What Is Expected Return?
The expected return is the average profit or loss an investor anticipates from an investment over a given period. It’s a probabilistic forecast — usually based on historical returns or scenario probabilities — not a guaranteed outcome. In finance it’s used to set reasonable performance expectations, compare investments, and feed portfolio-construction and pricing models such as modern portfolio theory (MPT) and the Capital Asset Pricing Model (CAPM).
Key Takeaways
– Expected return = a weighted average of possible returns, where weights are probabilities (or portfolio weights).
– Common formulas: E(R) = Σ p_i·r_i (scenario-based), E(R_p) = Σ w_i·E(R_i) (portfolio weighted), CAPM: E(R_i) = R_f + β_i·(E(R_m) − R_f).
– Expected return measures central tendency (mean); standard deviation measures dispersion (risk). Both are needed to evaluate an investment.
– Limitations: depends on historical data and assumptions about probabilities/correlations; subject to estimation and model risk.
Expected Return Theory
– In decision theory and finance, expected return is the expected value of an uncertain payoff.
– In portfolio theory (MPT), expected return and variance/standard deviation of returns are the two inputs used to select efficient portfolios.
– In option pricing and asset pricing, expected returns are central to pricing models (e.g., CAPM, Black–Scholes under certain risk-neutral assumptions).
Formula(s)
1) Scenario-based expected return (expected value):
E(R) = Σ (p_i × r_i)
where p_i = probability of scenario i, r_i = return in scenario i.
Example: 50% chance of +20%, 50% chance of −10%:
E(R) = 0.5×0.20 + 0.5×(−0.10) = 0.05 = 5%.
2) Portfolio expected return (weighted average):
E(R_p) = Σ (w_i × E(R_i))
where w_i = weight of asset i in the portfolio.
Example: portfolio value $1,000,000 with Google 50% (E=15%), Apple 20% (E=6%), Amazon 30% (E=9%):
E(R_p) = 0.50×15% + 0.20×6% + 0.30×9% = 11.4%.
3) CAPM (expected return relative to market risk):
E(R_i) = R_f + β_i × (E(R_m) − R_f)
where R_f = risk-free rate, β_i = asset i’s beta, E(R_m) = expected market return.
4) Standard deviation (measure of dispersion; population form):
σ = sqrt( (1/n) Σ (x_i − x̄)^2 )
(If using sample data, STDEV.S in Excel uses n−1 in the denominator.)
Limitations
– Historical-biased: Using past returns to predict the future assumes stationarity. Markets change, regimes shift.
– Estimation error: Probabilities, expected market return, and beta are estimated with error — small errors can materially change results.
– Correlation instability: Portfolio expected return uses asset returns; but risk depends on correlations, which vary over time.
– Non-normal returns and tail risk: Expected value ignores skewness and kurtosis — rare extreme outcomes can dominate investor welfare.
– Ignores fees, taxes, and liquidity constraints unless explicitly included.
Example(s)
1) Simple probabilistic example (above): 50% +20% / 50% −10% ⇒ E(R)=5%.
2) Portfolio weighted example (above): 50%×15% + 20%×6% + 30%×9% = 11.4%.
3) Same mean, different risk example: two investments both have expected return 8% over five-year histories; Investment A has standard deviation ≈ 11.26% and Investment B ≈ 2.28%. Even with equal means, Investment A is ~5× riskier (more volatile).
How Is Expected Return Used in Finance?
– Investment selection and comparison: Compare expected returns across assets, but always paired with risk measures (SD, beta, downside risk).
– Asset pricing and required returns: Models (CAPM, multifactor models) use expected return = risk-free + risk premium(s).
– Portfolio construction: Expected returns weight assets and feed optimization routines (mean–variance optimization).
– Performance measurement: Expected returns help calculate risk-adjusted metrics (Sharpe ratio = [E(R) − R_f]/σ).
– Scenario and stress testing: Combine expected returns with probabilities to evaluate outcomes under different macro scenarios.
