Regret-based Selection for \\Sparse Dynamic Portfolios
\fnms{David
\fnms{P. Richard
\fnms{Carlos M.
This paper considers portfolio construction in a dynamic setting. We specify a loss function comprised of utility and complexity components with an unknown tradeoff parameter. We develop a novel regret-based criterion for selecting the tradeoff parameter to construct optimal sparse portfolios over time. %This paper considers portfolio optimization under uncertainty, specifically when one desires a sparsified set of optimal portfolio weights. We introduce a utility function and corresponding stochastic optimization problem that produces a small portfolio from many potential assets while acknowledging uncertainty in asset returns and parameter estimates. We demonstrate the procedure using static and dynamic models for asset returns. %Our loss function used for selection is constructed by integrating over both dimensions of uncertainty and our optimization is structured as a LASSO minimization with penalized weights. The foundations of this paper are adapted from the decoupled shrinkage and selection (DSS) procedure of \cite{HahnCarvalho where statistical inference and the investor's preference for simplicity are separated.
DSSOpt-BA.tex
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\begin{document}

\begin{frontmatter}
\title{Regret-based Selection for \\Sparse Dynamic Portfolios}
\runtitle{Regret-based Selection}
%\thankstext{T1}{Footnote to the title with the ``thankstext'' command.}

\begin{aug}
\author{\fnms{David} \snm{Puelz}\thanksref{addr1}\ead[label=e1]{david.puelz@utexas.edu}\ead[label=u1,url]{http://faculty.mccombs.utexas.edu/David.Puelz/}},
\author{\fnms{P. Richard} \snm{Hahn}\thanksref{addr2}\ead[label=e2]{richard.hahn@chicagobooth.edu}}
\and
\author{\fnms{Carlos M.} \snm{Carvalho}\thanksref{addr1}
\ead[label=e3]{carlos.carvalho@mccombs.utexas.edu}}

\runauthor{Puelz, Hahn and Carvalho}

\address[addr1]{
	University of Texas McCombs School of Business
}

\address[addr2]{
University of Chicago Booth School of Business
}

\end{aug}

\begin{abstract}
This paper considers portfolio construction in a dynamic setting.  We specify a loss function comprised of utility and complexity components with an unknown tradeoff parameter. We develop a novel regret-based criterion for selecting the tradeoff parameter to construct optimal sparse portfolios over time.

%This paper considers portfolio optimization under uncertainty, specifically when one desires a sparsified set of optimal portfolio weights.  We introduce a utility function and corresponding stochastic optimization problem that produces a small portfolio from many potential assets while acknowledging uncertainty in asset returns and parameter estimates.  We demonstrate the procedure using static and dynamic models for asset returns.

%Our loss function used for selection is constructed by integrating over both dimensions of uncertainty and our optimization is structured as a LASSO minimization with penalized weights.  The foundations of this paper are adapted from the decoupled shrinkage and selection (\textit{DSS}) procedure of \cite{HahnCarvalho} where statistical inference and the investor's preference for simplicity are separated. 
\end{abstract}

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\end{frontmatter}

 
%This paper develops a method for dynamic sparse portfolio selection derived from expected regret foundations in decision theory.
%
%\textcolor{red}{this first problem is a little wrong .. should be function $f$ within the probability instead.}
%
%\begin{equation}\label{expregret}
%	\begin{split}
%		\min_{w} \max_{\lambda} p\left( \mathcal{L}_{\lambda}(w,\tilde{R}) - \mathcal{L}_{0}(w_0^{*},\tilde{R}) < 0 \right)
%	\end{split}
%\end{equation}where $w$ is a vector of portfolio weights, $\lambda$ is a penalty parameter governing portfolio complexity, $\tilde{R}$ is a vector of future asset returns, and $\mathcal{L}_{\lambda}(w,\tilde{R}) = f(w,\tilde{R}) + \lambda\Phi(w)$ is a loss function comprised of two components.  The first component $f(\cdot,\cdot)$ is the investor's negative utility (or loss) function, and $\Phi(w)$ is a complexity function that penalizes portfolios allocated to many assets.
%
%Problem \ref{expregret} is referred to as \textit{minimax} expected regret.  The intuition behind this decision-theoretic formulation is to minimize the worst case regret, which is defined as the expected difference between loss functions.


