Every investment comes with risk. If you’ve been invested in the stock market over the past few months then you have received the education of a lifetime on this basic point. I personally feel as though I have aged about 10 years as an investor! Naturally the question should come up about how to best measure the risk inherent in your investments. This brings us once more to the realm of Modern Portfolio Theory (MPT). We still call it that even though the foundations are pushing 60 years old so it’s not very “modern” at this point. Call it whatever you want, but a solid understanding of the basics of MPT can be very useful and can help you to make better (or at least more informed) decisions about how to invest your money.
In this post we will focus on two of the conical measures of risk that are essential to successfully deploying MPT in your portfolio: Sharpe Ratio and Beta. At the end of this post I have included two spreadsheets that you can download with the Beta and Sharpe Ratio of all of the stocks in the S&P 500 calculated using weekly data since June 2017. I plan to update these measures on at least a monthly basis so this information will continue to be available to you going forward.
First we need to learn what this stuff is so let’s dive in!
Sharpe Ratio: The Risk-Reward Trade-Off
The simplest way to think of the Sharpe Ratio is as a measure of the trade-off between the risk and return of an investment. First developed by William Sharpe in the 1960’s as part of his doctoral dissertation it has since become a cornerstone of the investment industry. The formula is as follows:
Where:
- Ri = Return of investment ‘i’
- rrf = Risk-free rate of return. Typically measured by the yield on short term T-Bills
- σi = Volatility/risk (i.e. standard deviation) of investment ‘i’
The Sharpe Ratio is typically calculated on an annualized, backward looking (ex-post) basis. The interpretation is straight forward: return per unit of risk. As you might imagine, the higher the ratio the better. When comparing two stocks the one with the higher Sharpe would be considered the better investment as you are receiving more return per unit of risk assumed.
The ratio needn’t be positive. Global stock markets have been in rolling bear markets for at least the last 2 years so if you download the accompanying spreadsheet you will find many stocks have had a negative Sharpe Ratio over the analysis period. You will also notice wide dispersion in the Sharpe of the companies listed in the S&P 500. Companies like Microsoft (MSFT) have performed extraordinarily well while others like Apache Corp (APA) have struggled significantly.
As a quick example, the long term historical annual return of the S&P has been ~10% with an annual volatility of ~16%. Over the past 10+ years the yield on short term Treasury Bills has been very close to zero. For simplicity we will plug in zero for the risk-free rate. The calculation for the Sharpe Ratio is carried out as follows:
You can use .625 as a very general rule of thumb when looking at your own investments. If the ratio of your picks in less than this then it might be a sign that you need to make some adjustments.
Beta: A Measure of Market Risk
Beta is a measure of the systematic risk of an investment. More simply, Beta measures the exposure of a stock to movements in the market index.
Let’s say that you own an ETF tracking the S&P 500. Modern Portfolio Theory states you have “diversified away” (i.e. eliminated) all of the risk unique to the individual underlying holdings (often called idiosyncratic risk) and are now exposed only to systematic risk (i.e. undiversifiable risk). Systematic risk is the risk inherent in any investment. It cannot be reduced further at it represents the risk of loss from economic variables to which every stock is exposed (growth, profits, inflation, geopolitics, etc.); Beta attempts to summarize this in a single measure.
MPT’s claim goes further and asserts that you should be compensated based only on the systematic risk of an investment. By this logic, you should not own individual companies as you will be taking on additional idiosyncratic risk without the expectation of greater returns.
The key result of MPT is the Capital Asset Pricing Model (CAPM) which uses beta to derive the return that you can expect on an investment given its level of systematic risk. Beta is calculated as follows:
Where:
- piM = Correlation of investment ‘i’ with market index ‘M’
- σi = Standard Deviation of investment ‘i’
- σM = Standard Deviation of the market index ‘M’
The Beta of the market index to which you are drawing a comparison is 1 by definition. Therefore, Beta has a very straight forward interpretation:
- βi < 1 Stock is less risky than the index. Expect lower returns as compensation for lower risk.
- βi = 1 Stock is as risky as the index. Expect returns similar to the market index.
- βi > 1 Stock is riskier than the index. Expect higher returns as compensation for assuming additional risk.
Now that we have calculated Beta, we can plug the result into the Capital Asset Pricing Model and derive our expected return. The formula for CAPM involves a couple of different parts. Let me present the formula in full and then breakdown each component:
Where:
- E(Ri) = Expected return on investment ‘i’
- rf = Risk free rate of return. Again, using the yield on short term T-Bills
- βi = Beta of investment ‘i’
- rM = Return on the market index
You’ll notice similarities between the variables of CAPM and the Sharpe Ratio previously discussed. The risk-free rate ‘rf’ is typically proxied by the yield on short term T-Bills but can also be proxied by the yield on the 10-Year Treasury. There is some academic debate about which is better, but in practice I tend to prefer the yield on T-Bills. The T-Bill yield will tend to vary less in the short term and to me is more intuitively appealing than the yield on bonds.
The term in parentheses is referred to as the Equity Risk Premium (ERP). The ERP represents the premium you can expect to extract for holding equity rather than Government notes. The premium is leveraged by the amount of systematic risk of the stock. Consequently, if the stock is high beta then your effective equity risk premium will be higher than the market average.
Let’s go through a quick example for Nucor Steel (NUE) to see how the formula works in practice. As you might surmise from the name, Nucor makes steel. As such NUE is significantly exposed to changes in the economic cycle. In good times, steel tends to be in higher demand due to increased construction activity (Nucor also manufacturers piping for oil and gas fracking which is another notorious boom and bust business), but in bad times demand for steel tends to be low. As such, Nucor has a Beta of ~1.10 which is greater than the market index (in this case the S&P 500) and implies it is riskier than the market overall. As such you would need to receive a greater return over time to compensate you for the additional risk you will take. Let’s see what that return is:
For this calculation I found the current yield on T-Bills of 25 bps (.0025 or .25%) and plugged in the long run average annual return for the S&P of ~10%. You can see that the resultant required return for Nucor is ~11.5%.
Before we conclude, it is important to bear in mind that CAPM gives you the expected return on a stock; “expected” being the operative word. This does not mean that at any point you will actually realize this return; in fact, you probably won’t, at least not over a 1-year time frame. Returns in the stock market are notoriously volatile and as such the risk that you assume is going to cause significant variations around this expectation. CAPM is an equilibrium theory and only over a very long period of time would you expect your realized return to converge to 11.5%.
Conclusion
In this post we have introduced two metrics that you can apply when evaluating your portfolio or potential investments. The first was the Sharpe Ratio which tells you the return per unit of risk that you take. The Sharpe Ratio is a measure of efficiency and can be used to help you differentiate among stocks which offer the best risk-reward trade off. The second was Beta which measures the systematic risk of an investment and is a key input to the Capital Asset Pricing Model. By using CAPM you can determine the return you expect to receive over the long run as compensation for risk.
Take at look at the holdings for your own portfolio and see how they measure up by downloading the Sharpe and Beta spreadsheets below. The stocks have been organized in ascending order.
DOWNLOAD SHARPE RATIO SPREADSHEET
Thanks for reading!
-Aric Lux.
