我对页眉中的命令感到困惑\rightmark。除了第 2 页之外，它在文档的其他部分运行良好，第 2 页上的子部分名称没有出现在页眉中，而它应该出现。我的问题是如何修复此错误，文档的页眉在第 2 页上是空的，而它不应该是空的。以下是我的 MWE：
% !TeX TS-program = xelatex
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\newtheorem{proposition}{Proposition}[section]
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\usepackage[dvipsnames,table]{xcolor}
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\fancyfoot[LO,LE]{ GROUP ASSIGNMENT}
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\usepackage{titletoc}
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\usepackage{longtable}
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\begin{document}
\setcounter{page}{1}
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%--------------------------
\begin{keyword*}
Security Market Line (SML), Capital Market Line (CML), Capital Asset Pricing Model (CAPM), Expected return, Volatility, Beta
\end{keyword*}
\section{Introduction}
Capital market theory expanded the concepts
introduced by Markowitz portfolio theory by
introducing the notion that investors could borrow
or lend at the risk-free rate in addition to forming
efficient portfolios of risky assets. This insight led
to the development of the capital market line
(CML), which can be viewed as a new efficient
frontier that emanates from the risk-free rate
and is tangent to the Markowitz efficient frontier.
The point of tangency is called the market
portfolio.
The CML’s main contribution is the relationship it
specifies between the risk and expected return of a
well-diversified portfolio. The CML makes it clear
that the market portfolio is the single collection of
risky assets that maximizes the ratio of expected
risk premium to portfolio volatility. The investment prescription of the CML is that investors
cannot do better, on average, than when they
divide their investment funds between the riskless asset and the market portfolio. The capital asset pricing model (CAPM) generalizes
the risk-return trade-off in the capital market line
(CML) to allow for a consideration of individual
securities as well as entire portfolios. To make this
extension, the CAPM redefines the relevant measure of risk as beta, which is the systematic component of a security's volatility relative to that of
the market portfolio.
\subsection{Definition of Key Terms}
\begin{definition}[Expected Return]
A computation for the return of a security
based on the average payoff expected.
\end{definition}
\begin{definition}[Volatility]
The standard deviation of a return.
\end{definition}
\begin{definition}[Capital Market Line (CML)]
When plotting expected returns
versus volatility, the line from the risk-free investment through
the efficient portfolio of risky stocks (the portfolio that has the
highest possible Sharpe ratio). In the context of the CAPM, it
is the line from the risk-free investment through the market
portfolio. It shows the highest possible expected return that
can be obtained for any given volatility
\end{definition}
\begin{definition}[Capital Asset Pricing Model (CAPM)]
An equilibrium model
of the relationship between risk and return that characterizes
a security’s expected return based on its beta with the market
portfolio.
\end{definition}
\begin{definition}[Beta]
The expected percent change in the excess return of a
security for a 1\% change in the excess return of the market (or
other benchmark) portfolio
\end{definition}
\begin{definition}[Security Market Line (SML)]
The pricing implication of the
CAPM, it specifies a linear relation between the risk premium
of a security and its beta with the market portfolio.
\end{definition}
\section{Comparison and Contrast of Graphical Features of the CML and SML}
\subsection{Capital Market Theory}
Capital market theory extends Markowitz model to a situation when a
risk free asset is introduced in the capital market. It also assumes that
investors are rational and mean variance optimizers as assumed by
Markowitz Portfolio Theory. The capital market line is given in the following equation:
\begin{align*}
E[R_{p}]&=R_{f}+\frac{\sigma_{p}}{\sigma_{m}}\left( E[R_{m}]-R_{f}\right)
\intertext{where}
E[R_{p}]&=\text{the expected return of a portfolio;}\\
R_{f}&=\text{the risk free interest rate};\\
E[R_{m}]&=\text{the expected return on market portfolio}\\
\sigma_{p}&=\text{standard deviation of the portfolio, and}\\
\sigma_{m}&=\text{standard deviation of the market portfolio}
\end{align*}
The capital market line shows that the return from a portfolio depends
upon the risk free rate, reward per unit of market risk and total risk of the
portfolio. The higher the risk the higher will be the expected return.
