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经济代写|计量经济学代写Econometrics代考|One-Factor Model

The fundamental equation common to most asset pricing models states that the price of a given asset $i$ at time $t$, denoted $P_{i, t}$, must be equal to the expected discounted value of its payoff (for more details see, for instance, Cochrane 2005; Ferson 2003; Smith and Wickens 2002):
$$
P_{i, t}=E_t\left[m_{t+1}\left(P_{i, t+1}+D_{i, t+1}\right)\right]
$$
where $D_{i, t+1}$ is the payment (interest or dividend) received at time $t+1$, and $m_{t+1}$ is a strictly positive random variable used to discount the future payoffs, called the stochastic discount factor $(\mathrm{SDF}) . E_t()=.E\left(. \mid \Omega_t\right)$ denotes the time-t conditional expectation given the information set $\Omega_t$.

Equation (1) assumes that the future payoff and discount factor ${ }^3$ are stochastic: both are uncertain at date $t$ and are contingent to future states of nature. It is worth noting that this equation holds for any investment horizon and type of asset (bond, share, option, real estate and so forth) and this without any specific assumption such as complete markets, financial market equilibrium, investor preference, or distribution of asset returns.
Equation (1) can be expressed in terms of returns:
$$
1=E_t\left(m_{t+1} R_{i, t+1}\right)
$$
where $R_{i, t+1}=\frac{P_{i, t+1}+D_{i, t+1}}{P_{i, t}}$ is the gross return of the asset at time $t+1$.
In the case of a risk-free asset whose gross return $R_f$ is known with certainty and assumed to be constant, Eq. (2) implies that the conditional expectation of the SDF is equal to the inverse of the risk-free rate:
$$
E_t\left(m_{t+1}\right)=1 / R_f
$$

经济代写|计量经济学代写Econometrics代考|Multiple Betas

The consumption CAPM and conditional CAPM are basically one-factor asset pricing models. Indeed, Eqs. (8) and (10) belong to the more general class of linear pricing kernel
$$
m_{t+1} \approx a_{0, t}+a_{1, t} F_{1, t+1}+\cdots+a_{N, t} F_{N, t+1}
$$
in which $F_{1, t+1}, \ldots, F_{N, t+1}$ are variables or factors that are good proxies for growth of marginal utility.

Many authors have considered the case where the stochastic discount factor can be represented as a linear function of $N$ factors of the form given by Eq. (26). For instance, in Eq. (10), which leads to a conditional CAPM, the return on the market portfolio is proxied by the return on a stock market index $R_{M, t+1}$, but this assumption is criticized by Roll (1977), who argues that this approximation neglects the human capital component in total wealth. Wang (1996) and other authors suggest that the growth rate of labor income can be a good proxy for the return on human capital. Under this hypothesis, one can consider an SDF with two factors: the return of a stock market index $R_{M, t+1}$ and the growth rate of labor income $\Delta y_{t+1}: m_{t+1} \approx$ $a_{0, t}+a_{1, t} R_{M, t+1}+a_{2, t} \Delta y_{t+1}$.

Considering Eq. (26), Ferson and Jagannathan (1996) show that in the case where the factors $F_{1, t+1}, \ldots, F_{N, t+1}$ are traded assets ${ }^{14}$ and if the coefficients are defined as follows ${ }^{15}$
$$
\begin{aligned}
a_{j, t} & =-\frac{E_t\left(\tilde{f}{j, t+1}\right)}{R_f \operatorname{var}_t\left(F{j, t+1}\right)}, \quad j=1, \ldots, N \
a_{0, t} & =\frac{1}{R_f}-\sum_{j=1}^N a_{j, t},
\end{aligned}
$$
where $E_t\left(\widetilde{f}{j, t+1}\right)=E_t\left(F{j, t+1}\right)-R_f$ denotes the (conditional) expected risk premium of factor $j$, then we obtain a multi-factor representation of the conditional CAPM:
$$
E_t\left(\widetilde{r}{i, t+1}\right)=\sum{n=1}^N \beta_{n, t} E_t\left(\tilde{f}_{n, t}\right)
$$

