投资组合优化Theory

投资组合优化Problems

投资组合优化问题涉及确定满足三个标准的投资组合：

最大程度地降低风险的代理。
Match or exceed a proxy for return.
满足基本可行性要求。

投资组合是构成资产宇宙的可行资产集中的点。投资组合指定资产宇宙中每个资产中的持有或权重。该约定是在权重方面指定投资组合，尽管投资组合优化工具也可以使用持有量。

一组可行的阿宝rtfolios is necessarily a nonempty, closed, and bounded set. The proxy for risk is a function that characterizes either the variability or losses associated with portfolio choices. The proxy for return is a function that characterizes either the gross or net benefits associated with portfolio choices. The terms “risk” and “risk proxy” and “return” and “return proxy” are interchangeable. The fundamental insight of Markowitz (see投资组合优化）是，投资组合选择问题的目标是寻求给定水平水平的最低风险，并寻求给定风险水平的最大回报。满足这些条件的投资组合是有效的投资组合，这些投资组合的风险和收益的图形形成了称为曲线efficient frontier。

投资组合问题规范

要指定投资组合优化问题，您需要以下内容：

Proxy for portfolio return (μ)
代理投资组合风险（σ)
Set of feasible portfolios (X），称为投资组合集

Financial Toolbox™ has three objects to solve specific types of portfolio optimization problems:

The文件夹object supports mean-variance portfolio optimization (see Markowitz [46], [47] at投资组合优化）。This object has either gross or net portfolio returns as the return proxy, the variance of portfolio returns as the risk proxy, and a portfolio set that is any combination of the specified constraints to form a portfolio set.
The文件夹CVaR对象实现所谓的条件价值危险投资组合优化（请参见Rockafellar和Uryasev [48]，[49]在投资组合优化），通常称为CVAR投资组合优化。CVAR投资组合优化可与均值方差投资组合优化相同的返回代理和投资组合集合，但使用投资组合返回的条件价值作为风险代理。
The投资组合object implements what is known as mean-absolute deviation portfolio optimization (see Konno and Yamazaki [50] at投资组合优化), which is referred to as MAD portfolio optimization. MAD portfolio optimization works with the same return proxies and portfolio sets as mean-variance portfolio optimization but uses mean-absolute deviation portfolio returns as the risk proxy.

返回代理

The proxy for portfolio return is a function $μ : X \to R$ on a portfolio set $X \subset R^{n}$ 这表征了与投资组合选择相关的奖励。通常，投资组合申报表的代理有两种通用表格，总表格和净投资组合申报表。两个投资组合退货表格分开无风险费率r₀so that the portfolio $x \in X$ 仅包含风险资产。

Regardless of the underlying distribution of asset returns, a collection ofSasset returnsy₁，...，，y_S具有资产回报的平均值

$m = \frac{1}{S} \sum_{s = 1}^{S} y_{s},$

和(sample) covariance of asset returns

$C = \frac{1}{S - 1} \sum_{s = 1}^{S} (y_{s} - m) {(y_{s} - m)}^{T} 。$

These moments (or alternative estimators that characterize these moments) are used directly in mean-variance portfolio optimization to form proxies for portfolio risk and return.

Gross Portfolio Returns

投资组合的总投资组合返回 $x \in X$ 是

$μ (x) = r_{0} + {(m - r_{0} 1)}^{T} x,$

where:

r₀是the risk-free rate (scalar).

m是the mean of asset returns (n向量）。

If the portfolio weights sum to1，无风险费率无关紧要。在文件夹object to specify gross portfolio returns are:

风险命中forr₀
资产form

净投资组合返回

投资组合的净投资组合返回 $x \in X$ 是

$μ (x) = r_{0} + {(m - r_{0} 1)}^{T} x - b^{T} 最大限度 {0, x - x_{0}} - s^{T} 最大限度 {0, x_{0} - x},$

where:

r₀是the risk-free rate (scalar).

m是the mean of asset returns (n向量）。

b是购买资产的比例成本（n向量）。

s是the proportional cost to sell assets (n向量）。

You can incorporate fixed transaction costs in this model also. Though in this case, it is necessary to incorporate prices into such costs. The properties in the文件夹指定网络投资组合返回的对象为：

风险命中forr₀
资产form
InitPortforx₀
BuyCostforb
卖方fors

Risk Proxy

投资组合风险的代理是一个功能 $σ : X \to R$ on a portfolio set $X \subset R^{n}$ that characterizes the risks associated with portfolio choices.

