Guest post by Shivam Patel

Ever hear a friend rave about a hotshot portfolio manager who doubled his money in a year? Chances are your friend had a bit too much drink or is stretching the truth. If he isn’t though, it still may not be a wise idea to hand over your hard earned money to this market whiz without doing some due diligence. Financial ratios like the Treynor ratio and Sortino ratio can help assess the performance of portfolio managers or even yourself. These ratios can be used in conjunction with the more popular Sharpe ratio in order to provide a more complete picture of investment performance. But first, we need to get some terminology out of the way in order to make sure we have a clear understanding of the following financial ratios.
Terminology

In order to fully understand the Treynor and Sortino ratios, we need to understand what a risk free asset is, what beta is, and what standard deviation is. Common examples of risk-free assets are U.S. Treasury Bonds. Buying these bonds pretty much means that the U.S. government is handing you an IOU. These bonds are backed by the U.S. government and are therefore commonly thought of as the closest thing to a rise free asset. Beta is a measure of the volatility that determines how much a security moves in comparison to the overall market. For example, if the market goes up 10% and a certain security goes up 20%, its beta would be 2. Another way analysts assess risk is by using standard deviation. Standard deviation tells us how much the investor’s performance deviated from the mean. We can compute the standard deviation by the following formula:

•    x is return of the portfolio over a year(we can calculate at the end of every month)
•    𝜇is the annual historical rate of return
•    𝛴is summation
•    N is the number of times we recorded the data (12 in this case)
The understanding of these concepts will allow us to examine the Treynor and Sortino ratios.

Treynor Ratio

The Treynor ratio was created by Jack Treynor, a respected economist who sought to analyze the performance of portfolio managers. The Treynor ratio attempts to isolate the returns that are a result of risk taking and dividing that by the beta of the portfolio. Mathematically, this would translate into

•    T is the Treynor ratio
•    ri is the portfolio return
•    rf is the return of the risk-free rate(normally Treasury Bills)
•    βi is the beta of the portfolio
The higher the ratio, the better a money manager has done in terms of risk-adjusted returns. The biggest drawback to the Treynor ratio is that it uses beta as a gauge for determining the volatility of an investment. This means that upside movement is penalized and contributes to a higher beta. In order to combat this a new ratio was created, the Sortino ratio.

Sortino Ratio

In order to only punish downside volatility, Dr. Frank A. Sortino created the Sortino Ratio. Another way that this ratio differs from others is that instead of using the risk free rate, this ratio utilizes the target rate of return, which is set by an individual investor. This way, an investor can set the target rate of return to a number that will allows them to meet certain financial goals (such as outperforming a benchmark). The equation for the Sortino ratio is as followed:
•    S is the Sortino ratio
•    ri is the expected rate of return for a given portfolio
•    rt is the target rate of return
•    σn is the downside standard deviation

Like the Treynor ratio, the higher the number the better.

Treynor/Sortino Ratio Example

Now, let’s go through an example of how the treynor ratio and sortino ratio can help us make a better investment decision.

Let’s, for a moment, assume you are an investor looking at two companies:

Amy’s AutoShop and Carl’s Coffeeshop. Carl’s Coffeeshop has higher returns over the past year than Amy’s AutoShop, but you notice that there are a lot of ups and downs in the stock price for Carl’s Coffeeshop. In order to understand how well the stock is doing when taking into account its volatility, we can use the treynor ratio and sortino ratio.

Let’s assume that return for Carl’s Coffeeshop and Amy’s Autoshop are 30% and 10% respectively. Let us also assume that the beta for Carl’s Coffeeshop is 2.5 and that the beta for Amy’s Autoshop is 0.5. Finally, let us assume that the risk free rate is 2%

We can then plug these values into the treynor ratio equation (shown above):

Note: Return values are converted to decimals for both the treynor ratio and sortino ratio. Ex. 30% will become 0.3 and 10% will become 0.1.

Carl’s Coffeeshop: T = (0.3 – 0.02) / 2.5 = 0.112

Amy’s Autoshop: T = (0.1 – 0.02) / 0.5 = 0.16

In this case, even though at first glance we would think Carl’s Coffeeshop would be the better investment since in has a higher rate or return, the treynor ratio is telling us that when accounting for volatility, Amy’s Autoshop is the better investment.

With treynor ratio: Amy’s Autoshop > Carl’s Coffeeshop

But what if the high beta of Carl’s Coffeeshop was caused by significant movements to the upside?

If this were the case, then the treynor ratio may have given us an incomplete picture by unnecessarily penalizing the huge movements of Carl’s coffeeshop to the upside. That’s where the sortino ratio comes in handy. Let us have the same assumptions as above and assume that the computed downside deviation for Carl’s Coffeeshop and Amy’s Autoshop are 15% and 10% respectively.

We can plug the values into the sortino ratio(shown above).

Carl’s Coffeeshop: S = (0.3 – 0.02) / 0.15 = 1.87

Amy’s Autoshop: S = (0.1 – 0.02) / 0.1 = 0.8

Wow! That is a very different result than what was given by the treynor ratio.

With sortino ratio: Carl’s Coffeeshop > Amy’s Autoshop

So which investment do we go with? Well, both ratios have their strengths and are used by finance professionals. In this case though, it is pretty clear that Carl’s coffeeshop is the better investment. Although the treynor ratio showed us that Carl’s Coffeeshop was underperforming compared to Amy’s Autoshop, the treynor ratio penalized Carl’s Coffeeshop for its big spikes upward. The sortino ratio took this into account and showed us that Carl’s Coffeeshop outperforms Amy’s Autoshop.

When used correctly, financial ratios can glean useful information on an asset manager’s performance. To learn about other financial ratios, check out this article on the Sharpe ratio.

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