Three Returns, Three Questions
A return forecast, a CAPM benchmark, and an observed investment result answer different questions. Expected return asks what return is anticipated before the investment period, accounting for possible outcomes and their probabilities. Required return asks what return CAPM assigns as the appropriate expected-return benchmark for an asset’s systematic risk. Realized return records what the investor actually earned over a completed period.
These quantities may be numerically close, or they may differ substantially. Their interpretation depends on timing: expected and required returns are ex ante quantities, formed before the period’s outcome is known; a realized return is ex post. In the CAPM equilibrium model, an asset’s expected return equals its CAPM-required return. That statement concerns an expectation, not a guarantee that the asset will earn that return in every period.
The CAPM-required return is a model benchmark, not a forecast of the asset’s exact outcome. In the standard one-period CAPM equilibrium setting, expected returns are related linearly to beta, with the risk-free rate as the return for zero market exposure and the market risk premium \(E[R_M]-r_f\) as the premium for a beta of one. The model uses systematic risk, not the asset’s total volatility, to set this expected-return benchmark.
The benchmark is meaningful only when its inputs match the investment being considered. The risk-free rate and market return must use the same time horizon and return convention; beta must apply to the relevant asset or equity claim. A CAPM-required return for one year cannot be compared directly with a forecast for a month without converting the rates consistently. The calculations below use one-period annual returns unless stated otherwise.
Expected Return and CAPM-Required Return
Expected return is a probability-weighted average of possible returns. If a one-period return has possible values \(R_1,\ldots,R_n\), with probabilities \(p_1,\ldots,p_n\), then \(E[R]=\sum_{j=1}^{n}p_jR_j\), where the probabilities sum to one. This definition applies to a forecast distribution, whether or not the forecast is produced using CAPM.
The CAPM-required return is calculated from the model inputs, not by averaging an asset’s possible outcomes. In equilibrium, the model predicts \(E[R_i]=r_i^{\mathrm{req}}\). A separate analyst forecast may differ from the CAPM benchmark. That difference tells the analyst the forecast and benchmark do not coincide; it does not, by itself, establish what an eventual realized return will be.
Worked Example: Comparing a Forecast with the CAPM Benchmark
A hypothetical infrastructure fund has a one-year forecast distribution: a return of 20% with probability 0.50, or a return of \(-4\)% with probability 0.50. Its estimated beta is 1.20. The annual risk-free rate is 4%, and the expected annual market return is 10%. Calculate the forecast expected return and CAPM-required return, then compare them.
Method and assumptions. Treat the two outcomes as the complete one-year forecast distribution, with the stated probabilities. Use the one-period CAPM with annual rates and the stated beta. The probabilities total \(0.50+0.50=1.00\).
Step 1: Calculate the forecast expectation. Weight each return by its probability:
Step 2: Calculate the market risk premium.
Step 3: Apply CAPM.
The forecast expectation of 8% is 3.2 percentage points below the CAPM-required return of 11.2%. This comparison identifies a difference between the forecast and the CAPM benchmark; it does not tell us that the fund will realize an 8%, 11.2%, or any other particular return.
Check. The expected return, 8%, lies between the two possible outcomes, \(-4\)% and 20%, as it should. The CAPM premium over the risk-free rate is \(11.2\%-4\%=7.2\%\), equal to beta \(1.20\) multiplied by the 6% market risk premium.
Common mistake avoided: Do not call 11.2% the fund’s forecast expected return simply because CAPM produces it. It is the required-return benchmark under the model; the stated probability distribution gives a separate forecast of 8%.
Why Expected Return Is Not a Promise
An expectation summarizes a distribution; it does not say which value will occur. For a finite set of outcomes, the expectation follows from averaging each outcome in proportion to its probability. Repeating that weighting yields the formula above. A single investment period produces one outcome from the distribution, not the weighted average itself. Even if an asset’s expected return equals its CAPM-required return, the realized return can be above or below both.
