Markowitz Portfolio Theory【Morden Portfolio Theory】

A rational person【理性人】

理性人应该对风险有着前后一致的态度。

Risk aversion【风险厌恶】：承担的相同的增量，要求更多的收益增量。
Risk neutral【风险中性】
Risk preference【风险偏好】

横轴对应风险，纵轴对应回报。

Mean Variance Model

Return: E®

Risk: σ【标准差】

用收益和方差概述了所有的金融产品。

Indifferent curve

markowitz认为大多数人都是风险厌恶者。

效用无差异曲线是一组凸【convex】函数。

箭头的方向是效用增加的方向。

Assumptions about a Markowitz investor

No transaction costs【无交易成本】
Assets are infinitely divisible【资产无限可分】
The absence of personal income tax【无税】
An individual cannot affect the price of a stock by his trading【价格接受者】
Investors make decisions solely in terms of returns and standard deviation of the returns【投资者只care 收益和标准差】
Unlimited short sales are allowed【可以无限做空】
Unlimited lending and borrowing at the riskless rate
All investors have identical expectations: μ σ ρ【投资者具有一致的预期/对同样的产资产有相同的看法】
All assets are marketable【所有的资产均可交易】

Assumption

for market :perfect

for investor

risk aversion and utility maximum

only care about E® and sigma

Effects of Correlation on Diversification Benefits

标准差一定为正，故而没有低二三象限；理性投资者不会承担风险而去亏钱，故没有第四象限。

$\rho=1$:完全正相关

$\rho=-1$:完全负相关

不同的相关系数对应不同的曲线。

$\rho$越小，在相同收益下，风险分散化效果越好，波动率越低。$\rho=-1$分散化效果最好。

Markowitz Efficient Frontier【有效前沿】

当仅存在两种资产A、B时，可供选择的资产组合在AB的连线上。

当存在三种资产时，可供选择的资产组合可以看作A，B之间的任意组合和资产C的组合。黄色区域即可供选择的资产组合区域。

当有无数种资产可供选择时，最终的组合区域是一个左侧光滑、右侧不规则的区域。

一个理性人在收益相同时，不会选择风险更大的资产组合，因此投资者只会选择左侧曲线上的资产组合。该曲线称为最小方差前沿【minimum variance frontier/curve】。

A点称为Global minimum variance portfolio.

一条双曲线。

一个理性人在风险相同时，不会选择收益更小的资产组合。因此投资者只会选择最小方差前沿的上半部分。该上半部分称为Markovwits Efficient Frontier

Minimum Variance Portfolio

Definition: the portfolio with the smallest variance among all possible portfolios on a portfolio possibilities curve. The shape of the portfolio possibilities curve is best described in two pieces:

The portion of the portfolio possibility curve that lies above the minimum variance portfolio is concave.
The portion of the portfolio possibility curve that lies below the minimum variance portfolio is convex.

Capital Market Line (CML)

相较于Markov wits理论只多了一个无风险资产的假设。目的和Markov wits一样，还是资产配置。

The point of tangency【切点】 – Portfolio P, is known as the market portfolio. When one can lent or borrow money use riskless rate, investor will hold a combination of the market portfolio and the risk-free asset.

CML理论认为投资者只需要持有市场组合和无风险利率的组合。

引入无风险资产【由威廉夏普提出】

Risk-averse investors will create lower risk portfolios by lending (i.e.,investing in the risk-free asset). More risk-tolerant investors will increase portfolio return by borrowing at the risk-free rate.

切点M：market portfolio；market portfolio 包含了市场上所有的证券，且该证券在market portfolio中权重就是该证券在整个市场中的权重。

$$
E\left(R_P\right)=R_f+\frac{E\left(R_M\right)-R_f}{\sigma_M} \sigma_P
$$

定义
$$
\frac{E\left(R_M\right)-R_f}{\sigma_M}
$$
为sharpe ratio

投资者的无差异曲线与CML相切的点就是最适合投资者的投资组合。

Systematic and Unsystematic Risk

对前面两种理论的深化。

Total risk= systematic risk+ unsystematic risk

在CML上投资不存在系统性风险。称CML上的组合为well-diversified portfolios.

