# Treynor ratio formula and calculator

As part of our free investments course, we have already covered two important investment performance measures: Jensen’s alpha and Sharpe ratio. In this post, we focus on the Treynor ratio, which is another popular risk-adjusted performance measure. In particular, we explain the Treynor ratio formula and offer an easy-to-use Treynor ratio calculator as well.

## Treynor ratio formula

The Treynor ratio formula for an asset i can be written as follows:

Treynor ratio for asset i = (Return on asset i − Risk-free rate) / Beta of asset i

Or, using a mathematical notation, we have:

Ti = (Ri − Rf) / βi

where Ti is the Treynor ratio of asset iRi is the return on asset iRf is the rate of return on a risk-free asset, and βi is the beta of asset i.

To give a specific example, suppose that ABC Inc. yielded a return of 12% last year. Over the same period, the yield on treasury bills, which we take as a proxy of the risk-free rate, was 3%. Let’s also assume that ABC’s beta was 2. Then, we can easily calculate the company’s Treynor ratio last year as:

Ti = (12% 3%) / 2 = 4.5%

This means that ABC yielded a return of 4.5% for each unit of systematic risk borne. We can interpret this as the “realized Treynor ratio.”

Now, let’s imagine that ABC’s beta will remain as 2, its expected return is 15%, and the risk-free rate for the next year is 5%. Then, the company’s “expected Treynor ratio” would be:

Ti = (15% 5%) / 2 = 5%

If another company called DEF Inc. has an expected Treynor ratio of 3%, then we can consider ABC Inc. as a better investment opportunity than DEF Inc. on the basis of their expected Treynor ratios. This is because ABC offers a higher reward than DEF (5% vs 3%) per unit of systematic risk carried.

## Treynor ratio interpretation

Treynor ratio is a measure of how much ‘excess return’ (i.e., return above the risk-free rate) a security (stock, bond, mutual fund, etc.) offers per unit of systematic risk, which is captured by beta. In that sense, it is a reward-to-risk ratio like the Sharpe ratio.

We can also interpret the Treynor ratio visually by means of a plot of (systematic) risk against return. In Figure 1, we have two risky assets (e.g., stocks or funds): A and B, and the risk-free asset, which yields Rf.

For both A and B, we can draw a line that connects the risky asset with the riskless one. Then, the Treynor ratio is simply the slope of that line. In this case, asset B has a higher Treynor ratio than asset A because of the steeper slope.

## Treynor ratio calculator

In order to use the Treynor ratio calculator, you would need to input:

• The return on the risky asset.
• The return on the riskless asset.
• The beta of the risky asset.

SUMMARY

The Treynor ratio, also known as the Treynor measure, was developed by the American economist Jack Treynor in the mid-1960s (you can find the reference for his seminal paper below). To this date, it remains one of the most popular reward-to-risk measures that is used by practitioners and academics alike. It tells us about the excess return an asset yields per unit of systematic risk borne. Therefore, the higher it is, the better.