business loses its lead
In two of the paper's further markets, beside a second insurer that learns and under a severity trend that no insurer projects, the plan route on the learned estimate grows more than the plan route with learning on new business.
| The difference | Beside a second insurer that learns | Under the severity trend |
|---|---|---|
| The plan route: learning on new business less the learned estimate | −8.4 | −7.2 |
| The Kelly route less the plan route, on the learned estimate | +0.3 | +7.2 |
| The Kelly route less the plan route, with learning on new business | −3.5 | −4.6 |
Beside a second insurer that learns, the Kelly route on the learned estimate ends within the equivalence margin of the plan route, a margin set after this difference too was measured. Under the severity trend, the reversal on the plan route and the Kelly route's shortfall with learning on new business do not hold at the level that allows for the number of tests.
Where learning and the Kelly decision add growth
In the main market, on the plan route, learning the loss adds growth on new business and takes it away on renewals, and the Kelly decision adds most on the rating plan's estimate.
Where learning the loss adds growth
On the plan route, on new business, where the insurer holds no file of its own: there the learned estimate adds 62.4 percent of terminal surplus over the rating plan's estimate. On renewals the rating plan's estimate, computed in part from the insurer's own claim file through the experience modification, grows surplus more. The rest of the rating plan stays: the rate level, the schedule credit on new business, and the provision for expenses and profit.
Where the Kelly decision adds most
On the rating plan's estimate, where the Kelly route ends with 21.8 percent more terminal surplus than the plan route. On the learned estimate the Kelly route reaches the plan route's growth within the equivalence margin, with a different book of accounts and a different course over the years. With learning on new business it falls short of the plan route.
In the main market, the plan route with learning on new business grows its surplus at 3.71 percent a year above the return on funds, against 1.93 percent for the benchmark in its place.
What the study does not show
The study measures one simplified market within one model. These are the main limits the paper states.
One market, simplified
One line of business, with no underwriting cycle, no reinsurance, no loss trend outside one further market, and no shock common to all accounts. The three conventional insurers respond only through their own pricing rules to a competitor that learns, and the buyer responds to price alone. The page reports the main market, two of the paper's further markets (its Section 6) in one section, and the market of one tenth of the accounts among these limits.
The weight on risk, in two markets
At the main market's surplus the variance term is equivalent to a charge of 0.0641 percent of the expected loss of an account at the bureau's scale of claim sizes, and removing it is shown not to move the growth of the Kelly route on the learned estimate by more than 0.010 in the logarithm. In the market of one tenth of the accounts the charge is 10.0 times as large, and removing it lowers that growth by 2.3 percent of terminal surplus, not at the level that allows for the number of tests. The study does not run a market whose accounts are large against an insurer's surplus.
A margin set after the measurement
The equivalence margin on the learned estimate, 0.10 percentage points of growth a year, was set with the difference already measured.
The cause of a shortfall
With learning on new business the paper does not establish why the Kelly route falls short of the plan route; the two books above come with the shortfall.
A known horizon
The insurer knows the horizon, the close of the 28 measured years, and growth is read there.
The precision of one model
Every difference on this page is measured within one model over 200 runs; the ranges on the page measure the precision of the simulation, not that of an observed market.