# standard deviation

## Standard Deviation as a measure of ‘risk’

Various technical terms are commonly referred to in finance. Standard Deviation is the term often used to statistically quantify the level of risk of an investment return. What is really being described is the volatility or fluctuations around a result (expected or actual) that the investor has experienced in deriving a particular outcome.

The standard deviation is a statistic that tells you how tightly all the various examples are clustered around the mean in a set of data. When the examples are pretty tightly bunched together and the bell-shaped curve is steep, the standard deviation is small. When the examples are spread apart and the bell curve is relatively flat, that tells you there is a relatively large standard deviation.

Computing the value of a standard deviation is complicated. However, graphically it is much easier to grasp what a standard deviation represents.

If you looked at normally distributed data on a graph, it would look something like the graph on the right. The x-axis (the horizontal one) is the value in question. In the case of investment returns, it is the percentage return p.a. The y-axis (the vertical one) is the number of data points for each value on the x-axis. In other words, the number of times x percent return is recorded in the sample period.

Now, not all sets of data will have graphs that look this perfect. Some will have relatively flat curves, others will be pretty steep. Sometimes the mean will lean a little bit to one side or the other. But all normally distributed data will have something like this same "bell curve" shape.

For data that is ‘normally distributed’, one standard deviation away from the mean in either direction on the horizontal axis (the red area on the above graph) accounts for somewhere around 68 percent of the occurrences within all the data points represented within the bell curve (or the statistical sample in question). Two standard deviations away from the mean (the red and green areas together) account for roughly 95 percent of the returns, and finally three standard deviations (the red, green and blue areas) account for about 99 percent of returns.

If this curve were flatter and more spread out, the standard deviation would have to be larger (and likely represent a broader range of percentage outcomes) in order to account for those 68 percent or so of the returns. Thus the standard deviation can tell you how spread out the results are, or how widely they fluctuate, from the mean.

Distribution of Returns applied to understand manager performance

Using statistics we can analyse the ‘value added of forecast-based funds management vs. chance (the randomness of the market returns over time) as plotted below:

While no system is perfect, capital markets overall do a good job of assimilating news and information and building it quickly into prices. This makes it hard for anyone to consistently “beat” the market, despite the bulk of forecast-based fund managers claiming to be able to do so.

The chart above reflects this by showing the results of a 28-year study of 4,549 US equity funds, analysing their value add relative to the market. The bars to the left of zero represent all the actual forecast-based or traditionally active funds that underperformed the market over this period. The bars to the right represent the funds that outperformed the market.

The light grey curve behind the bars shows a normal distribution or the distribution you expect to see if you were just randomly tossing coins to pick stocks.

As you can see, the actual distribution of forecast-based fund results is quite significantly skewed to the left of the curve. This means that, overall, forecast-based managers do worse than you would expect by chance. This is largely due to the fact that these managers charge high fees that detract from the market return you should receive.

Of course, you might hope to just pick the managers to the right of zero – the ones that outperform the market. But the study shows that managers with strong outperformance tend not to repeat, while managers with strong underperformance do repeat.

The other observation from this study is that the distribution of actual results is what academics call “fat-tailed”. That means forecast-based management adds another layer of uncertainty to the outcomes you can expect.

Using distribution of returns and probability analysis to project portfolio returns using a Monte Carlo simulator

A Monte Carlo simulator is a powerful tool for calculating probable returns based on a portfolio’s expected return and volatility (standard deviation) by way of modelling thousands of potential sequential outcomes over an estimated time period.

As the portfolios implemented by Axiome Consultants are comprised of defined asset class funds with statistically identifiable risk and return characteristics (rather than a forecasting based funds management approach), it is possible to model the impact of projected cash flows in, and drawings out of an investment portfolio over time in a meaningful way.

To be conservative, we discount the top 25% of possible returns (the shaded area in the chart below). I.e. projections will have at least a 75% probable chance of success.

Combinations of lump sums in and out, progressive contributions and potential drawings levels can be plotted over time (see example below) to ensure the portfolio delivers the investment objectives for each client’s circumstances. Updating the analysis at review points enables adjustments considered to portfolio risk, savings and consumption levels ensure the plan continues to deliver outcomes with the desired level of confidence.