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\title{Continuous-Time Mean--Risk Portfolio Selection \vspace*{2cm}}
\author{\href{http://www.se.cuhk.edu.hk/~hqjin}{{\green Jin Hanqing}}\\
\vspace*{0.5cm}
Based on joint work with \\
Prof. Harry Markowitz, Prof. Xun Yu Zhou and Prof. Yan Jia An
}

\newcommand{\cF}{{\cal F}}
\renewcommand{\R}{{\hbox{I{\kern -0.22em}R}}}
\newcommand{\sR}{{\hbox{\scriptsize I{\kern -0.6em} R}}}
\newcommand{\argmin}{\mbox{argmin}}

\begin{document}
\maketitle
%----------------------------------------------------------------- SLIDE --------------------------
\overlays{6}{
\begin{slide}{Outline}
\begin{itemstep}
\mitem{We will study the continuous-time optimal portfolio selection problem in a
{\blue mean-risk} framework }

\mitem{Weighted-mean-variance problem.

%{\psellipse[linecolor=blue](10cm, 0.5cm)(0.6cm, 0.3cm)
\rput(9.96cm, 0.45cm){\rnode{cwmvp}{\blue \large \smiley}}
\hspace*{0.3cm}{\blue Explicit solution}
}

\mitem{Mean-semivariance problem.

\rput(9.96cm, 0.45cm){\rnode{cmsp}{\red \large \frownie}}
\fromSlide*{3}{\nccurve[linecolor=blue,angleA=330,angleB=30]{->}{cwmvp}{cmsp}}
\hspace*{0.3cm} {\red No optimal solution}
}

\mitem{Mean-downside-risk problem.

%\psellipse[linecolor=red](10cm, 0.5cm)(0.6cm, 0.3cm)}
\rput(9.96cm, 0.45cm){\rnode{cmdrp}{\red \large \frownie}}
\fromSlide*{4}{\nccurve[linecolor=red,angleA=330,angleB=30]{->}{cmsp}{cmdrp}}
\hspace*{0.3cm}{\red No optimal solution}
}

\mitem{When a general mean-risk problem admits optimal solutions?

%{\psellipse[linecolor=blue](10cm, 0.5cm)(0.6cm, 0.3cm)}
\rput(9.96cm, 0.45cm){\rnode{cmrp}{\blue \large \smiley}}
\hspace*{0.3cm} {\blue Equivalent condition}
}

\mitem{How about the single period mean-semivariance?

%{\psellipse[linecolor=blue](10cm, 0.5cm)(0.6cm, 0.3cm)}
\rput(9.96cm, 0.45cm){\rnode{smsp}{\blue \large \smiley}}
\fromSlide*{6}{\nccurve[linecolor=red,angleA=180,angleB=180]{<->}{cmsp}{smsp}}
\hspace*{0.3cm}{\blue Optimal solution existing}
}
\end{itemstep}
\end{slide}
}

\overlays{6}{
\begin{slide}{Markowitz's (Original) Model}
\begin{itemstep}
\mitem{Single period
\hspace*{5cm} $t=0$ \pscircle(-0.6cm,-0.2cm){0.08cm}
\psline[linestyle=dotted](-0.52cm,-0.2cm)(1.5cm, -0.2cm) \pscircle(1.55cm,-.2cm){0.08cm}
\hspace*{1cm} $t=T$}
\vspace{0.3cm}
\mitem{$m(\ge 2)$ securities, each with return rate $R_j$
\hspace*{0.9cm}$S_j$ \psline[linestyle=dotted]{->}(0cm,0.1cm)(2cm, 0.1cm)  \hspace{2cm}   $S_jR_j$  }
\mitem{$ER_j=r_j$, ${\rm Cov}(R_i,R_j)=\sigma_{ij}$}
\mitem{An agent with fund $x_0$, and a targeted expected payoff $z$ at the end of the investment period}
\mitem{To find a portfolio $\pi=(\pi_1,\cdots, \pi_m)$ so as to \vspace{-0.2cm}
$$\begin{array}{ll}
{\rm Mimimize}& {\rm Var}(\sum_j\pi_jR_j)=\sum_{i,j=1}^m\pi_i\sigma_{ij}\pi_j \hspace{1.8cm} \mbox{(risk)}\\
\fromSlide*{5}{\mbox{subject to} & \left\{\begin{array}{l}
                    \sum_i \pi_i=x_0 \;\; \hfill \mbox{(budget constraints)}\\
                    E(\sum_j\pi_jR_j)=\sum_i r_i\pi_i=z \;\; \hfill\mbox{(targeted payoff)}\\

                    [\pi_i\ge 0  \;\; \hfill\mbox{(no shorting)}]
                    \end{array}\right.}
\end{array}$$}

\vspace*{-0.2cm}
\mitem{Given {\blue expectation} return level, minimizing the {\red risk}}
%The optimal portfolio $\pi^*$ is called an efficient portfolio corresponding to $z$}
\end{itemstep}
\end{slide}
}

