# 3 – M1L5 03 Log Returns V5

Quantitative analysts frequently work with a quantity related to but slightly different from the raw return, the natural logarithm of return. Remember how the return was defined as P sub t minus P sub t minus one divided by P sub t minus 1. Well, the log return is defined slightly differently, as the natural logarithm of P sub t divided by P sub t minus one. To see how to convert between these two quantities, note that if we add one to P sub t minus P sub t minus one divided by P sub t minus one, we can convert one to P sub t minus one, divided by P sub t minus one, and the expression simplifies to P sub t divided by P sub t minus one. So, if we denote the log return as capital R, and the raw return as little r, then capital R equals the log of little r plus one and little r equals e to the capital R minus one. Why the quantitative analysts use log returns? Well, it turns out that log returns have a number of appealing properties. Instead of listing them all now, we’re going to highlight them as they arise while we talk about the remaining content in this lesson. For one thing, if the value of R is small, or much less than one, then the natural log of one plus r approximately equals r. Since returns are typically small percentages, the values of log returns are typically close to the values of returns. To understand this intuitively, take a look at the graph of the natural log of x plus one, it goes through the point 00, and it has a slope of one there, so at 00 it’s tangent to the line y equals x. If we were to zoom into a tiny neighborhood around x equals 0, the two lines practically overlie each other, so the natural log of x plus one approximately equals X. That’s all that equation is saying.