From wealth-management
Evaluates investment performance on a risk-adjusted basis using Sharpe, Sortino, Information, Treynor, Calmar, Omega ratios, and upside/downside capture analysis.
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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.
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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.
1a — Realized Risk & Performance
Retrospective
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
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).
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.
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.
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.
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.
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).
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.
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.
| 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 |
| 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 |
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.
Given:
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.
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.
See scripts/performance_metrics.py for computational helpers.