The CML has the following features:
\begin{itemize}
\item CML shows a linear and a positive relationship between expected
return and risk of a portfolio.
\item It originates from $ R_{f} $ , hence the intercept of CML is $ R_{f} $.
\item The slope of CML is the reward to variability ratio i.e $$\frac{1}{\sigma_{m}}\left( E[R_{m}]-R_{f}\right)$$
\item CML is tangent to original efficient frontier at the Mean Variance Efficient Porfolio.
\item Only efficient portfolios consisting of risk free asset and the Mean Variance Efficient Porfolio lies on CML.
\item CML is upward sloping because price of a risk must be positive
since investors are risk averse.
\end{itemize}
\subsection{Capital Asset Pricing Model}
The capital asset pricing model (CAPM) is a model used to determine a
theoretically appropriate required rate of return of an asset, to make
decisions about adding assets to a well-diversified portfolio. It shows that
there is a linear and a positive relationship between expected return and
systematic risk. As per this model only systematic risk is priced,
unsystematic risk being diversifiable risk is not priced in the capital
market. Hence the Capital Market Line which shows the relationship
between and total risk should show a relationship between expected
return and systematic risk indicated by $ \beta $ factor. The linear relationship between the return required on an investment
(whether in stock market securities or in business operations) and its
systematic risk is represented by the CAPM formula:
\begin{align*}
E[R_{i}]&=R_{i}=R_{f}+\underbrace{\beta_{i}\times\left( E[R_{m}]-R_{f}\right)}_{\text{Risk premium for security } i}
\intertext{where $ \beta_{i}$ is the beta of the security with respect to the market portfolio}
\beta_{i}&=\frac{\overbrace{SD(R_{i})\times Corr(R_{i},R_{m})}^{\text{Volatility of $ i $ that is common with the market}}}{SD(R_{m})}=\frac{Cov(R_{i},R_{m})}{Var[R_{m}]}
\end{align*}
The $ \beta $\footnote{is a measure of the systematic risk in other words} of a security measures its volatility due to market risk relative to the market as a whole, and thus captures the security’s sensitivity to market risk.
\subsubsection{Security Market Line}\label{subsub2.2.1}
Security market line (SML) is the representation of the capital asset
pricing model. It displays the expected rate of return of an individual
security as a function of systematic, non-diversifiable risk. The risk of an
individual risky security reflects the volatility of the return from security
rather than the return of the market portfolio. The risk in these individual
risky securities reflects the systematic risk.
The SML essentially graphs the results from the capital asset pricing
model (CAPM) formula. The $ x$\textendash axis represents the risk (beta), and the $ y$\textendash axis represents the expected return. The market risk premium is
determined from the slope of the SML.
The relationship between $ \beta $ and required return is plotted on the securities
market line (SML), which shows expected return as a function of $ \beta $. The
intercept is the nominal risk-free rate available for the market, while the
slope is the market premium, $ E[R_{m}]− R_{f} $. The securities market line can
be regarded as representing a single-factor model of the asset price,
where Beta is exposure to changes in value of the Market. The equation
of the SML is thus:
\begin{align*}
SML=E[R_{i}]&=R_{f}+\beta_{i}(E[R_{m}]-R_{f})
\intertext{where}
E[R_{i}]&=\text{expected return on security}\\
E[R_{m}]&=\text{expected return on market portfolio}\\
\beta_{i}&=\text{non-diversifiable or systematic risk}\\
R_{m}&=\text{market rate of return}\\
R_{f}&=\text{risk free rate}
\end{align*}
It is a useful tool in determining if an asset being considered for a
portfolio offers a reasonable expected return for risk. Individual securities
are plotted on the SML graph. If the security's expected return versus risk
is plotted above the SML, it is undervalued since the investor can expect
a greater return for the inherent risk. And a security plotted below the
SML is overvalued since the investor would be accepting less return for
the amount of risk assumed.
%width=0.7\linewidth, height=.7\textheight
When used in portfolio management, the SML represents the
investment's opportunity cost (investing in a combination of the market
portfolio and the risk-free asset). All the correctly priced securities are
plotted on the SML. The assets above the line are undervalued because
for a given amount of risk (beta), they yield a higher return. The assets
below the line are overvalued because for a given amount of risk, they
yield a lower return.
