Linear vs. Logarithmic Returns

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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