Simple returns and log returns

The return we talk about is usually the simple return,

\(r(t)=\frac{s(t)}{s(t-1)}-1\)

where s(t) is today’s price and s(t-1) is yesterday’s. Because the simple return of a portfolio is the weighted sum of the simple returns of the assets,

\(r_p(t) = \sum_{i=1}^{N} w_i(t) \, r_i(t)\)

quantitative equity research uses this definition of return — it aggregates across positions. Daily returns have a bell-shaped distribution, so simple returns are commonly modeled as normally distributed and joint distributions of asset simple returns as multivariate normal. A theoretical problem is that since stock investors have limited liability and stock prices are bounded at 0, the distribution of simple returns cannot be strictly normal, since that implies a nonzero probability of negative prices.

Another drawback of simple returns is that to aggregate them over time you must account for compounding. The simple return from time t to time t+2 is

\((1 + r(t+1)) * (1 + r(t+2)) - 1\)

not just the sum of the simple returns. For example, if a stock rises 10% for two days in a row, the cumulative simple return is 1.10^2 - 1 = 21%, not 20%. If daily simple returns are independent and normally distributed, yearly simple returns, which compound about 252 of these returns, will have positive skew and not be normally distributed.

To avoid negative stock prices and have returns that simply add over time, the Black-Scholes option pricing model and related models assume that log returns

\(r(t) = \log\!\left(\frac{s(t)}{s(t-1)}\right)\)

are normally distributed. This means today’s price s(t) is related to yesterday’s price and the log return as

\(s(t) = s(t-1) \, \exp\!\big(r(t)\big)\)

Since the exponential of minus infinity is 0, stock prices are bounded at zero, and there is no probability of negative prices, as there was when simple returns were modeled as normally distributed. Another nice feature of log returns is that they are additive over time. The log return from 2 days ago to today is the sum of today’s log return and yesterday’s:

\(\log\!\left(\frac{s(t)}{s(t-2)}\right) = \log\!\left(\frac{s(t)}{s(t-1)}\right) + \log\!\left(\frac{s(t-1)}{s(t-2)}\right) \)

The problem with using normally distributed log returns to model stock indices is that empirically, log returns scaled by exponentially weighted (RiskMetrics) volatility are even more non-normal than simple returns are, as shown by the statistics below for daily returns of SPY from 1993-02-02 to 2025-10-22,

\(\begin{array}{lrr} \textbf{Return type} & \textbf{Skew} & \textbf{Excess kurtosis} \\ \text{log} & -0.662 & 3.145 \\ \text{simple} & -0.602 & 2.987 \end{array} \)

For the normal distribution skew and excess kurtosis are zero. On GitHub, at ReturnDistributions I fit various probability distributions to returns, finding that distributions with heavier tails than the normal, such as Student’s t, better fit the distribution of returns. At Non-lognormal Option Pricing I show that if the simple returns until expiration are modeled with a heavy-tailed distribution, you get Black-Scholes implied volatility curves that resemble those in the SPX options markets.