Return without risk is half a sentence. Five measures finish it, and although they are usually taught as a list to memorise, they are better understood as one numerator over five different denominators. The denominator is the entire argument.
Throughout, use a single data set: twelve quarters of portfolio and benchmark returns (Worked Example 4), from which the annualised figures are portfolio 10.80%, benchmark 9.13%, risk-free 4.00%, portfolio standard deviation 13.28%, benchmark standard deviation 9.35%, beta 1.3920, tracking error 4.51%.
The five measures, with formulas and numbers
1. Sharpe ratio = (RP − Rf) / σP = (10.80 − 4.00) / 13.28 = 0.512 (benchmark 0.549). Excess return per unit of total risk: the right measure when the portfolio is the investor's whole portfolio, because then every wobble matters. It treats upside and downside deviation identically, is meaningless when the numerator is negative (a more volatile loser scores higher), and is not comparable across measurement frequencies unless both sides annualise the same way.
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2. Treynor ratio = (RP − Rf) / βP = (10.80 − 4.00) / 1.3920 = 4.89% (benchmark 5.13%). Excess return per unit of systematic risk: the right measure when the portfolio is one sleeve inside a diversified whole, so idiosyncratic risk is diversified away by the other sleeves. It assumes a single-factor world (M3.02's CAPM), a beta that is neither stable nor unique across estimation windows, and a denominator that is nonsense near zero or negative beta.
3. Jensen's alpha = RP − [Rf + βP(RB − Rf)] = 10.80 − [4.00 + 1.3920 × (9.13 − 4.00)] = −0.35%. Not a ratio but a difference: realised return minus what the security market line demanded for that beta. This is M3.02's SML used as a scoreboard, and it is scale-dependent (double the positions and alpha doubles), so it is not comparable across portfolios without dividing by beta, which returns you to Treynor.
4. Information ratio = (RP − RB) / TE, tracking error being the standard deviation of the active return series = 1.667 / 4.515 = 0.369. Active return per unit of active risk: the measure of a benchmark-relative mandate, and the one institutional allocators live by. Near 0.5 is respectable, near 1.0 exceptional; a sustained IR above 1.0 should send you to check the benchmark before the manager. Two live conventions: some houses put a risk-free-adjusted excess return in the numerator, some compute tracking error about zero rather than about the mean. State yours.
One more collision is worth naming here, because it costs exam marks and client credibility. The portfolio literature, and this node, define the information ratio as active return over tracking error, a property of a portfolio against a benchmark. The signal-construction literature, and QD1.02 section 8, use the same two words for the mean information coefficient divided by the standard deviation of the information-coefficient series, a property of a forecast against subsequent returns. The two numbers are not close and are not convertible: 0.369 here is a twelve-quarter record of a mandate; an information coefficient ratio of the same size describes how stable a signal's forecasting power has been. Which one is meant is settled by what sits in the denominator, so read the denominator before quoting the number. The fundamental law of active management consumes the forecast version, and AA1.07 shows numerically what happens when the portfolio version is fed into it by mistake.
5. M² (Modigliani–Modigliani) = Rf + Sharpeₚ × σB = 4.00 + 0.512 × 9.35 = 8.79%. The Sharpe ratio translated into percentage points: what the portfolio would have returned levered or de-levered to the benchmark's volatility. It exists for communication: a trustee who cannot read "0.512 versus 0.549" understands "8.79% versus 9.13% at the same risk" instantly. Being a monotone transform of Sharpe, M² can never disagree with Sharpe in rank order; a table showing otherwise is wrong. The M²-alpha, M² − RB = −0.35 points, is the risk-matched shortfall.