L02 现代投资组合理论

It is the part of a wise man...not to venture all his eggs in one basket. —-Miguel de Cervantes

"Don't put all your eggs in one basket" is all wrong. I tell you "put all your eggs in one basket, and then watch that basket." -- Andrew Carnegie

"Harry M. Markowitz - Facts". Nobelprize.org. Nobel Media AB 2014. Web. 2 Sep 2014.

组合管理的主要思路

马科维茨与组合管理的小故事

"Harry M. Markowitz - Prize Lecture: Foundations of Portfolio Theory". Nobelprize.org. Nobel Media AB 2014. Web. 2 Sep 2014.

组合管理的均值方差分析

组合收益

收益

持有期收益

组合收益（Portfolio Return）

公式：
RP=∑i=1NwiRi,  ∑i=1Nwi=1R_P=\sum_{i=1}^Nw_iR_i,\ \ \sum_{i=1}^Nw_i=1

组合收益的风险

单个资产收益的方差与协方差

投资组合收益的方差

假设wiw_i为各资产占整个投资组合的权重，组合的收益为
RP=∑i=1NwiRiR_P=\sum_{i=1}^Nw_iR_i\nonumber

因此，其方差可以表示为
σP2=Var(RP)=Var(∑i=1NwiRi)=∑i,j=1NwiwjCov(Ri,Rj)=∑i=1Nwi2Var(Ri)+∑i,j=1,i≠jNwiwjCov(Ri,Rj)\begin{array}{lll} \sigma^2_P&=&Var(R_P)=Var(\sum_{i=1}^Nw_iR_i)\nonumber\\ &=&\sum_{i,j=1}^Nw_iw_jCov(R_i,R_j)\nonumber\\ &=&\sum_{i=1}^Nw_i^2Var(R_i)+\sum_{i,j=1,i\neq j}^Nw_iw_jCov(R_i,R_j)\nonumber \end{array}

由两个风险资产组成的组合

组合收益:
RP=w1R1+(1−w1)R2R_P=w_1R_1+(1-w_1)R_2\nonumber
组合风险:
σP2=Var(Rp)=Var(w1R1+w2R2)=w12Var(R1)+w22Var(R2)+2w1w2Cov(R1,R2)=w12σ12+w22σ22+2w1w2ρ12σ1σ2\begin{array}{lll} \sigma^2_P&=&Var(R_p)=Var(w_1R_1+w_2R_2)\nonumber\\ &=&w_1^2Var(R_1)+w_2^2Var(R_2)+2w_1w_2Cov(R_1,R_2)\nonumber\\ &=&w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{12}\sigma_1\sigma_2\nonumber \end{array}
因此，其标准差为
σP=w12σ12+w22σ22+2w1w2ρ12σ1σ2\begin{array}{lll} \sigma_P&=&\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{12}\sigma_1\sigma_2}\nonumber \end{array}

ρ12\rho_{12}取不同数值时组合收益与风险的关系

  1. ρ=1\rho=1
    σP=w12σ12+w22σ22+2w1w2σ1σ2=w1σ1+w2σ2\begin{array}{lll} \sigma_P&=&\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\sigma_1\sigma_2}=w_1\sigma_1+w_2\sigma_2\nonumber \end{array}

  2. −1<ρ<1-1<\rho<1
    σP=w12σ12+w22σ22+2w1w2ρ12σ1σ2<w1σ1+w2σ2\begin{array}{lll} \sigma_P&=&\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{12}\sigma_1\sigma_2}<w_1\sigma_1+w_2\sigma_2\nonumber \end{array}

  3. ρ=−1\rho=-1
    σP=w12σ12+w22σ22−2w1w2σ1σ2=∣w1σ1−w2σ2∣\begin{array}{lll} \sigma_P&=&\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2-2w_1w_2\sigma_1\sigma_2}=|w_1\sigma_1-w_2\sigma_2|\nonumber \end{array}