\overlays{6}{
\begin{slide}{Risk Measures}
\begin{itemstep}
  \item {\itsc What is \red risk? } Chance of bad consequences (Oxford Dictionary)
  \mitem{ A subjective notion as opposed to return}
  \mitem{ {\blue \itsc Variance / covariance} used to measure risk by Markowitz (1952)}
  \item Criticisms on using variance include
     \begin{itemize}
       \item penalty on {\red upside} return
       \item weight on upside and downside {\red equal} whereas asset return distribution generally asymmetric
     \end{itemize}
  \item {\itsc Semivariance} proposed where only the return below its mean or a target level counted as
        risk (Markowitz 1959: {\itsc \blue ``semivariance seems more plausible than variance as a measure of risk''})
  \item Generalization of semivariance: {\itsc Downside risk} (Fishburn 1977, Sortino and van der Meer 1991)
  \item Other risk measures: lower partial moment (Bawa 1975), mean-absolute deviation (Konno and Yamazaki, 1991),
  minimax measure (Cai, Teo, Yang and Zhou 2000) ......
\end{itemstep}
\end{slide}
}


\overlays{4}{
\begin{slide}{A Continuous-Time Market}
\begin{itemstep}
\mitem{A market in which ${\blue m}+1$ {\itsc securities} ({\itsc assets}) traded continuously}
\mitem{Market randomness described by a  complete filtered probability space
$(\Omega, \, {\cal F}, \, \{ {\cal F}_t\}_{t \geq 0}, \, P)$
along with an
\rnode{Nn}{$\blue m$}-dimensional, ${\cal F}_t$-adapted standard
Brownian motion $W(t) = (W^1(t), \cdots, \, W^m(t))'$
with $\{{\cal F}_t\}_{t \geq 0}$ generated by $W(\cdot)$}
\mitem{ A {\itsc bond (or bank account)} whose price process $S_0(t)$ satisfies \vspace{-0.2cm}
$$ dS_0(t)=r(t)S_0(t)dt,\ t\in[0,T]; \;\;\; S_0(0)=s_0>0,$$

\vspace{-0.2cm}
{\itsc where $r(\cdot)$:  interest rate}}
\mitem{\rnode{Nm}{$m$} {\itsc stocks} whose price processes $S_1(t),\cdots S_m(t)$
satisfy stochastic differential equation (SDE)\vspace{-0.2cm}
$$\left\{\begin{array}{l}
dS_i(t)=S_i(t)\big\{\mu_i(t)dt+\sum_{j=1}^m\sigma_{ij}(t)dW^j(t)\big\},\;  t\in[0,T];\\
S_i(0)=s_i>0,
\end{array}\right.
$$
where $\mu_i(t)$: {\itsc appreciate rate}; \qquad  $\sigma_{ij}(t)$: {\itsc volatility}
({\itsc dispersion}) {\itsc rate}
}
\end{itemstep}
\end{slide}
}

\overlays{5}{
\begin{slide}{Wealth and Portfolio}
\begin{itemize}
\mitem{Define {\itsc covariance matrix} $\sigma(t)$ and {\itsc excess rate of return process} $B(t)$ by
\vspace{-0.2cm}
 $$\sigma(t):=(\sigma_{ij})_{m\times m},\;\; B(t):=(\mu_1(t)-r(t),\cdots,\mu_m(t)-r(t))'$$}
 \vspace{-0.7cm}

\fromSlide*{2}{
\mitem{Basic Assumption:
  \begin{itemize}
    \item $r(\cdot), B_i(\cdot), \sigma_{ij}(\cdot)$ are all {\blue uniformly bounded}, $\cF_t$-adapted
    stochastic process
    \item $\blue \sigma(t)\sigma(t)'\ge \delta I$ for some $\delta>0$
  \end{itemize}
  }
}

\fromSlide*{3}{
\mitem{$\pi(t)=(\pi_1(t),\cdots,\pi_m(t))'$, where $\pi_i(t)$ is the {\blue capital amount}
invested in stock $i$, is called (monetary) {\itsc portfolio}}
}

\fromSlide*{4}{
\mitem{ $\pi(\cdot)$ is called {\itsc admissible} if it is $\cF_t$-adapted and
$E\int_0^T|\pi(t)|^2dt<+\infty$. \\ %}
}
}

\fromSlide*{5}{
\mitem{An investor's {\itsc wealth process} $x(t)$ follows {\itsc wealth equation}\vspace{-0.2cm}
\begin{equation}\label{systems}
\left\{\begin{array}{rl}
dx(t)&=[r(t)x(t)+B(t)\pi(t)]dt+\pi(t)'\sigma(t)dW(t)\\
x(0)&=x_0
\end{array}\right.
\end{equation}
}
}
\end{itemize}

\cdn{equation}{5}
\rcount{equation}{5}{1}
\end{slide}
}

\overlays{5}{
\begin{slide}{Continuous-time Mean-risk Problem}
Continuous-time \hyperlink{statop}{\hypertarget{cmvop}{mean--Risk}} portfolio selection is
formulated as the following optimization problem parameterized by a pair of scalar $(x_0, z)$:

