Expected Return

An expected return is the mean of the possible returns on an investment over a stated period, each weighted by its probability. It describes what an investor anticipates on average, assessed in advance, and it says nothing on its own about how widely the actual result may differ. Economists treat it as the unobservable quantity behind risk and pricing models: the equity risk premium, for example, is defined as the expected return on stocks in excess of the risk-free rate. The phrase has a separate, technical meaning in federal tax law, where "the expected return under the contract" in Internal Revenue Code section 72 is the total of the payments an annuity is expected to pay, used to work out how much of each payment is taxable.

A forecast, not a measurement. The return an investment actually earned can be read off a statement afterwards; the expected return cannot be observed at all, even in hindsight. It has to be estimated, and estimates differ. A 2015 Federal Reserve Bank of New York staff report on the equity risk premium, which is an expected-return concept, did not settle on one method: it combined twenty models used by practitioners and academics, and it noted that expected returns are themselves random over time "since expectations depend on the arrival of new information that has a random component not known in advance." An expected return is therefore always someone's estimate, made on a date, for a horizon.

How estimates are usually built. Three approaches recur. The first weights explicit scenarios by their probabilities, which is the definition applied directly. The second adds up building blocks: a risk-free rate, usually a Treasury yield for the same horizon, plus a risk premium the investment is thought to command. The capital asset pricing model is the best-known version, setting the premium by the asset's beta, although Eugene Fama and Kenneth French described the model's empirical record in 2004 as "poor enough to invalidate the way it is used in applications." The third uses long-run historical averages, which tell you what happened rather than what will, and shift depending on the years chosen.

The average that does not compound. An expected return is usually stated as an arithmetic mean: the average of the possible one-year results. Money, however, grows at the compounded (geometric) rate, and whenever returns vary from year to year the compounded rate is lower than the arithmetic average. The gap widens as volatility rises. That matters in planning: a projection that grows a balance at an arithmetic average return will tend to overstate the balance a volatile portfolio typically reaches, even though it matches the average across all possible outcomes, which a few very good results pull up. The compounded figure is what an annualized return measures after the fact.

What it leaves out. Two investments can share an expected return and carry very different risks, which is why expected return is always read alongside a measure of dispersion such as standard deviation, and why modern portfolio theory frames investment choice as a trade-off between the two. Expected returns are also usually quoted as nominal figures before inflation and before fees and taxes, each of which lowers what the investor keeps.

Computing an expected return from scenarios takes three steps: list the possible outcomes, assign each a probability that adds up to 100%, then multiply each outcome by its probability and add the results.

Consider an example with made-up figures for a one-year holding. Suppose an analyst assigns a 30% chance of a 10% loss, a 50% chance of an 8% gain and a 20% chance of a 25% gain. The expected return is (0.30 × −10%) + (0.50 × 8%) + (0.20 × 25%), which is −3% + 4% + 5%, or 6%. In none of the three scenarios does the investment actually earn 6%; the figure is the center of the range, not a likely result.

The difference between an average and a compounded return shows up over two years. Take $10,000 that gains 50% in year one and loses 50% in year two. The arithmetic average of the two years is 0%. The money goes from $10,000 to $15,000 and then to $7,500, a 25% loss over the two years, which works out to about −13.4% a year compounded (the square root of 0.75 is about 0.866). A planner using the 0% average would have overstated the ending balance by $2,500.

Pros

• Gives a single, comparable figure for what an investment is anticipated to earn on average.
• Makes the trade-off between reward and risk explicit when read alongside a measure of dispersion.
• Is a necessary input to retirement projections, savings targets and simulations.

Cons

• Cannot be observed, only estimated, and reasonable methods give different answers.
• Says nothing about how bad an individual year can be.
• Stated as an arithmetic average, it overstates how fast a volatile portfolio compounds.
• Is easily mistaken for a promise, especially when a single figure drives a long-range plan.