投資學3-資本資產定價模型（CAPM）

作者：由 Round face 發表于舞蹈時間：2020-07-14

Investment 3-CAPM and beyond

1。 Capital Asset Pricing Model （CAPM）

The market portfolio

The market capitalization of asset

$MCAP_j=(price\ \ per\ \ share)_j\times(\#of\ \ shares\ \ outstanding)_j$

The total market capitalization of all risky N assets is

$MCAP_M=\sum_{j=1}^{N}MCAP_j$

The market portfolio

is the portfolio of all risky assets weighted by their relative market captialization； i。e。 weight of asset j is

$W_M^j=\frac{MCAP_j}{\sum_{j=1}^{N}MCAP_j}=\frac{MCAP_j}{MCAP_M}$

整個市場可以看成一個投資組合，某個資產的權重就是其市場佔有比率

Equilibrium

There are

i=1， 2，。。。， I

investors and investor i invests in two funds：

$W_{i,0}$

in riskless asset and

$W_i-W_{i,0}$

in the tangency portfolio

$w_{tan}$

Money market equilibrium： risk-free assets is in zero net supply，

$\sum_{i=1}^IW_{i,0}=0$

Risky aeest equilibrium：

$\sum_{i=1}^I(W_i-W_{i,0})w_{tan}=MCAP_Mw_M$

無風險資產如現金、存款的淨投資值，即投入的財富總和為0，因為貨幣市場是借貸平衡的

風險資產的均衡是不同投資者的正切投資組合相加後的總投資為該資產的市場總量

把兩式相加，得到兩個結論：一是市場總量就是所有投資量，即

$\sum_{i=1}^IW_i=MCAP_M$

，二是正切投資組合（最優證券組合）就是市場投資組合，即

$w_{tan}=w_{M}$

Capital Market Line (CML)

The Capital Market Line goes through the riskless asset and the market portfolio in （σ，μ）-space and consists of mean-variance efficient portfolios

The expected return on any portfolio on the CML is given by

$\mu_e=R_0+\frac{\mu_M-R_0}{\sigma_M}\sigma_e$

That means the expected return consists of the risk-free rate and the risk premium

The risk premiumis the product of the market price of risk

$\frac{\mu_M-R_0}{\sigma_M}$

and the amount of risk

$\sigma_e$

The market price of risk is also sharpe ratio of the market portfolio

$\lambda=\frac{\mu_M-R_0}{\sigma_M}$

，

$\lambda$

determines compensation for each unit of risk

$\lambda$

就是CML的斜率

在均值方差投資組合中，投資者

的最優投資組合是

$w_i=\frac{1}{a_i}\Sigma^{-1}(\mu-R_01)$

，把所有投資者看成一個整體，其資產組合為

$w_M=\sum_{i=1}^I(\frac{W_i}{W}w_i)=\sum_{i=1}^I(\frac{W_i}{W}\frac{1}{a_i})\Sigma^{-1}(\mu-R_01)$

，W為所有投資者的總財富

變形

$w_M(\mu-R_01)=\frac{1}{\sum_{i=1}^I(\frac{W_i}{W}\frac{1}{a_i})}w_M$

或

$\lambda=\frac{\mu_M-R_0}{\sigma_M}=\frac{1}{\sum_{i=1}^I(\frac{W_i}{W}\frac{1}{a_i})}\sigma_M$

The market price of risk （slope of CML） is decreasing in the wealth-weighted average risk tolerance across investors and increasing in the risk of the market portfolio

The security market line (SML)

The

expected excess return

on any individual asset is proportional to

the risk premium on the market portfolio

times

the beta coefficient

The relationship can be portrayed graphically as the security market line （SML）

$\mu_i=R_0+\beta_i(\mu_M-R_0)$

where

$\beta_i=\frac{Cov(R_i,R_M)}{Var(R_M)}=\rho_{Mi}\sigma_i/\sigma_M$

is the coefficient in a regression of

， so

$\mu_i=R_0+\lambda \rho_{Mi}\sigma_i$

CAPM解釋了為什麼觀測到的市場投資組合是均值方差有效的，以及證券的beta值是如何衡量市場風險；市場投資組合的beta值是1，即

$\beta_M=1$

SML是beta值和expected return的線性圖，斜率是

$\mu_M-R_0$

$\lambda$

是風險的市場價格，

$\rho_{Mi}\sigma_i=d\sigma_M/dw_M^i$

是一單位資產i在市場投資組合中的權重的邊際增長所導致的市場風險的邊際增長，可以理解為資產i的系統性風險/不可分散風險的數量

Risk premium on asset i is the product of the market price of risk λ， and the amount of asset i’s systematic risk

對比SML和CML

CML描述了有效資產組合的風險溢價與和資產組合的標準差（也就是該portfolio的風險）之間的關係

SML描述了個人資產的風險溢價和資產的beta值之間的關係，beta值可以看成衡量了作為有效多元投資組合一部分的個人資產的風險

所有的資產都在均值方差的邊界以內，如果CAPM成立，也就是在投資組合是均值方差有效的情況下，所有的資產都落在一條SML上

The Security Characteristic Line (SCL)

