text:
\section{Introduction}
Investment analysis and portfolio choice theory are very important and challenging topics in finance and economics nowadays. Since Harry Markowitz (1952) presented his mean-variance paradigm  portfolio theory has become a fundamental tool for understanding the interactions of systematic risk and reward.

It is well known that the mean-variance optimization problem of Markowitz (1952) is equivalent to the expected exponential utility optimization under the normality assumption (see Merton (1969)). Unfortunately, his approach only gives an answer to the one-period portfolio selection problem in discrete time but it says nothing about the multi-period (long-run) setting. Therefore, it is of importance to investigate the multi-period portfolio optimization problem which is of great relevance for an investor as well.  The multi-period portfolio selection problem has been analyzed for different types of utility functions by, e.g., Mossin (1968), Merton (1969, 1972), Samuelson (1969), Elton and Gruber (1974), Brandt and Santa-Clara (2006), Basak and Chabakauri (2010).

The continuous case has already been solved for many types of utility functions in the one- and multi-period case by Merton (1969). A number of generalizations under weaker assumptions are given, among others, by A\"{i}t-Sahalia et al. (2009) and Skaf and Boyd (2009). Exact solutions in discrete time are even more difficult to obtain in the multi-period case. Mossin (1968) considers the case of one risk and one riskless asset. He derives conditions when the multi-period strategy becomes myopic or partial myopic. Frequently this can be achieved by demanding independent asset returns. However, the assumption of independence is unfortunately not fulfilled in many applications.

For an investor it would be very useful to have a closed-form solution of the discrete multi-period portfolio problem. Moreover, it is desirable that the optimal portfolio weights can easily calculated at each period. Analytical solutions of the multi-period optimal portfolio choice problems are hard to obtain and they are only available for some partial cases which are often derived under very restrictive assumptions on the distribution of the asset returns. For instance, a closed-form solution for the multi-period portfolio choice problem exists for the quadratic utility function under the assumption that the asset returns are independently distributed (see Li and Ng (2000), Leippold et al. (2004)).

In the present paper we consider an investor who invests into $k$ risky assets and one riskless asset with an investment strategy based on the exponential utility function
\begin{equation}\label{exp_utility}
U(W_t)=-e^{-\alpha W_t}\,.
\end{equation}
Here $W_t$ denotes the investor's wealth at period $t$ and $\alpha>0$ stands for the coefficient of absolute risk aversion (ARA), which is a constant over time for the exponential utility (CARA utility). The application of the exponential utility function is more plausible than the use of the quadratic utility since the first one is monotonically decreasing. That is why the exponential utility function is commonly used in portfolio selection theory. Moreover, the optimization of the expected exponential utility function leads to the well known mean-variance utility maximization problem and consequently its solution lays on the mean-variance efficient frontier.

We derive a closed-form solution of the multi-period portfolio choice problem with the exponential utility function (\ref{exp_utility}) under the assumption that the asset returns depend on certain predictable variables. The joint process consists of the asset returns and the predictable variables and it is assumed to follow a vector autoregressive (VAR) process. This approach is very popular in finance and it is often used for modeling the asset returns (see, e.g., Campbell (1991, 1996), Barberis (2000), Brandt (2010)).

The rest of the paper is organized as follows. In Section 2, the main result of the paper is given. In Theorem 1 an analytical expression of the portfolio weights is provided for each period. In Corollary 1, the case without a predictable vector is considered while independent asset returns are treated in Corollary 2. In Section 3 a short empirical study is presented. The performance of the derived strategy is compared with the one for independent asset returns. The comparison is performed in terms of the cumulative empirical distribution function of the investor's wealth at the end of the investment period. We find significant improvements if the dependence structure is taken into account. Section 4 contains a short summary.

\section{Multi-Period Portfolio Problem for an Exponential Utility}
In this section we derive the analytical solution of the multi-period portfolio choice problem for an exponential utility function assuming that the asset returns and the predictable variables follow a VAR(1) process.

There are only a few papers in literature where the exponential utility function is considered in the multi-period discrete time setting. For instance, \c{C}anako$\breve{\text{g}}$lu and \"{O}zekici (2009) solved the portfolio choice problem assuming that the stochastic market follows a discrete time Markov chain and all parameters of the asset returns, i.e., mean vector and covariance matrix, depend only on the current state of the stochastic market and not on the previous states which is equivalent to the assumption of independence in our settings. In the paper of Soyer and Tanyeri (2006) a Bayesian computational approach with the exponential utility was presented. The authors write that the solution of the multi-period portfolio choice problem with the exponential utility under the assumption of normality "\textit{...cannot be evaluated in closed form and the optimal portfolio cannot be obtained analytically}". In this paper, however, we present an exact solution assuming that the asset returns follow a vector autoregressive process with predictable variables.

 Let $\mathbf{X}_t=\left(X_{t,1},X_{t,2},\ldots,X_{t,k}\right)^{\prime}$ denote the vector of the returns of $k$ risky assets and let $r_{f,t}$ be the return of the riskless asset at time $t$. Let $\mathbf{z}_t$ be a $p$-dimensional vector of predictable variables. We assume that $\mathbf{Y}_t=(\mathbf{X}_t^\prime, \mathbf{z}_t^\prime)^\prime$ follows a VAR(1) process given by
\begin{equation}\label{VAR}
\mathbf{Y}_t=\tilde{\boldsymbol{\nu}}+\tilde{\mathbf{\Phi}}\mathbf{Y}_{t-1}+\tilde{\pmb{\varepsilon}}_t\,
\end{equation}
with $\tilde{\pmb{\varepsilon}}_t\sim \mathcal{N}(\mathbf{0}, \tilde{\bol{\Sigma}}(t))$, where $\tilde{\bol{\Sigma}}(t)$ is a positive definite deterministic matrix function. Let $\mathcal{F}_t$ denote the information set available at time $t$. Then $\mathbf{Y}_t|\mathcal{F}_{t-1}\sim \mathcal{N}_{k+p}(\tilde{\bol{\mu}}_t, \tilde{\bSigma}(t))$, i.e., the conditional distribution of $\mathbf{Y}_t$ given $\mathcal{F}_{t-1}$ is a $k+p$ dimensional normal distribution with mean vector $\tilde{\bol{\mu}}_t=E(\mathbf{Y}_t|\mathcal{F}_{t-1})=E_{t-1}(\mathbf{Y}_t)$ and covariance matrix $\mbox{Var}(\mathbf{Y}_t|\mathcal{F}_{t-1})=\tilde{\bSigma}(t)$.

