# Definitions

Simple Returns are defined as
$R^S_t = \frac{P_{t+1}}{P_t} - 1 = \frac{P_{t+1} - P_t}{P_t}$
Logarithmic Returns are defined as
$R^L_t = log (R^S_t + 1) = log (P_{t+1}) - log (P_t)$

# Main Properties

## Simple Returns

Simple returns are additive accross securities. If $(\omega_i)_{i=1 \ldots n}$ are the weights of different securities within a portfolio, with linear returns $(R^S_{t,i})_{i=1 \ldots n}$, the linear return of the overall portfolio is
$R^S_{t} = \sum_{i = 1}^n \omega_i R^S_{t,i}$

## Logarithmic Returns

Logarithmic returns are additive over time. If the logarithmic returns of a security over time are $(R^L_{t+i})_{i=0 \ldots T}$, the logarithmic return over the whole $[t, t+T]$ period is:
$R^L_{t->t+T} = \sum_{i = 0}^T R^L_{t+i}$
If a security follows a Brownian motion, its logarithmic returns are normally distributed.

## Difference

The Taylor series decomposition of log is as follows:
$log(1 + x) = \sum_{i=1}^\infty (-1)^{(n+1)}\frac{x^n}{n}, \qquad \forall x \in ]-1,1]$
So the error of the log-return vs. the simple return is in $o(|x|)$ for returns under 100%.
Typically, the error is less than 0.5% for returns below 10% in absolute value. However, it reaches several percentage points above 15% and more than 10% for returns above 50% or under -40%. The following 2 charts illustrates the extent of the error for small and large values.

## Moments

The mean of the log return can be approximated as follows: $\overline{R^L} \approx \overline{R^S} - 0.5\sigma^2_S$
The 2nd, 3rd and 4th moment are also affected.

# Conclusion

Whenever the terminal wealth is assessed or returns can be higher than a few percentage points, log returns can lead to significant estimation errors.

## References

• Hudson, Robert, Comparing Security Returns is Harder than You Think: Problems with Logarithmic Returns (February 7, 2010). Available at SSRN: http://ssrn.com/abstract=1549328
• Meucci, Attilio, Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management (May 1, 2010). GARP Risk Professional, pp. 49-51, April 2010 . Available at SSRN: http://ssrn.com/abstract=1586656
• Pini, Sergio, Approximations of Portfolio Returns: Are the Errors Really Small? (December 10, 2009). Available at SSRN: http://ssrn.com/abstract=1521442
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## One thought on “Linear vs. Logarithmic Returns”

1. Markus says:

Nice work. Very clear explanation. A valuable addition would be to explain the difference between “linear return”, “geometric return” and “geometric mean”.

My understanding (happy to be corrected if I’m wrong) is that, for example, looking at a series of three end-of-year prices, the geometric return pa is the same as the linear return over the two-year period (i.e. return over two years = P3/P1 – 1) annualised (i.e. annualised linear return = (P3/P1) ^ (1/2) – 1 = geometric return = (P2/P1 * P3/P2) ^ (1/2) – 1). The geometric mean, however, is not the same as the geometric return. If any of the annual returns (e.g. P2/P1 – 1) are negative, then the geometric mean is undefined. If any of the annual returns (e.g. P3 / P2 – 1) are zero, then the geometric mean is zero.

I’ve just finished looking at some university course notes (in an actuarial subject, so they should know better) where the terms “geometric return” and “geometric average” were used interchangeably. I also see this a lot on the internet. Of course, if my understanding is not correct, then I apologise profusely to whoever wrote the course notes.