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

#### Doug I. Jones

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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

$$P_{i, t}=E_t\left[m_{t+1}\left(P_{i, t+1}+D_{i, t+1}\right)\right]$$

$$1=E_t\left(m_{t+1} R_{i, t+1}\right)$$

$$E_t\left(m_{t+1}\right)=1 / R_f$$

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

$$m_{t+1} \approx a_{0, t}+a_{1, t} F_{1, t+1}+\cdots+a_{N, t} F_{N, t+1}$$

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