The Math Of Insurance

As a responsible and increasingly decrepit middle aged man I’ve been thinking about insurance lately. In particular about what it should cost.

The internet is less than helpful: some folks have realized that mathematically the insurance company has to make money in expectation. Thus, they claim, it’s a terrible deal for you. We’ll see that this is both true and false: you will lose money in expectation and you will like it.

Let’s assume pricing is roughly fair

For the remainder of the discussion we’re going to assume that insurance pricing is actuarially accurate plus some markup. This is an assumption and is likely not true for any particular person.

Adverse selection is the most obvious reason this is not accurate: if you tend to be particularly reckless you actually get a better deal on insurance. There’s some ability for insurers to counteract this (smokers pay more for health insurance, your car insurance rates go up when you have an accident or are a teenager) but there’s limits to what they can consider. Note that getting a better deal on insurance means more bad things happen to you so this is a dubious benefit. But this still stings for the responsible among us who implicitly subsidize the reckless.

But, never fear, we do get the last laugh. Because in practice certain types of insurance can actually be positive in expectation if optimally executed. Matt Levine writes hilariously about the asset management firm Apollo buying up life insurance on secondary markets. Why would they take the supposed losing end of this wager? You could imagine they were trying to take advantage of risk mispricing: finding folks who are more likely to die sooner than actuarially forecast. Turns out you don’t have to be that clever: almost anybody will do. A shockingly large proportion of life insurance ends up not paying out due¹ to failure to pay premiums. Similarly the forms and procedures for making claims are tailored to the diligent among us and some folks just don’t follow through. Companies know this and thus can offer below break-even pricing. Thus if you just manage to pay your premiums you might be getting a favorable deal.

Of course it’s quite hard to tell which of these effects holds stronger in any particular case. So we’re just going to assume a relatively fair actuarial price with a markup.

Digging into the math

So now the question becomes at what markup do you stop buying insurance. Some on the internet would argue 0 which is inconsistent with insurance being a viable market. Thus there has to be holes in this argument.

It turns out we can put this is terms that even a redditor understands: going uninsured on an asset gives you the exact same return distribution as writing a naked out-of-the-money put². To recall, a put is an options contract which gives another party the right to buy an asset from you at a certain strike price. But, of course, you can usually just buy the asset from the market so to be meaningful the price has to differ from the market price. And since folks probably don’t want to buy the asset at a higher price it has to be lower. You’ll want to be compensated for potentially having to sell at that price in the form of an upfront premium.

How much lower determines the premium. If prices fall substantially below the put’s price then as the seller you stand to lose a lot of money. But of course it becomes very unlikely at some point for the price to drop so low. Thus the premium drops in those case: these are called out-of-the-money since they’re so far away from the market price.

Lots of folks equate “very unlikely” with “impossible” and thus consider these premiums free money. Every now and again they realize their mistake after losing a ton of money: thus the idiomatic characterization of “picking up pennies in front of a steamroller”.

What’s interesting to us is the shape of these returns: to some approximation we can make it bimodal

\[ R = \begin{cases} \text{premium} & \text{probability } 1-p \\ \text{premium} - \text{cost} & \text{probability } p \end{cases} \qquad \text{premium} \ll \text{cost} \]

as you either win a small amount or lose a huge amount with some low probability. This is the same payout shape as not buying insurance!

So supposing we believe that you shouldn’t buy insurance with any markup then we can also assume that you shouldn’t buy options at any markup. This is a much harder claim to make as very sophisticated firms buy and sell options all the time: these aren’t financially naive rubes getting exploited by insurance salespeople. There must be something else going on here.

Your baseline matters

Entropic Thoughts has a great post on When Is Insurance Worth It which inspired this post. He makes the point that you generally shouldn’t care about individual returns but rather aggregate returns and frames insurance as a Kelly criterion bet.

One way to derive the Kelly criterion is a log utility function³. Instead of caring about every incremental dollar equally, we tend to care less and less about incremental dollars as we go. If this takes the form of a log function (among others) we then can derive our expected utility as

\[ E[U \mid \text{insured}] - E[U \mid \text{not insured}] = \log(W-P) - \Big[(1-p)\log W + p\log(W-c)\Big] \]

with current wealth \(W\), total premium \(P\), probability \(p\) the insured event occurs, and its uninsured cost \(c\), the value of insuring is the gap between the certain log-wealth of paying the premium and the expected log-wealth of going without:

Since \((1-p)\log W + p\log(W-c) = \log W + p\log(1-c/W)\) and \(\log(W-P) = \log W + \log(1-P/W)\), the \(\log W\) terms cancel and buying is worth it exactly when

\[ \frac{P}{W} < 1 - \left(1-\frac{c}{W}\right)^{p} \]

But by Bernoulli’s inequality, \((1-c/W)^p \le 1 - p\,c/W\), the right-hand side is always at least \(p\,c/W\) which is just the expected value of insurance! An individual with a log utility should be willing to pay more than fair value for insurance and still come out ahead (in utility terms).

