资本资产定价模型CAPM & 套利定价理论APT简介

资本资产定价模型（Capital Asset Pricing Model, CAPM）是由美国学者William Sharpe、John Lintner、Jack Treynor和Jan Mossin等人在现代投资组合理论的基础上建立起来的，广泛应用于投资决策和公司理财领域。资本资产定价模型中，所谓资本资产主要指的是股票资产，而定价则试图解释资本市场如何决定股票收益率，进而决定股票价格。

对于投资者而言，他们偏向更小的风险和更高的回报(risk averse).但是对于每个人来说，能够承受的风险的程度都是不一样的，如果作为一名投资者，你想最大化回报，最小化风险，那么有一个efficient frontier(效率前缘)的概念是允许你的投资组合达到这个状态的。投资组合理论中，有效前沿是指投资组合，它占据了风险收益谱的“有效”部分。 从形式上讲，这是一组满足以下条件的投资组合：不存在其他具有更高预期收益但具有相同标准差收益的投资组合。

在市场中，我们能做到的有：

–we can diversify away unique risk by forming portfolios

–we can only be compensated for market risk

–Market risk is the relevant risk for the investor

除此之外还有一个risk free asset的概念。什么是risk free asset?

Risk-free asset is an asset that:

–Is risk free; i.e. the standard deviation of its returns is zero.

–Has zero correlation with all other risky assets.

–Will lie on the vertical axis of a portfolio graph (since σ = 0)

•Is usually proxied by the rate of short-term government bonds, such as T-Bill in the US, Short Gilts on the UK etc.

那么CAPM可以概括为下面这样一个图：

CAPM的公式就是：ERi=Rf+βi(ERm−Rf)

ERi=expected return of investment

Rf=risk-free rate

βi=beta of the investment

(ERm−Rf)=market risk premium​

资本资产定价模型的图示形式称为证券市场线(SML)。它主要用来说明投资组合报酬率与系统风险程度β系数之间的关系，以及市场上所有风险性资产的均衡期望收益率与风险之间的关系证券市场线是资本资产定价模型的图示形式，它显示总体的期望报酬与系统风险之间的关系。当一个单一资产落在证券市场线的上方，代表在同样风险下有比较高的报酬，也就是资产被低估，应该买进；当落于证券市场线下方，代表资产被高估，应该卖出。

那么对于这个公式中的beta如何估计呢？

Beta= Covariance/Variance

Covariance=Measure of a stock’s return relative to that of the market

Variance=Measure of how the market moves relative to its mean

Covariance measures how two stocks move together. A positive covariance means the stocks tend to move together when their prices go up or down. A negative covariance means the stocks move opposite of each other.

Variance, on the other hand, refers to how far a stock moves relative to its mean. For example, variance is used in measuring the volatility of an individual stock's price over time. Covariance is used to measure the correlation in price moves of two different stocks.

Beta>1 股票的风险大于市场风险

Beta=1 相等

Beta<1 小于市场风险

Factor Model多因素模型

Factor model:asset pricing models that can be used to estimate the discount rate for the valuation of financial assets. They are generally extensions of the single-factor capital asset pricing model (CAPM).

R=R(bar)+U

=R(bar)+ m+ɛ

U--unexpected risk=m+ɛ;ɛ--company specific;m=market risk

Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset's returns can be predicted using the linear relationship between the asset’s expected return and a number of macroeconomic variables that capture systematic risk. It is a useful tool for analyzing portfolios from a value investing perspective, in order to identify securities that may be temporarily mispriced.

where:E(R)i​=Expected return on the asset

Rz​=Risk-free rate of return

βn​=Sensitivity of the asset price to macroeconomic factor n

Ei=Risk premium associated with factor i​

The beta coefficients in the APT model are estimated by using linear regression. In general, historical securities returns are regressed on the factor to estimate its beta.