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Three Ways To Invest Better Using Statistics

When I was offered the opportunity to guest post on Andrew’s website, I was thrilled. I wanted to start by thanking him and my readers – I hope you enjoy this post.

Andrew and I share a very similar investment strategy. From his website, he is “an avid value investor who loves dividend paying stocks.” This describes me perfectly – in fact, I could have written it myself.

I am also a big believer in the importance of implementing statistics while investing. Too often, I hear of investors getting burned because they don’t perform their proper due diligence. It isn’t enough to pick the highest-performing stocks – you must also focus on risk.

Statistics, though considered to be boring by some, are a fantastic tool to reduce risk both at the security and portfolio levels. This post will present three ways to invest better using statistics.

Number One: Reduce Volatility Using Standard Deviations

Portfolio volatility is often used as a proxy for investment risk.

I’ve often had trouble grasping this, since the underlying value of a business does not typically change very much on a day-to-day basis. Stock prices, on the other hand, definitely demonstrate some extreme volatility on occasion. I really like this quote from The Big Short:

“[Burry] gave a talk in which he argued that the way they measured risk was completely idiotic. They measured risk by volatility: how much a stock or bond happened to have jumped around in the past few years. Real risk was not volatility; real risk was stupid investment decisions.

‘By and large,’ he later put it, ‘the wealthiest of the wealthy and their representatives have accepted that most managers are average, and the better ones are able to achieve average returns while exhibiting below-average volatility. By this logic a dollar selling for fifty cents on day, sixty cents the next day, and forty cents the next somehow becomes worth less than a dollar selling for fifty cents all three days. I would argue that the ability to buy at forty cents present opportunity, not risk, and that the dollar is still worth a dollar.'”

One thing that cannot be argued is that volatility is a proxy for withdrawal risk – which is the risk that you are forced to sell an investment when it is trading cheaply. While having a long time horizon is important, sometimes exogeneous forces mean investors are muscled into selling their stocks at undesirable prices (think car repairs, an unexpected medical bill, etc.)

As investors, we are able to minimize this risk by investing in stocks that have low historical volatility. A typical example is Johnson & Johnson, a healthcare conglomerate with a fantastic record of raising dividend payments over time.

Take a look at their performance and volatility in recent years:

Invest Better by Calculating Standard Deviation

With higher returns and lower volatility than it’s benchmark, Johnson & Johnson has delivered fantastic returns on a risk-adjusted basis.

This trend is also evident in the wider stock market. The S&P Low Volatility index outperformed the S&P500 by 2.00% for the 20 years ending in 2011. That type of outperformance, compounded over time, makes a huge difference in investment returns.

Bottom line: Don’t underestimate the effect that volatility could have on your portfolio. Low and slow just might win the race! 

Related: Analyzing Stock Volatility

Number Two: Use Correlations for a Truly Diversified Portfolio

In investing, one of the key principles is that of diversification. By investing in a wide variety of companies across different industries, you can reduce your portfolio’s volatility.

But that begs the question – what is the best way to diversify your portfolio?

The answer lies in correlation statistics. Generally, the word correlation is used to describe a mutual relationship between two or more things. But statistically it has a more precise definition – correlation is a measure of the extent to which two things have a linear relationship.

Statistical correlations range from -1 to 1, which describes both the magnitude and direction of the relationship. Some examples:

  1. If Stock A and Stock B have a correlation of 0.5, then if Stock A increases by 10%, Stock B will increase by 5%.
  2. If Stock C and Stock D have a correlation of -0.5, then if Stock C increases by 50%, Stock D will decrease by 25%.
  3. If Stock E and Stock F have a correlation of 1, then they will move in tandem.

So how do we implement correlations into our investing? The goal is to construct of portfolio of stocks that have low (or negative) correlations for each stock-stock pair. This can involve a lot of number-crunching, and the best way to visualize these correlations is through a correlation matrix, especially if you can “heatmap” the matrix using the Conditional Formatting function on Excel.