For each security estimate a different Security Characteristic Line

$R_i-R_0=\alpha_i+\beta_i(R_M-R_0)+\epsilon_i$

where

$E[\epsilon_i]=0, Cov[R_M,\epsilon_i]=0,Var[\epsilon_i]=\nu_i^2$

The linear regression measures three security characteristics

$\beta_i$

measures systematic risk

$\nu^2_i$

measures idiosyncratic （diversifiable） risk

$\alpha_i$

measures excess premium （mispricing） w。r。t CAPM

The risk of asset i can be decomposed as systematic risk and idiosyncratic risk

系統風險就是

$\beta^2_iVar[R_M]$

，特別風險就是

$Var[\epsilon_i]$

，迴歸的

值就是系統風險佔總風險的比例

Estimating betas

When we estimate betas from rolling time series of data we observe large time-variation in betas， either the true betas change at high frequencies or the estimated betas change because of statistical errors

Bloomberg （and others） calculate adjusted betas

suppose the estimated beta from regression is

$\beta_{est}$

and the average firm beta is

$\beta_{avg}$

adjust：

$\beta_{adj}=w∗\beta_{est} +(1−w)∗\beta_{avg}$

where w is the weight put on the data

Bloomberg uses average beta as 1 and w = 2/3

This is called a shrinkage estimator

But

$\beta_{avg}=1$

is not always optimal

Better estimates of beta can be obtained by shrinking towards the average beta for the firm’s industry

If the firm is a conglomerate you can use a weighted average of industry betas

Example： Diversified Inc。 is composed of a 20% transportation division and 80% hotel division， equity beta should shrink towards

$β_{avg}$

= 0。2 x 1。17 +0。8 x 1。13 = 1。14

Short-sale constraints

In the CAPM， all investors hold the market portfolio。 In equilibrium， no investor sells any security short

Short-selling constraints is non-binding and equilibrium prices are unaffected by it

Applications of the CAPM

Portfolio choice （e。g。 the Treynor-Black or the Black-Litterman models）

Provides rationale for index funds （mutual funds and ETFs tracking broad market indices）

Shows what a “fair” security return is and provides a benchmark for security analysis （e。g。， identifying over and undervalued securities through their CAPM-alpha）

Provides a cost of capital （or required return） used in capital budgeting to

compute NPV of risky project or “hurdle rate” for IRR

Evaluation of fund manager performance

2。 Zero-Beta CAPM

在沒有無風險資產的情況下，投資者只能選擇風險資產進行投資，根據均值方差投資理論進行投資

Zero-Beta CAPM relies on two properties of portfolios on the minimum-variance frontier

Any minimum-variance portfolio can be written as a linear combination of two other minimum-variance portfolios

Every portfolio on the mean-variance efficient frontier has a “companion” portfolio on the inefficient portion of the minimum-variance frontier with which it is uncorrelated

CAPM-style relation with

is replaced with

$\mu_z$

， the expected return on the zero-beta portfolio：

$\mu=\mu_z1+\beta(\mu_M-\mu_z)$

where

$\beta=\frac{Cov(R,R_M)}{Var(R_M)}$

$\mu_z>R_0$

， security market line is flatter than in the regular CAPM

Short-sale constraints

In the standard CAPM， all investors hold the market portfolio and short sale restrictions are not binding

In the zero-beta CAPM， investors may want to hold any portfolio on the mean-variance efficient frontier。 For these portfolios to be attainable， short-selling of risky assets must be allowed

3。 Leverage CAPM

Leverage constraints： investors can‘t use leverage （pension funds， mutual funds， etc。）

Margin constraints： investors willing to use leverage are constrained by their margin requirements and may need to de-lever （hedge funds， proprietary traders， etc。）

CAPM with these constraints developed by Frazzini and Pedersen “Betting Against Beta”

For investor i：

$max_{w_i}(\mu_p-\frac{a_i}{2}\sigma_p^2)=max_{w_i}(R_0+w$

subject to constraint

當

等於1時，無槓桿；大於1時無槓桿且有cash限制；小於1時為margin限制

Set up the Lagrangian

When constraint is not binding，

$\lambda_i=0$

FOC：

$\partial w_i=0$

， so

$w_i=\frac{1}{a_i}\Sigma^{-1}(\mu-R_01-\lambda_i1m_i)$

，

$\lambda=\frac{1$

Unconstrained investor （

$\lambda_i=0$

） invests in tangency portfolio

Constrained investor invests in portfolio that is convex combination of

$w_{tan}, w_{min}, R_0$

根據凸最佳化知識，當限制條件取等號時，拉格朗日引數取0，即binding

Portfolio choice with margin constraints mi&；amp；lt；1

Portfolio choice with no leverage and cash-constraints mi &；amp；gt； 1

=1：不受限制的投資者持有正切投資組合T，並且借貸是無風險利率

>1：受到邊際限制的投資者持有的風險資產C透過槓桿達到

$\bar{C}$

<1：受到槓桿和現金限制的投資者持有的風險資產佔總財富比例從D變成D‘

Zero-Beta CAPM SML has a higher intercept and flatter slope than CAPM SML and the slope （intercept） decreases （increases） in