The stochastic model (\ref{VAR}) is described in detail by Campbell et al. (2003) who argued that the application of VAR(1) is not a restrictive assumption because every vector autoregression can be presented as a VAR(1) process through an expansion of the vector of state (predictable) variables. The idea behind this approach is to find a vector of predictable variables $\mathbf{z}_t$ which is mostly correlated with the asset returns and to build a VAR(1) process with respect to the asset returns $\mathbf{X}_t$ and the vector of predictable variables $\mathbf{z}_t$. The choice of $\mathbf{z}_t$ depends rather on the data and not on the utility function. Possible predictable variables are, e.g., the dividend yield (cf. Campbell at al. (2003)), the term spread (see, e.g, Brandt et al. (2006)) or another asset return.

From (\ref{VAR}) we obtain the following model for $\mathbf{X}_t$ expressed as
\begin{equation}\label{VAR11}
\mathbf{X}_t=\mathbf{L}\tilde{\boldsymbol{\nu}}+\mathbf{L}\tilde{\mathbf{\Phi}}\mathbf{Y}_{t-1}+\mathbf{L}\tilde{\pmb{\varepsilon}}_t=\boldsymbol{\nu}+\mathbf{\Phi}\mathbf{Y}_{t-1}+\pmb{\varepsilon}_{t}~~~\text{with}~~\mathbf{L}=[\mathbf{I}_k~ \mathbf{O}_{k,p}]\,,
\end{equation}
where $\mathbf{I}_k$ is a $k\times k$ identity matrix and $\mathbf{O}_{k,p}$ is a $k\times p$ matrix of zeros.
Consequently, $\mathbf{X}_t|\mathcal{F}_{t-1}\sim \mathcal{N}_k(\bol{\mu}_t, \bol{\Sigma}(t))$, where $\bol{\mu}_t=E(\mathbf{X}_t|\mathcal{F}_{t-1})=\boldsymbol{\nu}+\mathbf{\Phi}\mathbf{Y}_{t-1}$ and $\bol{\Sigma}(t)=\mbox{Var}(\mathbf{X}_t|\mathcal{F}_{t-1})=\mathbf{L}\tilde{\bSigma}(t)\mathbf{L}^\prime$.

 Let $\mathbf{w}_t=\left(w_{t,1},w_{t,2},\ldots,w_{t,k}\right)^{\prime}$ denote the vector of the portfolio weights of the $k$ risky assets at period $t$. Then the evolution of the investor's wealth is expressed as
\begin{equation}\label{wealth_Sec3}
W_t=W_{t-1}\left(1+r_{f,t}+\mathbf{w}_{t-1}^{\prime}(\mathbf{X}_t-r_{f,t}\mathbf{1})\right)=W_{t-1}\left(R_{f,t}+\mathbf{w}^{\prime}_{t-1}\breve{\bx}_t\right) \,,
\end{equation}
where $R_{f,t}=1+r_{f,t}$ and $\breve{\bx}_{t}=\mathbf{X}_{t}-r_{f,t}\mathbf{1}$ with $\breve{\bol{\mu}}_{t}=E_{t-1}(\breve{\bx}_{t})=\boldsymbol{\nu}+\mathbf{\Phi}\mathbf{Y}_{t-1}-r_{f,t}\mathbf{1}$.
The aim of the investor is to maximize the expected utility of the final wealth.

The optimization problem is given by
\begin{equation}\label{OP_Sec3}
V(0,W_0,\mathcal{F}_{0})=\max\limits_{\{\mathbf{w}_s\}_{s=0}^{T-1}}E_t[U(W_T)]\,
\end{equation}
with the terminal condition
\begin{equation}\label{exponent}
U(W_T)=-\exp(-\alpha W_T)~~~\text{for}~~\alpha>0.
\end{equation}
Following Pennacchi (2008) the optimization problem (\ref{OP_Sec3}) can be solved by applying the following Bellman equation at time point $T-t$
\begin{eqnarray}\label{BE_Sec3}
V(T-t,W_{T-t},\mathcal{F}_{T-t})
&=&\max\limits_{\mathbf{w}_{T-t}}E_{T-t}
\Big{[}\max\limits_{\{\mathbf{w}_s\}_{s=T-t+1}^{T-1}}E_{T-t+1}[U(W_T)]\Big{]} \\
&=&\max\limits_{\mathbf{w}_{T-t}}E_{T-t}\Big{[}V(T-t+1,W_{T-t}\left(r_{f,T-t}+\mathbf{w}^{*\;\prime}_{T-t+1}\breve{\bx}_{T-t+1}\right),\mathcal{F}_{T-t+1})\Big{]}\nonumber
\end{eqnarray}
subject to (\ref{exponent}), where $\mathbf{w}^{*}_{T-t+1}$ are the optimal portfolio weights at period $T-t+1$. Note that in contrast to the static case now the vector of optimal portfolio weights $\mathbf{w}_{T-t+1}$ is a function of the weights of the next periods, i.e., of $\mathbf{w}_{T-t+1}, \mathbf{w}_{T-t+2},\ldots,\mathbf{w}_{T-1}$, what is the consequence of the backward recursion method (see, e.g. Pennacchi (2008)).

For the period $T-1$ we get
{\small
\begin{eqnarray}\label{VT-1}
&&V(T-1,W_{T-1},\mathcal{F}_{T-1})\nonumber\\
&&=E_{T-1}\left[-\exp(-\alpha W_{T-1}(R_{f,T}+\mathbf{w}^{\prime}_{T-1}\breve{\bx}_{T}))\right]\nonumber\\
&&=-\exp(-\alpha W_{T-1}R_{f,T})E_{T-1}[\exp(-\alpha W_{T-1}\mathbf{w}^\prime_{T-1}\breve{\bx}_T)]\nonumber\\
&&=\exp(-\alpha W_{T-1}R_{f,T})\left(-\exp\left[-\alpha(W_{T-1}\mathbf{w}^\prime_{T-1}\breve{\bol{\mu}}_T-\frac{\alpha}{2}\mathbf{w}_{T-1}^\prime\bol{\Sigma}(T)\mathbf{w}_{T-1}W_{T-1}^2)\right]\right)\rightarrow\text{max}\,.
\end{eqnarray}
}
The last optimization problem is equivalent to
\begin{equation}\label{meanv}
W_{T-1}\mathbf{w}^\prime_{T-1}\breve{\bol{\mu}}_T-\frac{\alpha}{2}\mathbf{w}_{T-1}^\prime\bol{\Sigma}(T)\mathbf{w}_{T-1}W_{T-1}^2\rightarrow\text{max}~~~\text{over}~~\mathbf{w}_{T-1}\,.
\end{equation}
Taking the derivative and solving (\ref{meanv}) with respect to $\mathbf{w}_{T-1}$ we get the classical solution for the period $T-1$
\begin{equation}\label{w_t-1}
\mathbf{w}^*_{T-1}=\dfrac{1}{\alpha W_{T-1}}\bol{\Sigma}^{-1}(T)\breve{\bol{\mu}}_T=\dfrac{1}{\alpha W_{T-1}}(\mathbf{L}\tilde{\bSigma}(T)\mathbf{L}^\prime)^{-1}(\breve{\bn}_T+\mathbf{\Phi}\mathbf{Y}_{T-1})~~\text{with}~~\breve{\bn}_T=\boldsymbol{\nu}-r_{f,T}\mathbf{1}\,.
\end{equation}
In Theorem 1 the multi-period portfolio weights for all periods from $0$ to $T-1$ are given.

