# Certainty Equivalent

## What is 'Certainty Equivalent'

The certainty equivalent is a guaranteed return that someone would accept rather than taking a chance on a higher, but uncertain, return. To put it another way, the certainty equivalent is the guaranteed amount of cash that would yield the same exact expected utility as a given risky asset with absolute certainty.

## BREAKING DOWN 'Certainty Equivalent'

Investments must pay a risk premium to compensate investors for the possibility that they may not get their money back. If an investor has a choice between a U.S. government bond paying 3% interest and a corporate bond paying 8% interest, and he chooses the government bond, the payoff is the certainty equivalent. The company would need to offer this particular investor a potential return of more than 8% on its bonds to convince him to buy. A company seeking investors can use the certainty equivalent as a basis for determining how much more it needs to pay to convince investors to consider the riskier option. The certainty equivalent varies because each investor has a unique risk tolerance.

## Certainty Equivalent and Cash Flow

The idea of certainty equivalent can also be applied to cash flow. The certainty equivalent cash flow is the risk-free cash flow that an investor or manager considers equal to a different expected cash flow which is higher, but also riskier. The formula for calculating the certainty equivalent cash flow is as follows:

Certainty equivalent cash flow = expected cash flow / (1 + risk premium)

The risk premium is calculated as the risk-adjusted rate of return minus the risk-free rate. The expected cash flow is calculated by taking the probability-weighted dollar value of each expected cash flow and adding them up.

For example, imagine that an investor has the choice to accept a guaranteed \$10 million cash inflow or an option with the following expectations:

1. A 30% chance of receiving \$7.5 million

2. A 50% chance of receiving \$15.5 million

3. A 20% chance of receiving \$4 million

Based on these probabilities, the expected cash flow of this scenario is:

Expected cash flow = (30% x \$7.5 million) + (50% x \$15.5 million) + (20% x \$4 million) = \$10.8 million

Assume the risk-adjusted rate of return used to discount this option is 12% and the risk-free rate is 3%. Thus, the risk premium is (12% - 3%), or 9%. Using the above equation, the certainty equivalent cash flow is:

Certainty equivalent cash flow = \$10.8 million / (1 + 9%) = \$9.908 million.

Based on this, if the investor prefers to avoid risk, he should accept any guaranteed option worth more than \$9.908 million.