Should I Buy Insurance? Calculator

A Kelly-style framework for an old financial question

Insurance is not just about expected value. It is about surviving bad tails.

Sometimes you do not have a choice. Car insurance is often required by law, and home insurance is often required by mortgage lenders. In many other cases you do have a choice, but the decision gets made with habit, fear, or vibes rather than a quantitative framework.

The Kelly Criterion gives you one. At first glance, it might seem as if the answer should always be “no”. Insurance usually has a negative expected value: the premium exceeds the expected payout, otherwise the insurer would lose money. But people do not value money linearly. The first $1,000 matters far more when you only have $1,000 than when you already have $1,000,000. Once you account for that, some negative expected value insurance products are still rational to buy, while others are not.

Try the calculator See the intuition
Decision rule Choose the option with higher expected log wealth.
Core trade-off Small certain drag vs. large tail-risk reduction.
Best use case Risks that can meaningfully damage your balance sheet.

Why expected value is not enough

The same dollar means more when you have less of them.

Expected value asks whether the premium is lower than the expected loss. That is a useful pricing question, but it is not the same as a personal decision question. Kelly asks a different thing: which choice better preserves and compounds your wealth over time?

Log utility sharply penalizes large losses near the left tail of your balance sheet. That is why a small premium can be worth paying when the uninsured downside is severe relative to your wealth, and why the exact same policy can look wasteful to somebody much wealthier.

Practical reading: insurance is most attractive when the claim would move you to a much worse financial state, not merely when the expected loss is large in absolute dollars.

Visual 1

Diminishing marginal value of wealth

The curve steepens near low wealth.

Visual 2

Insurance narrows the outcome spread

You swap upside in the calm state for protection in the bad one.

The distribution view

Buying insurance usually lowers average dollars and raises resilience.

If the insurer is solvent, the premium must exceed the actuarial expected payout. That means insurance usually looks bad on pure expected value. But it can still improve your long-run financial trajectory if it meaningfully reduces the chance of landing in a painful or ruinous state.

This is exactly the kind of trade-off Kelly was built to reason about. It is not telling you to insure everything. It is telling you to insure the risks that matter to the survival and growth of your bankroll.

Buy more readily when the loss is large relative to liquid wealth.
Buy less readily when the premium mainly protects comfort, not solvency.
Skip the model entirely when insurance is required by law or contract.

Interactive calculator

Run your own insurance decision.

Enter your numbers below. Keep the units consistent. The default labels use dollars, but any currency works.

This is a simplified one-period model. It ignores taxes, claim friction, insurer default risk, and any legal requirement to buy the policy.

Kelly verdict

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Kelly max premium

Expected value gap

Claim-state wealth

Certainty-equivalent wealth

Visual 3

Log-wealth PDF (smoothed)

Kelly lives in log-wealth space, so the horizontal axis is logarithmic.

Model boundaries

When this framework is useful, and when it is not.

Useful for

Optional policies, deductibles, self-insurance decisions, and sizing protection against rare but material losses.

Too simple for

Correlated risks, business interruption, health shocks with many states, liability tails, and products with complex exclusions.

Important caveat

Kelly is a growth-and-survival framework. It is not a full measure of comfort, sleep quality, or the value of outsourced claims handling.