%We consider an investor who desires portfolio simplicity.  Using the $\ell_1$-norm on portfolio weights as a measure of portfolio complexity, this investor seeks to solve the following problem.
%
%\begin{equation} \label{problem}
%	\begin{split}
%		&\min_w \norm{w}_1
%				\\
%		\text{s.t.} \hspace{3mm} &p\left[ \mathcal{L}(w,\tilde{R}) < c\mathcal{L}(w_{0}^{*},\tilde{R}) \right] < \alpha
%	\end{split}
%\end{equation}

%We seek to solve the problem:
%
%\begin{equation} \label{problem}
%	\begin{split}
%		\min_{w,\lambda} \mathbb{E}\left[ \mathcal{L}(w) + \lambda\Phi(w) \right]
%		\\
%		\text{s.t.} \hspace{3mm} p\left[ \mathcal{L}(w) < \mathcal{L}(w_{0}^{*}) \right] > \alpha
%	\end{split}
%\end{equation}  

%Intuitively, problem \ref{problem} separates the choice of optimal action into two stages.  First, the expected loss is optimized for a range of decision makers indexed by $\lambda$.  Second, $\lambda$ is chosen to satisfy the constraint and thus specify the set of optimal preferences.  Note that the constraint will bind if the probability is monotone in $\lambda$.



\section{Introduction}

Practical investing requires balancing portfolio optimality and simplicity. In other words, investors desire well-performing portfolios that are \textit{easy} to manage, and this preference is driven by many factors. Managing large asset positions and transacting frequently is expensive and time-consuming, and these complications arise from both trading costs and number of assets available for investment.  For the individual investor, these challenges are strikingly amplified.  Their choice set for investment opportunities is massive and includes exchange-traded funds (ETFs), mutual funds, and thousands of individual stocks. This raises the question: \textit{How does one invest optimally while keeping the simplicity (sparsity) of a portfolio in mind?} Further challenges arise when sparse portfolio selection is placed in a dynamic setting. The investor will want to update her portfolio over time as future asset returns are realized while maintaining her desire for simplicity.  Institutions offering retirement accounts to employees represents a practical example of this problem (i.e., a 403(b) plan at public universities). An employee must choose from a large menu of funds to invest in her retirement account.  This paper takes the perspective of an individual investor who is faced with the task of selecting funds and constructing portfolios over time.

The focus of this paper are loss functions that balance utility and sparsity myopically for each time $t$:
\begin{equation} \label{overalllossfirst}
	\begin{split}
		\textbf{L}_{\lambda_t}(w_t,\tilde{R}_t) = \mathcal{L}(w_t,\tilde{R}_t) + \Phi(\lambda_t,w_t),
	\end{split}
\end{equation}where $\tilde{R}_t$ is a vector of future asset returns, $\mathcal{L}$ is the negative utility of an investor, $w_t$ is a vector of portfolio weights, $\Phi$ is a function that encourages sparsity in $w_t$, and $\lambda_t$ is a penalty parameter governing the degree to which the complexity function $\Phi$ is influential in the overall loss function.  Special attention must be paid to $\lambda_t$, the parameter that governs the utility-sparsity tradeoff.  If it is known a priori, the investor's optimal portfolio may be found for each time by routine minimization of the expectation of (\ref{overalllossfirst}).  By contrast, this paper considers the more challenging case where this parameter is unknown and may be thought of as part of the investor's decision. 

The interplay between dynamics, utility and portfolio simplicity in the investor's portfolio decision are viewed through the lens of regret. Assuming the existence of a desirable target portfolio, we define regret as the difference in loss (or negative utility) between the simple portfolio and the target; our investor would like to hold a simple portfolio that is ``almost as good as" a target. This paper distills a potentially intractable dynamic selection procedure into one which requires specification of only a single threshold of regret.  At the outset, the investor need only answer the question: \textit{With what degree of certainty do I want my simple portfolio to be no worse than the target portfolio?}  Put differently: \textit{What maximum probability of regretting my portfolio decision am I comfortable with?} 
 

%The contribution of this paper is the development a dynamic portfolio selection procedure that incorporates the investor's exogenously specified tolerance for regret.  

Once the regret threshold is specified, the investor's preference for portfolio simplicity will automatically adjust over time to accommodate this threshold.  In other words, the penalty parameter $\lambda_t$ continuously adjusts to satisfy the investor's regret tolerance.  In one period, her portfolio may only need to be invested in a small number of assets to satisfy her regret threshold.  However, that same portfolio may be far off from the target in the next period, requiring her to invest in more assets to accommodate the same level of regret.  This thought experiment illustrates that our procedure, although requiring a static regret tolerance to be specified at the outset, results in investor preferences for sparsity that are \textit{dynamic}. 






