How this calculator works
Expected utility weights the utility of each final-wealth outcome rather than applying utility to expected wealth. Square-root and logarithmic functions are concave and represent diminishing marginal utility in this illustration; linear utility is risk neutral. The certainty equivalent is the certain wealth giving the same utility. Values describe total wealth outcomes, not signed investment profits. Reverse modes solve a probability for exactly two distinct outcomes, or replace one specified outcome in a longer wealth list.
Formula
EU = sum(p_i u(W_i)); expected wealth = sum(p_i W_i); certainty equivalent = u^(-1)(EU); risk premium = expected wealth - certainty equivalent.
Worked example
Equal chances of $100 and $400 give square-root expected utility 15, expected wealth $250, certainty equivalent $225 and risk premium $25. With logarithmic utility, equal chances of $100 and $400 give a $200 certainty equivalent.
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Frequently asked questions
Why can expected utility differ from utility of expected wealth?
A nonlinear function treats changes in wealth differently at different wealth levels. Concavity makes the utility of the mean at least as large as mean utility.
Can I enter negative wealth?
These built-in functions use nonnegative wealth. Logarithmic utility needs strictly positive outcomes.
Are utility numbers measured in dollars?
No. Only expected wealth, certainty equivalent and risk premium have dollar units. Utility depends on the selected function.