What Are Historical Returns?
– Historical returns are past realized returns for a security or index. Analysts use historical returns to estimate expected returns, volatilities, and correlations.
– Practical caution: Past performance is not a guarantee of future results. Historical estimates must be adjusted for structural changes, fees, or survivorship bias.
How Does Expected Return Differ From Standard Deviation?
– Expected return = mean (central tendency) of the return distribution.
– Standard deviation = dispersion around the mean; a proxy for volatility/risk.
– Two investments can have identical expected returns but very different standard deviations (different risk profiles). Risk-averse investors prefer higher expected return per unit of risk (e.g., higher Sharpe ratio).
Practical Steps — How to Estimate and Use Expected Return
1) Define the investment horizon and objective
– Short-term vs long-term affects which historical data and scenarios you’ll use.
2) Choose an estimation approach
– Scenario/probability-based: assign plausible scenarios and probabilities (useful for macro-driven assets).
– Historical average: take arithmetic mean of past periodic returns (good for year-to-year expectation) or geometric mean for compounded multi-period expectation.
– Model-based: use CAPM or multi-factor models to derive expected return given measured betas and factor premia.
– Market-implied: derive expected returns from option prices or dividend yields when possible.
3) Gather and clean data
– Use total return data (price changes + dividends) over an appropriate lookback. Adjust for corporate actions, splits, and survivorship bias.
4) Compute the expected return
– Scenario method: E(R) = Σ p_i·r_i.
– Historical mean: E(R) ≈ average(returns). In Excel: =AVERAGE(range).
– CAPM: E(R) = R_f + β·(E(R_m) − R_f). Estimate beta via regression of asset returns on market returns.
5) Estimate uncertainty and risk
– Calculate standard deviation: =STDEV.S(range) in Excel for sample SD.
– Compute correlation matrix if using a portfolio.
– Consider downside measures: semi-deviation, max drawdown, Value-at-Risk (VaR), CVaR.
6) Combine into portfolio expected return
– Weighted average: E(R_p) = Σ w_i×E(R_i). Don’t forget to account for expected costs (fees, taxes).
7) Translate to decision metrics
– Sharpe ratio, Information ratio, Sortino ratio, and expected utility frameworks help decide if the expected return justifies the risk.
8) Perform sensitivity and scenario analysis
– Test how results change with different R_f, market premium, betas, or scenario probabilities. Consider stress scenarios.
9) Use simulation if appropriate
– Monte Carlo simulations let you model path-dependent outcomes, compounding effects, and tail risks.
10) Re-assess and update
– Regularly update estimates as new data and information arrive. Rebalance portfolios consistent with changes in expected returns/risks.
Practical Excel/Tools tips
– Expected (historical) return: =AVERAGE(returns_range)
– Standard deviation: =STDEV.S(returns_range)
– Portfolio expected return: =SUMPRODUCT(weights_range, expected_returns_range)
– Regression beta: use =SLOPE(asset_returns_range, market_returns_range)
– Monte Carlo: many add-ins or VBA macros; or use Python/R for more flexibility.
The Bottom Line
Expected return is a core concept for valuing investments and building portfolios. It provides a simple summary measure of what you might reasonably expect to earn, but it must be paired with measures of risk (standard deviation, downside risk, correlations) and treated as an estimate — not a guarantee. Use multiple estimation methods, stress-test assumptions, and translate expected return into risk-adjusted metrics (Sharpe, Sortino) to support investment decisions.
Sources and Further Reading
– Paige McLaughlin, “Expected Return,” Investopedia. https://www.investopedia.com/terms/e/expectedreturn.asp
– University of Washington, Chapter 1, Introduction to Portfolio Theory.
– University at Albany, SUNY, Financial Economics: Black–Scholes Option Pricing.
If you want, I can:
– Calculate expected return and standard deviation for a dataset you provide; or
– Build an Excel template (or Python script) that computes expected returns, portfolio expected return, beta, and Sharpe ratio from historical price data. Which would you prefer?