Each month, I provide an update to my four dividend growth model portfolios that includes portfolio beta and other volatility-adjusted metrics, such as the Sharpe Ratio. Recently, I was asked by a few readers in the Dividends & Income group to explain how I calculated beta and the other metric values. This article provides the information necessary for investors to calculate these values and some discussion about their use and limitations.
Inputs Required
Let's start with the data that is needed. For this example, I am working with my DG-Income Growth model and the S&P Index (SPY) for the last 3 months. The account values reflect the weekly values of these portfolios, which originally started with $300,000 on August 16, 2011. You could do daily or monthly tracking, depending on your preference. I chose weekly to get more data without resorting to daily updates. In my MBA course, we used 5 years of monthly values to calculate a stock's beta, which is what Yahoo Finance does, or at least my calculations matched Yahoo's when I did it this way.
In Excel, create columns for: the date, the benchmark value, your portfolio or stock value, and then columns for the percent of change for both of these portfolios. Other inputs include:
Input | Excel Formula |
The Total Return for each Portfolio | =(last cell / initial cell) - 1 |
The risk-free rate | Search for "Treasury Rate Curve" to get current and historical annualized values for your time period. I just eyeballed an average and adjusted it for the time period. |
Standard Deviation of Returns (%chg) | =stdevp ( first %chg cell : last %chg cell ) |
Variance of Returns (%chg) | =varp (first %chg cell : last %chg cell ) |
Co-Variance of Portfolio and Benchmark (%chg) | =covar ( portfolio %chg first cell : last %chg cell, benchmark %chg first cell : %chg last cell ) |
The table below shows the data from my portfolios for a 3-month period.
We now have the inputs necessary for our metrics. The sections below provide descriptions of each metric, the calculation method, and some discussion on usage and limitations.
Beta
I'll start with one that income investors should recognize, Beta. Beta measures the volatility, or systematic risk, of a stock or portfolio relative to a market benchmark, which has a beta of one. A beta higher than one means the portfolio change typically moves more than the market (i.e. more volatile), whereas a lower positive beta means it moves less than the market. A beta of zero indicates that the asset moves independently of the benchmark, and a negative beta means that it moves in the opposite direction of the benchmark. Risk-averse investors prefer a positive beta less than one, as they don't like large market swings, particularly in the negative direction. Those looking to hedge their portfolio may want some negative beta holdings. For the purposes of this discussion, I am interested in positive beta that is less than the market's beta of one, as dividend stocks should be less volatile than the overall market while still tracking the market's direction.
To calculate beta from the inputs, divide the portfolio's (or stock's) co-variance by the benchmark's variance. For the benchmark, these values are equal, hence its beta equals one. For my data, we get a beta of 0.71 for the model portfolio. As expected, the dividend portfolio had a lower beta.
Sharpe Ratio
The Sharpe ratio measures the excess return per unit of standard deviation in a stock or portfolio. Essentially it is evaluating how well an investor was compensated for each unit of risk taken; risk defined as price volatility. A higher Sharpe ratio is better, though this can be achieved either by obtaining higher absolute returns or by reducing the volatility. Because this is a unit ratio, a portfolio could have lower absolute returns than its benchmark, yet have a higher Sharpe ratio because of lower volatility. In financial theory, it is assumed that the investor can borrow at the risk-free rate and invest the funds into the portfolio until the risk level matches the market. In theory, the investor would have outperformed the market in total return under this scenario. The point is that a higher Sharpe ratio does not necessary mean you achieved a higher total return; this is a volatility-adjusted metric.
The formula for the Sharpe ratio is: ( total return - risk-free rate ) divided by the standard deviation. In my example, the SPY has a Sharpe ratio of 4.11, while the DG-Income Growth model's Sharpe ratio is 5.96. So the DG-Income Growth model offered more reward for each unit of volatility. It achieved this by having both higher total returns and lower volatility, which is what we want.
There are some limitations and oddities to the Sharpe ratio. The Sharpe ratio assumes that the asset's returns are normally distributed, which may not be the case. It is also a dimensionless ratio, which makes it difficult to know how much better a ratio of 0.6 is versus one of 0.4. Higher is better, but it would help for the portfolio's ratio to be relative to the benchmark's ratio (see M2 measure). Negative Sharpe ratios are also somewhat difficult to understand. Consider if two portfolios have the same negative return (-10%), but one has a standard deviation of 1% and other's is 2%. The Sharpe ratio for the first portfolio will be -10, while the second's will be -5, hence it has the higher ratio and is "better". This makes sense, as the higher volatility reduces the risk-adjusted return value, but one would think that the combination of negative returns with higher volatility would be bad. This web site does a better job of explaining the idea with graphics.
Treynor Ratio
The Treynor ratio is similar to the Sharpe ratio, but it measures excess returns relative to beta instead of the asset's volatility. This moves closer to linking the metric to the benchmark, as beta involves the covariance of the asset with the market benchmark . Again though, this is more of a ranking system, and the Treynor is meant for evaluating sub-portfolios of a broader, fully-diversified portfolio (e.g. comparing different mutual funds within your 401K account). Assuming that a dividend portfolio is a component of the overall investment portfolio, this metric can be used for comparisons.
The formula for the Treynor ratio is: ( total return - risk-free rate ) divided by beta. In my example, the SPY has a Treynor ratio of 9.2%, which is its excess return over the risk-free rate. The DG-Income Growth portfolio has a Treynor ratio of 13.7%, indicating its superior relative performance. Again, this does not necessarily mean higher absolute total return, as the higher ratio could be achieved through a low beta value.
Modigliani-Modigliani Measure (M2)
The Modigliani-Modigliani measure, or M2, measures the volatility-adjusted returns of a portfolio relative to a benchmark (e.g. the SPY). While similar to the Sharpe ratio, the M2 is represented in percent return terms. It effectively scales the excess return of the portfolio until the portfolio's risk level equals that of the benchmark. It then subtracts the benchmark's excess return to produce the M2 measure, which shows how much more or less percentage return the portfolio generated on a risk-adjusted basis.
The formula for the M2 is:
(Portfolio total return - risk-free rate)*(Benchmark std dev / Portfolio std dev) - (Benchmark total return - risk-free rate)
In my example, the SPY has a M2 of zero, as it is being compared to itself. The DG-Income Growth model yielded an M2 of 4.1%, showing it had superior risk-adjusted performance. If we used leverage to raise this model's volatility level to equal the SPY's, it would have produced a 4.1% higher total return compared to the SPY. As no leverage is being used, we have to live with the 0.6% higher total return, and take comfort that the model's volatility was about 30% less than the SPY.
What I like about the M2 is that it includes the benchmark in the calculation, so the result has more tangible meaning. It is easy to understand a 4% higher return. Again though, this is theoretical, as it assumes we leverage the portfolio to equate the risk level with the benchmark. But the goal here is to evaluate our portfolio relative to its volatility, not to compare absolute total returns, which we already know.
Summary
I hope you found this introduction to some of the volatility-adjusted metrics useful and that you are able to follow the formulas to create your own data in Excel. To aid with this, you can find an Excel template here with the data and calculations used in my example.