\fromSlide*{2}{
\begin{equation}\label{cmv}
\blue
\begin{array}{ll}
{\rm Minimize}  &J_{x_0,z}(\pi(\cdot)):=Ef(x(T)-Ex(T))\\ % {\rm Var}\; x(T)\equiv Ex(T)^2-z^2, \hspace{1cm}\\

{\rm subject\; to} &\left\{\begin{array}{l}
Ex(T)=z,\\
(x(\cdot),\pi(\cdot))\;\; { \rm  admissible \; pair},\\
x(0)=x_0
\end{array}\right.
\end{array}
\end{equation}
}

\fromSlide*{3}{where $f: \R\mapsto \R$ is a given risk function.}

\fromSlide*{4}{
\begin{itemize}
\item The continuous-time Markowitz model: $\blue f(x)=x^2$
  \begin{itemize}
  \item[-] studied extensively recently (Zhou and Li 2000, Lim and Zhou; Lim 2004;
          Heunis and Labbe 2004; Bielecki, Jin, Pliska and Zhou 2005 ......)
  \end{itemize}

\fromSlide*{5}{ %\vspace*{0.4cm}
\item Problem (2) is a {\red dynamic} optimization problem
}
\end{itemize}
}
\cdn{equation}{2}
\rcount{equation}{5}{1}
\end{slide}
}

\overlays{6}{
\begin{slide}{A Static Problem }
\begin{itemize}
\mitem{Define \hyperlink{defiTheta}{{\blue $\theta(t)=\sigma(t)^{-1}B(t)$}}
and $\blue \rho(t):=e^{-\int_0^t[r(s)+|\theta(s)|^2/2]ds-\int_0^t\theta(s)dW(s)}$}

\fromSlide*{2}{\item Wealth equation is equivalent to %\vspace*{-0.2cm}
$x(t)=\rho(t)^{-1}E[x(T)\rho(T)|\cF_t]$ %, \;\; \forall t\in [0,T]$

\item The market is {\blue complete}.
}
\onlySlide*{2}{{\it i.e.}, $\forall \; \xi\in L^2(\cF_T, \R)$, there exists an
admissible wealth-portfolio pair $(x(\cdot),\pi(\cdot))$ such that $x(T)=\xi$ ({\blue replication}).}
\fromSlide*{3}{
\item Budget constraint becomes $x_0=E[x(T)\rho(T)]$
}
\end{itemize}

\fromSlide*{4}
{Consider the following \hypertarget{statop}{{\itsc  static}} optimization problem: \vspace*{-0.3cm}
\begin{equation}\label{stat}
 \blue \begin{array}{ll}
       {\rm Minimize}  & Ef(X-EX), \\
       {\rm subject \; to} &
       \left\{\begin{array}{l}
       EX = z; \;\;\;\; E[\rho(T)X]=x_0; \;\;\; X\in L^2(\cF_T,\R)\\
       \end{array}\right.
\end{array}
\end{equation}
}

\fromSlide*{5}
{ \vspace*{-0.3cm} {\mtheo} {\itsc
  If $X^*$ is the optimal solution for the static optimization problem (3), then
  the optimal portfolio for the \hyperlink{cmvop}{{mean-risk problem}} (2)
  is the one {\red to replicate $X^*$}.

  On the other hand, if $(x^*(\cdot),\pi^*(\cdot))$ is the optimal
  wealth-portfolio pair, then $\red X^*=x^*(T)$.
}}

\fromSlide*{6}{
\vspace{0.2cm}
{\bf \small \black Problem: } Solve the {\red static} problem (3)}

\cdn{equation}{4}
\cdn{theo}{5}
\rcount{equation}{6}{1}
\rcount{theo}{6}{1}
\end{slide}
}

\overlays{3}{
\begin{slide}{Weighted Mean-Variance Model}
Let $\blue f(x)=\alpha x^2_++\beta x^2_-$, where $\alpha>0, \beta>0$.
\begin{itemize}
\item[-] $\blue Ef(X(T)-EX(T))$ the weighted variance of $X(T)$
\end{itemize}

\fromSlide*{2}{
\vspace{0.2cm}
Changing variable $\blue Y:=X-z$, the static problem (3) specializes to \vspace*{-0.2cm}
 \begin{equation*}%\label{wmv}
 \blue       \begin{array}{ll}
           {\rm Minimize}& E[\alpha Y^2_++\beta Y^2_-]\\
           {\rm subject \;\; to}&\left\{ \begin{array}{l}
                                   EY=0\\
                                   E[Y\rho]=y_0\\
                                   Y\in L^2(\cF_T,\R)
                                   \end{array}\right.
       \end{array}
  \end{equation*}

