# Log Return

## Formula

The logarithmic return is a way of calculating the rate of return on an investment. To calculate it you need the inital value of the investment V_i, the final value V_f and the number of time periods t. You then take the natural logarithm of V_f divided by V_i, and divide the result by t:

R = ln(V_f/V_i) / t xx 100%

This value is normally expressed as a percentage, so you also multiply by 100.

The calculated rate will depend on the value of t that you use. If t is the number of years, then you get an annual rate. This then gives you the continuously compounded annual interest rate that you would need to receive in order to match the return on this investment.

## Comparing Log Returns

Because the formula for log return takes the duration of the investment into account, it can be used to compare multiple investments that cover different lengths of time. Typically you would compare multiple investments using an annual rate, so t in the above formula will be the number of years.

The log return is less useful for comparing our investment with other investments that have a fixed interest rate, such as bank savings accounts, because these are normally quoted as a yearly compounded interest rate, and the log return is a continuously compounded rate.

## Example

Jenny is a property investor. She buys a house for 100000. Exactly 3 years later she sells it for 120000. The logarithmic return is then:

R_j = ln(120000/100000) / 3 xx 100%

R_j = 6.08%

(I've rounded the percentage up to 2 decimal places here.)

Jenny's friend Stan also buys a house. He pays 95000 and sells it 18 months (1.5 years) later for 105000. Jenny and Stan would like to compare their returns, so Stan also calculates the log return:

R_s = ln(105000/95000) / 1.5 xx 100%

R_s = 6.67%

From this we can see that Stan has got a better logarithmic rate of return than Jenny on his property investment.