Systematic risk is the only important ingredient in determining expected returns and that nonsystematic risk plays no role

承担非系统性风险不能带来收益补偿。

系统性风险才有收益补偿。

通过调整A的资产组合可以在保持收益不变的情况下降低风险。

投资证券越多，非系统性风险越少。当数量大于30时，非系统性风险几乎消失。

Security Market Line (SML)

假设同上。

承担的系统性和收益补偿的关系。SML描述了收益率和系统性风险之间的线性关系。

$$
E\left(R_P\right)= R_f+\left[E\left(R_M\right)-R_f\right]\beta_P \
$$

$$
\frac{\operatorname{COV}(\mathrm{P}, \mathrm{M})}{\sigma_{\mathrm{M}}^2}=\beta
$$

beta就是证券所包含的市场风险。

市场组合的beta为1.

$R_P$称为required rate of return,是承担系统性风险理应获得的回报。

Capital Asset Pricing Model (CAPM)

Assumptions about CAPM

Access to information for all market participants, meaning that all information is freely available and instantly absorbed.
No transaction costs, taxes, or other frictions.
Allocations can be made in an investment of any partial amount (i.e., perfect divisibility).
All participants can borrow and lend at a common risk-free rate.
Any individual investor’s allocation decision cannot change the market prices.

Capital Asset Pricing Model (CAPM)

$$
E\left(R_P\right)=R_f+\beta_P\left[E\left(R_M\right)-R_f\right] \
$$
$$
\beta_P=\frac{\operatorname{Cov}(P, M)}{\sigma_M^2}=\rho_{P, M} \frac{\sigma_P}{\sigma_M}
$$

$E\left(R_p\right)$ : expected return on risky asset

$R_f$ : risk-free rate

$E\left(R_M\right)-R_f$ : market portfolio risk premium【风险溢价】，系统性风险的单价

$\beta_p$ : systematical risk of asset $P$,承担的系统性风险的量

$\beta_{\mathrm{P}} \times\left[E\left(R_M\right)-R_f\right]$ : beta-adjusted market risk premium【beta调整的市场风险溢价】

例题

The expected return on the market is 15%, the risk-free rate is 8%, and the beta for stock A is 1.2. Compute the rate of return that would be expected (required) on this stock.

Answer: E(RA) = 0.08 + 1.2 × (0.15-0.08) = 0.164
Note: β_A>1 ⇒ E(R_A) > E(R_m)

The expected return on the market is 15%, the risk-free rate is 8%, and the beta for stock B is 0.8. Compute the rate of return that would be expected (required) on this stock.

Answer: E(RB) = 0.08 + 0.8 × (0.15-0.08) = 0.136
Note: β_B <1 ⇒ E(R_B) < E(R_m)

市场组合的beta等于1

答：4%+0.94*(9%)=12.46%>10%,投资不划算，股票被高估。

CML和SML的对比：

1.方程式不同

2.目的:sml用来估值valuation，计算应有的回报，不能用来资产配置；cml用来资产配置asset allocation。

3.在cml线上的组合称为well-diversified portfolios，sml线上的一个点对应的不是一个组合，而是无数多个系统性风险一定的组合，称为all portfolio/not well-diversified。

在sml线上，只知道一个组合的系统性风险所对应的required rate of risk ，但不知道其所蕴含的非系统性风险。

Performance Measures

In a world where the market is in equilibrium and is expected to remain in equilibrium, no investor can achieve an abnormal return. Of course, this is not the case in the real world.

Portfolio managers rely on indices to measure the performance of a given stock or portfolio relative to the CAPM equilibrium risk-return relationship.

The focus is on the three traditional measures of portfolio performance based on CAPM:

 the Sharpe reward-to-volatility ratio,

 the Treynor reward-to-volatility ratio,

 the Jensen performance index.

Sharpe Performance Index

Measures the ratio of the average rate of return E(R_P), in excess of the risk-free rate RF, to the absolute risk σ(R_P).