 \vspace*{-0.2cm}
where $\blue \rho:=\rho(T)$ and $\blue y_0:=x_0-zE\rho$.
}

\fromSlide*{3}{
\vspace*{0.2cm}
Introducing two {\itsc Lagrange multipliers} $\blue (\lambda, \mu)$ for the two constraints, one needs only to solve
\begin{equation}\label{lagwmv}
\blue \min_{Y\in L^2(\cF_T,\R)} E[\alpha Y^2_++\beta Y^2_--2(\lambda-\mu\rho)Y]
\end{equation}
And then determine $\blue (\lambda,\mu)$ by the two constraints $\blue EY=0, E[Y\rho]=y_0$.
}

\cdn{equation}{3}
\rcount{equation}{3}{1}
\end{slide}
}

\overlays{4}{
\begin{slide}{Solution to Weighted MV Problem}
{\mlemma} The optimal solution of Problem (4) is
$\tiny \red Y^*=\frac{(\lambda-\mu\rho)_+}{\alpha}-\frac{(\lambda-\mu\rho)_-}{\beta}$.
%is the unique optimal solution of Problem (5)

\fromSlide*{2}{
{\mlemma} For any $y_0$, there exists a unique pair $(\lambda, \mu)$ such that
the optimal solution $Y^*$ in Lemma 1 satisfies $\blue EY^*=0, E[Y^*\rho]=y_0$. Moreover,
$\red \lambda<0, \mu<0$ if $\blue y_0>0$; \quad $\red \lambda>0, \mu>0$ if $\blue y_0<0$;\quad
$\red \lambda=\mu=0$ if $\blue y_0=0$.
}

\fromSlide*{3}{
{\mtheo}{\itsc The unique optimal portfolio for the weighted MV problem corresponding to $(x_0,z)$
is the one to replicate the contingent claim \vspace*{-0.3cm}
{\tiny $$\red X^*=\frac{(\lambda-\mu\rho)_+}{\alpha}-\frac{(\lambda-\mu\rho)_-}{\beta}+z$$}

\vspace*{-0.6cm}
with $(\lambda,\mu)$ being the unique solution to the system of algebraic equations
\vspace*{-0.3cm}
{\tiny
$$ \blue   \frac{E(\lambda-\mu\rho)_+}{\alpha}-\frac{E(\lambda-\mu\rho)_-}{\beta}=0, \qquad
    \frac{E[\rho(\lambda-\mu\rho)_+]}{\alpha}-\frac{(\lambda-\mu\rho)_-}{\beta}=x_0-zE\rho.$$}

\vspace*{-0.4cm}
Moreover, the minimum risk %(weighted variance)
value is given by $\red J^*(x_0,z)=-\mu(x_0-zE\rho)$.}
}

\fromSlide*{4}{
\vspace*{0.3cm}
{\mremark} When the market coefficients are {\blue deterministic}, the optimal portfolio can be obtained
more {\red explicitly} via some {\blue \itsc Black-Scholes} type equation.}

\cdn{lemma}{1}
\cdn{lemma}{2}
\cdn{theo}{3}
\cdn{remark}{4}
\rcount{lemma}{4}{2}
\rcount{theo}{4}{1}
\rcount{remark}{4}{1}
\end{slide}
}

\overlays{5}{
\begin{slide}{Mean-Semivariance Model}
\begin{itemize}
\item Let $\blue f(x)=x^2_-$.   Mean-risk problem specialized to mean-semivariance problem \vspace{-0.2cm}
       \begin{equation}\label{cms}
      \blue  \begin{array}{ll}
           {\rm Minimize}& EY^2_-\\
           {\rm subject \;\; to}&\left\{ \begin{array}{l}
                                   EY=0\\
                                   E[Y\rho]=y_0:=x_0-zE\rho\\
                                   Y\in L^2(\cF_T,\R)
                                   \end{array}\right.
       \end{array}
  \end{equation}

\vspace{-0.4cm}
\fromSlide*{2}{
\item Define \vspace{-0.2cm}
$$ \blue
\rho_0:=\inf\{\eta\in \R: P(\rho<\eta)>0\}\qquad
\rho_1:=\sup\{\eta\in \R: P(\rho>\eta)>0\}
$$
}

\fromSlide*{3}{
\vspace{-0.6cm}
\begin{itemize}
\item[-] If $r(\cdot)$ and $\theta(\cdot)$ are {\blue deterministic} and $\blue \int_0^T|\theta(t)|^2dt>0$,
then $\blue \rho_0=0$ and $\blue \rho_1=+\infty$.
\end{itemize}
}
\end{itemize}

\fromSlide*{4}{
{\mtheo} The mean-semivariance problem does {\red not admit any optimal solution} so long as
$\blue z\neq \frac{x_0}{E\rho}$.
}

\vspace{0.3cm}
\fromSlide*{5}{
\mremark  $\blue z= \frac{x_0}{E\rho}$ is a trivial case, where the optimal portfolio is the {\red risk-free} one.
}

\cdn{equation}{1}
\cdn{theo}{4}
\cdn{remark}{5}
\rcount{equation}{5}{1}
\rcount{theo}{5}{1}
\rcount{remark}{5}{1}
\end{slide}
}

