The Curious Case Of Bad Math & Why It Does Not Matter
July 25, 2013 by Chris Burand
I have recently been reminded of some of the nonsensical insurance idiosyncrasies that veterans of the industry take for granted. Several of these involve math—and bad math in particular
1. Combined Ratios
Combined Ratio = Loss Ratio + Expense Ratio. A ratio is a fraction. Sometime around the fourth grade most kids learn that to add fractions the denominators must be common.
Loss ratio = losses/earned premium.
Expense ratio = expenses/written premium.
The denominators are not the same, so the sum of the ratios is effectively a random, meaningless number.
For example, if losses are $7 million and earned premiums are $10 million, the loss ratio is 70.
If expenses are $3 million and written premiums are $11 million, the expense ratio is 27.3.
The combined ratio is like saying Sally has 60 percent of the apples and John has 27.3 percent of the apples and oranges. Together they have 97.3 percent of what? Apples? Oranges?
We don’t know because we don’t know what percentage, if any, of the oranges Sally has, and we don’t know John’s ratio of apples to oranges.
A 97.3 ratio is pretty good. But if the denominators were equalized, quite possibly in this example the combined ratio would be 100-plus on an earned basis if this particular company is growing quickly.
Do three percentage points make a material difference?
Three percentage points may be quite material. For an insurance company with $100 million of premium, three extra percentage points are worth $3 million of profit.
If this were important, wouldn’t someone have addressed this basic math error by now?
Of course. Therefore, it’s not important. No insurance company would ever take advantage of the opportunity this provides to tout better profitability than its slower-growing peers. So just keep this in mind for your next insurance trivia game.
2. Functions In Calculus—And Insurance
While taking a calculus class many years ago, the concept of a function was explained to me like this: The idea is mechanical. The function of a gas pump is to pump gas. The function of a budget is to limit spending. The function of a contingency contract is, theoretically at least, to generate more profits.
Many companies must believe limiting spending is more important than making money. That must be why so many companies spend so much time budgeting every year. They would not waste time on anything that was not so critical. And this must also be why companies limit, through stability clauses in their contingency contracts, how much they will pay their agencies.
(Editor’s Note: The stability clause gives companies permission to set a maximum amount they will pay in contingencies in any given year. The company caps its contingency expense at some percentage of net premiums.)
I was trying to explain this to a rookie, and he just did not get it. He kept asking: “If the agencies collectively do a great upfront underwriting job, thereby creating enormous profits for the companies, why would the companies cap what they pay agents just so they do not exceed their contingency budget? Don’t they know the function of a contingency contract is to drive profits? Don’t they know that if agents create an extra 1 percent of profit, causing the company to exceed its contingency budget even by several tenths of a percent, that the company has made a great investment?”
Obviously he has taken calculus and understands functions. He just could not understand that making budget was a gazillion times more important to companies than making more money, keeping the respect of their best agencies and building for the future.
3. Some Things Just Don’t Add Up
Another mathematical issue I tried to explain was how two plus two actually equals 4.5 sometimes in this industry. The example is clusters.
Many insurance companies figured out—or agencies helped them understand—that if five agencies combine their volume, those agencies deserve to be paid more. (I have chosen five for the purposes of this example only, as it could be two agencies or one hundred agencies.)
The way it works is this: Assume for simplicity that each agency has $500,000 premium with a company. Some companies do not think $500,000 is adequate, but they look favorably upon $2.5 million.
So now that the company’s reports show an agency with $2.5 million rather than five agencies with $500,000, the premium is automatically more valuable. So if the company could only pay a total of 2 percent, on average, in bonuses before, they can now spend 3 percent because the report shows a bigger book of business. This math is like magic.
Moreover, sometimes loss ratios decrease. For example, as individual agencies, if one of those agencies has a large loss on a $500,000 book, that agency would not usually earn any contingency, but the other four still would. The carrier’s combined ratio on the $2.5 million total book (math problem not withstanding) might be 100 (example only).
But when the five combine to form an agency cluster, the cluster loss ratio is lower than the loss ratio for that one individual agency with the large loss. That means the large loss might not knock any of the five combined agencies out of a contingency. The carrier combined ratio could increase slightly to, say, 100.5 because the cluster loss is the same as the sum of the losses for the individual agencies, but the expense ratio increases while the loss ratio declines.
The agencies make more, and the companies are happy paying more for worse results because they know $2.5 million generated by five agencies on one report is worth much more than the same premiums and losses but shown as five different agencies.
Milo Minderbender (“Catch-22” financial genius) would be proud.
The insurance industry is fascinating.
It is as though the medieval alchemists who tried to turn lead into gold were several centuries ahead of their time and were searching the wrong industry. As these three simple examples prove, the folks running this industry have figured out why basic math does not apply and how by using the industry’s unique formulas and innovative concepts, they can turn losses into pluses no one thought possible.
