Measuring Asset Returns using Logarithms
August 7, 2006
We calculate return from investments every day, in a variety of situation and yet sometime fail to understand that calculating the return from an investment is not always a straightforward task. In a recent training session, we asked five fund managers this following question.
Say a stock was trading at $100 the day before yesterday. An investor bought one stock for $100. The next day, i.e. yesterday, the stock went up by $1 to reach a price of $101. How much did the investor make? He made a profit, or a return of one dollar. How much did he make in terms of percentage? His percentage profit, or return, was 1%. Now say, the stock came down today by $1 to settle at $100. Between yesterday and today how much did the investor lose? His loss, i.e. his negative return, was $1. How much was his negative return in percentage terms? Uh, uh, it comes out to be -0.99% (minus 0.99%).
Therefore, though the investor made one dollar on the way up and lost one dollar on the way down, yet his percentage return was different. In dollar terms the investor gained an lost the same amount bur in percentage terms his gain was slightly more than his loss. This is one of the fundamental problems of finance and lies at the heart of finance theory. Assets follow geometric Brownian motion and continuous time return should be measured using natural logarithms. If we had used natural logarithms we would see that the absolute value of the positive return is equal to the absolute value of the negative return:
All the five fund managers, trainees in our session, thought that this though mathematically true is not something they would use in their calculations.
Seldom do fund managers and investors measure return in this manner. They use the arithmetic measure and calculate return as the difference between the ending price and the beginning price divided by the beginning price. If you believe that the markets follow arithmetic Brownian motion then this is ok, and gives reasonably accurate result though in a continuous time geometric Brownian motion, one has to use the log measure. Traders use this estimate all the time to measure historical volatility of stock prices. Using the arithmetic measure of return can sometime lead to intractable situations from which there is no escape.
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