Expected Utility Calculator

Inputs

Utility Type Choose constant relative or absolute risk aversion or linear (risk neutral).

Risk Parameter r For CRRA use r ≥ 0 (r = 1 uses ln). For CARA use a > 0.

Display Decimals

Outcome x	Probability p	Utility U(x)	Action
		–
		–
		–

Tip: Probabilities should sum to 1. Use Normalize Probabilities to automatically rescale them.

Results

Expected Value E[x] –
Expected Utility E[U(x)] –
Certainty Equivalent (CE) –
Risk Premium (E[x] − CE) –

Exports include the dataset plus summary metrics.

Example Data Table

Scenario	Outcome x	Probability p
Loss	50	0.20
Base	100	0.50
Gain	200	0.30

Formula Used

The expected utility of a discrete lottery with outcomes x_i and probabilities p_i is

E[U(x)] = Σ p_i · U(x_i)

Supported utility functions:

Linear: U(x) = x.
CRRA: U(x) = x^1−r/(1−r) for r ≠ 1, and U(x) = ln x when r = 1 (requires x > 0).
CARA: U(x) = −exp(−a x) with a > 0.

Certainty equivalent (CE) is the sure x such that U(CE) = E[U(x)]. We invert U to obtain:

Linear: CE = E[x].
CRRA: CE = ((1−r)·E[U])^1/(1−r) for r ≠ 1; CE = exp(E[ln x]) for r = 1.
CARA: With U(x)=−exp(−a x), CE = −(1/a)·ln(−E[U]).

Risk premium is RP = E[x] − CE. Positive RP indicates risk aversion.

How to Use

Select a utility type and set the risk parameter (r for CRRA, a for CARA).
Enter each outcome and its probability. Add or remove rows as needed.
Click Normalize Probabilities if p’s do not sum to 1.
Click Compute to calculate E[x], E[U], CE, and RP. The chart updates automatically.
Export the dataset and results via Download CSV or Download PDF.

Domain notes: CRRA requires strictly positive x. CARA accepts any real x. Choose parameters consistent with your application.

FAQs

1) What is the difference between CRRA and CARA?
CRRA models relative risk aversion constant across wealth scales, while CARA models absolute risk aversion constant regardless of wealth level.

2) What if my probabilities do not sum to 1?
Use the Normalize button to rescale them proportionally. Alternatively, adjust the values manually until the total equals 1.

3) Why is expected utility negative under CARA?
Because U(x) = −exp(−a x) is negative for all x. Comparisons use relative magnitudes rather than absolute level.

4) Why does CRRA require positive outcomes?
CRRA uses powers or logarithms of x, which are undefined for nonpositive values. Shift outcomes or choose CARA if your model includes losses.

5) What does risk premium mean?
It is the difference between the expected value and the certainty equivalent. Larger premiums indicate stronger aversion to risk.

Quick Tips

Edit rows directly. Utilities appear after compute.
Hover the chart to see point values.
Try r = 1 for CRRA to use log utility.
CE below E[x] implies risk aversion.

About

Designed for courses in economics finance and decision theory and for practitioners evaluating risky projects lotteries insurance and investments.