Sharpe vs Treynor vs Information Ratio: How to Tell Which One the Question Wants

Risk-adjusted performance measures should be a gift on the CFA exam. The formulas are short, the arithmetic is trivial, and most questions are solvable inside ninety seconds. Candidates still drop marks on them, almost always for one reason: they compute the right ratio for the wrong question.

The distinguishing feature of each measure is its denominator. Get that straight and everything else follows.

The four measures

Sharpe ratio

(R_p − R_f) / σ_p

Excess return per unit of total risk. The denominator is portfolio standard deviation, capturing systematic and unsystematic risk together. Because it charges for all risk, it implicitly penalises poor diversification.

Treynor ratio

(R_p − R_f) / β_p

Excess return per unit of systematic risk. Identical numerator to Sharpe; the denominator switches to beta. It assumes unsystematic risk has already been diversified away elsewhere and therefore should not be charged for.

M-squared

M² = (R_p − R_f) × (σ_m / σ_p) + R_f

The Sharpe ratio expressed as a percentage return rather than a bare number. It answers: what would this portfolio have returned if levered or de-levered to match the market's volatility? Because it is a monotonic transformation of Sharpe, it always produces the same ranking. Its advantage is interpretability — "this portfolio is equivalent to earning 11.4%" communicates more than "its Sharpe ratio is 0.52".

Information ratio

(R_p − R_b) / σ_{(Rp − Rb)}

Active return per unit of active risk. The numerator is return relative to a benchmark, not the risk-free rate. The denominator is tracking error — the standard deviation of the active return series, not of the portfolio itself. Substituting R_f for R_b is the standard error here and produces a plausible-looking wrong answer.

Choosing the right measure

The selection rule turns on a single question: is this portfolio the investor's entire holding, or one component within a larger one?

Whole portfolio → Sharpe. The investor is exposed to all of its risk, systematic and unsystematic alike, so total risk is the relevant denominator. A poorly diversified portfolio should be penalised, and Sharpe does the penalising.

One sleeve of a diversified portfolio → Treynor. The unsystematic risk of a single component is largely diversified away at the total-portfolio level, so only its systematic contribution matters. This is the classic exam setup: an investor considering adding a fund to an already diversified holding. The phrase "already well diversified" in a vignette is effectively an instruction to use Treynor.

Manager measured against a benchmark → information ratio. Any question involving active management, tracking error, or a mandate to beat an index is an information ratio question. Sharpe and Treynor say nothing about benchmark-relative skill.

A worked comparison — and what disagreement means

Two funds, risk-free rate 3%:

• Fund A: return 11%, standard deviation 16%, beta 1.1
• Fund B: return 9%, standard deviation 9%, beta 0.9

Sharpe A = (11 − 3) / 16 = 0.50  |  Sharpe B = (9 − 3) / 9 = 0.67

Treynor A = (11 − 3) / 1.1 = 7.27  |  Treynor B = (9 − 3) / 0.9 = 6.67

The rankings disagree, and that disagreement is the entire point of the question. On total risk, B wins. On systematic risk, A wins.

Reading the disagreement

This is the part most treatments skip. When Treynor ranks a fund better than Sharpe does, the fund carries substantial unsystematic risk — its total risk is high relative to its market exposure. In other words, it is under-diversified.

Fund A has 16% total volatility on a beta of 1.1, while Fund B has 9% on a beta of 0.9. Scaling roughly, A carries far more risk than its market exposure explains. That residual is concentration, sector bets, or single-name exposure.

So the correct answer depends entirely on the investor:

• Standalone investor → Fund B. They bear the concentration risk with nothing to offset it.
• Investor slotting one fund into a diversified portfolio → Fund A. The concentration washes out at the total level, and A delivers more return per unit of the risk that survives.

Whenever a question supplies data for both measures, expect the rankings to conflict. Being asked to compute both and then choose is a test of whether you understand the denominators, not of arithmetic.

Traps worth knowing

Negative excess returns break the Sharpe ratio. When the numerator is negative, a larger standard deviation produces a less negative ratio. A riskier portfolio can therefore appear to rank better. The measure loses comparative meaning in this range, and questions do test whether you know it. If two portfolios both underperformed the risk-free rate, Sharpe cannot rank them.

Sharpe does not directly measure diversification. It charges for total risk, but a concentrated portfolio that happened to perform well can still post an attractive Sharpe. Comparing it against Treynor is what exposes the gap.

Tracking error is not portfolio standard deviation. The information ratio denominator is the volatility of the difference between portfolio and benchmark returns. A portfolio can be highly volatile in absolute terms while tracking its benchmark closely.

M-squared never re-ranks. If asked whether M² could order two portfolios differently from Sharpe, the answer is no. It rescales; it does not reorder.

Annualisation is not linear. A monthly Sharpe ratio is annualised by multiplying by √12, not by 12, because returns scale with time while standard deviation scales with the square root of time. Questions occasionally supply monthly data and expect an annual answer.

Treynor requires a meaningful beta. For a portfolio with negligible or unstable market correlation, beta is close to noise and the Treynor ratio becomes unreliable. Sharpe has no such dependency.

Where these appear across the three levels

Sharpe and Treynor are introduced in Level 1 Portfolio Management alongside CAPM, which supplies the beta that Treynor needs — which is why the two topics are usually studied together.

The information ratio becomes considerably more important at Levels 2 and 3, where it underpins the fundamental law of active management and the transfer coefficient. In that framework the ex-ante information ratio is approximately the transfer coefficient multiplied by the information coefficient and the square root of breadth — a result that only makes sense if you already understand what the information ratio measures.

Learning the distinction properly at Level 1 therefore pays twice, and the Level 3 material assumes you did.