## Wednesday, July 12, 2023

In the world of statistics, and particularly in investing circles, we cannot escape from the use of averages. A lot of investment philosophies, methodologies and models are very much based on using averages, or “mean” in statistic-speak. It is by far the easiest concept to grasp; many people can associate a given factual number and then extrapolate their projections from there. For example, supposedly the returns of the S&P 500 (using the SPY ETF) were around 7.54% annually for the past 25 years1, we were able to wrap our heads that, assume all things equal, we could get that returns every year for the next 25 years if we started on the S&P 500 now.

However, things are not that simple.

We had learnt back in school that average is the sum of the number of items/occurrences divided by the number of observations. The following is an example of a simple average math example that we may have encountered in our school days.

 Bob’s Diner Daily Profit/Loss for the Week Day Profit / (Loss) \$ Monday (\$300) Tuesday (\$200) Wednesday \$100 Thursday \$500 Friday \$1,000 Saturday \$2,500 Sunday \$2,000 Total Profit/Loss \$5,600 Average Daily Profit/Loss \$5,600 / 7 = \$800

Fig.1: Bob’s Diner average daily profit/loss for the week

From Figure 1, it is correct to assume that Bob’s Diner is earning \$800 per day over the course of a week. However, this \$800 number is taken from figures of the past seven days, and you could see that there is no “\$800” in any of the days. We could stretch it to two weeks or even more, and we might have a chance of getting \$800 somewhere in the extended timeframe.

This is where I will bring in the “not that simple” part; obviously, from Figure 1, the average is aggregated from the various amounts reported over the course of the week, and they swing rather wildly around the \$800 mark, with differences of between \$200 and \$1,700. If I had wanted to buy over Bob’s Diner with knowing only of the \$800 daily average, I would be in for a shock if I had known the actual daily profit or loss. You can imagine if the amounts in Figure 1 were translated from daily to annual investment returns.

Standard Deviation

To understand more on averages and their relationship with the actual numbers, a statistical tool known as standard deviation, or SD, is used. Standard deviation allows one to see the number of dispersals of the individual numbers with relative to the average. A low SD indicated that the numbers are quite close to the mean, while a high SD means (pun intended) that the numbers were dispersed further from the mean. For Figure 1, the SD is relatively high at 137%.

SD is seen as an indicator to determine if an investment is volatile; too much volatility will give the investor the psychological version of a roller coaster ride, and this puts off risk averse individuals who prefer to play it safe. With this, SD is also seen by investors as a risk metric. For information, the SPY ETF, with an annual return of 7.54% over 25 years, provided a SD of 15.63%1.

Think Long Term

Though seeing such swings on a daily, monthly or annual basis is rather nerve wracking, as investors we see things on a longer term. At the end of the long investment period, at which you are prepared to drawdown your portfolio, is where the average works best, for it is presented as a smoothed out, “looking-back” number. This brings us back to the point mentioned in the first paragraph of being the easiest to understand. The danger of using past averages for projection is mainly due to “past performance is not indicative of future results” clause, but, using the average rolling returns of SPY ETF between 3 and 15 years, it is roughly the same as the 25-year duration1, so the probability of the returns being repeated, ceteris paribus, in the future is relatively high.

1 – SPY statistics from Jan 1998 to Dec 2022. Portfolio Visualizer. https://www.portfoliovisualizer.com.