\begin{theorem}
Let $\mathbf{X}_{\tau}=\left(X_{\tau,1},X_{\tau,2},\ldots,X_{\tau,k}\right)^{\prime}$ be a random return vector of $k$ risky assets. Suppose that $\mathbf{X}_{\tau}$ and the vector of $p$ predictable variables $\mathbf{z}_{\tau}$ jointly follow a VAR(1) process as defined in (\ref{VAR}). Let $r_{f,\tau}$ be the return of the riskless asset. Then the optimal multi-period portfolio weights  are given by (\ref{w_t-1}) for period $T-1$,
\begin{equation}\label{weigthsT-2}
\mathbf{w}_{T-2}=\dfrac{1}{\alpha W_{T-2}R_{f,T}}\left(\mathbf{L}\tilde{\bSigma}^{-1}({T-1})\tilde{\bol{\mu}}^*_{T-1}-\mathbf{L}\mathbf{\Phi}^\prime\bol{\Sigma}^{-1}(T)(\breve{\bn}_T+r_{f,T}\mathbf{\Phi}\mathbf{L}^\prime\mathbf{1})\right)\,,
\end{equation}
and
{\footnotesize
\begin{align}\label{weightsT-t}
&\mathbf{w}^*_{T-t}=\dfrac{1}{\alpha W_{T-t}\prod\limits_{i=T-t+2}^{T}R_{f,i}}\left(\mathbf{L}\tilde{\bSigma}^{-1}(T-t+1)\tilde{\bol{\mu}}^*_{T-t+1}-\mathbf{L}\tilde{\mathbf{\Phi}}^\prime\tilde{\bSigma}^{-1}(T-t+2)(\breve{\bn}_{T-t+3}^*+r_{f,T-t+2}\tilde{\mathbf{\Phi}}\mathbf{L}^\prime\mathbf{1})\right)\\
&~~\text{with}~~\tilde{\bol{\mu}}^*_{T-t+1}=\tilde{\bol{\mu}}_{T-t+1}-r_{f,T-t+2}\mathbf{L}^\prime\mathbf{1} ~~\text{and}~~\breve{\bn}_{T-t+3}^*=\tilde{\bn}-r_{f,T-t+3}\mathbf{L}^\prime\mathbf{1}\nonumber\,,
\end{align}
for $t=3,\ldots,T$.
}
\end{theorem}

\begin{proof}
The value function at time point $T-2$ is obtained by substituting (\ref{w_t-1}) into (\ref{VT-1})
\begin{eqnarray}\label{VT-2}
&&V(T-2,W_{T-2},\mathcal{F}_{T-2})=-E_{T-2}\left[\exp{\left(-\alpha W_{T-1}R_{f,T}-\frac{1}{2}\breve{s}_T\right)}\right]\nonumber\\
&&=-\exp\left(-\alpha W_{T-2}R_{f,T-1}R_{f,T}\right)E_{T-2}\left[\exp{\left(-\alpha W_{T-2}R_{f,T}\mathbf{w}^\prime_{T-2}\breve{\bx}_{T-1}-\frac{1}{2}\breve{s}_T\right)}\right]\,,
\end{eqnarray}
where $\breve{s}_T=\breve{\bol{\mu}}_T^\prime\bol{\Sigma}(T)^{-1}\breve{\bol{\mu}}_T$. Second, according to the properties of VAR(1) processes we get that
\begin{equation}\label{sT}
\breve{s}_T=\breve{\by}_{T-1}^\prime\mathbf{\Phi}^\prime\bol{\Sigma}(T)^{-1}\mathbf{\Phi}\breve{\by}_{T-1}+2\breve{\by}_{T-1}^\prime\mathbf{\Phi}^\prime\bol{\Sigma}(T)^{-1}\breve{\bn}_T+(\breve{\bn}_T+r_{f,T}\mathbf{\Phi}\mathbf{L}^\prime\mathbf{1})^\prime\bol{\Sigma}(T)^{-1}(\breve{\bn}_T+r_{f,T}\mathbf{\Phi}\mathbf{L}^\prime\mathbf{1})\,,
\end{equation}
with $\breve{\by}_t=\mathbf{Y}_t-r_{f,t}\mathbf{L}^\prime\mathbf{1}$. This is a quadratic form with respect to the conditional normally distributed vector $\mathbf{Y}_{T-1}$. Moreover, using $\mathbf{L}\bL^\prime=\mathbf{I}_k$,
\begin{equation}\label{bwy}
\mathbf{w}^\prime_{T-2}\breve{\bx}_{T-1}=\mathbf{w}^\prime_{T-2}(\mathbf{L}\mathbf{Y}_{T-1}-r_{f,T}\mathbf{1})=\mathbf{w}^\prime_{T-2}\mathbf{L}\breve{\by}_{T-1}\,
\end{equation}
and (\ref{VT-2}), we get
\begin{align}\label{VT2}
&V(T-2,W_{T-2},\mathcal{F}_{T-2})=-\exp\left(-\alpha W_{T-2}R_{f,T}R_{f,T-1}\right)\nonumber\\
&\times E_{T-2}\left[\exp{\left(-\frac{1}{2}\breve{\by}^\prime_{T-1}\mathbf{B}(T)\breve{\by}_{T-1}-\mathbf{b}(\mathbf{w}_{T-2})^\prime\breve{\by}_{T-1}-c\right)}\right]\,,
\end{align}
where $\mathbf{B}_T=\mathbf{\Phi}^\prime\bol{\Sigma}(T)^{-1}\mathbf{\Phi}$, $\mathbf{b}(\mathbf{w}_{T-2})=\mathbf{\Phi}^\prime\bol{\Sigma}(T)^{-1}\breve{\bn}_T+\alpha W_{T-2}R_{f,T}\mathbf{L}^\prime\mathbf{w}_{T-2}+r_{f,T}\mathbf{B}(T)\mathbf{L}^\prime\mathbf{1}$ and $c=\frac{1}{2}(\breve{\bn}_T+r_{f,T}\mathbf{\Phi}\mathbf{L}^\prime\mathbf{1})^\prime\bol{\Sigma}(T)^{-1}(\breve{\bn}_T+r_{f,T}\mathbf{\Phi}\mathbf{L}^\prime\mathbf{1})$.

