The business of insurance, by various measures, is the largest industry on earth. Somewhat surprisingly, mainstream economics struggles to explain its existence. This post is about how ergodicity economics approaches the subject, which leads to a glimpse into complexity theory and the emergence of social structures of all sorts. It follows a recent editorial in Annals of Actuarial Science.

We will discuss the fundamental insurance puzzle – why do insurance contracts exist? Because this is such a basic question, we don’t need to go into great actuarial detail, but we do need to sketch what we mean by insurance. In essence, the story is this: you’re exposed to some risk, something that might happen in the future (a fire or a car crash etc). A bad event beyond your control with bad consequences. Insurance contracts allow you to mitigate the financial aspect of those consequences: you can buy an insurance policy for a fee, and you will receive a cash payout if the bad thing happens.

## Formal setup

We formalise this in purely financial terms as follows. You currently have some wealth x , and you’re exposed to the risk of losing an amount L , which happens with probability p over a future time period \Delta t . You can insure yourself against this loss by paying a fee F .

Intuitively, we all know that we will want to buy such contracts under certain circumstances. We may want to insure our house against fire, or our car against theft, provided the fee isn’t too high. So what’s the puzzle?

## The insurance puzzle

The puzzle arises when we consider that it takes two parties to set up an insurance contract – someone, some entity, has to be on the other side of the deal. Someone has to be willing to sell you the insurance at a fee which you are willing to pay. Clearly, the seller wants to charge as much as possible, and you want to pay as little as possible. So a fundamental question arises: is there any price at which both the seller and the buyer will be happy to sign the deal?

If the fee is such that you’re happy to pay it to get rid of the risk, then clearly, your assessment is that that’s better than keeping the fee and the risk. But why, then, would your counterparty (usually an insurance company) come to the different assessment that it’s better to receive the fee and the risk? If this difference in opinion doesn’t exist, then no contract will be signed — for a contract to emerge, both parties must feel they’re improving their position by signing.

Colloquially, we have established that there could be a problem. We will now show that economic theory formally encounters this problem when it analyses insurance contracts through the lens of expected value. Let’s write down the change in your expected wealth, which is brought about by signing the insurance contract. Compared to your wealth before signing, you lose the fee and gain the expected loss (because you’re no longer exposed to it). In other words, your expected future wealth changes by the amount

Eq.1 \Delta E(x_{\text{buyer}}) = + E(L) – F

You get rid of the expected loss, which is a positive contribution to your future expected wealth, and you pay the fee, which is a negative contribution. However, similarly computing the change in expected wealth for the seller of insurance yields

Eq.2 \Delta E(x_{\text{seller}}) = – E(L) + F .

Compared with Eq.1, this is just the negative of the result for yourself. In general, we have the symmetry

Eq.3 \Delta E(x_{\text{buyer}}) = – \Delta E(x_{\text{seller}}) ,

which makes insurance a zero-sum game in terms of expected wealth. That’s the problem: whatever the seller gains in expectation is what the buyer loses. Whatever fee we choose, one of the parties always loses in expectation and therefore has no reason to sign the contract.

**The insurance puzzle:** *according to expected-wealth theory, insurance contracts should not exist.*

At this level of analysis, insurance is a murky business: it seems that the game is for one party to outsmart the other, misrepresent risks or their probabilities, or look for unsophisticated or outright irrational buyers or sellers to exploit.

Classical economics doesn’t have a good answer to this problem, or it agrees with the murky-business interpretation. The classic answers are “asymmetric information” (I know something you don’t) or “asymmetric risk preferences” (I’m more or less comfortable with risks than you are). The first answer boils down to that murky business, and the second is really just a restatement of what we’re trying to explain — the real question, though, is: why do people have different risk preferences?

In the case of insurance, the immediately important quantity to focus on is wealth: how does personal wealth behave over time? How might we model it? As so often, it’s helpful to use the model of multiplicative growth, where equal percentage changes in your wealth have fixed probabilities over time. This reflects the fact that if you have a lot of wealth, you can invest a lot and stand to gain or lose large dollar amounts, and if you have little wealth, well, then you can’t invest much. The returns on investments you can make – whether financial investments in the stock market or investments in your health, education, or living conditions – scale to some extent with the level of your wealth.