例子：组合收益 vs. 组合风险

由多个风险资产组成的投资组合

E[Rp]=∑i=1NwiE[Ri],  σP2=(∑i=1Nwi2σi2+∑i,j=1,i≠jNwiwjσij2),  ∑i=1Nwi=1E[R_p]=\sum_{i=1}^Nw_iE[R_i],\ \ \sigma_P^2=\left(\sum_{i=1}^Nw_i^2\sigma_i^2+\sum_{i,j=1,i\neq j}^Nw_iw_j\sigma_{ij}^2\right),\ \ \sum_{i=1}^Nw_i=1\nonumber

假设wi=1N, ∀i=1,2,…,Nw_i=\frac{1}{N},\ \forall i=1,2,\dots,N，σˉ2\bar{\sigma}^2和Cov‾\overline{Cov}代表平均方差和平均协方差，我们有下式
σP2=σˉ2N+N−1NCov‾\sigma_P^2=\frac{\bar{\sigma}^2}{N}+\frac{N-1}{N}\overline{Cov}\nonumber
令ρ\rho为平均相关系数, 我们有
σP=σˉ2N+N−1Nρσˉ2\sigma_P=\frac{\bar{\sigma}^2}{N}+\frac{N-1}{N}\rho\bar{\sigma}^2\nonumber
有多于2个风险资产时，所有可行组合的风险和收益的关系不再能够由一条曲线表示。

均值-方差最优化问题

均值-方差最优化（MVO）

max⁡ω∈ΩμTωs.t.ωTΣω≤σmax2\begin{array}{lll} &\max_{\omega\in\Omega}& \mu^T\omega\nonumber\\ &s.t.&\omega^T\Sigma\omega\leq\sigma_{\text{max}}^2\nonumber \end{array}
或者
min⁡ω∈ΩωTΣωs.t.μTω≥μmin\begin{array}{lll} &\min_{\omega\in\Omega}& \omega^T\Sigma\omega\nonumber\\ &s.t.& \mu^T\omega\geq\mu_{\text{min}}\nonumber \end{array}
或者
max⁡ω∈ΩμTω−λωTΣω\begin{array}{lll} \max_{\omega\in\Omega} \mu^T\omega-\lambda\omega^T\Sigma\omega\nonumber \end{array}

集合Ω\Omega定义了资产权重wiw_i的所有可能取值, 比如:

均值-方差最优化（MVO）的解

有效前沿（Efficient Frontier）与最优风险组合

投资机会集合（Investment Opportunity Set）

最小方差投资组合与有效前沿}

一个无风险资产与多个风险资产

资本分配线（Capital Allocation Line）与最优风险组合（Optimal Risky Portfolio）

基金分离定理

两基金分离定理(two-fund separation theorem): 所有的投资者，不管其喜好、风险偏好、初始财富，都将只持有两个组合（或基金）：一个无风险资产和一个最优的风险资产。

最优的风险资产(组合)是什么呢?

最优风险组合

最优风险组合是市场组合（market portfolio）

风险规避（Risk Aversion）与组合选择

风险规避

对风险的态度

假设投资者面对以下两种选择：

投资者对风险的态度可分为以下几类：

效用理论与无差异曲线

效用与无差异曲线

无差异曲线与效用

无差异曲线与风险偏好

组合选择

资本分配线（Capital Allocation Line,CAL）

假设投资组合由一个无风险资产和一个风险资产组成。组合的收益和方差为：
E[RP]=w1Rf+(1−w1)E[Ri]σP=(1−w1)σi\begin{array}{lll} E[R_P]&=&w_1R_f+(1-w_1)E[R_i]\nonumber\\ \sigma_P&=&(1-w_1)\sigma_i\nonumber \end{array}
由此，我们可以得到\textbf{资本分配线(CAL)},
E[RP]=Rf+(E[Ri]−Rf)σiσPE[R_P]=R_f+\frac{(E[R_i]-R_f)}{\sigma_i}\sigma_P\nonumber
CAL类似消费者均衡分析中的预算线

假设投资者的效用函数为： U=E[r]−A2σ2U=E[r]-\frac{A}{2}\sigma^2.

