# Treynor ratio

The Treynor ratio measures the return differential that the portfolio or fund obtains over the risk-free asset per unit of risk, considering the risk by the Beta coefficient.

As a risk-free asset, the reference of a government bond is usually used, in the case of Europe the German Bund is used or in the United States the American Treasury bond.

## Treynor ratio formula

The Treynor ratio is calculated as follows:

Rp: Portfolio profitability.

Rf: Return on the risk-free asset.

βp: Beta of the portfolio.

It is therefore a ratio that measures excess return (defined as the difference between the portfolio's average return and the risk-free rate), earned per unit of systematic risk (beta).

The systematic risk is that risk that affects the market as a whole measured by the Beta, on the other hand, the unsystematic risk is that risk that affects the security or action in question.

Total risk = Systematic risk + Unsystematic risk

It should be noted that, if the calculation period decreases (for example from annual to monthly), the numerator of the Treynor Ratio will become smaller, but the denominator (beta) will remain the same, unchanged. Therefore, the relationship is direct, the Treynor ratio decreases as the calculation period decreases.

## Sharpe ratio and Treynor ratio

Both are performance or behavior ratios (to measure how an investment fund does it)* *and with them you can make rankings to choose if one portfolio is better than the other.

For well-diversified portfolios (after correct diversification, non-systematic risk is eliminated -in practice it is very difficult-), the ranking of the portfolios applying the Treynor Ratio should be the same as applying the Sharpe ratio. However, for undiversified portfolios, the ranking varies.

The Treynor ratio should not be used as a measure of *performance *in an independent way. In that case, the investment or portfolio in question must be valued for its total risk. That is, by the Sharpe Ratio and not by the Treynor Ratio, as this is appropriate when comparing well diversified portfolios.

## Treynor Ratio Example

Let's imagine that Pedro is the manager of an investment fund and has obtained a return of 14% during the past year, while Javier, manager of another investment fund, has achieved an 8% return in the same year.

At a glance, we can say that Pedro has managed the fund's assets more efficiently and has achieved higher returns (14% versus 10%).

We are going to find out which of the two has been better, for this we are going to use the Treynor ratio.

Assuming we are in Europe, the risk-free asset (Rf) that we are going to use is the German 10-year bond, which has obtained an average interest of 1.4%. We also need to know the Beta of both managers. If Pedro had a Beta of 1.2 last year and Javier a Beta of 0.6, their respective ratios are as follows:

Pedro: TR = (14-1.4) / 1.2 = 10.5

Javier: TR = (10-1.4) / 0.6 = 14.3

Based on these results, we can affirm that Javier has achieved higher profitability according to the risk taken. In fact, taking this ratio, it can be said that Javier "has played" to have less Beta (less exposure to the market) and above all, he has a higher ratio than Pedro.

Seen from another point of view, according to this ratio Javier has obtained more profitability bearing less risk.

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