Average Annual Returns

Suppose a stock has the following annual returns in the past years:

Arithmetic average

$$R_A=\frac{4.9+15.8+5.5-37.0+26.5}{5}=3.14\%$$

• Answers the question: “What was your return in an average year over a particular period?”
• Purpose: To estimate expected return for each year in the future
• If we have to predict what will be the rate of return next year, the best prediction would be $R_A$ return.

Geometric average

$$R_G=((1.049)(1.158)(1.055)(0.63)(1.265){)}^{\frac{1}{5}}-1=0.4231\%$$

• Answers the question: “What was your return in an average year over a particular period?”
• Purpose: To measure historical performance
• If we had invested 100 dollars two years ago, your portfolio balance will be now $100(1+R_G{)}^2$

We can use the two definitions above to answer the following questions:

1. Suppose you have invested $10,000 in this stock at the beginning of 2005, how much will your investment be worth at the end of 2009?

$$10,000\times (1.004231{)}^5=10,213$$

2. Suppose you plan to invest $10,000 in this stock for the next year, and you assume probabilities of having each of the past 5 years’ returns as equal. What will be the expected payoff of your investment of 10,000 next year?

$$10,000\times (1.0314)=10,314$$