# Treynor ratio

The Treynor ratio, developed by Jack Treynor in 1965, is a measure of the return on a portfolio in excess of the return on a risk-free investment in relation to systematic risk. It is similar to the Sharpe ratio, as discussed in February’s issue, except that it uses systematic risk, or beta, as a measure of volatility, rather than standard deviation. Systematic risk, also known as un-diversifiable risk or market risk, covers risks such as interest rates and recession that affect the entire market and cannot be mitigated by diversification, only by hedging.

### How is it calculated?

Like the Sharpe ratio, the Treynor ratio uses the excess return, which is the difference between the return on the portfolio in question and the return on a risk-free investment. This is then divided by the portfolio’s systematic risk:

• Return on portfolio, ${R}_{p}$.

• Return on a risk-free investment, ${R}_{f}$.

• Excess return, $\left({R}_{p}-{R}_{f}\right)$.

• Portfolio volatility, ${ß}_{p}$.

The measure of risk used, beta $ß$, is systematic risk. A portfolio’s beta is a measure of how susceptible it is to changes in the market as a whole. The price of a portfolio with a beta of 1, for example, will move with the market. A beta of 0.5 indicates a portfolio that is 50% less volatile than the market, and a portfolio with a beta of 1.5 will be 50% more volatile than the market.

### Example

A company wants to monitor the performance of its investment manager who has produced returns of 20%, by making an adjustment for the level of risk he has taken on. The beta of the portfolio is 1.4. The risk-free rate of return is 5%:

So, whilst the portfolio produced a good return, much of that was attributable to a relatively high level of risk rather than superior investment decisions.

### Points to consider

Since the Treynor ratio uses beta as its measure of portfolio volatility, it does not take into account any other risk besides market risk. It does not allow for unsystematic risk, for example, which is specific to each investment and must be mitigated through diversification. As a result of this limitation, Treynor went on to develop his ideas further with the Treynor-Black model.

### Treynor-Black model

The Treynor-Black model was developed by Jack Treynor in collaboration with Fischer Black in 1973. The purpose of this model is to determine the most advantageous asset allocation in an investment portfolio by assessing unsystematic risk in addition to systematic risk. Unsystematic risk is company or industry-specific risk that directly affects each investment in that company or industry. So, whilst the Treynor ratio neglects the issue of the need for a diversified portfolio, the Treynor-Black model provides a measure of return that, by taking unsystematic risk into account, rewards diversification.

The Treynor-Black model involves the investment manager being able to predict the abnormal performance of a security (alpha α), that is its return over and above that explained by its beta and the security market line. An optimal portfolio will consist of two parts: the passively managed or market portfolio based on the expected risk and return parameters of the market as a whole, and the actively managed portfolio which contains those securities for which the investment manager has made predictions about alpha.

Each security in the actively managed portfolio is weighted according to the ratio of its alpha to its unsystematic risk. This then allows the investment manager to calculate the ideal ratio of actively to passively managed assets in a portfolio. Treasurers can use the Treynor ratio (in conjunction with other metrics) in order to keep track of the level of their investments’ exposure to risk. Systematic or market risk, in particular, is a key consideration in volatile markets.