投资组合理论 / Portfolio Theory

本页结构

核心概念

两资产与多资产均值-方差优化 Two-asset and multi-asset mean-variance optimization
有效前沿、切点组合与资本市场线 Efficient frontier, tangency portfolio and capital market line
风险调整收益和定价因子直觉 Risk-adjusted performance and pricing-factor intuition

学习顺序

区分预期收益假设和协方差估计。 Separate expected return assumptions from covariance estimation.
说明分散化收益如何依赖相关性。 Show how diversification depends on correlation.
在信任优化组合前先讨论估计误差。 Discuss estimation error before trusting an optimized portfolio.

Portfolio Theory, pioneered by Harry Markowitz, provides the mathematical framework for constructing investment portfolios to maximize expected return for a given level of market risk, or equivalently, minimize risk for a given expected return.

投资组合理论由哈里·马科维茨（Harry Markowitz）首创，提供了构建投资组合的数学框架，以在给定的市场风险水平下最大化预期回报，或者同等地，在给定的预期回报下最小化风险。

Two-Asset Portfolio

两种资产组合

The core principle is that the risk of a portfolio is not simply the weighted average of the individual asset risks, but also depends on the correlation between the assets.

核心原则是，投资组合的风险不仅仅是单项资产风险的加权平均，还取决于资产之间的相关性。

For a two-asset portfolio with weights $w_1 = w$ and $w_2 = 1 - w$ :

对于权重为 $w_1 = w$ 和 $w_2 = 1 - w$ 的双资产投资组合：

Expected Return ( $\mu_p$ ):

预期回报 ( $\mu_p$ )：

\mu_p = w\mu_1 + (1 - w)\mu_2

Portfolio Variance ( $\sigma_p^2$ ):

投资组合方差 ( $\sigma_p^2$ )：

\sigma_p^2 = w^2\sigma_1^2 + (1 - w)^2\sigma_2^2 + 2w(1 - w)\rho\sigma_1\sigma_2

where $\rho$ is the correlation between the two assets. Diversification benefits are maximized when $\rho$ is low or negative.

其中 $\rho$ 是两种资产之间的相关性。当 $\rho$ 较低或为负时，多元化收益最大化。

The Efficient Frontier

有效前沿

The Efficient Frontier is the set of optimal portfolios that offer the highest expected return for a defined level of risk (standard deviation).

有效前沿是一组最佳投资组合，可在规定的风险水平（标准差）下提供最高的预期回报。

Optimization Problem: For a large number of assets, the problem is to find the weight vector $\mathbf{w}$ that solves:

优化问题：对于大量资产，问题是找到权重向量 $\mathbf{w}$ ，解决：

\min_{\mathbf{w}} \quad \mathbf{w}^\intercal \boldsymbol{\Sigma} \mathbf{w} \quad \text{subject to} \quad \mathbf{w}^\intercal \boldsymbol{\mu} = \mu_p \quad \text{and} \quad \mathbf{w}^\intercal \mathbf{1} = 1

where $\boldsymbol{\Sigma}$ is the covariance matrix of asset returns, and $\boldsymbol{\mu}$ is the vector of expected returns.

Interpretation: Any portfolio below the Efficient Frontier is sub-optimal, as a higher return could be achieved for the same risk, or lower risk for the same return.

其中 $\boldsymbol{\Sigma}$ 是资产收益的协方差矩阵， $\boldsymbol{\mu}$ 是预期收益的向量。

解释：任何低于有效边界的投资组合都是次优的，因为在相同的风险下可以获得更高的回报，或者在相同的回报下获得更低的风险。

The Sharpe Ratio

夏普比率

The Sharpe Ratio is the most widely used measure of risk-adjusted return, quantifying the excess return earned per unit of total risk (standard deviation).

夏普比率是最广泛使用的风险调整回报衡量标准，量化每单位总风险（标准差）获得的超额回报。

where $\mathbb{E}[R_p]$ is the expected portfolio return, $R_f$ is the risk-free rate, and $\sigma_p$ is the portfolio's standard deviation.

其中 $\mathbb{E}[R_p]$ 是预期投资组合回报， $R_f$ 是无风险利率， $\sigma_p$ 是投资组合的标准差。

Capital Market Line (CML) and Tangency Portfolio

资本市场线 (CML) 和切线投资组合

When a risk-free asset is introduced, the optimal investment strategy is to combine the risk-free asset with a single risky portfolio, known as the Tangency Portfolio (or Market Portfolio in the CAPM context).

当引入无风险资产时，最佳投资策略是将无风险资产与单一风险投资组合相结合，称为切线投资组合（或 CAPM 背景下的市场投资组合）。

CML: The line connecting the risk-free rate to the Tangency Portfolio on the mean-standard deviation plane. All efficient portfolios for an investor are combinations along this line.
Tangency Portfolio: The portfolio on the Efficient Frontier that has the highest Sharpe Ratio.

CML：在均值-标准差平面上将无风险利率与切线投资组合连接起来的线。对于投资者来说，所有有效的投资组合都是沿着这条线的组合。
切线投资组合：有效前沿上夏普比率最高的投资组合。

These models explain the expected return of an asset based on its exposure to systematic risk factors.

