What is actuarially fair insurance?

Actuarially fair insurance has an expected net pay-off of zero. From a consumer's point of view, an insurance contract is actuarially fair if the premiums paid are equal to the expected value of the compensation received. This expected value is, in turn, defined as the probability of the insured-against event occurring multiplied by the compensation received in the event of a loss.

Actuarially fair insurance is not necessarily complete. Complete insurance pays compensation equal to the harm caused by an accident, leaving the person just as well off as he would have been had the accident not occurred. For example, it might take $10,000 to compensate a car lover for the psychological harm caused by someone deliberately scratching the paintwork of his beloved antique MX-6. Complete insurance would cover that $10,000 harm. Actuarially fair insurance may or may not cover that harm, but it is still fair, as long as the premiums paid are equal to the probability of experiencing deliberate scratching of paintwork times the benefits paid out in the event of paintwork scratching.

Economists most frequently define actuarial fairness from the point of view of the consumer. Yet this definition implies that actuarially fair insurance can never exist. As Nicolson and Snyder's Intermediate Micro text informs students, "No insurance company can afford to sell insurance at actuarially fair premiums." Insurance companies have administrative costs. If they paid out everything that they received in premiums in compensation to customers experiencing losses, they could not cover their other costs of doing business.

The reason why economists define actuarial fairness from the consumer's point of view is that it yields useful predictions about consumer behaviour. For example, if actuarially fair insurance is available, a risk averse consumer - one who doesn't like risks -  will always purchase complete insurance, insuring herself fully against any losses. True, this type of actuarially fair insurance is an ideal that never exists in the real world. But like that other ideal, the perfectly competitive market, it is analytically convenient (translation: the math is easier), and provides a useful benchmark, a standard against which to judge real-world insurance contracts.

Yet when we define actuarially fair insurance from the consumer's point of view, we have no expression for an insurance contract that is actuarially fair from the insurer's point of view, one where the premiums paid by customers are equal to the expected cost of insurance claims plus administrative costs.

An insurance policy that provides the insurer with an expected payoff of zero has interesting properties. The insurance company makes zero economic profits. No one group of customers subsidizes any other group of customers, which means that the insurance company does not redistribute income ex ante. (Insurance still redistributes income ex post, from those who do not experience a loss to those who do). From a policy point of view, this alternative definition of actuarial fairness - actuarially fair from the point of view of the insurer - is frequently useful. After all, why spend time talking about insurance that is actuarially fair from the consumer's point of view, when we know that such insurance cannot exist?

Some writers do, in fact, define actuarial fairness from the point of view of the insurer. For example, Econport.org writes "for insurance to be actuarially fair, the insurance company should have zero expected profits." University of Toronto philosophy professor Joseph Heath seems to have this same definition in mind in his analysis of the moral implications of subsidies from one group of insurance companies to another. He writes, "it does not really matter whether it is just or unjust to charge actuarially fair premiums, it is necessary for insurers to do so if they wish to remain solvent." The fair-for-insurer definition of actuarial fairness is also implicit in statements such as this one: "An individual with spina bifida might be able to purchase an actuarially fair, albeit expensive insurance policy to provide care for the condition..."

It is surprisingly difficult to find clear definitions of actuarial fairness. There is no Wikipedia entry for "actuarially fair" or "actuarial fairness," perhaps because actuaries themselves appear to prefer betting language, talking about "fair odds". (Thinking about it, from an actuary's point of view, the adjective "actuarial" is redundant.) Where economists talk about actuarial fairness, insurers are more likely to draw a distinction between community rating, where all members of a community pay the same premium, even if some face a greater risk of loss than others, and individual risk rating, which is similar to what I have defined as "actuarially fair from the point of view of insurers."

Economics texts or articles often assume the reader knows what actuarial fairness means, and do not define it. Perhaps this reflects the profession's reluctance to talk about issues of fairness or equity - prices are what they are. The analysis of insurance and pension markets is one the few occasions when economists talk about "fair" prices. Even there, the "fair" price is the one that involves no ex ante redistribution.  

