We present a mathematical solution to the insurance puzzle. Our solution only uses time-average growth rates and makes no reference to risk preferences. The insurance puzzle is this: according to the expectation value of wealth, buying insurance is only rational at a price that makes it irrational to sell insurance. There is no price that is beneficial to both the buyer and the seller of an insurance contract. The puzzle why insurance contracts exist is traditionally resolved by appealing to utility theory, asymmetric information, or a mix of both . Here we note that the expectation value is the wrong starting point -- a legacy from the early days of probability theory. It is the wrong starting point because not even the most basic models of wealth (random walks) are stationary, and what the individual experiences over time is not the expectation value. We use the standard model of noisy exponential growth and compute time-average growth rates instead of expectation values of wealth . In this new paradigm insurance contracts exist that are beneficial for both parties .
Voluntary insurance contracts constitute a puzzle because they increase the expectation value of one party's wealth, whereas both parties must sign for such contracts to exist. Classically, the puzzle is resolved by introducing non-linear utility functions, which encode asymmetric risk preferences; or by assuming the parties have asymmetric information . Here we show the puzzle goes away if contracts are evaluated by their effect on the time-average growth rate of wealth. Our solution assumes only knowledge of wealth dynamics. Time averages and expectation values differ because wealth changes are non-ergodic. Our reasoning is generalisable: business happens when both parties grow faster .
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We present a mathematical solution to the insurance puzzle. Our solution only uses time-average growth rates and makes no reference to risk preferences. The insurance puzzle is this: according to
Voluntary insurance contracts constitute a puzzle because they increase
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wealth, buying insurance is only rational at a price that makes it irrational to sell insurance. There is no price that is beneficial to both the buyer and the seller of an insurance contract. The puzzle why insurance contracts exist is traditionally resolved by appealing to utility theory, asymmetric information, or a mix of both
one party's wealth, whereas both parties must sign for such contracts to exist. Classically, the puzzle is resolved by introducing non-linear utility functions, which encode asymmetric risk preferences; or by assuming the parties have asymmetric information
R
note that the expectation value is the wrong starting point -- a legacy from the early days of probability theory. It is the wrong starting point because not even the most basic models of wealth (random walks) are stationary, and what the individual experiences over time is not the expectation value. We use the standard model of noisy exponential growth and compute
show the puzzle goes away if contracts are evaluated by their effect on the
R
growth rates instead of expectation values of wealth . In this new paradigm insurance contracts exist that are beneficial for both parties
growth rate of wealth. Our solution assumes only knowledge of wealth dynamics. Time averages and expectation values differ because wealth changes are non-ergodic. Our reasoning is generalisable: business happens when both parties grow faster