\overlays{3}{
\begin{slide}{Idea of Proof}
\begin{itemize}
  \item View the mean-semivariance problem as the {\red limiting problem} of the weighted MV problem
        with $\blue \beta=1-\alpha$ and $\blue \alpha\rightarrow 0$
  \item Let $\blue Y(\alpha)$ be the optimal solution to the weighted MV problem with $\beta=1-\alpha$ and $\alpha>0$
\fromSlide*{2}{
  \item If $\blue y_0<0$, one can show %, applying the conclusion on the weighted mean-variance problem,
        that $\red E[Y(\alpha)_-]^2\rightarrow \frac{y_0^2}{E(\rho-\rho_0)^2}$ as $\alpha\rightarrow 0$.
        However, for any feasible solution $Y$ of (6), one can show via Cauchy-Schwartz's inequality that
       $\red EY_-^2>y_0^2/E(\rho-\rho_0)^2$. Hence there is {\red no optimal solution}.
  \item If $\blue y_0>0$, one can show that $\red E[Y(\alpha)_-]^2\rightarrow \frac{y_0^2}{E(\rho_1-\rho)^2}$
        as $\alpha\rightarrow 0$ ($\frac{y_0^2}{E(\rho_1-\rho)^2}$ is defined to be $0$ when $\rho_1=+\infty$).
        whereas $\red E[Y(\alpha)_-]^2>\frac{y_0^2}{E(\rho_1-\rho)^2}$ for any feasible solution $Y$.
        Again there is {\red no optimal solution}.
}
\end{itemize}

\fromSlide*{3}{
\vspace*{-0.1cm}
{\mremark} Although the mean-semivariance problem in general does not admit optimal solutions, the
infimum of the problem has been obtained explicitly, which is $\blue \frac{y_0^2}{E(\rho-\rho_0)^2}$
if $y_0<0$ and  $\blue \frac{y_0^2}{E(\rho_1-\rho)^2}$ if $y_0>0$. Moreover, {\red asymptotically optimal}
portfolios can be obtained by replicating $\red Y(\alpha)+z$ as $\alpha\rightarrow 0$.
}

\cdn{remark}{3}
\rcount{remark}{3}{1}
\end{slide}
}

\overlays{7}{
\begin{slide}{Mean-Downside-Risk Model}
\begin{itemize}
\item Let {\blue $f\ge 0$, left continuous at $0$, strictly decreasing on $\R^-$,
and for $\forall x\in \R^+$, $f(x)=0$} (an example: $f(x)=(x_-)^p$ for some
$p\ge 0$).

\fromSlide*{2}{
\item The corresponding risk $\blue Ef(X(T)-EX(T))$ only punish the downside part of
      the deviation of $X(T)$. ({\itsc downside-risk})
}
\end{itemize}

\fromSlide*{3}{
{\bf \green \scriptsize \hyperlink{citeassumb}{\hypertarget{assumb}{Assumption (B)}}}
For any $0\le M_1<M_2\le +\infty$,  $\blue P\{\rho(T)\in (M_1,M_2)\}>0$
and $\blue P\{\rho(T)=M_1\}=\{\rho(T)=M_2\}=0$.
}

\fromSlide*{4}{
{\mremark} This assumption is satisfied when, say, $r(\cdot)$ and $\theta(\cdot)$
are {\blue deterministic} and $\blue \int_0^T|\theta(s)|^2ds>0$.
}

\fromSlide*{5}{
{\mtheo}{\itsc When Assumption (B) hold, the mean-downside-risk model with $f(\cdot)$ as the risk
measure admits {\red no optimal solution} for any $\blue z\neq\frac{x_0}{E\rho}$. On the other hand,
if $\blue z=\frac{x_0}{E\rho}$, then the model has an optimal portfolio which
is the {\red risk-free portfolio}}.
}

\onlySlide*{6}{
\vspace{0.2cm}
{\bf \black \scriptsize Idea of proof: } Find out the infimum $\inf Ef(Y)$, and show it is {\red not attainable}.
}

\fromSlide*{7}{
\vspace*{0.2cm}
{\bf \red \small Question: }  When the optimal solutions exist for a general mean-risk problem?
}

\cdn{remark}{4}
\cdn{theo}{5}
\rcount{remark}{7}{1}
\rcount{theo}{7}{1}
\end{slide}
}

\overlays{5}{
\begin{slide}{General Mean-Risk Model}
\begin{itemize}
\item Let $f(\cdot)$ be {\blue convex}, and {\blue strictly convex at $0$}.