Following Mathai and Provost (1992, Theorem 3.2a.1) the expectation given in (\ref{VT2}) is the moment generating function of the quadratic form in normal variables at point $-1$. Hence, it holds that
{\begin{align}\label{mgfq}
&V(T-2,W_{T-2},\mathcal{F}_{T-2})=-\exp\left(-\alpha W_{T-2}R_{f,T-1}R_{f,T}\right)|\mathbf{I}+\mathbf{B}(T)\tilde{\bSigma}({T-1})|^{-\frac{1}{2}}\nonumber\\
&\times \exp{}\left[-\frac{1}{2}(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1})^\prime\tilde{\bSigma}(T-1)^{-1}(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1})-c\right.\nonumber\\
&+\frac{1}{2}\left(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}-\tilde{\bSigma}({T-1})\mathbf{b}(\mathbf{w}_{T-2})\right)^\prime \nonumber\\
&\times\left.(\mathbf{I}+\mathbf{B}(T)\tilde{\bSigma}(T-1))^{-1}\tilde{\bSigma}^{-1}(T-1)\left(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}-\tilde{\bSigma}(T-1)\mathbf{b}(\mathbf{w}_{T-2})\right)\right]\,,
\end{align}
}
where $E_{T-2}[\breve{\by}_{T-1}]=\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}$.
Thus, the optimization problem $V(T-2,W_{T-2},\mathcal{F}_{T-2})\rightarrow$ max is equivalent to $V^{*}(T-2,W_{T-2},\mathcal{F}_{T-2})\rightarrow$ max, where
{
\begin{align}\label{max}
&V^{*}(T-2,W_{T-2},\mathcal{F}_{T-2})=-\frac{1}{2}(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}-\tilde{\bSigma}({T-1})\mathbf{b}(\mathbf{w}_{T-2}))^\prime\nonumber\\
&(\mathbf{I}+\mathbf{B}(T)\tilde{\bSigma}({T-1}))^{-1}\tilde{\bSigma}({T-1})^{-1}\left(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}-\tilde{\bSigma}({T-1})\mathbf{b}(\mathbf{w}_{T-2})\right)\nonumber\\
&=-\frac{1}{2}(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}-\tilde{\bSigma}({T-1})\mathbf{b}(\mathbf{w}_{T-2}))^\prime \nonumber\\
&\times\left(\tilde{\bSigma}({T-1})+\tilde{\bSigma}({T-1})\mathbf{B}(T)\tilde{\bSigma}({T-1})\right)^{-1}(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}-\tilde{\bSigma}({T-1})\mathbf{b}(\mathbf{w}_{T-2}))\,.
\end{align}
}
Because the matrix $\tilde{\bSigma}({T-1})+\tilde{\bSigma}({T-1})\mathbf{B}(T)\tilde{\bSigma}({T-1})$ is positive definite the maximum of $V^{*}(T-2,W_{T-2},\mathcal{F}_{T-2})$ is attained at $\mathbf{w}^*_{T-2}$ for which
\begin{equation}\label{eq}
\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1}=\tilde{\bSigma}_{T-1}\mathbf{b}(\mathbf{w}^*_{T-2}).
\end{equation}
Using that $\mathbf{L}\bL^\prime=\mathbf{I}_k$ we obtain
\begin{equation}\label{weigths}
\mathbf{w}^*_{T-2}=\dfrac{1}{\alpha W_{T-2}R_{f,T}}\left(\mathbf{L}\tilde{\bSigma}^{-1}({T-1})(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1})-\mathbf{L}\mathbf{\Phi}^\prime\bol{\Sigma}^{-1}(T)(\breve{\bn}_T+r_{f,T}\mathbf{\Phi}\mathbf{L}^\prime\mathbf{1})\right)\,.
\end{equation}
Furthermore, the equality (\ref{eq}) leads to
{
\begin{align}\label{Vt2}
&V(T-2,W_{T-2},\mathcal{F}_{T-2})=-|\mathbf{I}+\mathbf{B}(T)\tilde{\bSigma}(T-1)|^{-\frac{1}{2}}\exp{(-c)}\nonumber\\
&\times\exp{\left(-\alpha W_{T-2}R_{f,T-1}R_{f,T}-\frac{1}{2}(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1})^{\prime}\tilde{\bSigma}^{-1}(T-1)(\tilde{\bol{\mu}}_{T-1}-r_{f,T}\mathbf{L}^\prime\mathbf{1})\right)}\\
&=-|\mathbf{I}+\mathbf{B}(T)\tilde{\bSigma}(T-1)|^{-\frac{1}{2}}\exp{(-c)}\nonumber\\
&\times\exp{\left(-\frac{1}{2}\breve{\by}_{T-2}^\prime\tilde{\mathbf{B}}(T-1)\breve{\by}_{T-2}-\tilde{\mathbf{b}}^\prime(\mathbf{w}_{T-3})\breve{\by}_{T-2}-\tilde{c}\right)}\,,
\end{align}
}
where $\tilde{\mathbf{B}}(T-1)=\tilde{\mathbf{\Phi}}^\prime\tilde{\bSigma}(T-1)^{-1}\tilde{\mathbf{\Phi}}$, $\tilde{\mathbf{b}}(\mathbf{w}_{T-3})=\tilde{\mathbf{\Phi}}^\prime\tilde{\bSigma}(T-1)^{-1}(\tilde{\bn}-r_{f,T}\mathbf{L}^\prime\mathbf{1})+\alpha W_{T-3}R_{f,T-1}\mathbf{L}^\prime\mathbf{w}_{T-3}+r_{f,T-1}\tilde{\mathbf{B}}(T-1)\mathbf{L}^\prime\mathbf{1}$ and $\tilde{c}=\frac{1}{2}(\tilde{\bn}-r_{f,T}\mathbf{L}^\prime\mathbf{1}+r_{f,T-1}\tilde{\mathbf{\Phi}}\mathbf{L}^\prime\mathbf{1})^\prime\tilde{\bSigma}(T-1)^{-1}(\tilde{\bn}-r_{f,T}\mathbf{L}^\prime\mathbf{1}+r_{f,T-1}\tilde{\mathbf{\Phi}}\mathbf{L}^\prime\mathbf{1})$.
Taking the conditional expectation from the value function (\ref{Vt2}) with respect to $\mathcal{F}_{T-3}$ we receive
{
\begin{align}\label{Vt3}
&V(T-3,W_{T-3},\mathcal{F}_{T-3})=-|\mathbf{I}+\mathbf{B}(T)\tilde{\bSigma}(T-1)|^{-\frac{1}{2}}\exp{(-c)}\nonumber\\
&\times E_{T-3}\left(\exp{\left(-\frac{1}{2}\breve{\by}_{T-2}^\prime\tilde{\mathbf{B}}(T-1)\breve{\by}_{T-2}-\tilde{\mathbf{b}}^\prime(\mathbf{w}_{T-3})\breve{\by}_{T-1}-\tilde{c}\right)}\right)\,.
\end{align}
}
Consequently, the value function for the period $T-3$ has a similar structure than $V(T-2,W_{T-2},\mathcal{F}_{T-2})$ with the only difference that $\mathbf{B}(T)$ is replaced by $\tilde{\bB}(T-1)$, $\mathbf{b}(\mathbf{w}_{T-2})$ by $\tilde{\mathbf{b}}(\mathbf{w}_{T-3})$ and $c$ by $\tilde{c}$. It follows immediately that the optimal portfolio weights at period $T-3$ are
{\footnotesize
\begin{equation}\label{weightsT-3}
\mathbf{w}^*_{T-3}=\dfrac{1}{\alpha W_{T-3}R_{f,T}R_{f,T-1}}\left(\mathbf{L}\tilde{\bSigma}^{-1}(T-2)(\tilde{\bol{\mu}}_{T-2}-r_{f,T-1}\mathbf{L}^\prime\mathbf{1})-\mathbf{L}\tilde{\mathbf{\Phi}}^\prime\tilde{\bSigma}^{-1}(T-1)(\breve{\bn}_T^*+r_{f,T-1}\tilde{\mathbf{\Phi}}\mathbf{L}^\prime\mathbf{1})\right)\,
\end{equation}
}
with $\breve{\bn}_T^*=\tilde{\bn}-r_{f,T}\mathbf{L}^\prime\mathbf{1}$.

