### 金融代写|金融计量经济学代写Financial Econometrics代考|ECOM40004

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• Statistical Inference 统计推断
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融计量经济学Financial Econometrics代考|Capital Market Line and Mean Variance Efficient Frontier

Likewise Mean Variance Efficient Frontier, the Capital Market Line (CML) is a graphical representation of all the portfolios that optimally combine risk and return. The basic different lies between the Mean Variance Efficient Frontier and Capital Market Line is that the Capital Market Line combines the risky assets with the non-risky assets.

In Fig. 4.4, the thin solid line represents the Capital Market Line. The Capital Market Line is the straight line drawn from the risk-free return rate (Y-axis) that cuts the Mean Variance Efficient Frontier (combinations of the risky assets) at the point $M$ (represents optimal portfolio) as shown in Fig. 4.4. Hence, slope of the Capital Market Line represents Sharpe ratio of the market portfolio. Risk of the portfolio increases on moving up along the CML while portfolio risk decreases on moving down along the CML. The Capital Market Line is represented by the following equation.
$$E\left(\mathrm{P}{r}\right)=\mathrm{r}{f}+\left[\frac{\mathrm{r}{m}-\mathrm{r}{f}}{\sigma_{m}}\right] * \sigma_{p}$$
where
$E\left(\mathrm{P}{r}\right)$ represents the expected portfolio returns $\mathrm{r}{f}$ is risk-free rate of return
$\mathrm{r}{m}$ is market return $\sigma{m}$ is the measure of standard deviation for market $\sigma_{p}$ is the measure of standard deviation for portfolio
$\left[\frac{r_{m}-r_{f}}{\sigma_{m}}\right]$ is Sharpe ratio of the market portfolio
Likewise, there is Security Market Line (SML) that visually represents the Capital Asset Pricing Model (CAPM). The Security Market Line is represented by the following Eq. (4.7).
$$S M L=\mathbf{r}{f}+\left[\beta *\left(\mathbf{r}{m}-\mathbf{r}_{f}\right)\right]$$
The Security Market Line (SML) represents all the securities present in the market along with their corresponding $\beta$ values as shown in Fig. $4.5$.
The Security Market Line is often refereed by the market participants to decide whether an asset is priced correctly or not? According to the principle of Security Market Line, price of the asset labelled as $A$ is overpriced while asset $B$ is underpriced as shown in Fig. 4.5.

## 金融代写|金融计量经济学Financial Econometrics代考|Key Topics Covered

Asset pricing studies deal with the pricing of assets where assets can be debts, equity, bonds, derivatives, and others. In general, asset prices follow the law of demand and supply. The asset pricing studies have basically two schools of thought, namely theoretical and empirical. The former deals with the theoretical aspects of asset prices, whereas empirical asset pricing deals more with the quantitative characteristics. Empirical asset pricing deals with the real market data resulting more preferred by the market participants. Pricing of the debt instruments is considered to be much easier as compared to the equity pricing. Presence of various risk factors and uncertainty make asset pricing quite a challenging task. Markowitz (1952) modern portfolio theory laid the theoretical foundation for analysing the risk and return relationship. Thereafter Sharpe (1964) developed a single factor asset pricing model often referred as the capital asset pricing model (CAPM). According to the CAPM, cross section of the asset returns depends only on the cross section of the asset $\beta$ s. Since then over 300 different risk factors have been identified by the various studies, resulting in several multifactor asset pricing models developing since the development of CAPM. Maiti $(2020 \mathrm{a})$ highlighted that the evolution process of risk factors and factor models seems to be an endless development. Several risk factors that drive the asset prices are continuously changing and evolving over time. In addition to that still numerous risk factors are yet to be identified. All of these safe bets jointly make asset pricing a topic of somewhat more than that of the class room coaching.

The capital asset pricing model (CAPM) is the primary successful formal model of market equilibrium. CAPM fundamentally describes the relationship between the expected returns and systematic risk for financial assets. Similarly, the consumption-based capital asset pricing model (CCAPM) uses consumption beta instead of the market beta. The capital asset pricing model (CAPM) is represented by the below Eq. 5.1:
$$E\left(r_{i}\right)=r_{f}+\beta *\left(r_{m}-r_{f}\right)$$
where
$E\left(\mathrm{r}{i}\right)$ represent expected return of the assets $\mathrm{r}{f}$ is risk free rate of return
$\mathrm{r}_{m}$ is market return
$\beta$ measures return volatility

