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The Sharpe Ratio: Measuring Risk And Return

In the world of investing, all analysis essentially falls into one of two categories: risk or return. And in order to properly invest, you need to understand both - one is pretty useless without the other.

In many ways, return is easier to understand. It is simply how much money you've made divided by how much you've invested.

Risk is much more difficult to quantify. On a high level, many people define risk as "the worst possible outcome weighted for the probability of the occurrence of that outcome." But that sentence is so full of jargon that I'm sure not even a lawyer could understand it! Practically, risk is measured by standard deviation, which is a measure of the volatility of the price of a stock.

But this brings us to one key problem in the world of investing - if we can measure return, and we can measure risk by looking at volatility, then how do we know if we are being compensated for the risk we are taking on?

The Sharpe Ratio is the answer.

The Sharpe Ratio: History and Calculation

The Sharpe Ratio was invented by an American economist named William Sharpe, who actually has a really interesting story and I would suggest you take some time to read about him. Being 82 now, William Sharpe has contributed a lot to the world of finance. Besides his obvious creation of the Sharpe Ratio, he also contributed to a method of valuing stock options (called the binomial method), a few techniques of asset allocation optimization and perhaps most importantly was one of the creators of the capital asset pricing model.

Nowadays, though, he is mostly known for creating the Sharpe ratio, which is the most commonly used metric for risk-adjusted returns in the financial world. In essence, the Sharpe Ratio is the investment's return above a risk-fee benchmark (typically the US treasury yield) divided by the standard deviation of that investment.

So to calculate it you would use the following formula:

Sharpe Ratio = [(Return) - (Risk-Free Return)]/[Standard Deviation]

As I said, we typically choose the risk-free rate of return to be the US treasury yield. But it doesn't necessarily have to be - it can be pegged as whatever you like. For example, some people believe that there is inherent risk to investing in US Government Bonds because of their high sovereign debt load and other macroeconomic factors. Thus, they might elect to use 0 as the risk-free rate of return, under the assumption that all non-cash assets are risky. Another example is Canadian investors, who might choose to use the 91-Day T-Bill return as their risk-free rate.

A Quick Example

Here, I'm going to walk you through how to calculate the Sharpe Ratio of a single stock and how to interpret it. I'll be using an approximation of the Canadian 91-day T-bill as my risk-free rate of return. After all, this is the Financial Canadian blog!

My example will be of Fairfax Financial Holdings Ltd. I'll first download historical price information since January 1, 2015 to January 1, 2016 from Yahoo! Finance. This gives us a full year of data to work with.

On Yahoo! Finance, you just need to enter the ticker of the security in question (for Fairfax, this happens to be FFH) and, on the left hand side of the screen, click on "Historical Prices". From there, it brings you to a table with tons of price data. Scroll to the bottom and click "Download to Spreadsheet":

The "Historical Prices" screen of Fairfax Financial on Yahoo! Finance, scrolled to the very bottom.

Similar to the Google Finance output that we saw in my post Analyzing Stock Volatility, the Yahoo! Finance output gives us opening price, intraday high, intraday low, and closing price. It also gives us volume data, along with "adjusted close" (not really sure what this is):

The Sharpe Ratio Excel Data

The only thing I really need is closing price and date, so I'm going to delete everything else and include a new column that calculate day-over-day price returns for Fairfax Financial stock. I use the following formula:

The Sharpe Ratio Day-over-Day Return

Selecting a Time Period

I'm looking to calculate a 1-Year Sharpe Ratio. The reason that I've chosen one year is because most investment returns are reported on an annualized basis, so it makes the most sense to follow this convention for Sharpe ratios too. To calculate the 1-Year return, I use the following formula:

The Sharpe Ratio Trailing 1-Year Return

Note that this is an array formula, so I need to input it with Ctrl+Shift+Enter rather than just pushing Enter alone. You can read more about array formulas and their uses in my post Analyzing Stock Volatility.

Next I need to determine what the risk-free rate of return is. As I mentioned earlier, I am going to be using an approximation of the 91-Day Canadian T-Bill yield. I know off of the top of my head that the average 91-Day yield for 2015 was approximately 0.80%, so that's the number I'm going to use for this analysis. Volatility (measured by standard deviation) is now my only missing variable, so I calculate it using the following formula:

You need to multiply by the square root of 252 to convert the daily volatility into an annualized volatility. This is because there is typically 252 trading days in a given calendar year.

I can easily calculate the Sharpe Ratio with one last formula:

Which gives the following result:

So we now know that the Fairfax Financial stock had a 1-year Sharpe Ratio of 0.36 for the calendar year of 2015. But what does this mean?

Interpreting the Sharpe Ratio

As you would expect, a positive Sharpe Ratio is a good thing. It means that I'm receiving compensation for the incremental risk that I'm taking on by investing in risky assets. But what is a "good" Sharpe Ratio?

Most people say that a Sharpe Ratio of 1 is acceptable, 2 is good, and 3 is great. It's very uncommon to see Sharpe ratios much higher than 3.

To give some perspective, Yahoo! Finance tells me that the SPDR S&P 500 Index ETF has an annualized Sharpe Ratio of 0.97 over the past three years.

By contrast, the iShares Core US Aggregate Bond (trades under the ticker AGG) has a Sharpe Ratio of 2.76. Bonds typically have much higher Sharpe Ratios than stocks because of their much lower volatility, so this isn't really a surprise.

Hopefully these two quotes can give you mental benchmarks for stocks and bonds when calculating Sharpe Ratios on your own investments now! So now that we know what a "good" Sharpe Ratio is, how do identify a bad one?

If a Sharpe Ratio is zero or negative, that's a not a good sign. This means that even though you are taking on some risk, you're not being paid for it. Because of this, a Sharpe Ratio less than or equal to zero is often called "return-free risk," since you're assuming investment risk for no reward.

So that's Sharpe Ratios in a nutshell. Hopefully by reading this post, you've gained a new tool that you can use in your personal financial analysis!

Readers, have any of you employed the Sharpe Ratio when evaluating your own investments? What's the highest Sharpe Ratio that you've ever seen? Let me know in the comments section!