This distinction is central to interpreting CAPM. The model’s equilibrium condition is about expected returns. It does not require \(R_i=r_i^{\mathrm{req}}\) in every period. Market movements, firm-specific events, and other sources of uncertainty affect the outcome actually observed. A single poor result is therefore not, by itself, evidence that the asset’s ex ante expected return violated the CAPM benchmark.
Worked Example: A Low Realization Despite an Expected Return at the Benchmark
A hypothetical renewable-energy security has annual return outcomes of \(-10\)% with probability 0.20, 10% with probability 0.50, and 30% with probability 0.30. Its beta is 1.50. The annual risk-free rate is 3%, and the expected annual market return is 9%. At the end of the year, the security realizes a return of \(-10\)%. Calculate all three returns and interpret the observation.
Method and assumptions. The three stated outcomes and probabilities define the one-year forecast distribution; probabilities sum to \(0.20+0.50+0.30=1.00\). Use the annual CAPM inputs for that same year. The observed \(-10\)% is the realized return.
Step 1: Find the forecast expected return.
Step 2: Find the CAPM-required return. The market risk premium is \(9\%-3\%=6\%\), so:
The forecast expected return equals the CAPM-required return: both are 12%. The realized return, however, is \(-10\)%. It is 22 percentage points below that expectation. This outcome is one of the possibilities in the stated distribution, assigned probability 0.20.
Check. The forecast probabilities sum to one, and the weighted outcomes sum to 12%. The CAPM calculation also gives 12%, using the stated risk-free rate, beta, and market risk premium.
Common mistake avoided: A realized loss does not show that the expected-return condition failed. Here, the given forecast expectation equals the CAPM benchmark even though the one-year result is negative.
Calculating a Realized Holding-Period Return
For a share held from the start to the end of one period, the realized total return includes both the change in price and any cash distribution received during the period. If the initial price is \(P_0\), the ending price is \(P_1\), and the distribution is \(D_1\), the holding-period return is \((P_1-P_0+D_1)/P_0\). It measures the total gain, including the distribution, as a proportion of the initial investment.
This formula follows by comparing the investor’s end-of-period wealth, \(P_1+D_1\), with the initial outlay \(P_0\). The gain is \((P_1+D_1)-P_0\); dividing that gain by \(P_0\) gives the return. Omitting a distribution understates the realized total return when the distribution is positive. The calculation also depends on the stated period: it is not automatically an annualized return if the holding period is shorter or longer than one year.
Worked Example: Distinguishing a Forecast, a Required Return, and a Share’s Realized Return
A hypothetical utility share has a one-year forecast distribution of 0% with probability 0.25, 8% with probability 0.50, and 20% with probability 0.25. Its beta is 1.20. The annual risk-free rate is 3%, and the expected annual market return is 8%. An investor buys one share for $80, receives a $4 dividend, and sells it for $88 one year later. Find the expected, required, and realized returns.
Method and assumptions. Use the stated probabilities as the one-year forecast distribution and apply one-period CAPM to the annual inputs. For the realized return, treat the $4 dividend as received during the same year and use the initial $80 share price as the investment.
Step 1: Calculate the expected return.
Step 2: Calculate the CAPM-required return. The market risk premium is \(8\%-3\%=5\%\). Therefore:
Step 3: Calculate the realized total return. The investor’s ending wealth from the share and dividend is \(88+4=\$92\). The gain over the initial $80 investment is \(\$92-\$80=\$12\). Thus:
The forecast expectation and CAPM benchmark both equal 9%, while the realized one-year total return is 15%. The realized return is 6 percentage points above the expected and required returns. This does not change what the ex ante forecast or model benchmark was.
Check. The price gain is $8 and the dividend is $4, for a total gain of $12. Dividing by the $80 initial price gives 15%. The expected return calculation gives 9%, as does CAPM.
Common mistake avoided: Using only the price change gives \((88-80)/80=10\%\), which omits the $4 dividend. The question asks for the total realized return, so the distribution must be included.