This lack of clarity is unfortunate, however, because insurance - for health, or for old age, in the form of pensions - is an issue of critical policy importance. If our language is imprecise, our thinking will be imprecise also.

HT to B., whose question forced me to clarify my own thinking about this topic. 

Comments

[K]

The first (in my view correct) definition is equivalent (roughly) to the marginal cost of production. But then nobody sells anything at that price. Is there something particularly industry specific here?

[Frances Woolley]

K "The first (in my view correct) definition is equivalent (roughly) to the marginal cost of production" -

This is a useful way of thinking about actuarial fairness when all administrative costs were fixed costs. If most administrative costs are fixed costs then, yes, "actuarially fair from the consumers point of view" is roughly equal to the marginal cost of production.

If most administrative costs are variable costs, then the second definition, i.e. cost of issuing one more insurance policy = expected claims on that policy + cost of administering that policy is closer to the marginal cost of production.

What is unusual is that a price=some kind of marginal cost is referred to as a "fair" price in this context. No where else in economics is that particular terminology used. Also I think there's more confusion about actuarial fairness than the average economic concept. Though that may be overly optimistic.

It's also really unusual to find a term that doesn't have a Wikipedia entry. I figure that this post should generate loads of hits for the blog from people looking for a definition of actuarial fairness.

[Kosta]

To "actuarially fair", there's the complimentary definition of "risk-neutral". Risk-neutral probabilities and prices incorporate investor risk-preference, with most investors being risk-averse, which in turn drives up the price of insurance. In your above formula of insurance company profits = premiums received + investment returns - (claims + administrative costs), the claims are driven by real-world probabilities, but pricing of the insurance is still based on risk-neutral probabilities.

The relationship between risk-neutral and real-world probabilities is the Radon-Nikodym derivative or less formally the "likelihood ratio".

Taking into account investor risk preference for risk-aversion, it's unclear if you can price insurance such that it is "actuarially fair", rather than fair to the demand of risk averse investors.

[JKH]

In addition to actuarially fair coverage and administrative costs (including employee compensation I presume), insurance companies must price their cost of debt and equity capital in their formulae for premium determination.

Finance professionals do not like to see required return on capital referred to as an administrative cost. And ignoring it altogether wouldn’t be financially fair, or complete.

[Frances Woolley]

Kosta: "there's the complimentary definition of "risk-neutral"" - doesn't this typically apply in situations where arbitrage is not possible, whereas insurance is about arbitraging risk?

JKH: "insurance companies must price their cost of debt and equity capital in their formulae for premium determination"

Yes, that thought occurred to me, too. However most of the economic literature on insurance models it in a static context, so premiums paid in = claims paid out and there's little need for capital. It's much harder to define things like actuarial fairness if you have multiple periods and the insurer and the person buying insurance face different interest rates. (There's a whole literature on "what is an actuarially fair pension").

[Alex Godofsky]

| Determinant: "Claims typically exceed premiums in all lines of insurance."
|
| Do you have any evidence whatsoever to support this claim?

Frances: I'm a pension, not insurance actuary so I don't have direct evidence, but I did sit for the life contingencies exam a few weeks ago so I think I can comment. I don't believe the term "actuarially fair" was ever used, but we only ever calculated two premiums, the "benefit premium" and the "expense" premium. These were the premiums whose expected present values were equal to the expected present values of benefits and expenses, respectively. Since premiums are always paid before benefits and the interest rate used for discounting is almost always positive, this means that the undiscounted sum of benefit premiums will necessarily be less than the undiscounted sum of benefits, in expectation.

[JKH]

"this means that the undiscounted sum of benefit premiums will necessarily be less than the undiscounted sum of benefits, in expectation"

aren't you saying basically that present value is less than future value, as those ideas apply specifically to the pension business?

always seemed to me that Buffet had a bit of a cowboy approach to the insurance business - invest premiums before you pay claims, and make out like a bandit

life contingencies still the 4th exam?