方差

投资组合返回投资组合的差异 $x \in X$ 是

$V a r i a n c e (x) = x^{T} C x$

whereC是the covariance of asset returns (n-经过-n阳性 - 半决赛基质）。Covariance是a measure of the degree to which returns on two assets move in tandem. A positive covariance means that asset returns move together; a negative covariance means they vary inversely.

The property in the文件夹指定投资组合返回方差的对象是AssetCovarforC。

尽管均值变化投资组合优化的风险代理是投资组合收益的差异，但经常报告并显示正方根，这是投资组合返回的标准偏差。此外，该数量通常称为投资组合的“风险”。有关详细信息，请参阅Markowitz（投资组合优化）。

有条件的价值风险

The conditional value-at-risk for a portfolio $x \in X$ , which is also known as expected shortfall, is defined as

$C V a R_{α} (x) = \frac{1}{1 - α} \int_{f (x, y) \geq V a R_{α} (x)} f (x, y) p (y) d y,$

where:

α是概率级别0<α<1。

f(x,y)是the loss function for a portfoliox和资产返回y。

p(y)是the probability density function for asset returny。

VaR_α是投资组合的价值x在概率级别α。

价值风险定义为

$V a R_{α} (x) = \min {γ : \Pr [f (x, Y) \leq γ] \geq α} 。$

An alternative formulation for CVaR has the form:

$C V a R_{α} (x) = V a R_{α} (x) + \frac{1}{1 - α} \int_{R^{n}} 最大限度 {0, (f (x, y) - V a R_{α} (x))} p (y) d y$

The choice for the probability levelα是typically 0.9 or 0.95. Choosingαimplies that the value-at-riskVaR_α(x)for portfoliox是the portfolio return such that the probability of portfolio returns falling below this level is (1–α）。给出VaR_α(x)for a portfoliox，投资组合的条件价值是危险的预期损失，投资组合的返回高于风险的回报率。

Note

Value-at-risk is a positive value for losses so that the probability levelα表明投资组合退货低于风险价值的负数的概率。

为了描述回报的概率分布，文件夹CVaRobject takes a finite sample of return scenariosy_s，和s=1，...，，S。Eachy_s是一个n载体包含每一个的回报nassets under the scenarios。This sample ofS方案被存储为大小的方案矩阵S-经过-n。然后，给定投资组合的CVAR投资组合优化的风险代理 $x \in X$ 和 $α \in (0, 1)$ , is computed as

$C V a R_{α} (x) = V a R_{α} (x) + \frac{1}{(1 - α) S} \sum_{s = 1}^{S} 最大限度 {0, - y_{s}^{T} x - V a R_{α} (x)}$

The value-at-risk,VaR_α(x），每当CVAR估计时，都会估计。损失功能是 $f (x, y_{s}) = - y_{s}^{T} x$ , which is the portfolio loss under scenarios。

在此定义下，根据给定的方案，VAR和CVAR是VAR和CVAR的样本估计器。更好的场景样本产生了更可靠的VAR和CVAR估计值。

For more information, see Rockafellar and Uryasev [48], [49], and Cornuejols and Tütüncü, [51], at投资组合优化。

Mean Absolute-Deviation

投资组合的平均吸毒偏差（MAD） $x \in X$ 是defined as

$M A D (x) = \frac{1}{S} \sum_{s = 1}^{S} | {(y_{s} - m)}^{T} x |$

where:

y_sare asset returns with scenarioss= 1,...S(S收集nvectors).

f(x,y)是the loss function for a portfoliox和资产返回y。

m是the mean of asset returns (n向量）。

such that

$m = \frac{1}{S} \sum_{s = 1}^{S} y_{s}$

有关更多信息，请参见Konno和Yamazaki [50]投资组合优化。