/performance-metrics

name: performance-metrics
description: "Evaluate investment performance on a risk-adjusted basis using industry-standard ratios and capture analysis. Use when the user asks about Sharpe ratio, Sortino ratio, Information Ratio, Treynor ratio, Calmar ratio, Omega ratio, or upside/downside capture. Also trigger when users mention 'risk-adjusted returns', 'return per unit of risk', 'M-squared', 'is this fund worth the volatility', 'how to compare two managers', 'capture ratio', or ask which investment performed better after accounting for risk."

Performance Metrics

Core Concepts

Sharpe Ratio

The most widely used risk-adjusted performance measure. It divides excess return (over the risk-free rate) by total volatility.

SR = (R_p - R_f) / sigma_p

• R_p: annualized portfolio return
• R_f: annualized risk-free rate
• sigma_p: annualized portfolio volatility (standard deviation of returns)

A higher Sharpe ratio indicates more return per unit of total risk. Typical benchmarks: SR < 0.5 is poor, 0.5-1.0 is acceptable, > 1.0 is strong, > 2.0 is exceptional.

**Annualization:** If computed from monthly data, SR_annual = SR_monthly * sqrt(12).

Sortino Ratio

Replaces total volatility with downside deviation, penalizing only harmful volatility (returns below a Minimum Acceptable Return).

Sortino = (R_p - R_f) / sigma_downside

where sigma_downside = sqrt((1/n) * sum(min(R_i - MAR, 0)^2)).

Common MAR choices: 0%, risk-free rate, or a target return. Always state which MAR is used, and use the same reference point in the numerator as in the downside deviation: if the MAR is not the risk-free rate, the numerator should be (R_p - MAR), not (R_p - R_f). Mixing reference points makes the ratio internally inconsistent.

Information Ratio

Measures active return (alpha) per unit of active risk (tracking error) relative to a benchmark.

IR = (R_p - R_b) / TE

where TE = std(R_p - R_b) * sqrt(N).

An IR above 0.5 is generally considered good; above 1.0 is exceptional and difficult to sustain.

Treynor Ratio

Measures excess return per unit of systematic risk (beta) rather than total risk.

Treynor = (R_p - R_f) / beta_p

Useful for evaluating diversified portfolios where idiosyncratic risk has been diversified away. For undiversified holdings, the Sharpe ratio is more appropriate.

Calmar Ratio

Relates annualized return to the worst peak-to-trough drawdown.

Calmar = CAGR / |MaxDrawdown|

A Calmar ratio above 1.0 means the annualized return exceeds the maximum drawdown. This ratio is popular among CTAs and hedge fund investors. Typically computed over a 3-year window.

Omega Ratio

A gain-loss ratio that considers the entire return distribution above and below a threshold tau.

Omega(tau) = integral from tau to +inf of [1 - F(r)] dr
             / integral from -inf to tau of F(r) dr

where F(r) is the cumulative distribution function of returns.

In practice, this is computed as:

Omega(tau) = sum(max(R_i - tau, 0)) / sum(max(tau - R_i, 0))

Omega > 1 means expected gains above tau exceed expected losses below tau. Unlike Sharpe, Omega captures the full shape of the distribution (skewness, kurtosis).

Upside and Downside Capture Ratios

Measure how the portfolio participates in benchmark up and down markets.

Up Capture   = R_p(in up months) / R_b(in up months) * 100
Down Capture = R_p(in down months) / R_b(in down months) * 100
Capture Ratio = Up Capture / Down Capture

Ideal profile: Up Capture > 100% and Down Capture < 100%, yielding a Capture Ratio > 1. "Up months" and "down months" are defined by the benchmark return being positive or negative, respectively.

M-Squared (Modigliani-Modigliani)

Expresses risk-adjusted return in the same units as return, by leveraging or deleveraging the portfolio to match benchmark volatility.

M^2 = R_f + SR_p * sigma_b
    = R_f + ((R_p - R_f) / sigma_p) * sigma_b

Interpretation: "If this portfolio were scaled to have the same volatility as the benchmark, it would have returned M-squared." This makes it directly comparable to benchmark returns.

Key Formulas

| Formula | Expression | Use Case | |---------|-----------|----------| | Sharpe Ratio | (R_p - R_f) / sigma_p | Return per unit of total risk | | Sortino Ratio | (R_p - R_f) / sigma_downside | Return per unit of downside risk | | Information Ratio | (R_p - R_b) / TE | Active return per unit of active risk | | Treynor Ratio | (R_p - R_f) / beta_p | Return per unit of systematic risk | | Calmar Ratio | CAGR / |MaxDD| | Return per unit of drawdown risk | | Omega Ratio | sum(max(R_i - tau, 0)) / sum(max(tau - R_i, 0)) | Full-distribution gain-loss ratio | | Up Capture | R_p(up) / R_b(up) * 100 | Participation in rising markets | | Down Capture | R_p(down) / R_b(down) * 100 | Participation in falling markets | | M-Squared | R_f + SR_p * sigma_b | Risk-adjusted return in return units |

Worked Examples

Example 1: Sharpe Ratio Calculation

**Given:** A fund returned 12% annualized, the risk-free rate is 4%, and the fund's annualized volatility is 15%.

**Calculate:** Sharpe Ratio.

**Solution:**

SR = (0.12 - 0.04) / 0.15
   = 0.08 / 0.15
   = 0.533

The fund earned 0.533 units of excess return per unit of risk. This is in the "acceptable" range but below 1.0.

Example 2: Comparing Funds with Sharpe and Sortino

**Given:**

• Fund A: Sharpe = 0.8, Sortino = 1.2
• Fund B: Sharpe = 0.7, Sortino = 1.5

**Calculate:** Which fund is better for a downside-averse investor?

**Solution:**

Fund A has a higher Sharpe ratio (0.8 vs 0.7), indicating better total-risk-adjusted performance. However, Fund B has a notably higher Sortino ratio (1.5 vs 1.2), meaning it delivers significantly more return per unit of downside risk.

The divergence implies Fund B's volatility is more skewed to the upside -- its total volatility includes more "good" volatility (gains), while its downside volatility is relati