The last step is to use mathematical induction with basis $T-3$ in order to receive the statement of Theorem 1.
\end{proof}

The results of Theorem 1 show us that the optimal portfolio weights at every period of time except the last one depend on the covariance matrices of the next two periods and the conditional mean vector of the next period. This property turns out to be very useful if we want to calculate the optimal portfolio weights for a real data set.

Note that the case without predictable variables is a special case of Theorem 1. In this case the following expressions are obtained.
\begin{corollary}
Let $\mathbf{X}_{\tau}=\left(X_{\tau,1},X_{\tau,2},\ldots,X_{\tau,k}\right)^{\prime}$ be a random return vector of $k$ risky assets which follows a VAR(1) process as defined in (\ref{VAR}) but without a vector of predictable variables $z_{\tau}$. Let $r_{f,\tau}$ be the return of the riskless asset. Then the optimal multi-period portfolio weights for period $T-1$ are given by
\begin{equation}\label{ohnez}
\mathbf{w}^*_{T-1}=\dfrac{1}{\alpha W_{T-1}}\bol{\Sigma}^{-1}(T)\breve{\bol{\mu}}_T=\dfrac{1}{\alpha W_{T-1}}\tilde{\bSigma}^{-1}(T)(\breve{\bn}_T+\mathbf{\Phi}\mathbf{Y}_{T-1})~~\text{with}~~\breve{\bn}_T=\boldsymbol{\nu}-r_{f,T}\mathbf{1}\,
\end{equation}
and for $t=2,\ldots,T$ by
\begin{equation}\label{weightsT-t}
\mathbf{w}^*_{T-t}=\dfrac{1}{\alpha W_{T-t}\prod\limits_{i=T-t+2}^{T}R_{f,i}}\left(\bol{\Sigma}^{-1}(T-t+1)\breve{\bol{\mu}}_{T-t+1}-\mathbf{\Phi}^\prime\bol{\Sigma}^{-1}(T-t+2)(\breve{\bn}_{T-t+2}+r_{f,T-t+2}\mathbf{\Phi}\mathbf{1})\right)\,,
\end{equation}
\end{corollary}
\begin{proof}
The results of Corollary 1 are obtained in the same way as the results of Theorem 1 by putting $\mathbf{L}=\mathbf{I}_k$ and $\mathbf{Y}_t=\mathbf{X}_t$ $(p=0)$.
\end{proof}

In Corollary 2 the return vectors are assumed to be independent.
\begin{corollary}
Let $\mathbf{X}_{\tau}=\left(X_{\tau,1},X_{\tau,2},\ldots,X_{\tau,k}\right)^{\prime}$ be a sequence of the independently and identically normally distributed vectors of $k$ risky assets, i.e., $\mathbf{X}_{\tau}\sim \mathcal{N}(\bol{\mu}, \bol{\Sigma})$. Let $r_{f,\tau}$ be the return of the riskless asset. We assume that $\bol{\Sigma}$ is positive definite. Then for all $t=1,\ldots,T$ the optimal multi-period portfolio weights for period $T-t$ are given by
\begin{equation}\label{iweightsT-t}
\mathbf{w}^*_{T-t}=\dfrac{1}{\alpha W_{T-t}\prod\limits_{i=T-t+2}^{T}R_{f,i}}\bol{\Sigma}^{-1}\breve{\bol{\mu}}~~~\text{with}~~\breve{\bol{\mu}}=\bol{\mu}-r_{f,T-t+2}\mathbf{1}\,.
\end{equation}
\end{corollary}
\begin{proof}
Corollary 2 immediately follows from Corollary 1 putting $\mathbf{\Phi}=\mathbf{0}$, $\bol{\Sigma}(t)=\bol{\Sigma}$ and $\boldsymbol{\nu}=\bol{\mu}$.
\end{proof}
The results of Corollary 2 can be obtained as a partial case of \c{C}anako$\breve{\text{g}}$lu and \"{O}zekici (2009), where the stochastic market was presented by a discrete time Markov chain. In that case the asset returns depend on the present state of the market and not on the previous ones which implies the independence of the asset return over time.

It is noted that the dynamics of the optimal portfolio weights in Corollary 2 is hidden in the coefficient of the absolute risk aversion $\alpha$  which is given by $\alpha_{\tau}=\left(\alpha W_{T-\tau}\prod\limits_{i=T-\tau+2}^{T}R_{f,i}\right)^{-1}$. Moreover, the expressions of the weights themselves are proportional to the weights of the so-called tangency portfolio (cf. Ingersoll (1987, p. 89), Britten-Jones (1999)). Because the
tangency portfolio is usually considered as a market portfolio in the single-period allocation
problem (see, e.g., Britten-Jones (1999)) we treat the weights given in (\ref{iweightsT-t}) as the weights of a benchmark portfolio in our empirical study presented in the next section.

