Expected return is the weighted-average outcome you anticipate earning on an asset or portfolio over a stated horizon—usually expressed as an annual percentage. It is the "gravity-center" of the distribution of possible future returns.
For a discrete distribution of possible returns with probabilities :
For a continuous distribution :
In practice we rarely know the true distribution, so we estimate from historical data or a forward-looking model.
| Method | Core Idea | Strengths | Drawbacks |
|---|---|---|---|
| Historical Mean | Average of past returns: . | Simple, transparent, data-driven. | Sensitive to sample window, ignores regime changes. |
| Exponentially-Weighted Mean | Recent returns get higher weight. | Reacts to trends. | Still backward-looking; picks decay factor heuristically. |
| CAPM Implied Return | . | Links return to priced market risk. | Relies on stable β and market premium. |
| Black–Litterman | Blends market-implied returns with subjective views. | Controls estimation error; consistent with equilibrium. | Requires equilibrium priors & subjective confidence. |
| Fundamental Factor Models | Expected return = sum of factor exposures × premia. | Incorporates earnings growth, value, momentum, etc. | Needs robust factor premia forecasts. |
Inside our implementation, PyPortfolioOpt's helper is called:
from pypfopt import expected_returns
mu = expected_returns.mean_historical_return(df, frequency=252)
Data: price level DataFrame df (already adjusted for splits & dividends).
Frequency: 252 → converts daily mean to annualised arithmetic mean:
Result: vector feeds directly into the Efficient Frontier optimiser and is also cached for reporting.
When viewing optimization results, you'll see the Expected Return (expressed as an annual percentage) displayed alongside Volatility and Sharpe Ratio. Keep these important points in mind:
Expected return is a statistical estimate, not a guarantee—it represents the center of the probability distribution of possible outcomes.
When comparing different optimization methods, focus on risk-adjusted metrics (like Sharpe or Sortino ratios) rather than maximizing expected return alone, as higher returns typically come with increased risk.
The expected return value serves as a key input for calculating various performance metrics (such as Treynor Ratio and Information Ratio) that help evaluate portfolio efficiency.
Markowitz, H. "Portfolio Selection." Journal of Finance (1952).
Black, F. & Litterman, R. "Global Portfolio Optimization." Financial Analysts Journal (1992).