1. How to measure return?

Return is the most fundamental concept in investment. It must be easy to measure. Let’s look at a simple example. Suppose we invest 1000 dollars. After the first year, the investment turns into 2000 dollars. After the second year, the investment turns into 1000 dollars. What is the average annual rate of return? 

For the first year, the investment doubles. The rate of return is 100%. For the second year, the investment halves. The rate of return is -50%. The average rate of return is therefore 25%. It is simple and clear. This is how standard investment theories, such as portfolio theory and CAPM (Capital Asset Pricing Model), measure investment returns.

Some of us may feel uncomfortable with the calculated rate of return. After two years of investment, we don’t make any money. How come the rate of return is 25%? This is the question asked by many people. 

About three hundred years ago, Daniel Bernoulli considered the same question. He proposed the concept of utility to resolve this issue. In his paper, he introduced the logarithm function to measure utility. Applied to our investment problem, the utility change for the first year is ln(2000)-ln(1000) = ln(2), and the utility change change for the second year is ln(500)-ln(1000) = ln(1/2). The average change of utility is 1/2*(ln(2)+ln(1/2))=0. From utility theory, there is no change in investment value. This is consistent with our intuition. 

Nowadays, we often use compound return, or geometric return, to measure investment performance. Using compound return, the return of our investment is 0. This is consistent with our intuition. Why we don’t use compound return in the standard theories of investment? 

When Markowitz developed the portfolio theory, he used arithmetic return and standard deviation to measure investment performance. This measurement was later adopted by Sharpe and others in developing CAPM. These theories have since become the standard in finance and economics. That is why we are stuck with arithmetic return, although it alone, or combined with standard deviation, doesn’t provide precise measurement of investment performance. 

Is there an investment theory that uses compound return to measure investment performance? The answer is yes. We will discuss that theory in the next section.

Behavioral finance 

0. Introduction 
1. How to measure return 
2. How to measure information 
3. Theory of investment 
4. Theory of asset pricing

5. How we process information 
6. Common psychological patterns 

7. Theory of judgment 
8. Investment behaviours and market patterns

9. Historical discussion 
10. Concluding comments