\section{Empirical Study}
In this section we apply the results of Section 2 to real data. In following we consider an investor who invests into an international portfolio. The portfolio consists of the capital market indices of five developed stock markets, namely Belgium, Germany, Japan, the UK, and the USA. We deal with weekly data of the MSCI (Morgan Stanley Capital International) indices for the equity market returns from the January 4,2002 to December 4,2009. A process is fitted to the return series
\begin{equation}\label{VARes}
\mathbf{X}_t=\boldsymbol{\nu}+\mathbf{\Phi}\mathbf{Y}_{t-1}+\pmb{\varepsilon}_t \quad \text{with} \quad \pmb{\varepsilon}_t \sim ii\mathcal{N}(\mathbf{0},\mathbf{\Sigma}_{\varepsilon})\,.
\end{equation}
We get
\begin{equation}\label{VARes_int}
\bol{\nu}=
\left[
\begin{array}{r}
4.83e-04\\
1.20e-03\\
6.74e-04\\
5.54e-04\\
2.79e-05\\
\end{array}\right], \mathbf{\Phi}=\left[
\begin{array}{rrrrr}
0.2011& -0.1592&  0.01892& -0.196& 0.455\\
0.3139& -0.1231& -0.00191& -0.511& 0.434\\
0.0487&  0.0888& -0.12131& -0.224& 0.343\\
0.1829 & -0.0889&  0.00988& -0.441& 0.382\\
 0.0766 & -0.0643& -0.03049& -0.114& 0.133\\
\end{array}\right],\; \text{and} \;
\end{equation}
\begin{equation*}
\mathbf{\Sigma}_{\varepsilon}=
\left[
\begin{array}{rrrrr}
0.0013085186 & 0.0010544496 & 0.0004365753 & 0.0009120373 & 0.0006781289\\
0.0010544496 & 0.0013833540 & 0.0005648237 & 0.0010218539 & 0.0008332314\\
0.0004365753 & 0.0005648237 & 0.0007994341 & 0.0004733366 & 0.0003667012\\
0.0009120373 & 0.0010218539 & 0.0004733366 & 0.0010176793 & 0.0006927251\\
0.0006781289 & 0.0008332314 & 0.0003667012 & 0.0006927251 & 0.0007242233\\
\end{array}\right].
\end{equation*}

It is remarkable that the last column of the matrix $\bol{\Phi}$ has the largest values which indicate on a strong positive correlation between the US market and the other markets. Moreover, it shows that the influence of the US market on the return indices is larger than those of the domestic ones. Following Campbell et al. (2003) we choose the stock index of the US market as a predictable variable $z_t$ in our empirical study.

Next, we calculate the weights of the two multi-period portfolio strategies for an exponential utility function. We want to compare the case of correlated return vectors given in Theorem 1 with the case of independent variables given in Corollary 2 which completely ignores the time dependence structure well documented for real data.

The performance of both strategies is compared with each other via an extensive simulation study based on $10^5$ independent repetitions. The multi-period portfolio strategies are constructed for $T \in \{13,26,52,104\}$ and for the coefficient of relative risk aversion (RRA) $\alpha_r \in \{0.8, 2\}$. The RRA $\alpha_r=\alpha W_0$ is chosen as a constant absolute risk aversion (ARA) $\alpha$ in this study (without loss of generality we put $W_0=1$). In order to compare the performance of these two strategies we determine the empirical cumulative distribution function (ECDF) of the investor's terminal wealth for each strategy.

The obtained results are presented in Figure 1 and 2. If we compare the performance of two portfolio strategies by their ECDFs, we should choose the strategy whose distribution function lies below the other because the probability of getting a larger wealth is larger for the strategy with a stochastically smaller distribution function. The strategy based on the weights given in Theorem 1 is denoted by EXP, while the notation EXP-iid is used for the method with the weights of Corollary 2.

Figure 1 presents the results for a smaller value of the coefficient of the relative risk aversion. We observe that EXP overperforms EXP-iid for all considered investment periods $T$. For instance, for $T=104$ the probability of getting a wealth between $60$ and $80$ is equal to roughly $25\%$ for EXP while it is almost zero for EXP-iid. For a small horizon $T$ there exists a small probability of bankruptcy for both strategies but it differs not significantly. For $T\geq52$ the probability of a loss tends to zero. The EXP and the EXP-iid strategies both improve as $T$ becomes larger what indicates their good performance in the long-run setting.

Similar results are obtained for larger values of the coefficient of the relative risk aversion (see Figure 2). The performance of the EXP strategy is better for all $T$. From the other side, the probability of obtaining a larger value of the wealth is for both strategies smaller in comparison to the results presented in Figure 1.

Using the results of both figures we can conclude that the EXP strategy has a higher performance for all $T$ and risk levels. Of course this is not surprising since more information about the distribution of the asset returns is taken into account. On the other hand, ignoring the time dependence of the asset returns weakens the results with respect to the final wealth but it does not influence the probability of being bankrupt at the end of the investment period. Moreover, it has to be noted that the comparison of the ECDFs of the expected utilities is not relevant in our study because both strategies give the maximum expected utility in most of the cases and do not differ significantly.

\section{Summary}
Although the first formulation of the multi-period portfolio choice problem was already provided by Markowitz (1952), there are only a few results on closed-form solutions available in literature. They are mostly derived under the assumption that the asset returns are independently distributed. Merton (1969) discovered that the maximization of the exponential utility function for normally distributed returns is equivalent to the maximization of the mean-variance utility function. \c{C}anako$\breve{\text{g}}$lu and \"{O}zekici (2009) obtained a closed-form solution for the exponential utility function under the assumption that the asset returns are independent.
In general, the derivation of an analytical solution of the multi-period portfolio choice problem with the exponential utility for discrete time was considered to be very difficult (see, e.g., Soyer and Tanyeri (2006)).

In the present paper we derive an exact solution of the multi-period portfolio selection problem for an exponential utility function which is obtained under the assumption that the asset returns and the vector of predictable variables follow a vector autoregressive process of order 1. Under the assumption of independence the obtained expressions of the weights are proportional to the weights of the tangency portfolio obtained as a solution in the case of a single-period optimization problem. We show that only the coefficient of absolute risk aversion depends on the dynamics of the asset returns in this case. The weights of the optimal portfolio derived without a vector of predictable variables are obtained as a partial case of the suggested general solution. In an empirical study we compare the derived multi-period portfolio strategies for real data taking five developed stock market indices. A very good performance of the general solution is observed which always overperforms the weights derived under the additional assumption that the asset returns are independent.

