What does the Sharpe ratio mean? The Sharpe ratio is one of those really useful metrics to assess either individual investments or a portfolio. Here is a definition.
The Sharpe ratio is a measure of the risk-adjusted return of an asset over the risk-free rate of return. It applies to individual assets and to a portfolio of assets.
The Sharpe ratio is not like many other formulas and equations in finance that often involve complex math that would make the hair on any sane person’s head stand on end. A good example of such a complex formula is the Black Scholes equation.
In contrast, the Sharpe ratio uses a delightfully simple formula.
- Ra is the return on the asset
- Rf is the risk-free return
- σa is the standard deviation of the excess returns on the asset
If you think about it, the Sharpe ratio is a pretty useful, you could even say fundamentally useful metric. It expresses in one figure both the expected returns and the risk of your investments in a single number.
There is a basic point here about risk that is both mathematically applicable and matches practical experience. If you take on riskier assets, you need to be compensated with higher returns.
Higher risks need higher returns
To illustrate why investing in higher-risk assets needs higher returns, let’s consider a simple example of two stocks, stock A and stock B
I’ve chosen annual returns here but the Sharpe ratio can just as well be applied to returns over any time period you chose.
Many experts say that it makes sense to use the risk-free rate for the same period. That makes total sense to me. So we are considering an annual risk-free rate of return of 4% which is a reasonably representative historical number to chose.
In this easy example, both stocks have an average return of 10%. But we’ve also made the math super simple so that the examples are intuitively graspable. We are only showing five years of returns with four of those years having the same returns and one year that is different.
In this comparison, stock A returns 10.25% for each of 4 years and then has one not-so-good year when it returns 9%. Stock B returns 15% for each of 4 years and then has one bad year when it loses 15%. Both have an average return of 10% but clearly, stock A is less risky while stock B is riskier.
Let’s also notice that stock A has a better compounded return of 61.04% while stock B has a compounded return of 57.41% even though the average returns are the same.
The standard deviation of stock A is just 0.5% while the standard deviation of stock B is 10%. Clearly, stock B has much higher variability than stock A.
There is no surprise that when we plug these numbers into the formula for the Sharpe ratio, stock A has a Sharpe ratio of 20 and stock B has a Sharpe ratio of just 0.6
Base reference for the Sharpe ratio
Before we find our feet with understanding different Sharpe ratios, we should consider what the Sharpe ratio would be for a risk-free investment. That is also straightforward. The numerator is
Where Rf is the risk-free rate of return. So the numerator is 0.
Then the denominator is the standard deviation of the risk-free asset, which is 0 because the return of the risk-free asset isn’t going to vary at least not for any defined time period.
So our formula is 0 divided by 0 which some mathematicians will tell you cannot be determined but anyone with an eye for the real world will tell us is equal to 1.
Straight away we can see that for an investment to be worth considering it should have a Sharpe ratio greater than 1. And from that referring back to our two examples, we can see that stock A is a pretty good investment while stock B isn’t a good idea at all.
Some other simple examples
Let’s expand on our simple examples and add a few more stocks with different numbers to get a sense of good Sharpe ratio numbers and less good Sharpe ratio numbers.
I think we should note that these numbers are mostly unrealistic but are shown to make a point.
I think the main point is that since we are keeping the average return fixed in these examples at 10% and the risk-free return is 4%, then the excess return is 6%.
So we find that we get a Sharpe ratio of 1 when the standard deviation of returns is also 6% because the excess returns are the same as the standard deviation of returns.
We can see in the table that this happens for Asset H and Asset I. In all cases, since we are keeping the average return constant at 10% when the standard deviation is less than 6% then the Sharpe ratio is greater than 1.
And when the standard deviation is minuscule, as is the case for Asset J and Asset K then the Sharpe ratio is over 100 which to all intents and purposes is off the chart.
Good Sharpe ratios and not so good Sharpe ratios
From what we have discovered so far we can conclude that a good Sharpe ratio is anything more than 1, and the higher it is the better. A bad Sharpe ratio is anything less than 1.
We could generalize this further with the following, rather obvious statement.
If we find an investment with an average return of X% we would want to see that the standard deviation of returns on that investment is less than X% minus the risk-free rate of return.
It’s time to face some mathematical facts here.
All this talk about the standard deviation of returns assumes that the returns on our assets follow a normal distribution. A normal distribution means that the returns are evenly dispersed around an average value, with many small deviations and a few larger deviations. The pattern of the normal distribution follows a bell curve and looks like this.
In reality, this is more for the sake of mathematical convenience rather than following what actually happens. The returns on most assets don’t follow a normal distribution. They tend to experience bigger drops and bigger jumps more frequently.
None of the examples above, stock A through stock J follow the normal distribution. They are all skewed either to the upside or to the downside.
