## Log Return The arithmetic return is defined as ${tex`\displaystyle r_a = \frac{p_{t+1} - p_t}{p_t}`}. It is expressed as a percentage.
The geometric return is defined as ${tex`\displaystyle r_g = \frac{p_{t+1}}{p_t}`}. It is expressed as a unitless ratio.
We observe that:
${tex`\displaystyle r_g = \frac{p_{t+1}}{p_t} = \frac{p_{t+1} - p_t + p_t}{p_t} = \frac{p_{t+1} - p_t}{p_t} + \frac{p_t}{p_t} = r_a + 1`}.
We also observe that ${tex`-100\% ≤ r_a < +\infty\enspace`} while ${tex`\enspace0 ≤ r_g < +\infty`}.
We denote the log return as ${tex`\lambda = \ln (r_g)`} and the series of log returns as ${tex`\Lambda`}.
This allows us to reformulate ${tex`r_g`} as ${tex`e^\lambda`}:
${tex`p_{t+1} = e^\lambda · p_t`}
This in turn allows us to express ${tex`p_{t+1}`} as an **Amplification** of ${tex`p_t`}:
${tex`p_{t+1} = {}^{×}{p_t}^\lambda = e^\lambda · p_t`}
**Amplification** should be thought of as a non-commutative multiplication.
${tex`\lambda`} is called the *coefficient of amplification*.
We reserve the conventional symbol ${tex`×`} for the commutative multiplication, and use the interpunct ${tex`·`} for its non-commutative counterpart (**Amplification**). Whenever two operands ${tex`x`} and ${tex`y`} have the same unit of measurement, ${tex`x × y`} is correct. Whenever they have different units, ${tex`e^{\ln{x}} · y`} or ${tex`e^{\ln{y}} · x`} should be used instead.
One should think of **Multiplication** as akin to calculating a *surface area*, while **Amplification** is akin to performing multiple additions of the same right operand, with a scaling factor for the last addition when the exponential of the coefficient of amplification is not an integer. For example:
${tex`3\text{m}\;×\;2\text{m} = 6\text{m}^2`}
${tex`3\;·\;2\text{m} = 2\text{m} + 2\text{m} + 2\text{m} = 6\text{m}`}
${tex`3.5\;·\;2\text{m} = 2\text{m} + 2\text{m} + 2\text{m} + 0.5\;·\;2\text{m} = 7\text{m}`}
Therefore, whenever ${tex`r_g \geq 0`}, any expression like ${tex`p_{t+1} = r_g · p_t`} should be written:
${tex`p_{t+1} = e^{\ln{r_g}} · p_t`}
This apparent violation of the principle of parsimony is actually essential for developing a proper understanding of the fundamental difference between **Multiplication** (commutative) and **Amplification** (non-commutative). Failure to do so usually results in the incorrect application of arithmetic measures to multiplicative variables, a cardinal sin of statistical analysis.
Finally, embracing the log return ${tex`\lambda`} offers the simpler expression:
${tex`p_{t+1} = e^\lambda · p_t`}
This really is the proper way of thinking about log returns, and what follows demonstrates that log returns should be used by default, as they properly capture the **idiomatic risk**, making it easier to isolate **idiosyncratic risk** as a residual.
As will be outlined below, the use of log returns is unquestionably mandatory. Nevertheless, even if it were not to be the case, log returns offer the following benefits over arithmetic returns:
- Additivity across time - Normality of distribution - Multiplicative compounding - Incorporation of the asymmetry between losses and gains - Incorporation of the skewness and kurtosis of stock returns - Interpretation in continuous time While arithmetic returns are simpler to understand and easier to compute, they are fundamentally incorrect, and modern computers and software applications make log returns easy-enough to handle. Unfortunately, many of the equations used with log returns by practitioners are incorrect, as they often omit the exponential. For example, the proper standard deviation for log returns is the exponential of the standard deviation of log returns. Omitting the exponential is not a matter of personal preference. It is incorrect.