The Kelly criterion
The Kelly criterion is a rule from betting and investing. When the odds are known, it sets each stake so that wealth grows as fast as possible over many rounds; an insurer does not know its odds and must estimate them. Here the stake is an account written at a price, and the wealth is the insurer's surplus. To price one account the rule needs four things: the expected loss; how much the account's loss varies, against the insurer's surplus; what keeping the account is worth in later years; and the chance that a quote wins it. The price the rule gives is the Kelly decision.
Learning
Learning means estimating from data the insurer already holds, and the paper's learning has two parts. Learning the loss estimates how often each account will have claims and how large they will be, from the insurer's claim files and from the loss runs brokers send with each submission, an account's claims of the past five policy years whoever wrote them. Learning the hold curve estimates the chance that a quote wins an account at each price, from the insurer's record of the accounts it won and kept and, when a quote it made loses on an account that went to market, the broker's figure for how far its quote was from the winning price. A rating plan supplies a version of the expected loss and none of the other three things.
The line of sight
filled: the section is about the Kelly criterionfilled: the section is about learningopen: the theme is not the section's subject
1
Introduction
- Establishes
- A rating plan gives a price for each account but does not say what the price is for. With the long-run growth of surplus as the objective, an insurer can learn much of what it needs to price from its own operations. In the main market, learning the loss grows surplus more than anything else the paper reports; learning the loss, priced through the rating plan, pays on new business, while on renewals the rating plan's estimate grows surplus more; and the better the estimate, the less the Kelly decision adds. The question joins research on the Kelly criterion, on how insurers price when the insurer that holds an account knows it better than its competitors do, and on pricing algorithms in competition.
- Leads to
- What does the objective require of a price, and where does an insurer find what it requires?
2
Growth as the Objective, and Learning as Its Source
- Establishes
- The objective is the fastest compound growth of surplus over the long run. For one account, Equation 1 gives what writing it at a price adds to the growth of surplus, from three of the four things the Kelly criterion needs: the expected loss, how much the loss varies against the insurer's surplus, and the value of keeping the account. The Kelly decision is the price that makes the chance of winning the account at that price, times that addition, as large as it can be. There are three estimates of an account's expected loss. The rating plan's estimate is the class loss cost, the bureau's loss cost for the account's class, adjusted by the experience modification, the plan's adjustment for the account's own claim history. The learned estimate is the insurer's own estimate from its data, used on every account. Learning on new business uses the learned estimate on new business, the accounts the insurer does not hold, and the rating plan's estimate on renewals. Each estimate is priced two ways: through the rating plan, with the estimate in place of the class loss cost and the experience modification and the rest of the plan unchanged; or by the Kelly decision, which moves the undeviated price, the price the rating plan gives the estimate, up or down within the filed band, 25 percent either way in the main market, to the price the rule gives. On any estimate, pricing through the plan is the plan route and pricing by the Kelly decision is the Kelly route. An insurer that prices by the Kelly decision learns the hold curve in every case: whichever estimate it prices, it learns the chance that a quote wins the account from its own quotes, and it reads the value of keeping the account from its own book.
- 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
- The market is middle-market general liability: 4,000 accounts, with growth read over 28 measured years, after a build-up of the books and 10 practice years. Four insurers compete for the accounts. Three of them are conventional insurers that price by the rating plan, each at its own rate level, the multiplier it applies to the bureau's loss costs, and they work the same way in every run of this market. The Bureau insurer refiles its rate level when its own experience moves far enough from the rating plan; the Loss-ratio insurer moves its rate level each year by the loss-ratio method, by how its book's losses ran against what it was priced to earn; the GLM insurer adds relativities, adjustments by class and size, from a generalized linear model of its own book. The fourth place in the market, the fourth seat, holds the insurer the paper tests. The paper tests each of the three estimates there, priced each of the two ways, one insurer at a time. The benchmark among them is a second Loss-ratio insurer, which prices the rating plan's estimate through the rating plan and learns neither the loss nor the hold curve. Everything else stays the same, so a difference between two insurers in the fourth seat comes from how they estimate and price, and from how the other three insurers respond. The holder of an account reads its own claim file, current to the quarter; a challenger, an insurer the broker invites to quote, sees the broker's loss run, the account's claims of the past five policy years. The holder therefore knows more about its accounts than a challenger does. The market is simulated 200 times, the paper's 200 seeds, each with its own random claims and the same for every insurer tested, and the results are averages over the seeds. This is the paper's main market; Section 6 changes it.
- Exhibits
- Figure 1, one year of the market
- Leads to
- What is learning the loss worth?
4
What Learning the Loss Is Worth
- Establishes
- In the main market, learning the loss makes the biggest difference of anything the paper reports. The benchmark grows its surplus at 1.93 percent a year beyond the return on funds. The plan route on the learned estimate grows at 3.39 percent a year and ends the 28 years with 49.2 percent more surplus than the benchmark. Learning how often an account has claims is worth more than learning how large they are. On the plan route, learning the loss pays on new business, where the insurer quotes from the broker's loss run: learning on new business alone ends with 62.4 percent more surplus than the benchmark. On renewals the insurer holds the account's own claim file, and the rating plan's estimate, computed in part from that file through the experience modification, grows surplus more: learning on renewals alone ends with 15.1 percent less surplus than the benchmark. The plan route with learning on new business grows at 3.71 percent a year, the most of any insurer the paper reports in the main market.