Also while a stock can double or triple in price within a period, so an increase of 100% or 200%, it can only drop to zero so it can only lose 100%.
Of course when an investment losses 100% nobody in their right mind is particularly happy, but the chances of a stock more than doubling does tend to compensate.
These two factors tend to result in returns that follow more a log-normal distribution. This is what a log-normal distribution looks like.
Sharpe ratio, real-life examples
Let’s take a look at the Sharpe ratio of some major stocks and ETFs using the annual returns from 2010 to 2020.
In all cases here, we are going to use the standard stock and ETF symbols.
To make this a representative selection, I propose to look at SPY, which tracks the Standard and Poor’s 500 Index, QQQ that tracks the tech-heavy NASDAQ 100 Index, AAPL, Apple Inc, F, the Ford Motor Company, GE, the General Electric Company, MSFT, Microsoft, WST, West Pharmaceutical Services Inc. and XOM Exxon Mobil Corporation.
Over this same ten-year period from 2010 to 2020 we also need the risk-free rate of return, which we can take to be the average return on T-bills.
If we just tabulate all the annual returns of the stocks and the T-bill from 2010 to 2020, calculate the average returns, the compounded returns, the standard deviation of the annual returns, and the Sharpe ratio for each ETF and stock, this is what we get.
1)Data source: T-Bill data: NYU Stern School of Business, stock, and ETF historical data: Yahoo Finance, all tables, charts, and calculations by Bad Investment Advice.
Now, these are starting to look more like realistic figures for the Sharpe ratio.
We can see straight away that the Standard and Poor’s 500 Index itself has a Sharpe ratio of 1.43 which is pretty good and only just beaten by the Qs, i.e. the NASDAQ 100 with a Sharpe ratio of 1.48. What is really interesting is that in spite of the stellar returns of Apple and Microsoft, because of the variability of those returns they each have Sharpe ratios much closer to 1, i.e. 1.03 and 1.12 respectively.
We should also note the rather poor Sharpe ratios of some industry stalwarts, namely General Electric and Exxon Mobil, with Sharpe ratios of 0.09 and 0.05 respectively, while the Ford Motor Company isn’t much better with a Sharpe ratio of only 0.23.
Sharpe ratio vs returns
Let’s take a look at the Sharpe ratios for each of these ETFs and stocks, plotted against the average and against the compounded returns.
So first the Sharpe ratios plotted against average annual returns.
And now the Sharpe ratios plotted against compounded returns.
There is an important point to absorb here.
Even though many of us would really like to have plowed all our money, and whatever we might have been able to borrow into Apple Inc stock about ten years ago and staying with it through thick and thin, the chances of even an astute investor doing that are in reality pretty remote.
There is always wisdom in hindsight. An investor could equally well have put all their nest egg into either General Electric or Exxon Mobil, then look where they would be today.
The value of diversification
One thing that this analysis does reinforce is the value of diversification. When a portfolio of investments is diversified, you may reduce the returns somewhat by moving out of highly volatile stocks or other investments, but by reducing the variability of the overall portfolio you can improve the risk-adjusted returns of your portfolio.
One of the usual goals of diversification is to reduce the variability of returns by a greater proportion than you reduce the average returns. The Sharpe ratio captures that through its simple formula.
Specific uses of the Sharpe ratio
Fund managers often use the Sharpe ratio to assess the impact of a change in a portfolio. That change could be to remove or add a new asset.
In either case, the fund manager will be looking to assess the impact on both the overall return of the portfolio and on the variability of the portfolio.
The investing public seeks out funds with high risk-adjusted returns. So by improving a fund’s Sharpe ratio, a fund manager can attract more investors.
Mr. William F. Sharpe has received many awards for his work in finance, including the Nobel prize for economics in 1990 for his work on the capital asset pricing model.
Questions and answers
Q. Do you want a low or a high Sharpe ratio?
A. You want the Sharpe ratio of your investments and your portfolio to be as high as possible. Any Sharpe ratio higher than 1 is attractive because your investments or portfolio will outperform the risk-free investment on a risk-adjusted basis. Any Sharpe ratio lower than 1 means your investment or portfolio is underperforming the risk-free investment on a risk-adjusted basis.
Q. What does a Sharpe ratio of 0.5 mean?
A. A Sharpe ratio of 0.5 means that the standard deviation of returns of your asset, or your portfolio is exactly twice the size of the excess returns of the asset or portfolio.
Q. What does a low Sharpe ratio mean?
A. A low Sharpe ratio means that the asset or the portfolio is underperforming risk-free investments on a risk-adjusted basis. A Sharpe ratio below 1 is low.
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|↑1||Data source: T-Bill data: NYU Stern School of Business, stock, and ETF historical data: Yahoo Finance, all tables, charts, and calculations by Bad Investment Advice.|