## 金融代写|金融计量经济学Financial Econometrics代考|What Is So Special About Fama and French

Fama and French estimated risk factors using the mimicking portfolio approach. The risk factors are estimated directly from the crosssectional asset returns sorted by the firm characteristics. Fama and French constructed their mimicking portfolios by using the single and double sorting techniques as shown in Table 5.1. Fama and French ranked the sample stocks every year in the month of June based on their market capitalization $(\mathrm{MC})$ and formed two portfolios, namely Big(B) and Small(S) using the NYSE median market cap breakpoint. Similarly,

three $\mathrm{BE} / \mathrm{ME}$ weighted portfolios, namely High(H), Neutral $(\mathrm{M})$, and Low(L), are constructed using the $30: 40: 30$ breakpoint. Then using the double sorting technique six portfolios are formed, namely $S / L, S / M$, $\mathrm{S} / \mathrm{H}, \mathrm{B} / \mathrm{L}, \mathrm{B} / \mathrm{M}$, and $\mathrm{B} / \mathrm{H}$. Portfolio $(\mathrm{S} / \mathrm{L})$ consists of the small $\mathrm{MC}$ stocks and low value $\mathrm{BE} / \mathrm{ME}$ stocks, whereas $(\mathrm{B} / \mathrm{H})$ consists of the big $M C$ stocks and high value $\mathrm{BE} / \mathrm{ME}$ stocks. The revision of portfolio formation is done every next year, and this process of portfolio revision continues till the end year.

SMB stands for the “small minus big” and represents risk mimicking portfolio for the size factor. Similarly, HML represents risk mimicking portfolio for the value factor. SMB and HML risk mimicking portfolios are estimated using the Eqs. $5.5$ and $5.6$.
$$\begin{gathered} \mathrm{SMB}=(S / L+S / M+S / H) / 3-(B / L+B / M+B / H) / 3 \ \mathrm{HML}=(S / H+B / H) / 2-(S / L+B / L) / 2 \end{gathered}$$
Subsequently, other notable multifactor asset pricing models that developed are shown below in equation numbers from $5.7$ to $5.9$ :

\begin{aligned} E\left(r_{i}\right)=& r_{f}+\beta_{i m} *\left(r_{m}-r_{f}\right)+\beta_{i s m b} *(\mathrm{SMB}) \ &+\beta_{i h m l} *(\mathrm{HML})+\beta_{i w m l} *(\mathrm{WML})+\varepsilon_{i} \end{aligned}
where WML is the Momentum factor
Fama and French (2015) five factor model (additional factor added are investment and profitability)
\begin{aligned} E\left(r_{i}\right)=& r_{f}+\beta_{i m} *\left(r_{m}-r_{f}\right)+\beta_{i s m b} *(\mathrm{SMB})+\beta_{i h m l} *(\mathrm{HML}) \ &+\beta_{i c m a} *(\mathrm{CMA})+\beta_{i r m w} *(\mathrm{RMW})+\varepsilon_{i} \end{aligned}
where CMA and RMW are the investment and profitability factors
$$\begin{gathered} \mathrm{CMA}=(S / C+B / C) / 2-(S / A+B / A) / 2 \ \mathrm{RMW}=(S / R+B / R) / 2-(S / W+B / W) / 2 \end{gathered}$$

## 金融代写|金融计量经济学Financial Econometrics代考|Capital Market Line and Mean Variance Efficient Frontier

r米是市场回报σ米是市场标准差的度量σp是投资组合标准差的度量
[r米−rFσ米]是市场投资组合的夏普比率

r米是市场回报
b衡量回报波动率

## 金融代写|金融计量经济学Financial Econometrics代考|What Is So Special About Fama and French

Fama 和 French 使用模拟投资组合方法估计风险因素。风险因素直接从按公司特征分类的横截面资产收益估计。Fama 和 French 使用表 5.1 所示的单次和双重排序技术构建了他们的模仿组合。Fama 和 French 在每年 6 月份根据其市值对样本股票进行排名(米C)并使用 NYSE 中值市值断点形成了两个投资组合，即 Big(B) 和 Small(S)。相似地，

SMB 代表“小减大”，代表规模因子的风险模拟投资组合。同样，HML 代表价值因子的风险模拟投资组合。SMB 和 HML 风险模拟投资组合使用方程式进行估计。5.5和5.6.

Carhart (1997) 四因子模型（添加的附加因子是动量）

Fama 和 French (2015) 五因子模型（添加的附加因子是投资和盈利能力）

C米一个=(小号/C+乙/C)/2−(小号/一个+乙/一个)/2 R米在=(小号/R+乙/R)/2−(小号/在+乙/在)/2

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