The obtained results can be further extended by taking into account the uncertainties about the parameters of the data generating process. The analytical expressions of the weights can be used to derive the expected mean vector and the covariance matrix of the estimated weights which provide us the starting point for the detailed analysis of their distributional properties. This problem is not treated in the present paper and it is left for future research.

redpajama_set_name: RedPajamaArXiv


Press release 12 June 2017.
Employer research* conducted by GRiD shows that caring issues are a main cause of workforce absence. Home and family issues (which are often related to caring issues or the breakdown of care arrangements), plus more specific issues with providing elder and child care are resulting in absence from work and lost productivity.
They can also be a contributory factor in the development of mental illness or the deterioration of mental health. Indeed, mental ill-health is another major cause of absence.
An increasing number of people are affected by caring responsibilities, be this for children, elderly relatives or other dependents, and it is a problem that's isn't going to go away. It's not just the direct responsibilities that are taking people away from their work, it's the knock-on effects of stress and deteriorating mental health that results in more absence.
In terms of what employers are doing to reduce absence and improve attendance, flexible working comes out as the top-rated solution, with 36% of employers citing this as one of the measures they have put in place.
* Group Risk Annual Employer Research, conducted in September 2015 among 501 UK businesses with between 5 and 1,000 employees. Research was conducted by Lightspeed Research for Group Risk Development (GRiD).


Baby shower cakes can be based on a number of themes like, teddy bears, duckies, infant booties, baby bottles, storks, Nursery Rhymes, or fairy tales to make them appear cute and lovable. The relaxation of the gifts can be based on the concept of the infant shower cake.Simple board games are fun for toddlers, and they offer a great opportunity to teach colours. If you don't currently have a board game that involves moving to certain colored spots, make your personal. Attract a path around a piece of construction paper or poster board, and color every square a different colour. Have your kid transfer a vehicle, motion figure, or something else they like to a certain colour sq..Socially, this is also a time when kids start to value skills that other people admire, such as throwing a football or reading a long guide. School-age children want to be favored and accepted by their peers; the draw back is that anyone who doesn't conform might get observed - a child with frizzy hair, a kid whose parents don't reside with each other, and a kid who speaks one language at home and another at school. Children might tease to specific their uneasiness with these differences or, even worse, to seek energy over the kids who appear different.It can also be very assist to iphone apps for kids using meals. For instance, if the child is eating red jello, say "Your jello is crimson. Can you say 'red'"? You can repeat this with green beans, red strawberries, etc.Knex is accessible in theme sets as well. So, if you have purchased such a established, you will find manuals inside them. These manuals can be very useful in providing you an idea about how to proceed. In reality, they generally have a comprehensive direction as to how to create the predesigned constructions. Nevertheless, it is usually good to apply your creativeness and experiment with the structures. Some you can maintain on improvising with the structures. This is particularly helpful if you don't have the instructions with you.Because the paper towel beads have big holes, you don't need to worry about tying knots in the string. Even with knots, the toddler will nonetheless be able to use the Crimson White and Blue Cardboard Bead Threading Craft.Have the toddler thread the beads primarily based on colors. Assist the toddler discover a red bead. Then allow the toddler string that bead. Do this for other colours as well. This will assist the toddler learn colors.It's not exactly a hard factor to understand. Most Learn Colors from older surfers just by talking while in the line-up. They share tips and wisdom like when and how to paddle back again out and maintain the spirit of browsing on the right track.From the Capitol to the White House to the Supreme Court to the Smithsonian there's all sorts of methods to stimulate your thoughts and assist your Learn Colors issues the could never discover in a classroom. From the Nationwide Zoo to the Worldwide Spy Museum there are much more enjoyable issues for both grownups and kids to do than it seems possible to pack into just one city.3) Also by 3DAL is another of our favorites, Preschool Songs. It has the same colourful and easy graphics as Preschool Adventure but focuses on songs. You'll find a easy keyboard exactly where kids can compose a tune that's then recited by a cheerful parrot, an undersea percussion and rhythm sport, and a foolish group of parrots that sing in a spherical. They'll be as fascinated as you are irritated by the repetition of musical seems.If your toddler loves music which most do then this is certainly the toy for them. Produced study just for toddler use. Your toddler can be their own DJ with this music box. Features: Songs, games, dance, colourful lights, songs slider and a microphone for singing alongside. This toy will teach your toddler all about letters, figures, shapes, colours, matching and songs along with devices. This is better then the normal sing alongside toys simply because their is no cassette tape for your toddler to pull out and break.One of the very best methods to discover how to be develop up successfully is teaching your kid how to interact with the globe. Consider them with you when you go out. The world is a place to discover. They can learn human interaction. Consider them to the grocery store so they can discover intelligent shopping methods this kind of as buying fruits and vegetables. Consider them to the library to discover much more books and how studying can be so a lot enjoyable. Consider them to the park and show them how to perform with other children.It helps if your story fits in with the theme of the session. Consequently, any story actions on the same theme can improve your concept. You may want to pause and perform a short sport or sing a tune. Nursery Rhymes are fantastic during story occasions if related to the story and theme. They really assist to include the kids. You can even borrow a tune and include your own phrases- you don't have to be a professional musician to do this as children are fairly forgiving! When interspersed with brief actions in this way, a tale can be much longer than if it is simply just informed all in one go.


Anyone who says the Christian life is unadventurous hasn't hung out much with the Apostle Paul. You can feel the tension rising as he goes to bat for the things Jesus has done in his life. You can hear him momentarily go off-roading in order to defend the message of the Gospel and pursue those whom he loves. Let's not forget–Christianity is not meant to be a bunch of "do's and don't's"–it's meant to be relational!
From a distance, it appears obvious that these leaders did not have the best interest of others in mind. They were selfish and wrongly motivated. However, it is so easy to get entangled with others (even in the church) who persuasively influence you or impact your thinking. When you are looking for a place to belong and people to accept you…it is surprising what you will put up with. It may take some personal introspection to examine your friendships and relationships in order to determine whether or not they are Christ-centered.
What word best describes the other person: arrogance or humility?
Do they tear you down, or build you up?
Do they talk over you, putting you in your place? Or do you consider them a place of refuge and comfort?
Do they brag about their self-accomplishments? Or, their weakness that was made strong in Jesus?
Do they want to hurry you on out the door? Or are they patient with you, filled with grace?
Do they make every problem "your" problem? Or, do you find them relatable and human, willing to put themselves in your shoes?
Do they point you to Jesus above everything else?
Paul was a committed leader who was willing to endure incredible hardship and suffering in order to pursue the heart of the ones he loved. He leveraged his talents, abilities, knowledge, lineage, education, and career to steer the believers back to Jesus. We need more people like this today! People who are willing to take risks for others. People who will humbly take a stand, regardless of the fallout. People who are fearless faith-walkers to defend the truth.
Why is it necessary to stand up for the truth? Why do we find it so difficult to do so?
Apostle Paul. One can never get enough of the knowledge God passed to us through him. Staggering.
I'm so glad it blessed you!
AmenThanks for taking the time to comment.
Amen! That's my comment. Paul was relentless in his pursuit of Sharing the good news of our Savior with others, relentless on himself.
I have so much more to learn from the Apostle Paul. I hope to be a trailblazer too and love people like he did.
AmenGodinterestadmin@godinterest.com Godinterest LinkedIn Twitter Facebook Godinterest.com a place to talk about God, Culture, Life and all that other stuff.
Good point. I mean hanging out with him in Scripture. I always like to jump into the text, put myself in the story, and find relevancy for today. Have a good one!


