Fri 5 Jan 2007

**Who:**This post is for people who want a way to quantify reward vs. risk in their investments or for people who know what a Sharpe Ratio is and want to learn more about it.**
What:** This post will discuss the meaning, calculation, and the usefulness of the Sharpe Ratio. The Sharpe Ratio is a measure of the reward to risk of a given investment.

**What is this ratio of which you speak?**

The Sharpe Ratio is a measure of how well an investment compensates an investor for the risk he/she takes on. First, I will show you the formula for the Sharpe Ratio and then I will go through a simple example. After that, I will discuss the conceptual meaning behind the ratio.

R is the expected return on the asset that we are evaluating. Rf is the risk-free rate which can be characterized in many ways. basically Rf refers to the rate of return one can achieve without taking on any significant (i.e. extremely low) risk. The big letter that we call E represents an expected value. In other words, the numerator of this ratio is the expected return that an asset is expected to give you *above and beyond* the risk free rate Rf. Finally, the denominator is the standard deviation of R. More specifically, the standard deviation of R is the square root of the variance of R, a measure of the volatility of the returns of the asset we are evaluating. I know that was a mouthful, but an example should illustrate how easy this calculation actually is. Don’t understand? It’s ok, check out the Sharpe Ratio Calculator.

**A Financial Revolution Loves Examples**

Assume we have a mutual fund that we *expect* to return 15% over the course of the next year. The mutual fund’s volatility is 10%, and the risk free rate that I can get at Emigrant Direct is 5.05%. *Note: For you picky folks, yes, we’re purposely ignoring tax consequences here, for simplicity.*

The Sharpe Ratio = (15.00% - 5.05%) / 10.00%

The Sharpe Ratio = 9.95% / 10.00%

The Sharpe Ratio =0.995

Voila, a piece of cake. What does 0.995 mean? By itself, not much. The Sharpe Ratio is meant as a method for *comparing* different risk/reward options. So, let’s try it out. If I offered you the investment that we highlighted in our example or another investment option which has a 20% volatility (double the volatility) and a 27% return (less than double the return). Which investment would you pick? Let’s find out.

The Sharpe Ratio = (27.00% - 5.05%) / 20.00%

The Sharpe Ratio = 21.95% / 20.00%

The Sharpe Ratio =1.098

As you can see, the second investment has a higher sharpe ratio. What this is saying is that the second investment, while riskier, provides a better risk-to-reward ratio than the first invesment does. *In general, the higher the sharpe ratio, the better an investment compensates it’s investors for risk taken*.

**Is it Magic? Do I Wield Unstoppable Power?**

Unfortunately, no. The Sharpe Ratio, like most things in life, is far from perfect. While I am a big fan of the Sharpe Ratio, I would be remiss if I did not mention the drawbacks to using the ratio and the caveats that one must keep in mind when using it as a part of an investment decision.

- The Sharpe Ratio should not be used as a blanket approach to picking investments. In our example, even though the 2nd investment has a higher Sharpe ratio, a risk-averse investor may still want to invest in the first choice. Sharpe Ratios should be used to compare investments that fit WITHIN your risk and return profiles.

- The effectiveness of the Sharpe Ratio is based on the (hotly debated) effectiveness of standard deviations as a measure of volatility. The mathematics behind it is rather complicated, but many argue that the standard deviation of an investment’s returns are not necessarily a good measure of it’s volatility. This is not to say that there aren’t many, many, individuals who have strong arguments as to why the standard deviation *is* a good measure for volatility

- An exact calculation of the Sharpe Ratio is not forward looking, and forward looking calculations of the Sharpe Ratio are estimates and projections. To calculate a Sharpe Ratio for the past performance of a fund, one must use the return and standard deviation over the previous time period in order to calculate the exact Sharpe Ratio. The values of these two quantities are unknown for the future, so any forward-looking Sharpe Ratio will be based on projections of the returns and standard deviations, which are subject to large variance.

**Basically, this post was useless?**

Not at all! Even though the Sharpe Ratio has some shortcomings, it does have many uses and advantages over other metrics.

- The Sharpe Ratio is an excellent tool to compare multiple investments that are all within your risk tolerance. Comparing the Sharpe Ratios of such investments will give you an excellent idea of how much each investment is compensating you for your risk as compared to the other investments.

- The Sharpe Ratio allows comparisons of investments over multiple sets of assumptions. There may be instances in which some part of an investment have a given risk-free rate and another part has a different risk-free rate. The inclusion of a risk-free rate parameter allows one to use the Sharpe Ratio to compare these options in different “universes” or circumstances.

