Treynor Ratio

The Treynor ratio asks the same question the Sharpe Ratio does — how much return am I getting for the risk I'm taking? — but it measures risk differently. Sharpe uses total volatility. Treynor uses only beta, the slice of risk that moves with the market and that diversification can never remove. The idea is that once you hold a well-spread portfolio, the wobble unique to any one stock washes out, and the only risk you're really left carrying is market risk. So that is what Treynor grades you on.

The formula is a simple ratio:

$$T = \frac{R_p - R_f}{\beta_p}.$$

Here $R_p$ is the portfolio's average return, $R_f$ is the risk-free rate (what you'd earn holding cash or T-bills), so $R_p - R_f$ is the excess return — the reward for taking risk at all. The denominator $\beta_p$ is the portfolio's Beta (β): how sensitive it is to the market. A beta of 1.2 means the portfolio tends to swing 20% more than the market. Treynor is the excess return earned per unit of that market sensitivity.

Geometrically, Treynor is the slope of the line from the risk-free rate to your portfolio when you plot return against beta. A steeper line means more reward per unit of market risk — a better Treynor.

Worked example

Two funds, with the risk-free rate at 2%.

• Fund A: average return 12%, beta 1.2. Treynor $= (12 - 2)/1.2 = 10/1.2 = 8.3\%$.
• Fund B: average return 9%, beta 0.6. Treynor $= (9 - 2)/0.6 = 7/0.6 = 11.7\%$.

Fund A earned the bigger raw return, but Fund B delivered far more excess return per unit of market risk — its Treynor is higher. If you were going to blend either fund with cash or leverage to hit a target risk level, B is the better engine: lever B up to A's beta and it would out-return A. That is exactly the comparison Treynor is built to make, and it is the same logic behind the The Capital Asset Pricing Model (CAPM) security market line — everything gets priced by its slope from the risk-free rate.

Where it misleads

• Beta must be trustworthy and diversified. Treynor only makes sense for a portfolio whose stock-specific risk is already diversified away. Run it on a single stock or a three-name book and you're ignoring most of the real risk; use Sharpe there instead.
• Small betas blow it up. If $\beta_p$ is near zero the ratio explodes, and a market-neutral book (beta ≈ 0) has an almost meaningless Treynor.
• Sign traps. A negative excess return divided by a negative beta produces a positive Treynor that looks good but isn't. Always check the raw excess return's sign before reading the ratio.
• It's a one-factor view. Beta captures market risk only. A portfolio loaded on size, value, or momentum carries risks the single market beta never sees (see The Fama-French Factor Models).