Was the return worth the market risk?
A fund that returned 12% sounds better than one that returned 9%, but not if it took far more risk to get there. Risk-adjusted performance measures divide the extra return by a measure of risk. The Treynor ratio, introduced by Jack Treynor in 1965, uses beta, the sensitivity of the portfolio to market movements:
Treynor ratio = (Rp − Rf) ÷ β
- Rp is the portfolio return over the period.
- Rf is the risk-free rate over the same period, such as Treasury bills.
- β is the portfolio’s beta against a market index. A beta of 1.2 means the portfolio tends to move 1.2% for each 1% move in the market.
Jensen’s alpha
Michael Jensen proposed a related measure in 1968. The capital asset pricing model (CAPM) predicts the return a portfolio should earn for its beta:
Required return = Rf + β × (Rm − Rf)
Jensen’s alpha is the actual return minus this required return. A positive alpha means the manager added value beyond what market exposure alone would have delivered.
Worked example
A portfolio returned 12% with a beta of 1.2, while the market returned 10% and Treasury bills paid 4%.
- Treynor ratio: (12 − 4) ÷ 1.2 = 6.67% per unit of beta.
- Market’s Treynor ratio: 10 − 4 = 6.00%, since the market’s beta is 1.
- Required return under the CAPM: 4 + 1.2 × 6 = 11.20%.
- Jensen’s alpha: 12 − 11.2 = 0.80%.
The portfolio earned more per unit of market risk than the market itself, so it outperformed on a risk-adjusted basis, though by a modest margin.
Treynor, Sharpe or alpha?
| Measure | Risk used | Best for |
|---|---|---|
| Sharpe ratio | Total volatility | A portfolio that is your whole investment |
| Treynor ratio | Beta (market risk) | A diversified portfolio that is part of a larger one |
| Jensen’s alpha | Beta, as a return difference | Judging a manager against a benchmark |
Limits
- Beta must be positive and estimated over the same period, usually from 3 to 5 years of monthly returns against a relevant index such as the S&P 500.
- A portfolio with low diversification carries specific risk that beta ignores, so the Treynor ratio flatters it.
- One period of outperformance may be luck. Alpha needs to persist over many years, and after fees, to be convincing.
This calculator gives estimates for education and planning. It is not investment, tax or legal advice; for decisions about your own situation, talk to a qualified financial professional.