How to Interpret Differences
A careful interpretation names the quantity being compared and keeps its timing clear. A forecast expected return can be compared with a CAPM-required return to assess whether that forecast coincides with the model benchmark. A realized return can be compared with either to describe how the period turned out. The comparison alone does not make these quantities interchangeable.
| Quantity | When known | What it answers |
|---|---|---|
| Expected return \(E[R_i]\) | Before the period, based on a forecast or probability distribution | What is the probability-weighted return anticipated? |
| CAPM-required return \(r_i^{\mathrm{req}}\) | Before the period, from CAPM inputs | What expected-return benchmark does CAPM assign to this beta? |
| Realized return \(R_i\) | After the period | What return actually occurred? |
If expected and required returns differ, first check whether the forecast and CAPM use the same horizon, currency, and return basis. Then interpret the difference as a disagreement between the stated expectation and the CAPM benchmark. It does not reveal the future realization. If they are equal, the equality still does not promise that the asset will earn that amount in the period.
A single realized return cannot establish whether CAPM was satisfied because one outcome does not reveal the underlying expected return. A high realization can occur when an asset’s expectation equals its CAPM-required return, and a low realization can occur under the same condition. Assessing an expectation requires information about the distribution or forecast, along with the relevant CAPM inputs and assumptions; the observed outcome alone is insufficient.
Common Exam Traps
- Treating the required return as a guaranteed return. CAPM provides an expected-return benchmark, not a promise about the result of one holding period.
- Calling a realized return an expectation. The return observed after the period is one outcome, not a probability-weighted average of possible outcomes.
- Using CAPM to calculate an independently specified forecast. Calculate the expected return from the given outcome distribution; calculate the required return separately from the CAPM inputs.
- Comparing inconsistent periods. Match the risk-free rate, market expected return, beta interpretation, and asset return to the same horizon and return convention.
- Ignoring cash distributions. A total holding-period return includes both price change and distributions received during that period.
Check Your Understanding
For each problem, keep the forecast expectation, CAPM benchmark, and realized return distinct. All rates and outcomes refer to one year.
- A hypothetical data-storage firm has forecast returns of 4% and 10%, each with probability 0.50. Its beta is 0.80, the risk-free rate is 2.5%, and the expected market return is 7.5%. Find its expected and CAPM-required returns, and state their difference. (Answer: Expected return 7.0%; required return 6.5%; expected return is 0.5 percentage points higher.)
- A hypothetical port operator’s forecast outcomes are \(-5\)% with probability 0.20, 5% with probability 0.50, and 25% with probability 0.30. Its beta is 1.00, the risk-free rate is 3%, and the expected market return is 9%. If it realizes \(-5\)% this year, give the expected, required, and realized returns and explain what the realization establishes. (Answer: Expected return 9%; required return 9%; realized return \(-5\)%; the single outcome does not establish whether CAPM’s expected-return condition holds.)
- A hypothetical share has forecast returns of 0% with probability 0.25, 10% with probability 0.50, and 20% with probability 0.25. Beta is 0.75, the risk-free rate is 4%, and the expected market return is 10%. The share is bought for $50, pays a $1.50 dividend, and is sold for $54.50. Calculate all three returns. (Answer: Expected return 10%; required return 8.5%; realized return 12%.)
- A hypothetical materials company has forecast returns of 5% with probability 0.60 and 20% with probability 0.40. Its beta is 1.40, the risk-free rate is 4%, and the expected market return is 9%. The realized return is 20%. Calculate expected and required returns, then compare the realized result with each. (Answer: Expected return 11%; required return 11%; realized return 20%, which is 9 percentage points above both.)
- A hypothetical medical-device share has forecast returns of \(-10\)% with probability 0.20, 5% with probability 0.40, and 20% with probability 0.40. Its beta is 1.00, the risk-free rate is 2%, and the expected market return is 7%; the realized return is 5%. Calculate each return and identify whether the forecast equals the CAPM benchmark. (Answer: Expected return 8%; required return 7%; realized return 5%; the forecast exceeds the CAPM benchmark by 1 percentage point.)