[Jeff Graver]

I did a 3 year stint as a budding actuary many years ago, and based on that experience (at a large Property&Casualty insurer and a small WorkersComp/Commercial insurer), I'd make a couple of points.

First, I think it's fair to say that actuaries wouldn't talk about actuarially fair prices in the sense you suggest because there is nothing actuarially interesting about administrative expenses. Whether fixed or variable, G&A costs are relatively well known and occur relatively quickly - not what actuaries are interested in.

On the other hand, I recall that when preparing prices for underwriters to use, we routinely calculated prices that were actuarially fair in the consumer sense (expected underwriting wash), then factored in an extra small percentage for actuarial uncertainty. That sounds like a more interesting sense of actuarial fairness from the producer's point of view.

Second, although I might be misinterpreting you, I think you are mistaken to mix up the notions of actuarially fair prices with community-rating vs individually risk-adjusted prices. The categories of risk that are measured ex ante are a separate consideration from whether the price is adequate to cover the risk. Or to put it the other way, you can underprice community-rated insurance just as easily as you can underprice individually risk-adjusted insurance.

The difference is that you need a somewhat higher adjustment for uncertainty when you use less data in doing the pricing. So there is a bigger difference between consumer-oriented vs producer-oriented actuarial fairness in insurance that prices relatively less information ex ante. But otherwise, they are separate things.

I haven't talked actuary in years. I hope I'm not completely unclear.

[Kosta]

" doesn't this typically apply in situations where arbitrage is not possible, whereas insurance is about arbitraging risk?"

Insurance is about having market participants willing to hold risk, hold that risk for a fee, which is actually quite similar to most financial transactions, and especially most financial derivatives. When a commodities broker buys a forward contract from a farmer on the price of wheat to delivered in the fall, the broker is accepting the risk of the price of wheat falling. When AIG sold CDS on subprime mortgage securities, they accepted the risk of those mortgages falling in price. When an investor buys a share or a bond of a company, they are accepting the risk that that company will fail and not pay back the value of the share or bond.

The same is true for insurance. The actuarially fair price of insurance, which is, I believe, the real-world price of insurance, would compensate the insurer exactly for expected claims of the insurance. But there is also the cost of having the insurer accept the risk involved in the insurance. This is where risk-neutral pricing arises.

For instance, consider my car. Let's say it is worth $50,000 and it has a 1%/year chance of being involved in an accident which destroys is completely. Would you be willing to insure me for $500/year? That's the actuarially fair number, but very few people would be willing to accept a payment of $500/year with the 1/100 chance of having to pay out $50,000? No, most people would demand a premium for accepting that risk. Incorporation of this premium leads to risk-neutral pricing.

In your discussion of actuarially fair pricing for insurances, I think it is important to keep in mind that it is the insurance companies that are bearing the risk. Bearing this risk has a cost, and demands a premium. I suppose that government intervention could regulate this risk premium, but I imagine that in some circumstances, perverse outcomes could result.

[Frances Woolley]

Kosta, "Let's say it is worth $50,000 and it has a 1%/year chance of being involved in an accident which destroys is completely. Would you be willing to insure me for $500/year? That's the actuarially fair number, but very few people would be willing to accept a payment of $500/year with the 1/100 chance of having to pay out $50,000? No, most people would demand a premium for accepting that risk."

It's interesting how different people approach things.

Typically the assumption made in economic analyses of insurance market is that insurers are risk-neutral, so don't demand a premium for accepting risk. See, for example, this passage from a typical economics paper "From textbook economic theory we know that if there exists a risk neutral agent who is not privately informed about s, this agent is willing to fully insure the risk averse agent at an actuarially fair rate." (The paper then goes onto assume the existence of risk neutral agents, and derive a number of theorems about insurance markets).

So economists' characterization of insurance markets is very different from the way that people who work in the financial industry see these markets.

[Frances Woolley]

Jeff: "we routinely calculated prices that were actuarially fair in the consumer sense (expected underwriting wash), then factored in an extra small percentage for actuarial uncertainty"

That idea of uncertainty - unknown unknowns - as opposed to risk - known unknowns - is one that is typically lost in economic discussions. But I think it's a very important one.