衡量了超过无风险利率RF的平均收益率E(R_P)与绝对风险σ(R_P)之比。

$$
S R=\frac{E\left(R_P\right)-R_F}{\sigma\left(R_P\right)}
$$

the slope of R_fP is SR.

due to the incompletence of market,we may realize a portfolio above/under CML .

Widely used for measuring portfolio performance that are not very diversified.

A better method for measuring historical performance.

难以预测非系统性风险，推测未来效果差。

Suitable for evaluating the performance of a portfolio that represents an individual’s total investment.

适用于投资组合，不太适合单只股票。

Treynor Performance Index

Treynor ratio is equal to the risk premium divided by beta (systematic risk)
$$
T R=\frac{E\left(R_P\right)-R_F}{\beta_P}
$$

非系统性风险=0，只有系统性风险。

More appropriate for comparing well-diversified portfolios and a more forward-looking【预测未来】 measure.

Jensen’s Performance Index

Jensen’s Performance Index (also called Jensen’s alpha) is the asset’s excess return over the return predicted by the CAPM
$$
E\left(R_P\right)-R_F=\alpha_P+\beta_P\left[E\left(R_M\right)-R_F\right]
$$

$$
E\left(R_P\right)-(R_F+\beta_P\left[E\left(R_M\right)-R_F\right])=\alpha_P
$$
Jensen’s alpha称为超额回报率excess return.

双alpha策略：一个组合既做多又做空，从而产生两个超额回报。

beta策略：通过提高投资组合的beta来增加回报。

Most appropriate for comparing portfolios that have the same beta and can be used to rank portfolios within peer groups.

对于具有相同β系数的投资组合进行比较并可用于对同行群体中的投资组合进行排名，最适合使用的方法是"同行群体分析"。

 An alternative approach, adopted by many professionals and investors, is to measure performance relative to a target portfolio or benchmark.

 There are three different measures of performance relative to a benchmark are discussed:

 tracking error

 information ratio

 sortino ratio.

Tracking Error

针对主动型基金经理，先规定一个基准，然后尽量比基准做的更好。

核心双动力基金：既买成长股又买价值股

The tracking error (TE) measures the difference between a portfolio’s returns and those of a benchmark. The first way to calculate TE is:

TE本身无法衡量投资组合所承担的风险。

$$
𝑅_𝑃 − 𝑅_𝐵
$$

Another way to measure TE is to calculate the standard deviation of the differences in the portfolio and the benchmark returns over time (N is the number of return periods measured):
$$
\sqrt{\frac{\sum\left(R_P-R_B-\overline{R_P-R_B}\right)^2}{N-1}}
$$

tracking error votality(TEV)

TE既可能是标准差也可能是差值。

Information Ratio

针对主动型基金经理

The information ratio measures the ratio of the residual return of the portfolio compared with its residual risk (tracking error).

$$
I R=\frac{E\left(R_P\right)-E\left(R_B\right)}{\sigma\left(R_P-R_B\right)}=\frac{\alpha_P}{\sigma\left(\alpha_P\right)}=\frac{\text { active return }}{\text { active risk }}
$$

IR=TE/TEV

IR越大，表现越好，是衡量基金表现的主要指标。

不同行业之间的IR可比性很差，只适用于同行业比较。

To check that the risk taken by the manager, in deviating from the benchmark, is sufficiently rewarded.

Sortino Ratio

MAR (minimum acceptable return【保底收益率】) is the return below which the investor does not wish to drop.

国家取消刚性兑付后，保底收益率不能公开讨论。MAR可以是R_F

Sortino ratio measures the ratio of the average rate of return E(RP), in excess of the risk-free rate RF, to the semi-standard deviation, which considers only data points that represent a loss.

$$
\text { Sortino Ratio }=\frac{E\left(R_P\right)-M A R}{\sqrt{\frac{1}{N-1} \sum_{t=1}^N\left(R_{P t}-M A R\right)^2}}\left(\mathrm{R}_{\mathrm{Pt}}<\mathrm{MAR}\right)
$$

Where N is the number of observed losses.

The Sortino ratio is more relevant【更有效】 than the Sharpe ratio when the return distribution is skewed to the left【左偏，损失数据更多】.