How do Sharpe ratio, Treynor ratio, and information ratio differ as risk-adjusted performance measures?

All three ratios measure risk-adjusted performance, but they differ in what risk they measure and what return they evaluate. Understanding when each is appropriate is critical for the CFA Level III exam.

The Three Measures:

1. Sharpe Ratio = (R_p - R_f) / σ_p

• Numerator: Excess return over the risk-free rate
• Denominator: Total risk (standard deviation of portfolio returns)
• Measures: Reward per unit of total risk
• Best for: Evaluating a standalone portfolio or the investor's entire wealth

2. Treynor Ratio = (R_p - R_f) / β_p

• Numerator: Excess return over the risk-free rate
• Denominator: Systematic risk only (beta)
• Measures: Reward per unit of market risk
• Best for: Evaluating a portfolio that is part of a broader diversified allocation (unsystematic risk is diversified away)

3. Information Ratio = (R_p - R_b) / σ(R_p - R_b)

• Numerator: Active return (alpha, excess over benchmark)
• Denominator: Tracking error (volatility of active returns)
• Measures: Consistency of alpha generation
• Best for: Evaluating active managers relative to their specific benchmark

Side-by-Side Comparison:

Numerical Example:

Manager at Dalton Partners earned 12.0% with σ = 15%, β = 1.1. Benchmark returned 10.0% with σ = 12%. Risk-free rate = 3.5%. Tracking error = 5.2%.

• Sharpe = (12.0% - 3.5%) / 15% = 0.567
• Treynor = (12.0% - 3.5%) / 1.1 = 7.73%
• Information = (12.0% - 10.0%) / 5.2% = 0.385

Interpretation:

• The Sharpe ratio of 0.567 says Dalton earned 56.7 bps of excess return per unit of total risk
• The Treynor ratio says Dalton earned 7.73% per unit of beta — compare to the market's 6.5% (market Treynor = (10% - 3.5%)/1.0)
• The IR of 0.385 means Dalton generated 38.5 bps of alpha per unit of tracking error — decent but not exceptional (top quartile active managers typically show IR > 0.5)

Exam Trap: The exam often asks which measure is most appropriate. The key is the context — total portfolio use Sharpe, part of a diversified allocation use Treynor, evaluating an active manager's skill use Information Ratio.

Sharpen your performance evaluation skills in our CFA Level III community.