Why insurance exists

A recent issue of Annals of Actuarial Science features a concise essay by Ole Peters, founder of the London Mathematical Laboratory, discussing the concept of insurance – and explaining why it is surprisingly paradoxical from the viewpoint of mainstream economics. The discussion is clear, but as I read it, a confusing thought entered my head concerning risk and my own conception of insurance, all linked to the well-known concept of “risk aversion,” which I think is somewhat misunderstood. What does that term really mean?

In case other readers experience similar confusion, I thought it might be useful to go through the logic of Ole’s article and consider the origin of my brief confusion over risk aversion – and how it finds an elegant resolution through the ideas of ergodicity economics.

An insurance contract is a conceptually simple agreement – a party A agrees to pay another party B some money up front, say on a yearly basis, if the second party B in return agrees to make a significant payment back to A in certain circumstances. Example: I make a yearly payment to an insurance company which in turn agrees to give me a large payment in the event that my house should burn down. It seems obvious why we engage in such familiar arrangements: we willingly pay a small amount (not so small, actually!) so as to avoid the potentially catastrophic consequences of bad events that would be beyond our capacity to manage, and would cause irreversibly negative consequences for our lives.

Weirdly, however, the mere existence of insurance contracts turns out to be somewhat tricky to understand using the machinery of traditional economics concerning decision making under uncertainty. As Ole explains, that perspective approaches such situations by supposing that risk-neutral agents (more on what that means shortly) should make the best decisions over uncertain outcomes by making choices that maximize their expected return. That is, they consider all possible outcomes (the house burning down, or not burning down), the probabilities of those events, and they calculate the weighted average over those possibilities of how well they will do. They then make the choice that gives the best or highest return. This idea is common in the theory of optimal behaviour in games of chance, and is the standard approach in economics in many fields.

The problem arises in the setting of insurance due to the bilateral nature of the arrangement. Quite simply, if the expected value for the insurance company is positive, then they expect to get more money from me than I will get in return. Why would I want to enter into a contract where I am the loser? No matter how the contract might be arranged, whatever one party gets, the other pays – so there is no way for both to do well. It seems, naively, that insurance shouldn’t exist.

In his essay, Ole mentions discussions of this issue with economist Ken Arrow, who explained how economists generally get around this problem, and noted that “Ken was dissatisfied with the most common answers.” One of these proceeds by invoking so-called “risk aversion,” which posits that different people have different appetites for risk and the possibility of absorbing losses, and parties with differing aversion might take the two different sides. One party accepts a bad deal largely for psychological reasons, because they have a massive aversion to the possibility of large losses. [Note: the term I used earlier, risk-neutral, would in this picture refer to someone who has essentially no risk aversion, being entirely indifferent to the potential of large gains or losses – they care solely about what happens on average].

Here is where I found myself having some conceptual difficulty, which eventually resolved itself as I read further. I wondered why Ken Arrow would be troubled by this idea of risk aversion, as it seems highly intuitive to me. In some visceral sense I do believe there’s something to the risk aversion story. I have house insurance, for example, and simply accept that the expected value of this is negative – I’m going to pay a considerable amount, and probably not get back anything. But, I’m willing to do this because a reasonable sized negative is to me so much more acceptable than a catastrophically large loss such as my house burning to the ground. The decisive factor in my decision is the magnitude of the potential loss, which could be essentially life-ruining (or would certainly feel that way).

Continuing through the essay, I eventually realised that there’s no problem at all with my intuition. Risk aversion is a real feeling that most people have. The interesting question is: does it makes sense to take this feeling as part of an explanation for how people behave, and, in particular, why we have insurance? Or, might it make more sense to see that feeling of risk aversion as a natural consequence of people having learned to act intelligently in a risky world? As it turns out, this latter view seems much more natural.

Indeed, the origin of this feeling and way of behaving finds an elegant explanation in ergodicity economics. In contrast, it finds no explanation at all in the standard approach based on agents having different attitudes to risk aversion. Here’s why.

Ergodicity economics approaches decisions under uncertainty not by averaging over parallel conceptual worlds of different outcomes, but by averaging over outcomes which occur sequentially through time (Ole’s article explains this clearly and in mathematical detail). It turns out that doing this for agents facing uncertainties and accruing gains and losses in a multiplicative fashion – as in standard models of the returns in financial markets – leads to an effective asymmetry between the time-averaged consequences of gains and losses. In part because losses can lead to the chance of permanent, irreversible bankruptcy, it pays an agent in the long run to act aggressively to avoid large losses. In the context of the insurance puzzle, as Ole has shown, this means that paying for an insurance policy – and accepting an almost certain long run cost as a result – can actually raise the time-averaged growth rate an agent experiences.

So while paying for insurance seems like a dead loss from the standard expected value approach of economics, it appears as a clear wealth enhancing strategy from the more realistic perspective of the actual time average an agent experiences. This implies that insurance contracts do make sense – they can be beneficial to both the buyer and seller, as the agreement raises the time-average growth rates for both parties.

The beauty of this perspective is that this conclusion tumbles out without the need to invoke any extra assumptions about psychological attitudes and risk aversion. Indeed, this perspective actually offers a plausible explanation for the fact that people do experience a feeling of being adverse to large risks. In my case, I must understand implicitly that avoiding a massive loss is actually crucial to doing well in the long run. That stirs in me a deep feeling of risk aversion, a kind of psychological and emotional fear of big risks. But it is actually just part of having a good time averaged strategy. Not to have such a feeling would be to have an almost psychopathic indifference to potential disasters. Indeed, as Ole mentioned to me in an email, an actual “risk neutral” agent who behaved in such a way would be guaranteed to go bankrupt in the long run (not actually a surprising outcome for someone who ignores all risks!).

In this sense, that feeling we have of being cautiously risk averse (especially for big risks), which just seems so obvious and sensible, might be the emotional embodiment of practical wisdom most of us have learned by living in a world with multiplicative dynamics. Perhaps we’re even genetically hard-wired to some degree based on our ancestors’ experiences.

In connection with his essay, Ole mentioned to me that he and Benjamin Skjold, also at LML, are currently working on a number of other related insurance questions. For example: If insurance generally makes it mutually beneficial for small entities to pay more than expected loss to large entities, what does this imply for societies? Naively, it seems that larger entities should gradually gather in all the resources and then collapse – until some new entity again becomes big. Is that really the case? Or could more subtle effects interfere? Look for more on this in the near future from Ole and Benjamin.

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