Certainty Equivalents Explained: Balancing Risk and Return

What Is the Certainty Equivalent?

The certainty equivalent represents the amount of guaranteed money an investor would accept now rather than take a risk to get more money at a future date. It varies between investors based on their risk tolerance. A retiree could have a higher certainty equivalent because they may be less willing to risk their retirement funds after they stop working.

The certainty equivalent is closely related to the concept of risk premium, or the amount of additional return an investor requires to choose a risky investment over a safer one. It helps in comparing risk premiums and making investment decisions.

Insights Gained from Certainty Equivalents

Investments pay a risk premium to compensate investors for the chance they may not recover their money. Higher risk means an investor expects a higher premium over the average return.

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 differential is the certainty equivalent. The company must offer this investor a return of over 8% on its bonds to persuade them to invest.

A company can use the certainty equivalent to decide how much extra it must offer to attract investors to a riskier option. The certainty equivalent varies because each investor has a unique risk tolerance.

The term is also used in gambling, to represent the amount of payoff someone would require to be indifferent between it and a given gamble. This is called the gamble’s certainty equivalent.

Investment Applications of Certainty Equivalents: A Practical Example

The idea of certainty equivalent can be applied to cash flow from an investment. 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:

$\text{Certainty Equivalent Cash Flow} = \frac{\text{Expected Cash Flow}}{\left(1\ +\ \text{Risk Premium} \right )}$Certainty Equivalent Cash Flow=(1 + Risk Premium)Expected Cash Flow​

The risk premium is the risk-adjusted rate of return minus the risk-free rate. To find expected cash flow, multiply each cash flow by its probability and sum 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:

• A 30% chance of receiving $7.5 million
• A 50% chance of receiving $15.5 million
• A 20% chance of receiving $4 million

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

$\begin{aligned} \text{Expected Cash Flow} &= 0.3\times\$7.5\text{ Million}\\&\quad + 0.5\times \$15.5\text{ Million}\\&\quad + 0.2\times\$4\text{ Million}\\ &=\$10.8 \text{ Million} \end{aligned}$Expected Cash Flow​=0.3×$7.5 Million+0.5×$15.5 Million+0.2×$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:

$\begin{aligned} \text{Certainty Equivalent Cash Flow} &= \frac{\$10.8 \text{ Million}}{\left(1 + 0.09 \right )} \\ &=\$9.908 \text{ Million} \end{aligned}$Certainty Equivalent Cash Flow​=(1+0.09)$10.8 Million​=$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.

The Bottom Line

The certainty equivalent provides a guaranteed return alternative to potentially higher, but riskier, returns. Understanding the certainty equivalent can help investors to quantify their risk tolerance and consider guaranteed returns versus risky investments. Additionally, it can aid companies in determining the necessary return rates they must offer to persuade investors to opt for riskier options. The certainty equivalent applies to other fields as well, such as gambling.