\onlySlide*{2}{ \psellipse[linecolor=red](4.7cm, 0.5cm)(1.4cm, 0.3cm)

\vspace{0.3cm} \hspace*{0.2cm }
{\it i.e.} {\blue $kf(x)+(1-k)f(y)>f(0)$ for any $k\in (0,1), kx+(1-k)y=0$})

\psline[linestyle=dotted]{->}(1.5cm, -2.5cm)(5cm, -2.5cm)
\psline[linestyle=dotted]{->}(3cm, -3cm)(3cm, -0.5cm)
\psline[linecolor=blue]{-}(3cm, -2cm)(4cm, -2cm)
\pscurve[linecolor=blue]{-}(2cm, -1cm)(2.5cm,-1.75cm)(3cm, -2cm)
\psline[linecolor=red, linestyle=dashed]{-}(2.5cm, -1.75cm)(3.5cm, -2cm)
}

\fromSlide*{3}{
\item Define the subdifferential $\partial f(x)$ in the sense of convex analysis
$$\blue \partial f(x):=\{x^*\in \R: f(y)-f(x)\ge x^*(y-x),\; \forall y\in \R\}$$
}
\end{itemize}

\fromSlide*{4}{
\vspace{-0.4cm}
We maintain the \hyperlink{assumb}{\hypertarget{citeassumb}{{\green Assumption (B)}}}. And define
$$\blue \left\{\begin{array}{l}
g(y):=\argmin_{x\in \R: y\in \partial f(x)}|x|\\
\Lambda:=\{\lambda\in \R: \exists \mu=\mu(\lambda)\in \R
          \mbox{ \rm so that }g(\lambda-\mu(\lambda)\rho)\in L^2(\cF_T,\R),\\
            \hspace{1cm } Eg(\lambda-\mu(\lambda)\rho)=0,\rho g(\lambda-\mu(\lambda)\rho)\in L^1(\cF_T,\R)\;\}\\
\tilde g(\lambda)=E[\rho g(\lambda-\mu(\lambda)\rho)],\; \forall \lambda\in \Lambda\\
\bar \lambda=\sup_{\lambda\in \Lambda}\lambda, \;\;\;\; \underline \lambda=\inf_{\lambda\in \Lambda}\lambda\\
\underline y:=\lim_{\lambda\uparrow \bar \lambda}\tilde g(\lambda),\;\;\;\;
\bar y:=\lim_{\lambda\downarrow \underline \lambda}\tilde g(\lambda)
\end{array}\right.
$$
}

\fromSlide*{5}{
{\mremark} All these definition can be calculated {\blue offline}
}

\cdn{remark}{5}
\rcount{remark}{5}{1}
\end{slide}
}

\overlays{3}{
\begin{slide}{Solution of the General MR problem}
{\mtheo}{\itsc
One has the following conclusions regarding the solution to the mean-risk
portfolio selection problem with the general $f$:

\fromSlide*{2}{
\begin{itemize}
\item [(i)] Assume that $\blue \cup_{x\in \sR} \partial f(x)=\R$. Then the
problem admits an optimal solution {\red if and only if $x_0-zE\rho\in A\cup B$},
where
$$\blue A=\left\{\begin{array}{lll}
[\underline y,0],&&\mbox{\rm if } \bar \lambda\in \Lambda\\
(\underline y,0],&&\mbox{\rm if } \bar \lambda\notin \Lambda
\end{array}\right.,\qquad \qquad
B=\left\{\begin{array}{lll}
[0,\bar y],&&\mbox{\rm if } \underline \lambda\in \Lambda\\
(0,\bar y),&&\mbox{\rm if } \underline \lambda\notin \Lambda
\end{array}\right.
$$

\fromSlide*{3}{
\item [(ii)] Assume that there exists $M_1, M_2\in \R$ such that
$\blue \cup_{x\in \sR}\partial f(x) \subset [M_1,M_2]$. Then the problem admits
an optimal solution {\red if and only if $z=x_0/E\rho$.}
}
\end{itemize}}
}

\cdn{theo}{1}
\rcount{theo}{3}{1}
\end{slide}
}

\overlays{2}{
\begin{slide}{Solution of General MR problem (Cont'd)}
{\itsc
\begin{itemize}
\item [(iii)] Assume that either $\blue \cup_{x\in \sR} \partial f(x)=(-\infty,\bar k]$
or $\blue \cup_{x\in \sR}\partial f(x)=(-\infty, \bar k)$ for some
$\bar k\in \R$. Then $\underline \lambda=0\in \Lambda$.
If $\blue \bar \lambda\notin \Lambda$, then the problem
admits an optimal solution {\red if and only if $x_0-zE\rho\in (\underline y,0]$}.
If $\blue \bar \lambda\in \Lambda$, then the problem admits an optimal
solution {\red if and only if
$x_0-zE\rho\in \{\tilde g(\bar \lambda)\}\cup (\underline y,0]$}. If in
addition $\bar \lambda<\bar k$, then $\tilde g(\bar \lambda)=\underline y$

\fromSlide*{2}{
\item [(iv)] Assume that either $\blue \cup_{x\in \sR} \partial f(x)=[\underline k,\infty)$
or $\blue \cup_{x\in \sR}\partial f(x)=(\underline k, \infty)$ for some
$\underline k\in \R$, then $\bar \lambda=0\in \Lambda$.
If $\blue \underline \lambda\notin \Lambda$,
then the problem admits an optimal solution {\red if and only if
$x_0-zE\rho\in [0,\bar y)$}.
If $\blue \underline \lambda\in \Lambda$, then the problem admits
an optimal solution {\red if and only if
$x_0-zE\rho\in \{\tilde g(\bar \lambda)\}\cup [0,\bar y,)$}. If in
addition $\underline \lambda>\underline k$, then
$\tilde g(\underline \lambda)=\bar y$
}
\end{itemize}
}
\end{slide}
}

\overlays{6}{
\begin{slide}{Examples}
{\mexam} $\blue f(x)=|x|$ ({\itsc mean-absolute-deviation model} ).