Writing the premium as a markup over fair value, \(P=(1+\pi)pc\), the buying condition becomes \((1+\pi)\,pc/W < 1-(1-c/W)^p\), so the maximum tolerable markup is

\[ \pi_{\max} = \frac{W}{pc}\left[1-\left(1-\frac{c}{W}\right)^{p}\right] - 1 \]

which by the Bernoulli bound above is always \(\ge 0\).

Note that this depends upon the fraction of your wealth that the loss represents. When the loss is miniscule \(\pi_{\max} \rightarrow 0\) quite rapidly which supports the general consensus that small value “extended warranties” are mostly a waste of money.

This solves the general mystery but let’s dig further.

Distributions matter

For instance our analysis above didn’t take the shape of the returns into account!

Generalize the fixed cost \(c\) to a random loss \(L \ge 0\) and the earlier expression becomes

\[ E[U \mid \text{insured}] - E[U \mid \text{not insured}] = \log(W-P) - \mathbb{E}[\log(W-L)] \]

which reduces to the two-point case above when \(L=c\) with probability \(p\) and \(L=0\) otherwise. Since \(\log\) is concave, Jensen’s inequality gives

\[ \mathbb{E}[\log(W-L)] \le \log(W - \mathbb{E}[L]) \]

There is a complication however when we consider very large losses. You can consider two types of insurance: bounded and unbound. Bounded losses are capped at some \(M\) (say your car’s cash value). However bad the distribution of \(L\) gets within \([0,M]\), its damage to \(\mathbb{E}[\log(W-L)]\), and so the value of insuring it, is capped by a constant fixed by \(M/W\).

Liability coverage though has no such cap: a judgment for someone else’s medical bills, lost income, or wrongful death can run arbitrarily high relative to your net worth. Indeed it can go beyond your net worth and there lies the problem: whenever there’s any probability \(q=P(L\ge\ell)>0\) of a claim at or beyond some level \(\ell\):

\[ \mathbb{E}[\log(W-L)] \le q\log(W-\ell) + (1-q)\log W \xrightarrow[\ell\to W]{} -\infty \]

so \(E[U \mid \text{insured}] - E[U \mid \text{not insured}]\to\infty\): which implies that no finite premium is too high to pay for your insurance. Note this doesn’t depend on the mean, this is purely a property of the tail risk: log utility goes to negative infinity as it goes close to bankruptcy and is undefined afterwards.

This of course makes no sense. We need a more realistic⁴ utility function like a power utility:

\[ U(W) = \frac{W^{1-\eta} - 1}{1 - \eta} \]

which for \(\eta < 1\)⁵ degrades to a finite loss as \(W \rightarrow 0\). Unfortunately for us folks tend to set \(\eta > 1\) as we turn out to be quite risk-averse.

We can save this by adding another realistic twist: a court can’t collect more than you have, and some assets (retirement accounts, a homestead exemption, future earning power) are typically exempt from collection entirely. Thus we can model that as a floor \(w_{\min}\ge0\) your wealth can’t be pushed below:

\[ W_{\text{eff}} = \max(W-L,\ w_{\min}) \]

This gives us an intuitive finding that the previous model can’t: some folks are judgement proof. Redoing the liability calculation above with \(W_{\text{eff}}\) in place of \(W-L\):

\[ \mathbb{E}[U(W_{\text{eff}})] = \mathbb{E}\big[U(W-L)\,\mathbb{1}_{L<W-w_{\min}}\big] + U(w_{\min})\cdot P(L\ge W-w_{\min}) \]

Writing \(A=\max(W-w_{\min},0)\) for the wealth actually at risk, this is equivalent to \(W_{\text{eff}} = W-\min(L,A)\): once you’re at the floor, additional liability costs nothing more, so the loss is effectively bounded at \(A\) no matter how large or fat-tailed \(L\).

As long as \(w_{\min}>0\) (or \(\eta<1\) if you insist on \(w_{\min}=0\)), \(U(w_{\min})\) is finite, so this stays bounded below no matter how fat the tail of \(L\) is or how much probability mass sits beyond \(W\). Being wiped out is bad but it isn’t infinitely bad.

With this generalization we find that buying insurance is worth it whenever

\[ P < W - \Big(\mathbb{E}\big[W_{\text{eff}}^{1-\eta}\big]\Big)^{1/(1-\eta)} \]

i.e. whenever the premium is less than the gap between actual wealth and the certainty-equivalent wealth.

Fortune favors the brave

All of this has been assuming a relatively static state of the world. However instead of individual bets we need to consider the portfolio⁶.

Buying insurance can free up capital for productive purposes: you no longer need to keep a large buffer of riskless investments just to self-insure.

