A quantity that compounds multiplicatively across steps can be made additive by applying the log function to it, which lets the sum-based statistical toolbox apply to it

For example, if we would like to calculate daily variance from an hourly chart, given a sample 3-hour period:

• T1: $100
• T2: $110
• T3: $121

Normally, if we’d like to calculate the ratios, we could simply do:

$$\frac{110}{100}=1.10$$

$$\frac{121}{110}=1.10$$

So naively it looks like we’ve just grown 20%, but have we? We can’t just add those - we have to multiply.

$$\$100\times 1.10\times 1.10=\$121$$

However, if we log the returns, this additiveness works:

$$\log (1.10)+\log (1.10)=\log (1.21)$$

Given we can add the ratios now, it opens the possibility of calculating variance by activating a property of it (assumes the terms are independent):

$$\mathrm{Var}(a+b)=\mathrm{Var}(a)+\mathrm{Var}(b)$$

No such clean property exists for the product.

Other examples:

• The mean/expectation
• The Central Limit Theorem
• Autocorrelation / variance-ratio tests