\fromSlide*{2}{\vspace*{0.1cm}
$f(\cdot)$ is {\blue strictly convex at $0$}, and $\blue \cup_{x\in \sR}\partial f(x)=[-1,1]$. {\red (Case (ii)).}}
\fromSlide*{3}{Thus the continuous-time mean-absolute-deviation model admits an optimal solution
{\red if and only if $z=x_0/E\rho$}, in which case the optimal portfolio is
simply the {\red risk-free} one.}

\fromSlide*{4}{
\vspace*{0.1cm}
{\mexam} $\blue f(x)=e^{-x}$ ({\itsc more sensitive to large loss} ).
}

\fromSlide*{5}{
\vspace*{0.1cm}
$f$ is {\blue strictly convex}, $\blue \cup_{x\in\sR}\partial f(x)=(-\infty, 0)$.{\red (Case (iii)).}}
\fromSlide*{6}{The MR  problem admits an optimal solution {\red iff
$x_0-zE\rho\in [(E\rho)(E\ln \rho)-E(\rho\ln \rho), 0]$} or,
equivalently,
$\red z\in [\frac{x_0}{E\rho},\frac{x_0-(E\rho)(E\ln\rho)+E(\rho\ln\rho)}{E\rho}]$.
When the problem does admit an optimal solution, the optimal portfolio
is the one {\red replicating the claim $z-\ln (-\lambda+\mu\rho)$},  where
$\blue (\lambda, \mu)$ is the unique solution pair to the following algebraic
equation ( which must admit a solution):
$$\blue \left\{\begin{array}{l}
E\ln(-\lambda+\mu\rho)=0\\
E[\rho\ln(-\lambda+\mu\rho)]=zE\rho-x_0
\end{array}\right.$$
}
\cdn{exam}{1}
\cdn{exam}{4}
\rcount{exam}{6}{2}
\end{slide}
}

\overlays{3}{
\begin{slide}{Examples (Cont'd)}
{\mexam} $\blue f(x)=(x-1)_-^2$ ({\itsc shift of the mean-semivariance model}).

\fromSlide*{2}{
\vspace*{0.2cm}
$f$ is {\blue not strictly convex} everywhere; but it is indeed {\blue strictly convex at $0$}.
$\blue \cup_{x\in \sR}\partial f(x)=(-\infty,0]$. {\red (Case (iii)).}}
\fromSlide*{3}{ The original portfolio
selection problem admits an optimal solution {\red if and only if
$x_0-zE\rho\in [E\rho-E\rho^2/E\rho,0]$} or, equivalently,
$\red z\in [\frac{x_0}{E\rho},\frac{x_0}{E\rho]}+\frac{E\rho^2}{(E\rho)^2}-1]$.
When the problem does admit an optimal solution, the optimal portfolio
is the one {\red replicating the claim $z+1+\frac{\lambda-\mu\rho}{2}$} where
$\blue (\lambda,\mu)$ is the unique solution pair to the following linear
algebraic equation:
$$\blue \left\{\begin{array}{l}
\lambda-\mu E\rho=-2\\
\lambda E\rho-\mu E\rho^2=2x_0-2(1+z)E\rho
\end{array}\right.$$
}
\cdn{exam}{1}
\rcount{exam}{3}{1}
\end{slide}
}

\overlays{3}{
\begin{slide}{Asymptotic Optimal Portfolios}
\begin{itemize}
\item In all the cases, no matter the optimal portfolios exist or not,
       the {\red infimum of the risk is finite} ($\blue Ef(X-EX)\ge f(E[X-EX])=f(0)$).

\vspace{0.2cm}
\fromSlide*{2}{\item When optimal portfolios do not exist, consider a perturbed risk function
$$f_\alpha(x)=f(x)+\alpha x^2, \;\; \alpha>0$$}

\fromSlide*{3}{
It can be shown:
\begin{itemize}
\vspace{0.2cm}
  \item The mean-risk portfolio selection problem with risk function $f_\alpha$ {\red must} admit
  an optimal solution

\vspace{0.2cm}
  \item The corresponding optimal portfolio $\red \pi_\alpha$ is {\red asymptotically optimal} for the original
  problem when $\alpha\downarrow 0$
\end{itemize}
}
\end{itemize}
\end{slide}
}

\overlays{6}{
\begin{slide}{Back to Single Period}
\begin{itemize}
\item{Continuous-time mean-semivariance problem admits no optimal solution.
{\red How about its single period counterpart}?}

\fromSlide*{2}{
\item{$m(\ge 2)$ securities, each with return rate $R_j$, $ER_j=r_j$, ${\rm Cov}(R_i,R_j)=\sigma_{ij}$}

