How to use the treynor, jensen alpha and information ratio calculator
- Choose summary figures if you already have annual numbers, or return series if you have a list of returns for each period.
- For summary figures enter the portfolio return, beta and market return. For series paste the portfolio and benchmark returns in percent and pick the periods per year.
- Enter the risk-free rate as a yearly percent.
- Read the Treynor ratio and Jensen alpha, and with a series the beta, M-squared, tracking error and information ratio.
Formula
Treynor = (portfolio return - risk-free rate) / beta
Jensen alpha = portfolio return - (risk-free + beta x (market return - risk-free))
Tracking error = standard deviation of (portfolio - benchmark) returns
Information ratio = average active return / tracking error
M-squared = risk-free + (portfolio return - risk-free) x benchmark volatility / portfolio volatility
With a series, annual returns are the average period return times the periods per year, and annual deviations are the period deviation times the square root of the periods per year.
A 10% portfolio with beta 1.2
The portfolio returned 10%, the market returned 8%, the risk-free rate is 3% and beta is 1.2. The Treynor ratio is (0.10 - 0.03) / 1.2 = 0.0583.
The CAPM expected return is 3% + 1.2 x (8% - 3%) = 9%, so the Jensen alpha is 10% - 9% = 1.00%. The portfolio beat the return its risk level called for by one percentage point.
Four ways to read a return against risk
The Treynor ratio divides the extra return over the risk-free rate by beta, so it measures the reward per unit of market risk. It suits a portfolio that is one part of a diversified whole. Jensen alpha is the return left over after the CAPM expected return, in percent, and a positive value means the portfolio did better than its beta alone explains.
Tracking error and the information ratio compare a portfolio with a benchmark. Tracking error is the volatility of the gap between them, and the information ratio is the average gap per unit of that volatility, so it rewards consistent outperformance rather than a large gain from a few lucky periods.
Cautions with historical measures
These numbers describe the past. A short series gives wide error in beta and alpha, and the answer depends on the benchmark and on the risk-free rate you choose. Compare portfolios over the same period and against the same benchmark.
M-squared restates the portfolio's return as if it had taken the same volatility as the benchmark, which makes it easy to read in percent. It is informational and not a forecast or a recommendation.
Assumptions and limits
- Returns are simple periodic returns in percent. The risk-free rate is annual.
- Deviations are sample standard deviations. Annual figures scale the averages and deviations as described above.
- Beta from a series is the covariance with the benchmark divided by the variance of the benchmark.
- The example figures are not market data. Nothing is uploaded.
Frequently asked questions
What is a good information ratio?
Higher is better, and it depends on the period and strategy. A value near 0.5 over many years is often regarded as good, but compare against peers rather than a fixed number.
Why does the series need at least three returns?
Beta and the deviations need a spread of values. With fewer points the estimates are not meaningful.
Can the Treynor ratio be negative?
Yes. It is negative when the portfolio returns less than the risk-free rate with a positive beta. A negative beta flips the sign again, so read it with care.
Does the tool predict future performance?
No. All measures use past returns only.