$\psi$

where

$\psi=\sum_{i=1}^I(\frac{W_i}{W}\frac{a}{a_i}\lambda_im_i)$

For given

$\psi$

， alpha is decreasing in β

Positive for low-beta （β < 1） assets

Negative for high-beta （β > 1） assets

For given β，（absolute） alpha is increasing in

$\psi$

4。 Liquidity CAPM

Liquidity and liquidity risk

Level of liquidity impacts prices

Less liquid stocks trade at lower prices and have higher expected returns

Liquidity varies over time and is correlated across assets

Risk-averse investors may require a compensation for being exposed to liquidity risk

Liquidity CAPM developed by Acharya and Pedersen （2005） “Asset pricing with liquidity risk”

Liquidity CAPM

Assume that there is a cost

associated with trading security i and the cost is stochastic giving risk to liquidity risk

Gross return on an asset is

$R_t^i=\frac{P_t^i+D_t^i}{P_{t-1}^i}$

Return net of transaction costs is

$\tilde{R_t^i}=\frac{P_t^i+D_t^i-C_t^i}{P_{t-1}^i}=R_t^i-c_t^i$

CAPM holds for net returns

$E_t[\tilde{R}^i_{t+1}]=R_0+\gamma_t\frac{cov_t(\tilde{R}^i_{t+1},\tilde{R}^M_{t+1})}{var_t(\tilde{R}^M_{t+1})}$

where

$\gamma_t=E_t[\tilde{R}^M_{t+1}-R_0]$

is the expected net excess return on the market

Rewriting the one-beta CAPM for net returns in terms of gross returns， gives a （conditional） liquidity CAPM for gross returns

$E_t[{R}^i_{t+1}]=R_0+E_t[c_{t+1}^i]+\gamma_t(\frac{cov_t({R}^i_{t+1},{R}^M_{t+1})}{var_t(\tilde{R}^M_{t+1})}+\frac{cov_t({c}^i_{t+1},{c}^M_{t+1})}{var_t(\tilde{R}^M_{t+1})}-\frac{cov_t({R}^i_{t+1},{c}^M_{t+1})}{var_t(\tilde{R}^M_{t+1})}-\frac{cov_t({c}^i_{t+1},{R}^M_{t+1})}{var_t(\tilde{R}^M_{t+1})})$

The expected excess return is the sum of

expected relative illiquidity cost

Four betas （or covariances） times the market risk premium： market beta （as in the standard CAPM） and three additional liquidity betas

Three liquidity betas

$cov_t(c_{t+1}^i,c_{t+1}^M)$

： Expected return increases with the covariance between the security’s illiquidity and market illiquidity

$cov_t(R_{t+1}^i,c_{t+1}^M)$

： Expected return decreases with the covariance between the security’s return and market illiquidity

$cov_t(c_{t+1}^i,R_{t+1}^M)$

： Expected return decreases with the covariance between a security’s illiquidity and market return

5。 Intertemporal CAPM， consumption CAPM， and beyond

Intertemporal CAPM

CAPM is a one-period model； investors have short horizons and follow myopic investment strategies

In general， myopic strategies are optimal if

1。 Investors have only short horizons

2。 Future investment opportunities are the same as today’s

In practice

1。 Investors do invest over long horizons

2。 Investment opportunities do change over time

Assets that help to hedge against deteriorating investment opportunities are attractive and have higher prices and lower expected excess returns

Expected excess returns depend on covariance with investment opportunities

Consumption CAPM

Expected excess returns depend on covariance with consumption

Assets with negative covariance， pay off when consumption is low and marginal utility is high。 These are attractive， have relatively high prices and low expected returns

Assets with positive covariance， pay off when consumption is high and marginal utility is low。 These are not so attractive， have relatively low prices and high expected returns

Popular in academics， not so much in practice

Consumption data is infrequent （monthly or quarterly） and is measured imprecisely

Beyond the CAPM

Individuals have imperfect information and heterogeneous beliefs about the characteristics of assets and their variation over time

Market portfolio is unobservable （Roll critique）。 In theory， the market portfolio should include all types of assets that are held by anyone – including privately held businesses， foreign assets， real estate， human capital， etc。

Investors are taxed and belong to different tax brackets。 Affects portfolio choice since taxes differ depending on whether income is from dividends， interest， or capital gains， while some assets are tax-exempt

Trading is costly and investors do not optimally rebalance their portfolios due to transaction costs

Variance of returns is not a complete measure of risk

Investors may have more general utility functions and time-varying risk aversion

標簽： market risk CAPM beta Portfolio

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