How to Calculate the Certainty Equivalent
The certainty equivalent is the guaranteed sum whose utility matches a gamble's expected utility, found by running that expected utility back through the utility function.
The Certainty Equivalent formula
Calculator
Enter the two payoffs, their probabilities and the utility exponent to get the certainty equivalent and the risk premium.
Wealth in the good state.
How often the good state happens. The two probabilities should total 100%.
Wealth in the bad state.
Chance of the bad state.
0.5 is the square-root utility used in most textbook problems. Set it to 1 for a risk-neutral person.
A guaranteed $625 feels exactly as good as this gamble, so any sure offer above it wins.
- Expected value of the gamble
- $700
- Expected utility
- 25
- Risk premium
- $75
- Attitude to risk
- Risk averse
Weighting the dollar payoffs gives $700, which is what the gamble pays on average.
The gamble scores 25 utils. That number is the input to the inversion, never the answer in dollars.
The gap between the average payout and the certainty equivalent is $75, the most this person would pay to shed the risk.
A certainty equivalent below the expected value is risk aversion, equal to it is risk neutrality, above it is risk loving.
How to calculate Certainty Equivalent, step by step
- 1Convert each payoff to utility. Apply the utility function to every possible wealth level the gamble can leave you with.
- 2Work out expected utility. Multiply each utility by its probability and add the products together.
- 3Invert the utility function. Solve utility of CE equals expected utility for CE. With the square root of wealth that means squaring the expected utility; with a natural log it means exponentiating it.
- 4Compare it with the expected value. Subtract the certainty equivalent from the expected value. The difference is the risk premium, the amount given up to be rid of the uncertainty.
- 5Read off the attitude to risk. Below the expected value is risk averse, level with it is risk neutral, above it is risk loving.
Worked example: Certainty Equivalent
A venture pays $1,600 if the launch works, which happens 25% of the time, and $400 if it does not. With utility equal to the square root of wealth the payoffs are worth 40 and 20 utils, so EU = 0.25 × 40 + 0.75 × 20 = 25. The wealth whose square root is 25 is 25 × 25 = $625, the certainty equivalent. Expected value is 0.25 × 1,600 + 0.75 × 400 = $700, so the risk premium is 700 − 625 = $75.
Certainty Equivalent questions
Is the certainty equivalent the same as expected utility?
No, and mixing them up is the usual lost mark. Expected utility is measured in utils; the certainty equivalent is measured in dollars. You get from one to the other by inverting the utility function, which turns an expected utility of 25 into $625 under square-root utility, not into $25.
How do you invert a utility function that is not a square root?
Undo whatever the function does. For utility = wealth^r, raise the expected utility to the power 1 ÷ r. For utility = the natural log of wealth, the certainty equivalent is e raised to the expected utility. The step is always the same idea: solve utility of CE equals expected utility for CE.
Why does the certainty equivalent sit below the expected value?
Because a risk-averse person's utility curve is concave, so losses in the bad state hurt more than equal gains in the good state help. That pulls expected utility below the utility of the average payoff, and inverting it lands on a dollar figure below the average payoff.
What does the risk premium tell you?
It is the most someone would pay to swap the gamble for a sure thing, which is why insurers can charge more than the expected loss and still find willing buyers. A larger premium means a more sharply bending utility curve and a more cautious decision maker.
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