1.93 percentgrowth a year beyond the return on funds: the benchmark, which learns neither the loss nor the hold curve
3.39 percentgrowth a year beyond the return on funds: the plan route on the learned estimate
+62.4 percentof surplus at the end, against the benchmark: learning on new business alone
−15.1 percentof surplus at the end, against the benchmark: learning on renewals alone
- Exhibits
- Table 1, the insurers in the fourth seat · Figure 2, growth by insurer and seed · Table 2, what learning the loss is worth
- Leads to
- What does the Kelly decision add on each estimate?
5
The Kelly Decision on Each Estimate
- Establishes
- In the main market, the better the estimate, the less the Kelly decision adds. On the rating plan's estimate the Kelly route ends with 21.8 percent more surplus than the plan route. On the learned estimate it trails the plan route, by a difference the 200 seeds tell from zero, in every year from year 1 through year 23 of the 28 measured years, and ends with growth within 0.10 percentage points a year of the plan route's. With learning on new business it ends with 9.6 percent less surplus than the plan route, and it trails in every measured year. On the learned estimate the Kelly route prices renewals above its undeviated price and new business below it. This has the shape of what the literature calls “lowballing”: a low price to win an account, and a higher one once the insurer knows the account better than its competitors do. With learning on new business the Kelly route prices both above its undeviated price and writes less business: at the end it holds 30 percent of the market's accounts, against 65 percent for the plan route.
+21.8 percentof surplus at the end: the Kelly route against the plan route, on the rating plan's estimate
−9.6 percentof surplus at the end: the same comparison, with learning on new business
30 against 65 percentshare of the market's accounts at the end, with learning on new business: the Kelly route against the plan route
- Exhibits
- Table 3, the three estimates priced two ways · Table 4, the two routes with learning on new business · Figure 3, the Kelly route against the plan route with learning on new business
- Leads to
- Do these results hold when the market changes?
6
Other Settings and Other Markets
- Establishes
- The other settings change four of the main market's dimensions, one at a time: the threshold at which the Bureau insurer refiles; how accurately underwriters read an account; the filed band, at 40 percent either way against 25 in the main market (the wider band); and how strongly buyers respond to price. The four further markets have smaller insurers; one tenth of the accounts; a second insurer that learns, in the GLM insurer's seat; and a severity trend, claim sizes rising 6.8 percent a year, that no insurer projects. The Kelly route, with learning on new business and on the learned estimate, was run at the wider band, at both other settings of the buyers' response, and in the four further markets. The plan route with learning on new business grows more than the benchmark wherever the two were run, ending with 9.3 to 78.9 percent more surplus. It grows more than the plan route on the learned estimate at the other settings and with smaller insurers, by 2.2 to 14.0 percent of surplus at the end, and ends below it beside a second insurer that learns, by 8.4 percent of surplus at the end, and under the severity trend, by 7.2 percent. With one tenth of the accounts the 200 seeds do not tell the two apart. On the learned estimate the Kelly route's difference from the plan route changes sign from setting to setting, from 19.7 percent less surplus at the end at the wider band to 3.1 percent more with buyers less responsive to price. In the four further markets the Kelly route grows more than the plan route, by up to 12.3 percent of surplus at the end with one tenth of the accounts, or ends with growth within 0.10 percentage points a year of the plan route's. With learning on new business the Kelly route trails the plan route wherever the 200 seeds are enough to tell the two apart, by 3.5 to 22.2 percent of surplus at the end.
+14.0 percentof surplus at the end: the plan route with learning on new business against the plan route on the learned estimate, smaller insurers
−8.4 percentof surplus at the end: the same comparison, beside a second insurer that learns
−22.2 percentof surplus at the end: the Kelly route against the plan route with learning on new business, at the wider band, 40 percent either way
- Exhibits
- Table 5, learning on new business in each setting and market · Table 6, the Kelly route against the plan route in each setting and market
- Leads to
- What follows for an insurer?
7
What Follows for an Insurer, and Conclusion
- Establishes
- In the main market, an insurer gains from learning the loss through its rating plan, without a decision computed at quote time. On new business it puts its learned estimate in place of two inputs, the class loss cost and the experience modification, and keeps the rest of the plan: the underwriter's schedule modification, which multiplies the estimate as before; its rate level; its schedule credits, which it gives new business from the winning prices it has seen; and its provision for expenses and profit. On renewals it changes nothing. In the main market this grows surplus at 3.71 percent a year, against 1.93 percent for the benchmark. Beside a second insurer that learns, and under the severity trend, the plan route on the learned estimate grows more than the plan route with learning on new business. In the main market the Kelly decision is worth most to an insurer that prices on its rating plan's estimate. Section 7.2 states the limits of the measurement. The paper's learning uses standard statistical methods; any estimator of an account's loss and of the chance of winning it, a neural network among them, could be held to the same measure.
3.71 percentgrowth a year beyond the return on funds: the plan route with learning on new business
1.93 percentgrowth a year beyond the return on funds: the benchmark
- 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