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# The Log returns simply explained

Starting with the principle that ln(a/b) equals ln(a) - ln(b), log returns are known for their precision and the convenience of being additive over time. This is in contrast to arithmetic returns, which lack this property.

Another notable aspect of log returns is their propensity to follow a normal distribution, a result of the Central Limit Theorem.

In finance, log returns can be considered as the sum of many small and independent changes in price. The CLT states that the sum of a large number of independent and identically distributed random variables, regardless of their original distribution, will approximate a normal distribution.

Therefore, due to this theorem, log returns tend to be more normally distributed, which is especially helpful in statistical modeling and risk management in finance.

This normal distribution is especially evident if stock returns follow a geometric Brownian motion, a common stochastic process used to model stock prices. This normality facilitates more straightforward statistical and financial modeling.

To illustrate, take an asset that increases in value from \$100 to \$110 and then to \$120. The log returns calculated for each period can be summed to give the total return over both periods. In comparison, arithmetic returns would be 10% and approximately 9.09% for each respective period, but these can't be directly summed to give the total return from \$100 to \$120.

Log returns also inherently guard against the possibility of implying negative asset prices, as they are based on natural logarithms of price ratios. This keeps financial analyses and models anchored in real-world scenarios.

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