How to calculate the Treynor ratio

What is the Treynor ratio?

The Treynor ratio is a calculation that measures the possible return against the systematic risk of a portfolio. The formula for the Treynor ratio values the average rate of return compared to the risk of the portfolio itself. The ratio’s product is known as the risk-adjusted return. Traders and investors prefer to use the Treynor ratio for well-diversified portfolios or low-risk assets like corporate bonds.

The higher the Treynor ratio of an asset or entire portfolio, the better the risk-to-reward of that security. Generally, you at least want a Treynor ratio above one. A Treynor ratio below one is considered a low risk-to-reward ratio.

What is the Treynor ratio used for?

The Treynor ratio is one way to judge the risk-to-reward ratio of a portfolio. It uses a simple formula to compare the excess rate of return with the standard deviation of the portfolio. The excess rate of return refers to whatever average return is expected beyond the average return of a very-low-risk asset like a gilt.

In this way the Treynor ratio may seem similar to the Sharpe ratio. However, the Treynor ratio compares average returns to a slightly different type of portfolio risk.

Treynor ratio vs Sharpe ratio

Both the Treynor ratio and the Sharpe ratio measure the reward-to-risk ratio of a portfolio, but the Treynor ratio measures systematic, or beta, risks instead of volatility. The beta risk is the minimum amount of risk inherent to a specific portfolio or market.

The Treynor ratio is more useful than the Sharpe ratio when inspecting a well-diversified portfolio. A balanced portfolio structured to with excess risk from volatility already removed is only exposed to systematic risk.

Treynor ratio formula

To calculate the Treynor ratio, take the portfolio’s expected rate of return, R(p) and subtract from it the rate of return for a low-risk asset, R(f), like a gilt. Then divide the difference by the beta risk of the portfolio, B(p).

The formula for the Treynor ratio looks like this: [R(p) – R(f)] / B(p)

Treynor ratio example

For an example of the Treynor ratio in use, let’s compare two hypothetical portfolios. Portfolio A has a return of 7% and portfolio B has a return of 5%. Assume the beta of Portfolio A is 0.9 and the Beta of Portfolio B is 0.7; the current return on government bonds is 3%.

Portfolio A’s Treynor ratio = (7% - 3%) / 0.9 = 0.044

Portfolio B’s Treynor ratio = (5% - 3%) / 0.7 = 0.057

The results show Portfolio B has a slightly higher ratio than Portfolio A. This means Portfolio B has a better risk-to-return ratio and is theoretically a better investment.

What is beta risk?

The beta risk is just the systematic risk of the portfolio, which is typically compared to a market representative. For stocks, this could mean the volatility level of the S&P 500.

To calculate beta risk, divide the standard variation of the security by the standard variation of the market, then multiple the quotient by the correlation of returns between the security and the market.

Beta = Correlation x (standard variation of security / standard variation of market)

Advantages of the Treynor ratio

The main advantage of the Treynor ratio is that it more accurately gauges the risk-to-reward ratio of well-balanced portfolios. Compared to more common risk and reward measurements, the Treynor ratio can provide an accurate measurement when the Sharpe cannot.

Disadvantages of the Treynor ratio

The main disadvantage of the Treynor ratio is that it compares past returns to the current level of risk. There is no way to guarantee a portfolio’s future return, so the Treynor ratio should not be thought of as 100% accurate.

The best way to use the Treynor ratio with this limitation in mind is to use appropriate benchmarks to set the expected rate of return and standardized risk. For some markets, like forex, it can be difficult to find an appropriate beta, or proxy, to pull a standardised risk level from.

One solution is to look at other types of securities with high correlations to the currency, or currencies, you’re trading such as currency indices or stock indices.