performance-metrics

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Performance Metrics

绩效指标

Purpose

用途

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.

使用行业标准比率和捕获分析，基于风险调整评估投资绩效。本技能涵盖Sharpe、Sortino、Information、Treynor、Calmar比率、Omega比率、上行/下行捕获率以及M-squared。这些指标允许对具有不同风险特征的策略进行公平比较。

Layer

层级

1a — Realized Risk & Performance

1a — 已实现风险与绩效

Direction

方向

Retrospective

回顾性

When to Use

使用场景

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

评估投资或策略相对于其所承担风险的表现
基于风险调整比较基金经理、基金或策略
计算Sharpe、Sortino、Information Ratio、Treynor或Calmar比率
针对基准进行上行/下行捕获分析
计算给定阈值收益下的Omega比率
评估M-squared（Modigliani-Modigliani）风险调整绩效

Core Concepts

核心概念

Sharpe Ratio

Sharpe比率

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).

应用最广泛的风险调整绩效衡量指标。它将超额收益（超过无风险利率）除以总波动率。

SR = (R_p - R_f) / sigma_p

R_p: 年化投资组合收益
R_f: 年化无风险利率
sigma_p: 年化投资组合波动率（收益的标准差）

Sharpe比率越高，表明单位总风险带来的收益越多。典型基准：SR < 0.5为较差，0.5-1.0为可接受，>1.0为优秀，>2.0为卓越。

年化处理： 如果从月度数据计算，SR_annual = SR_monthly * sqrt(12)。

Sortino Ratio

Sortino比率

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.

用下行偏差替代总波动率，仅惩罚有害波动（低于最低可接受收益的收益）。

Sortino = (R_p - R_f) / sigma_downside

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

常见MAR选择：0%、无风险利率或目标收益。需始终说明使用的MAR。

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.

衡量相对于基准的主动收益（alpha）与主动风险（跟踪误差）的比值。

IR = (R_p - R_b) / TE

其中TE = std(R_p - R_b) * sqrt(N)。

IR高于0.5通常被认为良好；高于1.0则为卓越且难以维持。

Treynor Ratio

Treynor比率

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.

衡量单位系统性风险（beta）而非总风险带来的超额收益。

Treynor = (R_p - R_f) / beta_p

适用于评估已分散非系统性风险的多元化投资组合。对于非多元化持仓，Sharpe比率更合适。

Calmar Ratio

Calmar比率

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.

将年化收益与最严重的峰谷回撤关联。

Calmar = CAGR / |MaxDrawdown|

Calmar比率高于1.0意味着年化收益超过最大回撤。该比率在CTA和对冲基金投资者中较为流行。通常基于3年窗口期计算。

Omega Ratio

Omega比率

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).

一种损益比率，考虑阈值tau上下的整个收益分布。

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

其中F(r)是收益的累积分布函数。

实际计算方式为：

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

Omega > 1表示tau以上的预期收益超过tau以下的预期损失。与Sharpe不同，Omega捕获了分布的完整形态（偏度、峰度）。

Upside and Downside Capture Ratios

上行和下行捕获率

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.

衡量投资组合在基准上涨和下跌市场中的参与度。

上行捕获率   = R_p(上涨月份) / R_b(上涨月份) * 100
下行捕获率 = R_p(下跌月份) / R_b(下跌月份) * 100
捕获比率 = 上行捕获率 / 下行捕获率

理想特征：上行捕获率>100%且下行捕获率<100%，使得捕获比率>1。“上涨月份”和“下跌月份”分别由基准收益为正或负定义。

M-Squared (Modigliani-Modigliani)

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.

