# Blog #58 The Shape of Stock Returns II

I can’t tell you for sure where the stock market is heading next week, but I can show you a formula on what return you can expect will earn if you follow a buy and hold strategy over the long term. There two key phrases in my sentence: “buy-and-hold” and “long-term”. The formula estimates the average return on your investment assuming you are a patient investor expecting a long-term reward from the stock market. It is not a speculator’s formula.

Here is the formula:

g ≅ µ – (1/2)σ²

On the left-hand side, g is the average compound rate of return on a buy-and-hold investment (say a stock portfolio). g is also known as the geometric mean rate of return.

The word “compound” implies that one simply allow short-term (say, monthly) returns to snowball into something bigger. The “snowball effect” is aided by your investment’s simple average return, also known as the arithmetic mean. The arithmetic mean is denoted by mu (µ). All things equal, a high µ means a high g. Unfortunately, the second term (1/2)σ² spoils the party. σ² is the variance of the stock portfolio. It is defined as the square of standard deviation; both are measures of risk. More volatile stocks have higher variance while stable stocks have lower variance.

The wavy symbol ≅ means that g is approximately equal to µ – (1/2)σ². Thus, the formula gives an approximation for g, not the exact solution. Still, approximations are useful because it saves us a lot of manual calculations! More importantly, the formula says that there is “tug-of-war” between µ and (1/2)σ². Everybody likes a stock that has high µ and low σ² but in general, this isn’t possible because risk and return go hand in hand (I have more to say about this interesting topic but let’s save it for another day).

Now let’s answer two important questions: what use is the formula and how to check whether it “works”?

If the formula is accurate, it should give us a close approximation of the geometric mean return earned on a buy-and-hold investment. To see if this is so, I collected a long time series of monthly stock returns to calculate (a) the exact g and (b) the approximate g given by the formula. If the formula is good, (a) should be close to (b).

My monthly returns pertain to two portfolios consisting of stocks traded in the US from 1962 to 2013. The portfolios are “Low-Vol” and “High-Vol”.  Low-Vol primarily consists of low volatility stocks. These are stocks whose prices don’y jump around too much from month to month.  High-Vol on the other hand comprises mainly high-volatility stocks. This portfolio has a disproportionate share of firms in sectors like semiconductors, computer software, digital equipment, aerospace, and precision manufacturing among others. You can download the spreadsheet here.

For each portfolio, I compute and present the following statistics: arithmetic mean, variance, exact g and approximate g. Here are the results:

Focus on the last two rows. You see that for each portfolio, the approximate g is pretty close to the exact g, which confirms that the formula does indeed work well. Best of all, it achieved this feat using just two variables, µ and σ². In physics, it is often said that the most powerful equations are the simplest (think of Einstein’s E=mc²). We don’t have many simple yet powerful equations in finance, but I’m proud to say that g ≅ µ – (1/2)σ² is one of them.

This is well and good, but can we use the formula to predict future returns?  My qualified answer is yes, provided you get your µ and σ² forecasts right. How then does one get good forecasts for µ and σ²? It doesn’t hurt to start with historical estimates of µ and σ² covering a long sample period (to prevent cheery picking particular stretches of stock market history that one fancies). You can then shade these historical estimates higher or lower if you think that past performance is unlikely to faithfully copy the past (this is admittedly a tricky exercise that calls for some expert knowledge in economics). To give you some broad perspective, historically, stock markets have rewarded investors with inflation-adjusted or real arithmetic mean returns of between 4 and 6 percent annually. If you add back the historical global inflation of 4% a year, this implies a historical range of 8% to 10% for nominal average returns (see Blog #30 Why I Love History). Some experts believe that going forward, a globally diversified equity portfolio will probably deliver average returns in the range of 5 to 7% as the low-hanging fruits have mostly been picked, and the world faces challenges due to aging populations, declining fertility, and slower productivity growth. Using a middle rate of 6% and assuming an annual standard deviation of 15%, our little formula implies that you can expect an annual compounded rate g of 4.88% in nominal terms. Even at this modest rate of return, every \$100,000 invested over 30 years will compound to a median wealth of nearly \$420,000. Such is the power of compounding over long horizons.