joellewis/performance-metrics

Performance Metrics

Purpose

Evaluate investment performance on a risk-adjusted basis using industry-standard ratios and capture analysis. This skill covers the Sharpe, Sortino, Information, Treynor, and Calmar ratios, the Omega ratio, upside/downside capture ratios, and M-squared. These metrics allow fair comparison across strategies with different risk profiles.

Layer

1a — Realized Risk & Performance

Direction

Retrospective

When to Use

• Evaluating how well an investment or strategy performed relative to the risk it took
• Comparing managers, funds, or strategies on a risk-adjusted basis
• Computing Sharpe, Sortino, Information Ratio, Treynor, or Calmar ratios
• Performing upside/downside capture analysis against a benchmark
• Calculating the Omega ratio for a given threshold return
• Assessing M-squared (Modigliani-Modigliani) risk-adjusted performance

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.

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 relatively contained.

For a downside-averse investor, Fund B is preferable because the Sortino ratio better captures the risk they care about (losses), and Fund B's superior Sortino indicates better downside risk management.

Example 3: Information Ratio

Given: A portfolio returned 10% annualized, its benchmark returned 8%, and the tracking error is 4%.

Calculate: Information Ratio.

Solution:

IR = (0.10 - 0.08) / 0.04
   = 0.02 / 0.04
   = 0.50

The manager generated 0.50 units of active return per unit of active risk. This is generally considered a good IR, suggesting consistent alpha generation relative to benchmark deviations.

Common Pitfalls

• Annualizing Sharpe incorrectly: The Sharpe ratio scales by sqrt(N) where N is the number of periods per year. SR_annual = SR_monthly * sqrt(12), not * 12. The excess return and volatility must be in consistent units before dividing.
• Using wrong risk-free rate frequency: If computing monthly Sharpe, use the monthly risk-free rate (annual rate / 12), not the annual rate directly.
• Sortino MAR ambiguity: The Sortino ratio result changes significantly depending on whether MAR = 0, MAR = risk-free rate, or MAR = some target return. Always state the MAR assumption explicitly.
• Small sample sizes making ratios unreliable: Ratios computed from fewer than 36 monthly observations are statistically unreliable. A Sharpe ratio from 12 months of data has a standard error of approximately sqrt((1 + SR^2/2) / 12), which is very wide.
• Comparing Sharpe ratios across different time periods: A Sharpe of 1.0 in a low-vol environment is not the same as 1.0 in a high-vol environment. Performance ratios are period-specific and not directly comparable across different market regimes.

Cross-References

• historical-risk (wealth-management plugin, Layer 1a): Provides the risk measures (volatility, drawdown, downside deviation, tracking error) used as denominators in these performance ratios.
• forward-risk (wealth-management plugin, Layer 1b): Forward-looking risk measures (VaR, CVaR) complement retrospective performance assessment by estimating future potential losses.
• volatility-modeling (wealth-management plugin, Layer 1b): Volatility forecasts from GARCH or EWMA can be used to compute forward-looking or conditional Sharpe ratios.

Reference Implementation

See scripts/performance_metrics.py for computational helpers.