Not investment advice. Educational reading. See Disclaimer.
L.9 · INTERMEDIATE · 3 MIN
Jensen's Inequality and Precautionary Saving
Why does a lifelong investor care about Jensen's inequality? Because it is the mathematical foundation of every risk-averse decision you make -- from buying insurance, to holding bonds alongside stocks, to maintaining a cash buffer instead of investing every dollar. The same curvature that bends utility downward at the margin is what makes households save against rainy days, what makes risk premiums positive in capital markets, and what explains why a certain return often beats a higher expected uncertain return.
Jensen's inequality: for a concave function f, E[f(X)] <= f(E[X]). In plain English, the expected utility of a random outcome is less than the utility of the expected outcome. The gap is the cost of uncertainty -- and the size of that gap is exactly what makes you willing to pay for insurance, accept a lower return for safety, and save more when the future is uncertain.
§ 02How utility shape drives investor behavior
Utility Shape
Investor Behavior
Real-World Example
Concave (risk-averse)
Declines fair gambles; pays for insurance; holds precautionary cash
Most households; most institutional investors with liability constraints
Linear (risk-neutral)
Indifferent to fair gambles; never pays insurance premium; holds no precautionary buffer
Rare in practice; sometimes assumed for very large, well-diversified funds
Convex (risk-loving)
Accepts fair gambles; pays for the chance to gamble; under-saves
Lottery players and certain speculative traders -- usually inconsistent with long-horizon investing
§ 03Worked example: the certainty equivalent gap
Worked example with round fictional numbers. Suppose your utility of wealth is the square root function, and you have $100M of wealth. A fair coin flip would pay you +$36M or -$36M with equal probability. Expected wealth after the flip is still $100M, but expected utility is 0.5 sqrt(64) + 0.5 sqrt(136) = 0.5 8 + 0.5 11.66 = 9.83. That is LESS than sqrt(100) = 10, the utility of certain wealth. The wedge -- 10 minus 9.83 -- is the certainty equivalent gap. You would rather have around $96.6M for sure than the fair gamble. That difference is what you would pay to avoid the uncertainty.
§ 04Recognize your own precautionary saving
Pick a year of your own life when income was unusually uncertain (job loss, business launch, between contracts). Did your spending fall and your saving rise compared with a stable-income year? That instinct is precautionary saving in action -- a behavioral echo of Jensen's inequality you are already running.
§ 05What makes precautionary saving rise
Precautionary saving rises with TWO things: the variance of expected income AND the degree of concavity (technically, the third derivative of utility, called prudence). A household facing 20 percent income variance with high prudence saves dramatically more than the same household facing 5 percent variance. This is why portfolio cash buffers should grow when career or business income becomes lumpier -- not because returns on cash improved, but because the wedge between expected utility and utility of expected wealth widened.
§ 06Jensen's inequality and the equity risk premium
Jensen's inequality is also why the risk-free rate is structurally lower than the expected return on risky assets. Concave-utility investors will pay a premium for certainty -- that premium is the equity risk premium. Without Jensen, there is no reason a riskless Treasury yield should sit below the expected return on stocks; with Jensen, a positive gap follows naturally from risk aversion. Treat the DIRECTION as theory and the SIZE as an open empirical question — the observed premium is famously larger than curvature alone predicts (the Mehra-Prescott 'equity premium puzzle'), because the premium compensates covariance with bad times, not concavity by itself.
Check your understanding
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Sit with the ideas.
Two investors face identical wealth of $100,000. Investor X has a concave utility function (diminishing marginal utility of wealth); Investor Y has a linear utility function (constant marginal utility). Both are offered a fair gamble that doubles wealth or zeroes it with equal probability. Which investor will accept, and what does this reveal about precautionary saving in households facing uncertain future income?