通过杠杆化或去杠杆化投资组合以匹配基准波动率，用收益的相同单位表示风险调整收益。

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

解释：“如果将该投资组合调整至与基准相同的波动率，其收益将为M-squared。”这使其可直接与基准收益比较。

Key Formulas

关键公式

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

公式	表达式	使用场景
Sharpe比率	(R_p - R_f) / sigma_p	单位总风险收益
Sortino比率	(R_p - R_f) / sigma_downside	单位下行风险收益
Information Ratio	(R_p - R_b) / TE	单位主动风险的主动收益
Treynor比率	(R_p - R_f) / beta_p	单位系统性风险收益
Calmar比率	CAGR /	MaxDD
Omega比率	sum(max(R_i - tau, 0)) / sum(max(tau - R_i, 0))	全分布损益比率
上行捕获率	R_p(上涨) / R_b(上涨) * 100	上涨市场参与度
下行捕获率	R_p(下跌) / R_b(下跌) * 100	下跌市场参与度
M-squared	R_f + SR_p * sigma_b	收益单位下的风险调整收益

Worked Examples

示例计算

Example 1: Sharpe Ratio Calculation

示例1：Sharpe比率计算

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.

已知： 某基金年化收益为12%，无风险利率为4%，基金年化波动率为15%。

计算： Sharpe比率。

解答：

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

该基金每单位风险获得0.533单位的超额收益，处于“可接受”区间但低于1.0。

Example 2: Comparing Funds with Sharpe and Sortino

示例2：用Sharpe和Sortino比较基金

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.

已知：

基金A：Sharpe = 0.8，Sortino = 1.2
基金B：Sharpe = 0.7，Sortino = 1.5

计算： 对于厌恶下行风险的投资者，哪只基金更优？

解答：

基金A的Sharpe比率更高（0.8 vs 0.7），表明其总风险调整表现更好。但基金B的Sortino比率显著更高（1.5 vs 1.2），意味着其单位下行风险带来的收益明显更多。

这种差异表明基金B的波动率更多偏向于上行——其总波动率包含更多“良性”波动（收益），而下行波动率相对可控。

对于厌恶下行风险的投资者，基金B更优，因为Sortino比率更能捕捉他们关注的风险（损失），而基金B更优的Sortino比率表明其下行风险管理更好。

Example 3: Information Ratio

示例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.

已知： 某投资组合年化收益为10%，基准年化收益为8%，跟踪误差为4%。

计算： Information Ratio。

解答：

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

该基金经理每单位主动风险获得0.50单位的主动收益，这通常被认为是良好的IR，表明相对于基准偏差，其持续产生alpha。

Common Pitfalls

常见误区

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.

Sharpe比率年化错误： Sharpe比率按sqrt(N)缩放，其中N为每年的期数。SR_annual = SR_monthly * sqrt(12)，而非*12。在相除前，超额收益和波动率必须保持一致的单位。
使用错误频率的无风险利率： 如果计算月度Sharpe比率，应使用月度无风险利率（年化利率/12），而非直接使用年化利率。
Sortino比率MAR模糊性： Sortino比率结果会因MAR=0、MAR=无风险利率或MAR=某目标收益而显著不同。需始终明确说明MAR假设。
样本量过小导致比率不可靠： 基于少于36个月度观测值计算的比率在统计上不可靠。基于12个月数据计算的Sharpe比率标准误差约为sqrt((1 + SR^2/2)/12)，误差范围非常大。
跨不同时间段比较Sharpe比率： 低波动环境下的Sharpe比率1.0与高波动环境下的1.0并不相同。绩效比率具有时效性，不能直接跨不同市场周期比较。

Cross-References

交叉引用

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.

historical-risk（财富管理插件，层级1a）：提供这些绩效比率分母所用的风险指标（波动率、回撤、下行偏差、跟踪误差）。
forward-risk（财富管理插件，层级1b）：前瞻性风险指标（VaR、CVaR）通过估算未来潜在损失，补充回顾性绩效评估。
volatility-modeling（财富管理插件，层级1b）：GARCH或EWMA的波动率预测可用于计算前瞻性或条件Sharpe比率。

Reference Implementation

参考实现

See

scripts/performance_metrics.py

for computational helpers.

计算辅助工具请参见

scripts/performance_metrics.py

。