The arithmetic mean return adds periodic returns and divides by their count: A = (1/N) Σrᵢ. It summarizes average periodic performance, but does not directly describe compounded wealth. section 5.2
For historical stock analysis, compute percentage price changes first, then apply pandas mean(). Multiplying a daily mean by 252 produces an annualized estimate. Missing first-period returns are ignored by mean(), although later calculations may require dropna(). section 3.1.1
Use the mean with these distinctions in mind:
- Compare it with the geometric average: arithmetic averaging treats returns additively, while compounded wealth depends on multiplying 1 + r for each period. section 5.2
- Account for volatility: a higher return variance increases the gap between arithmetic and geometric averages, approximately as G = A − ½σ². section 5.2
- Use arithmetic averages as inputs to return simulations, rather than substituting them directly into a compounded future-value calculation. section 5.13 section 5.2
- Treat historical means cautiously: optimization weights can be highly sensitive to expected returns, and the chosen historical period can materially change the estimate. section 3.1.1
Annualizing daily stock returns
For AAPL, FE, and WMT, calculate adjusted-close percentage changes and run stock_returns.mean()*252. The resulting historical annualized means are 30.7%, 10.0%, and 8.1%, respectively. section 3.1.1
Listing 5.3 Computing arithmetic and geometric average returns
import yfinance as yf
SP500 = yf.download('^GSPC', start='1971-12-31', end='2021-12-31') #1
SP500 = SP500.resample('Y').last() #2
SP500['ret'] = SP500['Close'].pct_change() #3
print('Arithmetic Average: ', SP500['ret'].mean())
print('Standard Deviation: ', SP500['ret'].std())
print('Geometric Average: ', (SP500['Close'].iloc[-1]/SP500['Close']
.iloc[0])**(1/(len(SP500)-1))-1)
print('Geometric Average Approximation: ', SP500['ret'].mean()
- 0.5 * (SP500['ret'].std())**2)
Compare arithmetic and geometric averages, standard deviation, and the volatility-based approximation using annualized S&P 500 data. section 5.2