"Or to put it the other way, you can underprice community-rated insurance just as easily as you can underprice individually risk-adjusted insurance."

True. But even actuarially fair community-rated insurance will redistribute income from high risks to low risks within the pool, but individually risk adjusted will not, unless the insurer has made a mistake in estimating the risk.

[Kosta]

Frances, I was actually having a debate with a colleague of mine on how to determine the real-world probability of default from CDS prices, when she rebutted with the "risk-neutral" probability arguments. Then I read your post, and I presented her arguments to you. Thanks for the pointer that insurers are risk-neutral, and their willingness to insure risk-averse agents at actuarially fair rates.

As you noted in your post, "actuarially fair" is not always clearly defined. I've taken it to mean that an "actuarially fair" price matches that of a contract derived from real-world probabilities. I wonder if the "risk-neutral" insurer position is always true? Specifically, real-world (or actuarially-fair) probabilities can not be determined ex ante, but only estimated. The estimated probabilities can be inferred by the pricing of the insurance. I wonder how often these ex ante probabilities reliably estimate ex post results? Or from a pricing perspective, how often premiums reliably match claims? When the inferred probabilities diverge from the results, is it because the probabilities changed, or because the insurers were either more or less willing to accept risk?

Switching back to the financial world, CDS contracts on Greek debt (or U.S. mortgage debt), a form of insurance, were underpriced in the mid 2000s (with a much lower inferred probability of default than in reality) in part because people didn't understand the potential risks in the system, but also in part because the insurers (that is the CDS issuers) had a greater appetite for risk. The reverse is probably true now.

Coming back to your actuarially fair insurance, I wonder if a similar dynamic might interfere with the ability to price insurance fairly, albeit on a much smaller scale. While noting that insurers are risk-neutral in theory, estimating claim probabilities a priori may not always be straightforward. Could this uncertainty lead to variability in pricing and make it difficult to generate actuarially fair prices?

[Jim Rootham]

Huh? With community rating isn't the low risk person paying a higher price than an individually risk adjusted price would be? And vice versa for the high risk person? Doesn't that make the transfer from the low risk person to the high risk person?

[Jim Rootham]

Ah, I see the confusion. I was assuming everybody actually buys (or is required to buy) the insurance. In your scenario the low risk person doesn't buy insurance.

This is, of course, why the individual mandate is part of the health care reform in the States.

[reason]

"For example, it might take $10,000 to compensate a car lover for the psychological harm caused by someone deliberately scratching the paintwork of his beloved antique MX-6."

Is this a not so subtle dig at Nick Rowe?

[reason]

"This lack of clarity is unfortunate, however, because insurance - for health, or for old age, in the form of pensions - is an issue of critical policy importance. If our language is imprecise, our thinking will be imprecise also."

I think this is a GENERAL problem with economics. Consider "savings" or "efficiency" for example.

[Max]

Kosta, you are talking about risk shifting. That is very different from normal insurance, which is mainly about risk diversification, not risk shifting. There is generally no point to buying "insurance" on a financial asset - why not just sell it if you don't want the risk?

[Kosta]

Max, let's say I own a dividend paying stock, but I am concerned that with the crisis in Europe, the stock might lose value. I could sell the stock, but then I'll lose the dividends. So instead, I could buy a put option on the stock, and thereby ensure that I have guaranteed at least a base price for my stock if something untowards happens in Europe.

Or Max, let's say I am a large financial institution, and I have regulatory requirements that I must hold a certain amount of XXX bonds. Now, I like the yields of French bonds, but I am worried about the default risk of those bonds. Well, I could by the bonds, and then purchase CDS insurance against the default risk.

I'm not sure there's a need to have to sell the asset in all circumstances, especially when an economical way to insure or hedge the asset exists. It may make much more economical sense to insure than it does to sell.

[Frances Woolley]

Jim: "Huh? With community rating isn't the low risk person paying a higher price than an individually risk adjusted price would be? And vice versa for the high risk person? Doesn't that make the transfer from the low risk person to the high risk person?"