In multiplicative growth, the expected value is no good guide to the evolution of the wealth of an individual, as canonically exemplified by the infamous coin toss or explained in this video. Instead of expected wealth – because of the ergodicity breaking in multiplicative growth – time-average growth rates of wealth better capture what happens to the actual wealth of an individual over time.

## The ergodicity solution

Considering the insurance problem in terms of time-average growth rates solves the insurance puzzle, as detailed in arXiv1507.04655 (2015). While the symmetry in Eq.3 holds for expected wealth, it does not hold for time-average growth rates: both the buyer and the seller of insurance can gain over time by signing insurance contracts, even though this is not possible in the expected-wealth picture favoured by mainstream economic theory. Using the temporal perspective of ergodicity economics breaks the symmetry in Eq.3. Not because of asymmetric information, not because of irrational perspectives on risk, but simply because the agents may have different wealth situations.

## The buyer’s perspective

We can now revisit the insurance puzzle in terms of changes in time-average growth rates. For the buyer, we consider the time-average growth rate of the buyer’s wealth, both with insurance,

Eq.4 g_{\text{buyer}}^{\text{with}} = \frac{1}{\Delta t} \ln\left(\frac{x_{\text{buyer}}(t) – F}{x_{\text{buyer}}(t)}\right)

and without insurance,

Eq.5 g_{\text{buyer}}^{\text{without}} = \frac{1}{\Delta t} p \ln\left(\frac{x_{\text{buyer}}(t)-L}{x_{\text{buyer}}(t)}\right).

Both growth rates are always negative in our setup because the buyer’s wealth is guaranteed to drop, or stay unchanged in the best case. We can also see that the growth rates diverge negatively as either the fee or the potential loss approach the wealth of the buyer. This reflects the fact that the multiplicative dynamic we are investigating has a natural boundary at zero: losing everything in a multiplicative game means we cannot recover because even if we multiply zero with an arbitrarily large gain we will still stay at zero.

Next, we use these growth rates to compute the maximum fee at which it is still beneficial to buy insurance, F_{\text{buyer}}^{\text{max}}. This is the value where the time-average growth rate without insurance, g_{\text{buyer}}^{\text{without}}, equals the time-average growth rate with insurance, g_{\text{buyer}}^{\text{with}},

Eq.6 \underbrace{\frac{1}{\Delta t}\ln\left(\frac{x_{\text{buyer}}(t)-F_{\text{buyer}}^{\text{max}}}{x_{\text{buyer}}(t)}\right)}_{g_{\text{buyer}}^{\text{with}}}= \underbrace{\frac{1}{\Delta t}p \ln\left(\frac{x_{\text{buyer}}(t)-L}{x_{\text{buyer}}(t)}\right)}_{g_{\text{buyer}}^{\text{without}}}.

Rearranging and substituting from Eq.5, we find

Eq.7 F_{\text{buyer}}^{\text{max}}=x_{\text{buyer}}(t)\left[1-\exp(g_{\text{buyer}}^{\text{without}}\Delta t )\right].

Staring at this equation, let’s remember that g_{\text{buyer}}^{\text{without}} diverges negatively as the potential loss approaches the wealth of the insurance buyer. Eq.7 tells us that someone facing the potential loss of everything he owns is well advised to pay his entire wealth as an insurance fee even if the loss will only happen with a small probability. A fee far greater than the expected loss can be extracted from an individual facing an existential risk, and an individual in such dire straights is in a sense well advised to pay such a fee.

## The seller’s perspective

Next, we follow a similar argument to arrive at the minimum fee at which it is long-term beneficial to sell insurance, F_{\text{seller}}^{\text{min}}. Again, we write down the time-average growth rates with and without insurance, this time for the wealth of the seller

Eq.8 g_{\text{seller}}^{\text{with}}=(1-p)\frac{1}{\Delta t}\ln\left(\frac{x_{\text{seller}}(t)+F}{x_{\text{seller}}(t)}\right)+p \frac{1}{\Delta t}\ln\left(\frac{x_{\text{seller}}(t)+F-L}{x_{\text{seller}}(t)}\right)

and

Eq.9 g_{\text{seller}}^{\text{without}}=0 .