计量经济学代考

经济代写|计量经济学代写Econometrics代考|One-Factor Model

大多数资产定价模型共同的基本方程表明，给定资产$i$在$t$时间(表示$P_{i, t}$)的价格必须等于其支付的预期贴现值(更多细节参见，例如，Cochrane 2005;Ferson 2003;Smith and Wickens, 2002):
$$
P_{i, t}=E_t\left[m_{t+1}\left(P_{i, t+1}+D_{i, t+1}\right)\right]
$$
其中$D_{i, t+1}$是在$t+1$时间收到的支付(利息或股息)，$m_{t+1}$是用于贴现未来支付的严格正随机变量，称为随机贴现因子$(\mathrm{SDF}) . E_t()=.E\left(. \mid \Omega_t\right)$表示给定信息集$\Omega_t$的时间t条件期望。

方程(1)假设未来收益和贴现因子${ }^3$是随机的:它们在日期$t$都是不确定的，并且取决于未来的自然状态。值得注意的是，这个等式适用于任何投资期限和资产类型(债券、股票、期权、房地产等)，而不需要任何特定的假设，如完全市场、金融市场均衡、投资者偏好或资产回报分配。
式(1)可以用收益表示:
$$
1=E_t\left(m_{t+1} R_{i, t+1}\right)
$$
其中$R_{i, t+1}=\frac{P_{i, t+1}+D_{i, t+1}}{P_{i, t}}$是资产在$t+1$时刻的总回报。
在无风险资产的情况下，其总收益$R_f$是确定的，并假设为常数，Eq.(2)表明，SDF的条件期望等于无风险利率的倒数:
$$
E_t\left(m_{t+1}\right)=1 / R_f
$$

经济代写|计量经济学代写Econometrics代考|Multiple Betas

消费CAPM和条件CAPM基本上都是单因素资产定价模型。的确，等式。(8)和(10)属于更一般的一类线性定价核
$$
m_{t+1} \approx a_{0, t}+a_{1, t} F_{1, t+1}+\cdots+a_{N, t} F_{N, t+1}
$$
其中$F_{1, t+1}, \ldots, F_{N, t+1}$是代表边际效用增长的变量或因素。

许多作者都考虑过这样一种情况，即随机折现因子可以表示为公式(26)所示形式的$N$因子的线性函数。例如，在导致条件CAPM的Eq.(10)中，市场投资组合的回报由股票市场指数$R_{M, t+1}$的回报来代表，但这一假设受到Roll(1977)的批评，他认为这种近似忽略了总财富中的人力资本成分。Wang(1996)等认为劳动收入增长率可以很好地代表人力资本回报率。在这个假设下，我们可以用两个因素来考虑SDF:股票市场指数的收益率$R_{M, t+1}$和劳动收入的增长率$\Delta y_{t+1}: m_{t+1} \approx$$a_{0, t}+a_{1, t} R_{M, t+1}+a_{2, t} \Delta y_{t+1}$。

考虑到Eq. (26)， Ferson和Jagannathan(1996)表明，在因素$F_{1, t+1}, \ldots, F_{N, t+1}$为交易资产${ }^{14}$的情况下，如果系数定义如下${ }^{15}$
$$
\begin{aligned}
a_{j, t} & =-\frac{E_t\left(\tilde{f}{j, t+1}\right)}{R_f \operatorname{var}t\left(F{j, t+1}\right)}, \quad j=1, \ldots, N \ a{0, t} & =\frac{1}{R_f}-\sum_{j=1}^N a_{j, t},
\end{aligned}
$$
式中$E_t\left(\widetilde{f}{j, t+1}\right)=E_t\left(F{j, t+1}\right)-R_f$表示因子$j$的(有条件的)预期风险溢价，则得到条件CAPM的多因素表示:
$$
E_t\left(\widetilde{r}{i, t+1}\right)=\sum{n=1}^N \beta_{n, t} E_t\left(\tilde{f}_{n, t}\right)
$$

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