\section{Similarities Between CML and SML }
\subsection{Graphical Similiarities}
The similarities between the CML and SML as models of the risk-return tradeoff are that both are a graphical representation of risk-return combinations. Both the CML and SML intersect the vertical axis or the $ y $\textendash axis at the risk-free rate point.
\subsection{Risk-Return Tradeoff Modeling}
Both the CML and SML is used to model the risk-return tradeoff for portfolios though the SML is used for both individual securities and portfolios of securities.
\subsection{Market Portfolio}
The market portfolio is assumed to be the same portfolio in both models. Both models combine a risk-free asset and offer opportunities to leverage the risky asset portfolio by borrowing at the risk-free rate.
\section{Differences Between CML and SML}
\subsection{Risk Coefficients}
While standard deviation, $ \sigma $ is the measure of risk in CML, $ \beta $
coefficient determines the risk factors of the SML. In SML, the risk is defined as total risk and is measured by $ \sigma $. In SML, risk is defined as the systematic risk and is measured by $ \beta $.
\subsection{What Each Line Defines}
While the Capital Market Line graphs define efficient portfolios, the
Security Market Line graphs define both efficient and non-efficient
portfolios. Thus, the CML is valid for efficient portfolios only while the SML is valid for all portfolios and all individual securities. As a result of this all efficient portfolios lie along the CML while SML applies to individual securities, efficient portfolios and inefficient portfolios.
\subsection{Superiority When Measuring Risk Factors}
The Capital Market Line is considered to be superior than the Security Market Line when measuring
the risk factors. The CML is considered to be superior to the efficient frontier\footnote{A set of optimal portfolios that offers the highest expected return for a defined level of risk or the lowest risk for a
given level of expected return.
}
since it takes into account the
inclusion of a risk-free asset in the portfolio. The Capital Asset Pricing Model (CAPM) demonstrates
that the market portfolio is essentially the efficient frontier. This is achieved visually through the
Security Market Line (SML). All
points along the CML have superior risk-return profiles to any portfolio on the efficient frontier,
with the exception of the Market Portfolio, the point on the efficient frontier to which the CML is the
tangent. From a CML perspective, this portfolio is composed entirely of the risky asset, the market,
and has no holding of the risk free asset ,i.e., money is neither invested in, nor borrowed from the
money market account.
\subsection{What Each Line Determines}
Where the market portfolio and risk free assets are determined by the
CML, all security factors are determined by the SML. The CML is derived by drawing a tangent line from the intercept point on the efficient frontier to the
point where the expected return equals the risk-free rate of return. The tangency point M represents the market portfolio, so named since all rational
investors (minimum variance criterion) should hold their risky assets in the same proportions as their
weights in the market portfolio.
\subsection{What Each Line Depicts}
The CML is a line that is used to show the rates of return, which
depends on risk-free rates of return and levels of risk for a specific
portfolio. SML\footnote{which is also called a Characteristic Line} is a graphical
representation of the market’s risk and return at a given time. The CML is performed by mixing the market portfolio with the risk free asset while the SML depicts the trade-off between systematic risk and the expected return for all assets. The CML specifies the equilibrium relationship between expected return and total risk for efficient diversified portfolios while the SML specifies the equilibrium relationship between expected return and systematic risk.
\subsection{Model to Which Each Line forms a Basis}
The CML is based on the Capital Market Theory while the SML is based on the CAPM. Capital Market Line (CML) is a line based on the Capital Market Theory used in the Capital Asset Pricing Model to illustrate the rates
of return for efficient portfolios depending on the risk-free rate of return and the level of risk
(standard deviation) for a particular portfolio while the Security market line (SML) is the representation of the Capital Asset Pricing Model. It displays the
expected rate of return of an individual security as a function of systematic,
non-diversifiable risk (its beta).
The SML essentially graphs the results from the capital asset pricing model (CAPM) formula.
