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Chinese

Capital Asset Pricing Model (CAPM)

资本资产定价模型（CAPM）

Overview

概述

CAPM (Sharpe, 1964; Lintner, 1965) establishes a linear relationship between systematic risk and expected return. The model states that the expected return on any asset equals the risk-free rate plus a premium for bearing market risk, scaled by the asset's beta.
CAPM（Sharpe，1964；Lintner，1965）建立了系统性风险与预期收益之间的线性关系。该模型指出，任何资产的预期收益等于无风险利率加上市场风险溢价，并乘以资产的beta值。

When to Use

适用场景

  • Estimating required rate of return for equity valuation
  • Calculating cost of equity in WACC
  • Comparing asset risk via beta
  • Evaluating portfolio performance against the Security Market Line (SML)
  • 为股权估值估算必要收益率
  • 在加权平均资本成本（WACC）中计算股权成本
  • 通过beta比较资产风险
  • 对照证券市场线（SML）评估投资组合表现

When NOT to Use

不适用场景

  • When the asset has significant exposure to size, value, or other factors beyond market risk
  • For illiquid or non-traded assets where beta estimation is unreliable
  • When market portfolio proxy is questionable (Roll's critique)
  • 当资产面临显著的规模、价值或市场风险以外的其他因素敞口时
  • 针对流动性差或非交易性资产（此类资产的beta估算不可靠）
  • 当市场投资组合代理存在疑问时（Roll的批评）

Assumptions

假设条件

IRON LAW: CAPM only prices SYSTEMATIC risk — diversifiable (unsystematic)
risk earns NO premium. An asset's expected return depends solely on its
beta with the market portfolio.
Key assumptions:
  1. Investors are mean-variance optimizers with homogeneous expectations
  2. A risk-free asset exists for unlimited borrowing and lending
  3. Markets are frictionless — no taxes, transaction costs, or short-selling constraints
  4. All assets are infinitely divisible and publicly traded
IRON LAW: CAPM only prices SYSTEMATIC risk — diversifiable (unsystematic)
risk earns NO premium. An asset's expected return depends solely on its
beta with the market portfolio.
核心假设：
  1. 投资者是具有同质预期的均值-方差优化者
  2. 存在可无限借贷的无风险资产
  3. 市场无摩擦——无税收、交易成本或卖空限制
  4. 所有资产均可无限分割且公开交易

Methodology

方法步骤

Step 1 — Identify Inputs

步骤1 — 确定输入参数

  • Risk-free rate (Rf): government bond yield matching investment horizon
  • Market return E(Rm): historical average or forward-looking estimate
  • Beta: regression of asset returns against market returns
  • 无风险利率（Rf）：与投资期限匹配的政府债券收益率
  • 市场预期收益E(Rm)：历史平均值或前瞻性估算值
  • Beta：资产收益对市场收益的回归值

Step 2 — Compute Expected Return

步骤2 — 计算预期收益

E(Ri) = Rf + Bi x (E(Rm) - Rf). See 
references/derivation.md
for the derivation from mean-variance optimization.
E(Ri) = Rf + Bi x (E(Rm) - Rf)。均值-方差优化的推导过程请参见
references/derivation.md
。

Step 3 — Plot on Security Market Line

步骤3 — 在证券市场线（SML）上绘图

Assets above the SML are undervalued (positive alpha); below are overvalued (negative alpha).
位于SML上方的资产被低估（正alpha）；位于下方的资产被高估（负alpha）。

Step 4 — Interpret and Decide

步骤4 — 解读与决策

  • Beta > 1: amplifies market moves, higher risk-higher expected return
  • Beta < 1: dampens market moves, lower risk-lower expected return
  • Beta = 0: returns equal the risk-free rate
  • Beta > 1：放大市场波动，高风险对应高预期收益
  • Beta < 1：减缓市场波动，低风险对应低预期收益
  • Beta = 0：收益等于无风险利率

Output Format

输出格式

⚠️ Decimal vs percent: When passing values to or from the bundled script, all rates (
risk_free
, 
market_return
, 
beta_contribution
, 
expected_return
, 
alpha
) are decimals — 
0.05
means 5%, NOT 
5.0
. The narrative report below renders them as percentages for humans, but never mix the two in the same JSON object.
markdown
undefined
⚠️ 小数与百分比: 向捆绑脚本传递数值或从脚本获取数值时，所有利率 (
risk_free
、
market_return
、
beta_contribution
、
expected_return
、
alpha
)均为 小数 — 
0.05
代表5%，而非
5.0
。下方的叙述报告以百分比形式呈现供人类阅读，但在同一个JSON对象中切勿混用两种格式。
markdown
undefined

CAPM Analysis: [Asset / Portfolio]

CAPM分析：[资产/投资组合]

Inputs

输入参数

ParameterValueSource
Risk-free rate (Rf)x%[source]
Market return E(Rm)x%[source]
Betax.xx[estimation method]
参数数值来源
无风险利率（Rf）x%[来源]
市场预期收益E(Rm)x%[来源]
Betax.xx[估算方法]

Expected Return

预期收益

  • E(Ri) = Rf + B x (E(Rm) - Rf) = x%
  • E(Ri) = Rf + B x (E(Rm) - Rf) = x%

SML Assessment

SML评估

  • Alpha = Actual return - Expected return = x%
  • Interpretation: [undervalued / overvalued / fairly priced]
  • Alpha = 实际收益 - 预期收益 = x%
  • 解读：[被低估/被高估/定价合理]

Limitations in This Context

本次分析的局限性

  • [Note any assumption violations]
undefined
  • [注明任何违反假设的情况]
undefined

Gotchas

注意事项

  • Beta is backward-looking; future beta may differ from historical estimates
  • Choice of market proxy matters enormously (Roll's critique, 1977)
  • CAPM assumes a single risk factor; empirical evidence supports multi-factor models
  • Risk-free rate selection (T-bill vs T-bond) affects results significantly
  • Beta estimation is sensitive to return frequency (daily vs monthly) and sample period
  • CAPM fails to explain the low-beta anomaly (low-beta stocks outperform predictions)
  • Beta具有滞后性；未来beta可能与历史估算值不同
  • 市场代理的选择至关重要（Roll的批评，1977）
  • CAPM假设单一风险因素；实证证据支持多因素模型
  • 无风险利率的选择（短期国库券vs长期国债）对结果影响显著
  • Beta估算对收益频率（每日vs每月）和样本周期敏感
  • CAPM无法解释低beta异象（低beta股票表现优于预测）

Scripts

脚本工具

ScriptDescriptionUsage
scripts/capm.py
Compute CAPM expected return and alpha
python scripts/capm.py --help
Run 
python scripts/capm.py --verify
to execute built-in sanity tests.
脚本描述使用方法
scripts/capm.py
计算CAPM预期收益和alpha
python scripts/capm.py --help
运行
python scripts/capm.py --verify
执行内置的完整性测试。

References

参考文献

  • Sharpe, W. (1964). Capital asset prices. Journal of Finance, 19(3), 425-442.
  • Lintner, J. (1965). The valuation of risk assets. Review of Economics and Statistics, 47(1), 13-37.
  • Roll, R. (1977). A critique of the asset pricing theory's tests. Journal of Financial Economics, 4(2), 129-176.
  • Sharpe, W. (1964). Capital asset prices. Journal of Finance, 19(3), 425-442.
  • Lintner, J. (1965). The valuation of risk assets. Review of Economics and Statistics, 47(1), 13-37.
  • Roll, R. (1977). A critique of the asset pricing theory's tests. Journal of Financial Economics, 4(2), 129-176.