You can naturally see this with liability insurance. Suppose you are a doctor and it’s conceivable that you have some small chance of being sued for malpractice for \$1 million. That would be disastrous so you either pay for malpractice insurance or self-insure by setting aside \$1 million to cover it. That’s a ton of money to just have sitting around for a low probability event.

Why does self-insuring require holding nearly all of \(S\) in the first place, rather than some smaller cushion (that ideally grows)? You don’t know when you’re going to get sued so any reserve \(r < S\) still leaves you short if you get sued (which we say happens with probability \(p\)). This is true even if you want less than certain protection \(p_{survive} > 1 - p\): you still have to reserve the entire \(S\) to achieve that rate. The same thing occurs if you invest in risky assets: you’re subject to sequence-of-returns risk if you have a down market at the same time you have to pay out.

An insurer, on the other hand, is pooling \(n\) basically independent⁷ copies of the same risk so they don’t have to reserve as much. For them the central limit theorem starts working so a reserve of

\[ R_n = npS + zS\sqrt{np(1-p)} \]

covers claims with probability \(\approx\Phi(z)\) for whatever tolerance \(p_{survive}\) the insurer targets. Per policy that’s \(R_n/n = pS + zS\sqrt{p(1-p)/n} \to pS\) as \(n\to\infty\): by pooling the insurer’s reserve requirements approach the actuarial expectation. With a pool of \(n=10{,}000\) policies, and \(z=3\) (\(p_{survive}\approx99.9\%\)), per-policy reserve is \(\approx S(0.05+3\sqrt{0.0475/10{,}000})\approx0.057S\) about a 18x reduction. Thus even if the insurance company had a log utility (which it doesn’t) both parties would still have a price where they can make this bet.

This is the cool or perhaps disconcerting thing about modern finance: just by restructuring things you can create value out of seemingly nothing by shifting around abstract values of risk.

As an individual you now benefit because instead of keeping your reserve you can transfer that risk to the insurance company. This, in turn, frees you up to take more risks! Instead of sitting in safe assets that self-insurance fund could be put into higher returning assets.

Model both options acting on a pot of capital \(S\), re-upped every year: a growth portfolio compounding at \(g\), a safe reserve compounding at \(r_f<g\), and an annual probability \(p\) of a malpractice suit that year. Take \(g=7\%\) and \(r_f=2\%\) real — plausible long-run figures for equities versus a safe reserve — a \(p=5\%\) annual chance of a malpractice suit, and a markup \(\pi\) over fair value, so the annual premium is \(P=(1+\pi)pS\).

Self-insuring parks the whole \(S\) in the safe reserve for the year, certain to cover the payout and remain solvent:

Writing \(\mathbb{1}_{\text{sued}}\) for the Bernoulli(\(p\)) indicator that you’re sued this year,

\[ \text{Self-insure} = S(1+r_f) - S\cdot\mathbb{1}_{\text{sued}}, \qquad \mathbb{E}[\text{Self-insure}] = S(1+r_f) - pS \]

Insuring frees \(S-P\) for the growth portfolio with no uncertainty:

\[ \text{Insure} = (S-P)(1+g) \]

Subtracting and dividing by \(S\):

\[ \frac{\text{Insure}-\mathbb{E}[\text{Self-insure}]}{S} = \big(1-(1+\pi)p\big)(1+g) + p - (1+r_f) \]

Solving for the breakeven markup we have

\[ \pi_{\max} = \frac{1}{p}\left[1-\frac{(1+r_f)-p}{1+g}\right] - 1 \approx 0.87 \]

so on a year-by-year basis insuring stops paying off above roughly an 87% markup for this stylized model.

Interestingly this doesn’t involve \(S\): this is purely about opportunity cost because we’re considering the expected value. Switching back to power utility we can rework this to find

\[ \pi_{\max}(x) = \frac{1}{px}\left\{1-\Big[(1-p)+p(1-x)^{1-\eta}\Big]^{1/(1-\eta)}\right\}-1 \]

which reduces to the log-utility formula above in the limit \(\eta\to1\), but stays finite at \(x=1\) for any \(\eta<1\). Taking \(\eta=0.5\) as an illustrative choice:

\(S/W\)	10%	50%	90%	100%
\(\pi_{\max}\)	2.5%	16.3%	49.4%	95%

So in addition to the capital-efficiency argument you also benefit from the convexity of utility and start accepting higher and higher markups the larger the potential loss is relative to your assets.

This only works this cleanly because the payout and the capital’s return are close to statistically independent. For many risks this is true: A crashing market doesn’t make you more likely to die⁸, and dying doesn’t crash the market⁹. I’m not sure about malpractice insurance: folks might be more inclined to sue in down markets and you might be more likely to make mistakes if you’re stressed about finances. In that case you’d need to account for the correlation, but that direction just makes insurance more attractive as the loss arrives right when your ability to pay it is diminished. Of course the best case would be if you had anticorrelated losses and returns where your ability to pay waxes at the very moment you need it most.