You're right. That whole sentence made no sense because community rated insurance isn't actuarially fair for individual plan members. I was tired.

reason: "Is this a not so subtle dig at Nick Rowe?" No, the psychological harm Nick would suffer from having his car deliberately scratched is much higher than $10,000 ;-)

Kosta - "I wonder how often these ex ante probabilities reliably estimate ex post results? Or from a pricing perspective, how often premiums reliably match claims?"

One argument for government provision of things like flood insurance is the markets don't work well for low probability/high loss events like natural disasters. Yet an article like this Journal of Econ Perspectives one on the economics of flood insurance happily use (without defining) the term actuarially fair.

There are people who argue that the conventional economic analysis of insurance fails in the event of this kind of uncertainty, e.g. http://www.ivm.vu.nl/en/Images/Naturaldisasters06_tcm53-109886.pdf. This is a reason to think question the meaningfulness of ideas like actuarial fairness - if the probabilities can't be calculated, is actuarially fair a meaningful term?

Max, yes, the difference between risk pooling and risk shifting is at the heart of the standard economic analysis of insurance.

[Phil Koop]

I've taken it to mean that an "actuarially fair" price matches that of a contract derived from real-world probabilities.

That is the usual interpretation.

real-world (or actuarially-fair) probabilities can not be determined ex ante, but only estimated.

Yes! Exactly so. And depending on your philosophy of probability, maybe real-world probabilities don't even exist. In finance we need only risk-neutral probabilities, and avoid asking what the real probabilities are. That is possible because finance is merely about relative pricing, and the entire field of financial mathematics is essentially a kind of glorified interpolation. That is why finance is so boring to economists: it is purely descriptive, and does not explain anything.

On the other hand, you do read some odd things when economists cross over into finance. The difference between real-world probabilities and risk-neutral probabilities is accounted for by "the market price of risk." In a representative agent model, this would be the price that the RA is charging to bear risk. If this price were zero, there would be no motivation to trade; the quantity traded would be zero. So any paper that starts by assuming that there are agents trading in the market who are risk-neutral with respect to real-world probabilities has pretty much gone off the rails and blown up at square zero. In most markets there would be unlimited interest in trading against such an agent.

When the inferred probabilities diverge from the results, is it because the probabilities changed, or because the insurers were either more or less willing to accept risk?

Economists usually prefer the former assumption because they do not like to assume that human characteristics change. In finance we usually speak in terms of the latter case because we do not like to rely on real-world probabilities.

[Frances Woolley]

Phil: "That is why finance is so boring to economists: it is purely descriptive, and does not explain anything."

Precisely! Though having said that, there are a fair number of people working in economics departments who find mathematics fun and enjoyable, and explaining the world mundane. For them, finance has many attractions. But then there is this disconnect with undergrad students who are studying finance because they want jobs as financial advisors etc.

"So any paper that starts by assuming that there are agents trading in the market who are risk-neutral with respect to real-world probabilities has pretty much gone off the rails and blown up at square zero."

Oh well, that's a large part of the economics of insurance literature done for.

[Phil Koop]

I see you disagree :-) Well, no doubt I misunderstood you.

Here is a slide presentation of the concepts of financial probability by Peter Carr (a worthwhile Canadian connection!); who is both a finance practitioner and a respectable academic: http://www.math.nyu.edu/research/carrp/papers/pdf/finprob.pdf.

Carr calls this type of error - substituting real probabilities for risk-neutral - "Type A error (A for arbitrage)".

The symmetric error, substituting risk-neutral probabilities for real (which was the sin you were contemplating), he calls type B (for Bankruptcy.)

[Frances Woolley]

Phil: "I see you disagree"

Not at all - I found your comments very worthwhile. And find it really striking that the analysis of insurance in public economics (e.g. talking about flood insurance, employment insurance, health insurance etc - this is what I do, and the type of people this post was aimed at) is divorced from the type of developments in thinking about insurance that you mention.