Browse businesses looking to expand in and around Winterville.
Located in the Peach State, as of the 2010 census, Winterville had a population of 1,019.
Are you looking to own your own franchise business in, or around, Winterville, GA? Well, you are in luck! Browse any of these multiple franchises, shown below, to request FREE information about the opportunities that interests you or click here to see the full list of franchises looking to expand throughout Georgia.
Listed below you will find population, income and education related statistics, as well as other pertinent information, for Winterville, GA.
When compared to the state's average median age of 35.8 years, Winterville's median age is 8.3 years older.
When compared to the median income of individuals working in Georgia, Winterville's average income is $3,681 more than that of the state.
Winterville has 13% fewer citizens, with only high school diploma and/or GEDs, than the general population's average, in Georgia.


text:
Giuseppe Armani Aurora
African American Artists ›
African American Art ›
Gilbert Young Gallery
SHOO BOP DOO WOP Giclee On Canvas
or 3 equal layaway payments in store, with a credit card of $2,500.00 over 60 days.
SHOO BOP DOO WOP $7,500.00
with a credit card of $2,500.00 over 60 days.
1/20/2022 $2,500.00 1st payment
2/19/2022 $2,500.00 2nd payment
3/21/2022 $2,500.00 3rd & final payment
Gilbert Young SHOO BOP DOO WOP Giclee On Canvas is eligible for 3 equal layaway payments in store, with a credit card of $2,500.00 over 60 days.
As an option you may also pay for Gilbert Young SHOO BOP DOO WOP - Giclee On Canvas using Paypal or with your Amazon Account(*select items). Please note that all orders must be delivered to a physical address verified by Paypal or Amazon. These options are not applicable for orders to be delivered to Military or International destinations.
SHOO BOP DOO WOP - GYSHOBOP Giclee On Canvas
NOTES: "Shoo Bop Doo Wop" shows those magical times, back in the day, when the brothers who could sing would gather on the corners and do the damn thang! Those were the one who had the prettiest girls, and we all thought they were "too cool for school"--as the saying went. I feel like this is the best painting I've ever done. .
SHOO BOP DOO WOP Giclee On Canvas by Gilbert Young is signed by the artist and comes with a certificate of authenticity.
Gilbert Young bio
Gilbert Young's support of various community activities and events in his native town of Cincinnati, Ohio spans over two decades. He was active there in both the artistic and scholastic arenas. Young served as Director and Conservator of the Fine Arts Collection for the University of Cincinnati for twenty-six years. He also served as Artist-in Residence for the University in 1994-1995. Dating back to 1975, this multi-talented, multi-faceted, internationally known artist has lent the use of his images and time to many community projects and organizations. Young served as a consultant to the Harriet Beecher Stowe House Project, a renovation and restoration project for the Harriet Beecher Stowe house, creating a cultural arts center for the visual and performing arts in Cincinnati. In 1991, when the Cincinnati branch of the National Council of Negro Women hosted its first "Black Family Reunion", Gilbert was asked to provide the image that would become the official poster of the event. He also lent the use of his image, "Inside Daddy's World" , to The Black Male Coalition of Greater Cincinnati, Inc. in 1995 as the cover of their brochure and one of his most profound works, a commissioned portrait of civil rights leader Dr. martin Luther King hangs in the lobby of University Hospital.
Gilbert Young Camille
GY001
Gilbert Young PASSING THE TORCH
GYPASSTORC
Gilbert Young FOR YOUR LOVE
GYFORLOVE
Gilbert Young HOT TEA & BISCUIT
GYHOTTEA
Gilbert Young As I See It: Woman in the Mirror
GYMIRRORP
GYPASSTORP

redpajama_set_name: RedPajamaCommonCrawl


This unique mandolin was built by the instrument maker Ulf Kloo in 1997 and are the source for this sample library. Tremolo, picked, flam, flageolets & FX.
This unique mandolin was built by the instrument maker Ulf Kloo in 1997 for the musician Leif Isebring. Kloo Manolin has an extended range from G2 to D6.
Kloo Mandolin has 2 different picked notes programs with fast attack and slow attack. The SoundFont version has only one specially edited "Picked note" program.
1 flam notes program with a more "blurry" articulation, suitable for chord playing and faster drills, also good for "rounding off" tremolo runs.
1 tremolo program with typical mandolin tremolo notes.
All programs are chromatically sampled, every half note in the whole tonal range, with 2 velocity levels. Additionally,there are flageolets, fret noises and a huge selection of noises, knocks, scrapes and scratch sounds.
C1 = Picked 1, C#1 = Picked 2, D1 = Flam, D#1 = Tremolo, E1 = Flageolets, F1 = FX 1, F#1 = FX 2 G#1 = FX 3.
The total size of the library compressed is 177 MB.


text:
'use strict';

import GL from "gl-react";
import React, { Component }  from "react";

import {
    GPUImageFragShaderPredefineString,
    GPUImageFilter
} from './GPUImageFilter';

export const GPUImageTwoInputFragShaderPredefineString = GPUImageFragShaderPredefineString + `
uniform sampler2D u_texture_1;
uniform float u_texture_1_samplingScale;
`;

const GPUImageTwoInputFragShaderString = GPUImageTwoInputFragShaderPredefineString + `
void main() {
    vec4 texColor0 = texture2D(u_texture_0, v_texCoord);
    vec4 texColor1;
    if (u_texture_1_samplingScale > 1.0) {
        texColor1 = texture2D(u_texture_1, fract(v_texCoord * u_texture_1_samplingScale));
    } else {
        texColor1 = texture2D(u_texture_1, v_texCoord);
    }
    gl_FragColor = texColor0 + texColor1;
}
`;

export const GPUImageTwoInputFilter = GL.createComponent(
    ({ children, frag, input2nd, input2ndScale, uniforms }) => {
        return (
            <GPUImageFilter
                frag={frag || GPUImageTwoInputFragShaderString}
                uniforms={{
                    u_texture_1: input2nd,
                    u_texture_1_samplingScale: input2ndScale,
                    ...uniforms,
                }}>
                {children}
            </GPUImageFilter>
        );
    }, {
        defaultProps: {
            input2ndScale: 1.0,
        }
    }
);

redpajama_set_name: RedPajamaGithub