# Treynor Ratio

Treynor ratio is a measure of investment return in excess of the risk-free rate earned per unit of systematic risk. It is calculated by finding the difference between the average portfolio return and the risk-free rate and dividing it by the beta coefficient of the portfolio.

Treynor ratio is also called Treynor measure. It is a modification of the Sharpe ratio. It has the same numerator as the Sharpe ratio, i.e. portfolio return minus risk free rate. However, Treynor ratio measures excess return with reference to the systematic risk (i.e. beta coefficient) instead of total risk (i.e. standard deviation).

## Formula

Treynor ratio is calculated as follows:

$$ Treynor\ Ratio\\=\frac{Average\ Portfolio\ Return-Risk\ Free\ Rate}{Portfolio\ Beta\ Coeffient} $$

Average portfolio return is calculated as the average of individual investment returns based on their respective weights.

Risk free rate is the rate of Treasury bonds whose maturity matches average maturity of the portfolio assets.

Portfolio beta is also a weighted average of individual investment betas.

## Example

You have three stocks in your portfolio:

Stock | Value | Return | Beta |
---|---|---|---|

Stock A | $20,000 | 8% | 1 |

Stock B | $35,000 | 12% | 1.5 |

Stock C | $25,000 | 4% | 0.75 |

Calculate your portfolio’s Treynor ratio if the risk-free rate is 3.5%.

We first need to find weights of each stock in the portfolio:

$$ Weight\ of\ Stock\ A\\=\frac{$20,000}{$80,000}=25\% $$

Weights of Stock B and C work out to 43.75% and 31.25%.

Average return on the portfolio is the weighted average of individual stock returns:

$$ Portfolio\ Return\\=25\%\times8\%+43.75\%\times12\%+31.25\%\times4\%\\=8.5% $$

Portfolio beta is the weighted average of the individual stock betas:

$$ Portfolio\ Beta\\=25\%\times1+43.75\%\times1.5+31.25\%\times0.75\\=1.14 $$

Now, we have all the inputs needed to calculate Treynor ratio:

$$ Treynor\ Ratio\\=\frac{8.5\%-3.5\%}{1.14}\\=4.39\% $$

Treynor’s ratio is better than Sharpe ratio because it excludes the unsystematic (i.e. diversifiable) sources of risk and only considers the systematic risk.