Consider the following correlation matrix for a proxy of my personal portfolio:

Invest Better Using Correlation Statistics

Correlations are trailing 5-year price correlations ending on August 30, 2016.

This matrix was created using the =CORREL() function on Microsoft Excel.

The presence of five negative correlations in the matrix is very promising for the purpose of providing diversification. Though low correlations have no effect on performance, since this basket of stocks has performed well in recent years, the portfolio has fared well:


The “mixed portfolio” was creating by investing an equal amount of money in each stock on August 30, 2011. It’s high return and low volatility means it is a great investment on a risk-adjusted basis (more on that later).

Bottom line: The best way to reduce portfolio volatility is to invest in a basket of stocks with low correlations for each stock-stock pair.


  1. Using Correlations to Diversify Your Portfolio
  2. Creating a Truly Diversified Portfolio Using Correlations: Part 1
  3. Creating a Truly Diversified Portfolio Using Correlations: Part 2

Number Three: Analyze Risk-Adjusted Returns w/ The Sharpe Ratio

So far, I have emphasized the importance of minimizing volatility. In the first section, I wrote about searching for individual stocks with low volatility. Then I described how to create a basket of different stocks in a way that reduces the volatility of the aggregate portfolio. Now I will describe how to measure the ability of your portfolio to generate adequate returns for the risk it is assuming.

Sometimes, while chasing returns, investors become negligent of their risk, instead focusing only on performance. Luckily, there’s a metric that combines both risk and return.

This metric is called the Sharpe Ratio, and is the #1 proxy for what’s called “risk-adjusted return.” The Sharpe Ratio answers the important question “Am I being compensated for the risk I am assuming?”

It is also very easy to calculate. In words, the Sharpe ratio is simply the excess performance above some risk-free rate of return, divided by the standard deviation of the investment.


Sharpe Ratio = (Investment Return – Risk-Free Return)/Standard Deviation

Here I will show you to calculate a 1-Year rolling Sharpe Ratio, using Apple as an example. I will use 2% as the risk-free rate of return as a proxy for the 10-year US Government bond yield.

The source that I use for total return data is Yahoo! Finance, because they provide closing prices that are adjusted for stock splits and dividends (called “Adjusted Close”).

The Yahoo! Finance output for Apple looks like this:

Three Ways To Invest Better Using Statistics

And since I only need adjusted close (“Adj Close”), I’ll trim out the rest of the unnecessary data. Then, I need to insert five more columns:

  1. Daily Return
  2. Trailing 1-Year Returns
  3. A column of 2% for the risk free rate of return
  4. Trailing 1-Year Standard Deviation
  5. Trailing 1-Year Sharpe Ratio

The resulting data set (with example formulas above) looks like this:


You’ll notice that the number 252 comes up multiple times in these formulas. We multiply the standard deviation by the square root of 252 to annualize the standard deviation of daily returns. We also find the product of 252 daily returns to find annualized return. This is because there are, on average, 252 trading days in a calendar year.

As you can see, calculating a Sharpe ratio is relatively simple. Having a sense of how well your investments are performing on a risk-adjusted basis can be very helpful. Graphing a trailing 1-Year Sharpe ratio can be very insightful. Let’s take a look at the graph for Apple:


With a Sharpe Ratio of nearly 6 in 2009, Apple clearly has a fantastic history of rewarding shareholders on a risk-adjusted basis.

Related: The Sharpe Ratio: Measuring Risk and Return

Concluding Remarks

While there are lots of statistical techniques that can be applied to the world of investing, these three are the ones I most frequently use. I believe they strike a nice balance between being extremely useful while still rather simple.

With these statistics, my hope is that you can:

  1. Select individual stocks with low volatility
  2. Construct a portfolio with lower volatility than it’s benchmark
  3. Use these low volatilities in conjunction with high-returning stocks to outperform on a risk-adjusted basis

Readers, do any of you make use of these statistical techniques in your investing? Have any questions about how to properly implement them? Feel free to send me an email at author@financialcanadian.com – or better yet, come visit my site.