Equating Eq.8 and Eq.9, we again find the fee where signing the contract makes no different to the time-average growth rate and arrive at the expression

Eq.10 \left(\frac{x_{\text{seller}}(t)+F_{\text{seller}}^{\text{min}}}{x_{\text{seller}}(t)}\right)^{(1-p)}=\left(\frac{x_{\text{seller}}(t)+F_{\text{seller}}^{\text{min}}-L}{x_{\text{seller}}(t)}\right)^{(-p)}.

Eq.10 does not have a neat closed-form solution but is easily solvable numerically.

From the perspective of ergodicity economics, a mutually beneficial range of insurance fees exist, and agents optimizing time-average growth rates will sign insurance contracts, whenever

Eq.11 F_{\text{seller}}^{\text{min}} <F_{\text{buyer}}^{\text{max}}.

## A simple simulation

So does this actually work as an explanation for the emergence of insurance? In other words, do agents who use the time-average growth rate to decide whether to buy or sell insurance outperform agents who are trapped in expected-value thinking and only sign contracts which increase their expected wealth?

This question was recently addressed numerically in Annals of Actuarial Science (2023), 17, 215–218, and we reproduce the findings here. Agents are split into two groups with different responses to risk. Agents in group A sell and buy insurance contracts from each other when this increases their time-average growth rate. Agents in group B are fully informed rational expected-wealth optimizers and don’t sign insurance contracts as a result of the insurance puzzle. Group A acts according to the ergodicity-economics model, group B acts according to expected-wealth theory. For maximum simplicity, we have just 2 agents in each of the two groups, all starting with the same wealth, x(0) = 1 , and in each time step, we randomly choose one agent in each group to face a risk. To keep the comparison between the two groups as fair as possible, we make sure that if, for example, agent 1 of group A faces a risk, then it’s also agent 1 in group B who faces a risk. In the simulation below, you can set the risk that the agents face by changing the loss fraction — that’s the potential loss as a proportion of current wealth — as well as the loss probability. By default, the loss fraction is 0.95, so that L=0.95 x_i(t), and the loss probability is p=0.05, but feel free to play around with the parameters. Again, we keep the comparison between the groups fair: if agents 1 are facing a risk, then either agents 1 in both groups suffer a loss or neither does.

Results are shown in the app below. Hit “Simulate” a few times to get a feeling for the typical performance of agents in the two groups.

The first thing we notice is that over time, the expected wealth (red dashed line) is an unachievable fiction for all agents. Next, we see the main result, namely that when the risk is high, in the long run, the 2 agents (blue) who judge insurance by time-average growth and often set up contracts exponentially outperform the 2 agents (orange) who judge insurance by expected wealth and always reject it, despite the identical loss fraction and loss probability in both groups.

## Open systemic questions

Looking at these results systemically, some interesting questions emerge, which are the subject of our currently ongoing research. For example, rich entities tend to insure the poor ones creating a flow of wealth from poor to rich via the fees that are being paid. This is already interesting: without coercion, purely because it is long-time optimal for both parties, the poor pay the rich. This raises the question what kind of an ecology of rich and poor emerges. What sort of wealth distribution do these dynamics give rise to? When we look at the simulations above, it seems that Group A, where insurance contracts are signed, has less inequality than Group B, despite the flow of wealth from poor to rich. One effect we’ve observed is that while the richest agent can dominate for a long time (depending on the parameters of the simulation), it can never fully break away from the herd. As the richest agent begins to dominate systemically, it eventually runs out of willing insurers, and abrupt shifts in the wealth hierarchy can occur. This interplay between poor and rich is an interesting question relating to all kinds of real-world happenings, from financial crises to political revolutions. We will share what we find out about these questions here on the blog in the future, and of course, you are invited to run your own simulations and join us in our explorations.

References:

O. Peters, Insurance as an ergodicity problem. Annals of Actuarial Science (2023), 17, 215–218.

O. Peters and A. Adamou, Insurance makes wealth grow faster. arXiv1507.04655 (2015).

The code for the simulations in the AAS article can be downloaded here.

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