- The Sharpe Ratio is arguably more versatile than other metrics such as Alpha and Beta (these will be discussed in a future post). While Beta needs to be based off a specific index and the precision of Alpha is based on a high R^2 (also to be discussed in a future post) value, the Sharpe Ratio has no such restrictions. A Sharpe Ratio can be used to compare different types of investments (stock funds and bond funds) using the same assumptions.

So there you have it! Does anyone have other financial metrics that they would like covered on A Financial Revolution? Any thoughts on the Sharpe Ratio or anything I may have missed or omitted? Sound off and let me hear about it.

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### 17 Responses to “Sharp Minds Love the Sharpe Ratio”

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January 6th, 2007 at 11:39 am

I hadn’t heard about the Sharpe Ratio till I came across your article. Good analysis. Like you say, such ratios should be taken with a pinch of salt and should be used as a general guide only.

January 8th, 2007 at 8:34 am

[…] Sharp Minds Love the Sharpe Ratio posted at A Financial Revolution. […]

January 8th, 2007 at 1:31 pm

[…] Do you understand the Sharpe Ratio? […]

January 8th, 2007 at 5:53 pm

[…] Do you understand the Sharpe Ratio? […]

January 8th, 2007 at 8:50 pm

I would emphasize for the less mathematically inclined that the standard deviation of volatility is not a boundary on how much the returns can vary, just an indicator of the most probable range.

A standard deviation of 20% on an expected return of 27% still leaves a significant chance of losing lots of money. That’s where the risk tolerance gets into it — if your investment failed would you end up on the street?

January 8th, 2007 at 9:41 pm

EMF, thanks for the important caveat.

To expand upon it a little bit, mathematically speaking, one standard deviation represents the middle 68% of possible returns.

If an investment has a mean return of 10% and a standard deviation of 8%, this means that on AVERAGE, the asset’s return will be between 2% and 18% roughly 68% of the time.

As EMF astutely pointed out, there is another 32% of the time where the assets return can be outside of this range.

Also, as I hinted to in this post, many people do not believe the standard deviation is a good measure of risk. More on that in a future post. Remember, no metric is perfect :-)

January 9th, 2007 at 4:12 pm

hey, great post. i learned all of those measurements (sharpe, jensen, treynor, etc.) in class last semester but i never really understood the practicality of it until i read this. keep up the great work. i am eagerly anticipating your posts on alpha and beta.

January 16th, 2007 at 12:22 am

Jan - Week #2 - Round Up Of Carnivals And Festivals!…This week we started publishing our posts at Carnival of Investing in addition to our regular publications at Carnival of Personal Finance and Festival of Frugality. Without much ado let us get down to covering them.

…

January 22nd, 2007 at 3:23 am

THANK YOU FOR YOUR POST!

Would you give me a real example of the Sharp Ratio’s application in the financial market?

Thank you very much!

March 7th, 2007 at 4:54 am

Did you hear about the sharper ratio than the sharp ratio? The Omega ratio!

/ Return distributions are more often than not non-normal.

Thus; the Omega ratio which sums up the area under the cumulative return function each seperated by a return threshold, and divides the area above the threshold to the area below the threshold. Hence, all particularities of the return distribution are taken into account and apples & oranges become comparable groups.

April 11th, 2007 at 6:36 pm

As a investment professional I use the sharpe ratio constantly and feel it’s important to understand and this site does a wonderful job of explaining it. I found it very useful to send to friends and clients.

June 18th, 2007 at 2:23 am

[…] A Financial Revolution » Sharp Minds Love the Sharpe Ratio Who: This post is for people who want a way to quantify reward vs. risk in their investments or for people who know what a Sharpe Ratio is and want to learn more about it. http://www.afinancialrevolution.com/2007/01/05/sharp-minds-love-the-sharpe-ratio/ […]

June 18th, 2007 at 3:25 am

[…] William Wallets’ best article at A Financial Revolution is all about The Sharpe Ratio, a means of calculating the risks and rewards of a given investment. […]

July 9th, 2007 at 8:16 am

Great article about the sharpe ratio. The sharpe isn’t perfect but provides a good way of comparing risk adjusted returns between investments or portfolios.

July 16th, 2007 at 2:00 am

Hi , really good and simple in explaining the complex theory. Thanks a lot

November 23rd, 2007 at 6:44 am

Why is it necessary to deduct the risk free rate (is it not constant for all expected returns)?

Does using the expected return (of past return)/stdev not give the same effect?

February 6th, 2008 at 3:26 am

Are you familiar with the risk measurment tool Value at Risk (VaR)? Which tool is better in measuring risk return, VaR or Sharpe ratio? VaR used volatility of the market price while Sharpe uses the volatility of the yield of the portfolio?

Thanks