\item The single period Mean-semivariance problem\vspace{-0.2cm}
\begin{equation}\label{spms}
\begin{array}{ll}
{\rm Mimimize}& E[(\sum_j\pi_jR_j-E\sum_j\pi_jR_j)_-]^2\\
{\rm subject\; to} & \left\{\begin{array}{l}
                    \sum_i \pi_i=x_0 \\
                    E(\sum_j\pi_jR_j)=\sum_i r_i\pi_i=z\\
                    \end{array}\right.
\end{array}
\end{equation}
}

\fromSlide*{3}{\vspace{-0.3cm}
\item Vast literature on single-period mean-semivariance models}

\fromSlide{4}{  \mitem{Concentrate on numerical solution (as analytical solution impossible) and comparison with
         mean-variance model}
}
\fromSlide{5}{
  \mitem{Existence of efficient portfolios/frontier {\red not addressed}}
}
\fromSlide{6}{  \mitem{Technically non-trivial as the feasible region generally {\red unbounded}, and the objective
  {\red not coercive}}

  ({\blue $f: \R^d\mapsto \R$ called {\it coercive} if $\lim_{|x|\rightarrow +\infty} f(x)=+\infty$})
}
\end{itemize}
\cdn{equation}{2}
\rcount{equation}{6}{1}
\end{slide}
}

\overlays{4}{
\begin{slide}{A Lemma}
Consider
\begin{equation}\label{spmsv}
\blue \min_{x\in\sR^d} E[(A+B'x)_-]^2,
\end{equation}
where $\blue B\equiv (B_1,\cdots, B_d)'$, and $\blue A, B_i$ are random variables with $\blue EA^2<+\infty$,
$\blue EB_i^2<+\infty, \;\; i=1,\cdots, d$.

\fromSlide*{2}{
{\mlemma} If $\blue EB_i=0,\; i=1,\cdots,d$, then problem (\ref{spmsv}) {\red admits optimal solutions}.
}

\fromSlide*{3}{
\vspace*{0.2cm}
{\mremark} Assumption that $EB_i=0\; i=1,\cdot, d$ is crucial.}

\fromSlide*{4}{
\vspace{0.2cm} {\red Counter example:}
Let $\blue A=-1, B=(e^{W_1},\cdots, e^{W_d})'$, where
$(W_1,\cdots,W_d)$ follow $N(0, I_d)$. For any $0\neq x\in \R^d_+$,
$\red \lim_{\alpha\rightarrow +\infty} E[(A+B'(\alpha x))_-]^2=0$. This implies the optimal value of (\ref{spmsv})
is $\red 0$. However, this value {\red cannot be achieved} since $\red E[(A+B'(\alpha x))_-]^2>0$ for any $x\in \R^d$.
}
\cdn{equation}{1}
\cdn{lemma}{2}
\cdn{remark}{3}
\rcount{equation}{4}{1}
\rcount{lemma}{4}{1}
\rcount{remark}{4}{1}
\end{slide}
}
%\end{document}

\overlays{3}{
\begin{slide}{Existence for Single-Period M-S}
{\mtheo} For any $x_0\in \R$ and $z\in \R$, Problem (\ref{spmsv}) admits optimal solutions {\red if and
only if it admits feasible solutions}.

\fromSlide*{2}{
\vspace{0.2cm}
{\itsc Idea of Proof.} Let $\blue \xi_i=R_i-r_i$. After eliminating $\pi_1$ and $\pi_2$ from the constraints,
one gets the following equivalent problem \vspace{-0.3cm}
$$\red \min_{(\pi_3,\cdots, \pi_m)\in \R^{m-2}}E[(A+\sum_{i=3}^m\pi_iB_i)_-]^2,$$

\vspace{-0.4cm}
where \vspace{-0.3cm}
$$\blue A=x_0\xi_1+\frac{z-x_0r_1}{r_2-r_1}(\xi_2-\xi_1), \;\; B_i=\xi_i-\xi_1-(r_i-r_1)\frac{\xi_2-\xi_1}{r_2-r_1}.$$

\vspace{-0.3cm}
Then Lemma 3 applies.
}

\fromSlide*{3}{
\vspace{0.4cm}
{\red Sharply contrast}  \quad Continuous-time mean-semivariance \; Vs \;  Single-period mean-semivariance
}

\cdn{theo}{1}
\rcount{theo}{3}{1}
\end{slide}
}

\overlays{2}{
\begin{slide}{Further Research}
\begin{itemstep}
  \item {\itsc Incomplete} market (the replicating problem becomes significant)

\vspace*{1cm}
\begin{itemize}
\item[-] Some work has been done on the mean-variance problem. ...
\end{itemize}
   \vspace*{1cm}

  \item Other risk measures: safety first, Var, minimax, ......
\end{itemstep}
\end{slide}
}


\begin{slide}{END}
\vspace{2cm}
\begin{center}
  \small
  Comments and questions are appreciated ... ...

  \vspace{2cm}

  Thank you very much!
\end{center}
\end{slide}

\end{document}