\subsection{Equation of Each Line}\label{sub4.7}
The CML equation is given by
\begin{align*}
E[R_{p}]&=R_{f}+\frac{\sigma_{p}}{\sigma_{m}}\left( E[R_{m}]-R_{f}\right)
\intertext{where}
E[R_{p}]&=\text{the expected return of a portfolio;}\\
R_{f}&=\text{the risk free interest rate};\\
E[R_{m}]&=\text{the expected return on market portfolio}\\
\sigma_{p}&=\text{standard deviation of the portfolio, and}\\
\sigma_{m}&=\text{standard deviation of the market portfolio}
\intertext{while the SML equation is }
SML=E[R_{i}]&=R_{f}+\beta_{i}(E[R_{m}]-R_{f})
\intertext{where}
E[R_{i}]&=\text{expected return on security}\\
E[R_{m}]&=\text{expected return on market portfolio}\\
\beta_{i}&=\text{non-diversifiable or systematic risk}\\
R_{m}&=\text{market rate of return}\\
R_{f}&=\text{risk free rate}
\end{align*}
Now the intercept of each line is the risk free rate denoted $ R_{f} $. On the other hand, the slope of the CML is given by $$\frac{1}{\sigma_{m}}\left( E[R_{m}]-R_{f}\right)$$ while the slope of the SML is given by $$E[R_{m}]-R_{f}$$
\subsection{CML and SML slopes in Portfolio Evaluation}
From Subsection \ref{sub4.7} we indicated the slopes of the CML and SML and in this subsection, we present the use of slope in portfolio evaluation. The slope of the CML is gives us the Sharpe ratio of the market portfolio while the slope of the SML is the Treynor ratio of the market portfolio since $ \beta=1 $. Following from the Efficient Market Hypothesis (EMH)\footnote{Efficient market hypothesis (EMH) asserts that financial markets are “informationally efficient”, which means one
cannot consistently achieve returns in excess of average market returns on a risk-adjusted basis, given the information
available at the time the investment is made.}, all assets should have both the Sharpe ratio and Treynor ratio less than or equal to that of the market. All of the portfolios on the CML have the same Sharpe ratio as that of the market portfolio, and all of the portfolios on the SML have the same Treynor ratio as does the market portfolio.
\section{Conclusions}
Like the CML, the SML still confirms that the optimum portfolio is the market portfolio. Because the return on a portfolio (or security) depends on whether it follows market prices as a whole, the closer the correlation between a portfolio (security) and the market index, then the greater will be its expected return. Finally, the SML predicts that both portfolios and securities with higher beta values will have higher returns and vice versa. Unfortunately, the CML only calibrates total risk $ \sigma_{p} $ not all of which is diversifiable. Fortunately, the SML offers investors a lifeline, by discriminating between non-systemic and systemic risk. The latter is defined by a beta factor that measures relative (systematic) risk, which explains how rational investors with different utility (risk-return) requirements can choose an optimum portfolio by borrowing or lending at the risk-free rate. Like the CML, the security
market line (SML) shows that the relationship
between risk and expected return is a straight
line with a positive slope. The SML provides investors with a tool for judging whether securities are
undervalued or overvalued given their level of systematic (beta) risk.
\end{document}
答案1
可以使用包extramarks 和\fancyhead[RE]{\lastrightmark}
它已经包含在fancyhdr发行版中，因此无需下载或安装任何东西。
\usepackage{fancyhdr}
\usepackage{extramarks} % added<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
\fancypagestyle{fancy}{%
\fancyhf{}
\fancyhead[OL]{\leftmark}
\fancyhead[RE]{\lastrightmark} % changed<<<<<<<<<<<<<<<<<<<<
\fancyfoot{}
\fancyfoot[RE,RO]{\thepage}
\fancyfoot[LO,LE]{ GROUP ASSIGNMENT}
\renewcommand\headrule
{{ \nointerlineskip
\color{black}\rule{1\headwidth}{2.5pt}%
}}
}
extramarks它的多种用途在文档的第25节中进行了描述fancyhdr。
答案2
也许下面能满足你的要求
\fancyhead[ER]{%
\IfMarksEqualTF{2e-right-nonempty}{top}{first}
{\rightmark}
{\FirstMark{2e-right-nonempty}}%
}
这需要最新的 TeX 发行版。有关更多信息，请参阅ltmarks-doc.pdf