Log returns are a type of return calculation that assumes a continuously compounding rate of return. They help the investor to quickly calculate the profit or loss they have received from a given investment. Log returns can be easily calculated using a simple mathematical formula or a simple formula in Excel. If other methods are preferable to you, it's also possible to code log returns in programming languages like R and Python.
Calculating Returns for Logs
To assess the effectiveness of any investment, the investor will calculate the return. Returns are defined by the United States Securities and Exchange Commission (SEC) as "the profit or loss on an investment." According to the writers for the Corporate Finance Institute, the two most common ways of calculating the rate of return are annual compounding and continuous compounding. Annual compounding calculates the return on a given investment on a yearly basis, using the below formula:
Principal x (1 + interest rate)^Number of years
In other words, if the annual interest rate is 10 percent and the initial investment is $50,000, with interest earned over a five-year period, the calculations would occur as follows. You would first raise 1.10 (one plus the interest rate) to the fifth power, which gives you 1.61. You would then multiply that number by the initial investment of $50,000, giving you $80,500. As such, the return on this investment is $30,500.
Shorter Periods for Compounding
When interest is compounded over a shorter period of time, such as monthly, weekly or daily, the total return ends up being higher. This leads to continuous compounding, wherein the interest on the investment is constantly calculated and reinvested into the account. Interest then accumulates not just on the principal, but also on the interest that has accrued on the principal and has been added to the balance of the investment. You can calculate a continuously compounding return using the below formula:
Principal x e^(interest rate x number of years)
The e in the formula is the mathematical constant known as Euler's number, which equals about 2.71828. In the above example, therefore, you would multiply the interest rate of 10 percent times five years, giving you .5. Raising e to the .5 power gives you 1.65. You would then multiply that number by the initial investment of $50,000, giving you $82,500. The return on this investment is, therefore, $32,500.
Log Return Excel Formula
Where does a log return come into these calculations? A logarithm is the inverse of exponentiation, which is the function whereby a number is raised to the power of another number (e.g., 4 raised to the fifth power is 1,024). The "logarithm base 4" of 1,024 is 5, because you have to raise the number 4 to the 5th power to get 1,024.
The natural logarithm, used to calculate log returns and expressed as ln, has Euler's constant, e, as its base. Given that e must be exponentiated to determine the dollar amount of the return on investment, the natural logarithm can then be used to calculate the rate of return as a percentage.
Log returns in Excel are calculated using the simple formula =LN(X), where X is equal to the ending value divided by the beginning value. For an investment with a fixed interest rate, X would equal the interest rate plus 1, thereby calculating the continuously compounded rate of return. The team of writers at Microsoft offers a brief explanation of the LN function in Microsoft Excel. This is a simple way to determine the log return.
Consider also: The Average Real Rate of Return of Common Stock