这些模型根据资产面临的系统性风险因素来解释其预期回报。

1. Capital Asset Pricing Model (CAPM)

1. 资本资产定价模型（CAPM）

CAPM states that the expected return of an asset is linearly related to its systematic risk ( $\beta$ ) and the expected return of the market portfolio ( $R_m$ ).

CAPM 指出，资产的预期回报与其系统风险 ( $\beta$ ) 和市场投资组合的预期回报 ( $R_m$ ) 线性相关。

Systematic Risk ( $\beta$ ): Measures the sensitivity of the asset's return to the market's return. It is calculated as $\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$ .
Security Market Line (SML): The graphical representation of CAPM, plotting expected return against $\beta$ .
Alpha ( $\alpha$ ): The intercept term in the empirical CAPM regression:

系统性风险（ $\beta$ ）：衡量资产回报对市场回报的敏感性。其计算方式为 $\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$ 。
证券市场线 (SML)：CAPM 的图形表示，根据 $\beta$ 绘制预期回报。
Alpha ( $\alpha$ )：经验 CAPM 回归中的截距项：

R_i - R_f = \alpha_i + \beta_i (R_m - R_f) + \epsilon_i

$\alpha$ represents the excess return achieved by the asset or portfolio that is not explained by the market risk. It is the primary metric sought by active portfolio managers (alpha generation).

$\alpha$ 代表资产或投资组合所获得的超额回报，而市场风险无法解释。这是主动投资组合经理（阿尔法一代）寻求的主要指标。

2. Arbitrage Pricing Theory (APT)

2.套利定价理论（APT）

APT is a multi-factor model that suggests an asset's expected return is a linear function of its sensitivity to multiple systematic risk factors.

APT 是一个多因素模型，表明资产的预期回报是其对多个系统风险因素敏感性的线性函数。

where $\beta_{ij}$ is the sensitivity of asset $i$ to factor $j$ , and $\lambda_j$ is the risk premium for factor $j$ . Unlike CAPM, APT does not specify the factors; they must be identified empirically.

其中 $\beta_{ij}$ 是资产 $i$ 对因子 $j$ 的敏感性， $\lambda_j$ 是因子 $j$ 的风险溢价。与 CAPM 不同，APT 没有指定因素；它们必须凭经验来识别。

3. Fama-French 3-Factor Model

3. Fama-French 三因素模型

An empirical extension of CAPM that incorporates two additional factors found to explain cross-sectional stock returns better than $\beta$ alone:

CAPM 的实证扩展，包含两个额外因素，比单独使用 $\beta$ 更好地解释横截面股票收益：

SMB (Small Minus Big): The return of a portfolio of small-cap stocks minus the return of a portfolio of large-cap stocks (Size factor).
HML (High Minus Low): The return of a portfolio of high book-to-market stocks (Value stocks) minus the return of a portfolio of low book-to-market stocks (Growth stocks) (Value factor).

SMB（小减大）：小盘股投资组合的回报减去大盘股投资组合的回报（规模因素）。
HML（高减低）：高账面市值比股票（价值股票）投资组合的回报减去低账面市值比股票（成长股票）投资组合的回报（价值因子）。

Estimation Error: MVO is highly sensitive to errors in estimating expected returns and the covariance matrix. Small changes in inputs can lead to drastically different, often unstable, optimal portfolios.
Black-Litterman Model: A practical approach that combines the market equilibrium (CAPM) with an investor's subjective views to produce more stable and intuitive portfolio allocations than pure MVO.
Risk Parity: An alternative portfolio construction method that focuses on allocating capital such that each asset or risk factor contributes equally to the total portfolio risk, often leading to more diversified and robust portfolios than MVO.

估计误差：MVO 对估计预期收益和协方差矩阵的误差高度敏感。输入的微小变化可能会导致截然不同的、通常不稳定的最优投资组合。
Black-Litterman 模型：一种实用方法，将市场均衡 (CAPM) 与投资者的主观观点相结合，产生比纯 MVO 更稳定、更直观的投资组合配置。
风险平价：另一种投资组合构建方法，侧重于分配资本，使每种资产或风险因素对总投资组合风险的贡献相等，通常会导致比 MVO 更多元化、更稳健的投资组合。

补充讲解

期望收益估计噪声很大

Expected returns are noisy

均值-方差优化对期望收益估计极其敏感。实务中必须配合约束、收缩估计和稳健输入。

Mean-variance optimization is highly sensitive to expected-return estimates. In practice, constraints, shrinkage, and robust inputs are essential.

分散化取决于协方差

Diversification depends on covariance

分散化不是简单持有很多资产，而是持有残余风险不一起波动的风险暴露；关键在相关性和协方差结构。

Diversification is not about holding many assets; it is about holding exposures whose residual risks do not move together.

优化必须纳入摩擦成本

Optimization must include frictions

换手、流动性、卖空约束、税费、融券成本和冲击成本，可能完全吞噬无约束优化器给出的理论改进，因此必须进入约束或目标函数。

Turnover, liquidity, short constraints, taxes, borrow costs, and impact can dominate the theoretical improvement from an unconstrained optimizer.

投资组合理论