最优投资者组合（Optimal Investor Portfolio）

Mathematics of MPT

Quadratic Programming (QP)

Primal, Lagrangian Function, & Dual

Optimality Conditions for QP

The vectors x∈Rn\mathrm{x} \in \mathbb{R}^n and (x~,y,s)∈Rn×(\tilde{\mathbf{x}}, \mathrm{y}, \mathrm{s}) \in \mathbb{R}^n \times Rm×Rp\mathbb{R}^m \times \mathbb{R}^p are optimal solutions to primal and dual problem respectively if and only if Qx=Qx~\mathbf{Q} \mathbf{x}=\mathbf{Q} \tilde{\mathbf{x}} and
Qx+c−A⊤y−D⊤s=0Ax−b=0Dx−d≥0s≥0(Dx−d)isi=0,i=1,…,p.\begin{aligned} \mathbf{Q x}+\mathbf{c}-\mathbf{A}^{\top} \mathbf{y}-\mathbf{D}^{\top} \mathbf{s} &=\mathbf{0} \\ \mathbf{A} \mathbf{x}-\mathbf{b} &=\mathbf{0} \\ \mathbf{D} \mathbf{x}-\mathbf{d} & \geq \mathbf{0} \\ \mathbf{s} & \geq \mathbf{0} \\ (\mathbf{D} \mathbf{x}-\mathbf{d})_i s_i &=0, i=1, \ldots, p . \end{aligned}
For a QP in standard form, the optimality conditions can be written as follows:
−Qx+A⊤y+s=cAx=bx≥0s≥0xisi=0,i=1,…,n.\begin{aligned} -\mathbf{Q} \mathbf{x}+\mathbf{A}^{\top} \mathbf{y}+\mathbf{s} &=\mathbf{c} \\ \mathbf{A} \mathbf{x} &=\mathbf{b} \\ \mathbf{x} & \geq \mathbf{0} \\ \mathbf{s} & \geq \mathbf{0} \\ x_i s_i &=0, \quad i=1, \ldots, n . \end{aligned}

The Basic Mean-Variance Models

max⁡xμ⊤x−12γ⋅x⊤Vx1⊤x=1\begin{array}{ll} \max _{\mathbf{x}} & \boldsymbol{\mu}^{\top} \mathrm{x}-\frac{1}{2} \gamma \cdot \mathbf{x}^{\top} \mathbf{V} \mathbf{x} \\ & \mathbf{1}^{\top} \mathbf{x}=1 \end{array}

min⁡xx⊤Vx s.t. μ⊤x≥μˉ1⊤x=1,\begin{array}{ll} \min _{\mathbf{x}} & \mathbf{x}^{\top} \mathbf{V} \mathbf{x} \\ \text { s.t. } & \boldsymbol{\mu}^{\top} \mathbf{x} \geq \bar{\mu} \\ & \mathbf{1}^{\top} \mathbf{x}=1, \end{array}

max⁡xμ⊤x s.t. x⊤Vx≤σˉ21⊤x=1\begin{array}{cl} \max _{\mathbf{x}} & \boldsymbol{\mu}^{\top} \mathbf{x} \\ \text { s.t. } & \mathbf{x}^{\top} \mathbf{V} \mathbf{x} \leq \bar{\sigma}^2 \\ & \mathbf{1}^{\top} \mathbf{x}=1 \end{array}

Analytical Solutions

Generalizations & Implementations

Common Constraints

Leverage constraints such as long-only, or 130/30 constraints

Turnover constraints: a constraint on the total change in the portfolio positions.

Maximizing the Sharpe Ratio

Putting Portfolio Theory to Work}

We will cover most of the above topics in later lectures.

课后阅读与练习

课后阅读与练习

参考文献

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