Measuring risk and return
Higher returns usually come with higher risk. Measuring both precisely is the starting point of investment theory. This lesson is educational, not personal investment advice.
Return
An investment's return over a period is \(\frac{\text{income} + (\text{ending value} - \text{beginning value})}{\text{beginning value}}\). Expected return is the probability-weighted average of possible returns. Historical average returns are often used as rough estimates of expected returns, though the future may differ.
Risk as variability
In finance, risk usually means the variability of returns: how far actual results may differ from expectations. Standard deviation measures this spread. A savings account has almost no variability; a single small company's stock may swing wildly.
Historical returns
Long-run US data compiled by researchers such as Roger Ibbotson and later Duff & Phelps and Kroll show that, over the past century, large company stocks returned about 10% a year on average, long-term government bonds about 5-6%, and Treasury bills about 3%, with inflation about 3%. Stocks had much higher standard deviation, roughly 20% a year versus under 10% for bonds.
The risk premium
The equity risk premium is the extra return investors have earned, and expect, for holding risky stocks rather than risk-free assets. Its size is debated; estimates for the future often range from about 3% to 6% a year. Without a premium, investors would have no reason to bear the extra risk.
The normal distribution
If returns are roughly normally distributed, about 68% of outcomes fall within one standard deviation of the average and 95% within two. With a 10% average return and 20% standard deviation, a stock would lose more than 10% in about one year in six. Real returns have 'fat tails': extreme events happen more often than the normal curve predicts.
Risk aversion
Most investors are risk-averse: they prefer a certain outcome to a gamble with the same expected value, and require extra expected return to accept extra risk. The degree of risk tolerance depends on a person's time horizon, financial situation, and temperament.
Arithmetic and geometric averages
The arithmetic average is the simple mean of yearly returns; the geometric average is the compound annual growth rate, which is lower when returns vary. A gain of 50% followed by a loss of 50% has an arithmetic average of 0% but leaves you with 75% of your money, a geometric average of about -13.4% a year.
Worked example
An investment has a 30% chance of returning 25%, a 50% chance of returning 10%, and a 20% chance of losing 10%. Expected return = 0.3 x 25 + 0.5 x 10 + 0.2 x (-10) = 7.5 + 5 - 2 = 10.5%. Its variability, measured by standard deviation, is about 12.1%.
Key takeaways
Return combines income and price change; expected return is probability-weighted. Risk is variability, measured by standard deviation. Over the past century, US stocks averaged about 10% a year with much higher volatility than bonds and bills. The equity risk premium rewards bearing risk. Returns are roughly normal but have fat tails. Investors are generally risk-averse. Geometric averages, reflecting compounding, are lower than arithmetic averages when returns vary.
Flashcard
(Income + price change) / beginning value
Check your understanding
A stock gains 50% one year and loses 50% the next. What happens to 100 dollars invested?