%We tackle this question from a new statistical and decision-theoretic perspective.  
%Portfolio optimization requires inputs that are parameters from a statistical model (typically the mean and covariance of asset returns) which are inherently uncertain, and a Bayesian estimation of these parameters yields posterior distributions quantifying this uncertainty.  Because the optimal portfolio depends on these parameters \citep{markowitz1952portfolio}, the optimality criterion (typically characterized by a ratio of portfolio return and risk and known as the Sharpe ratio) is also uncertain (see \cite{PandV} and \cite{jacquier2012asset} for important Bayesian treatments of uncertainty in finance).  In this paper, we propose a way to study how much ``optimality" we lose ex ante for varying levels of portfolio simplicity.  We compare simple portfolios with their more complex counterparts while acknowledging statistical uncertainty.  One key result of our approach is that very little ex ante is lost (in terms of Sharpe ratio) by an optimal portfolio built on a small subset of assets when many assets are available for investment. We also find (through out-of-sample tests) that the ex post performance of these ``small portfolios" exceed that of their ``full portfolio" counterparts.  In other words, if the deterioration in the predicitive Sharpe ratio for increasingly simpler portfolios is negligible up to statistical uncertainty \textit{and} they perform well out-of-sample, why not hold a simpler and easier to manage portfolio?
%
%\textcolor{red}{\textbf{change here -- add DLM sparsity and cross validation stuff ...}}



The regret-based selection approach presented in this paper is related to the \textit{decoupling shrinkage and selection (DSS)} procedure from \cite{HahnCarvalho} and \cite{puelz2016variable}.  In both papers, loss functions with explicit penalties for sparsity are used to summarize complex posterior distributions.  Posterior uncertainty is then used as a guide for variable selection in static settings.  This paper expands on these notions by developing a regret-based metric for selection and placing it within a dynamic framework. The important innovations presented herein are (\textit{i}) The use of investor's \textit{regret} for selection and (\textit{ii}) The development of a principled way to choose a \textit{dynamic} penalty parameter $\lambda_t$ (and thus select a portfolio) for a time-varying investment problem.

A key finding of this paper is that sparse portfolios and their more complex (or dense) counterparts are often very similar from an \textit{ex ante} regret perspective. More surprisingly, this similarity often persists \textit{ex post}.  This gives credence to a common piece of investing advice: ``Don't worry about investing in a variety of funds; just buy the market."


%  Their paper defines a new framework for model selection where statistical inference, regardless of prior and likelihood specifications, is separated from the model selection process itself. In general, the \textit{DSS} paradigm involves a model fitting step (inference) and a selection step (optimization). For the selection step, one specifies a loss function on which a particular model is graded, often by predictive loss.  The two steps are married by integration over uncertainty, and the final loss function is derived by integrating over the predictive and posterior distributions of the future observables and model parameters from the inference, respectively.  Then, this integrated loss function can be optimized. \cite{HahnCarvalho} develop the \textit{DSS} procedure for linear and gaussian graphical models and use variation explained and excess error heuristics for model selection. Extending this work, we apply the \textit{DSS} approach to the problem of portfolio selection and optimization.  The portfolio optimization loss function is defined as the portfolio return mean minus the portfolio return variance, and the portfolio Sharpe ratio is chosen as the model selection heuristic.   

%\textcolor{red}{\textbf{-----------------------------}}
%\vspace{-7mm}
%\\
%
%\textcolor{red}{\textbf{-----------------------------}}
 
%In this paper, we focus on the problem of investing in exchange-traded funds (ETFs). ETFs have gained popularity over the past two decades as a low-cost way for individual investors to hold large baskets of stocks.  These funds are mandated to closely follow (track) the returns of stock market indices.  For example, the ETF SPY tracks the S\&P 500 index which is comprised of 500 large market-capitalization companies.  An individual investor has increasing numbers of ETFs to choose from, including funds that invest in currencies, foreign equities, bonds, real estate, and commodities.  Given this large choice set and modest means by which individuals can invest, we utilize our approach to optimally select a small subset of ETFs.  Several financial startups are now providing ``ETF selection" services, including Betterment and Wealthfront.  After answering a series of questions to determine an investor's risk tolerance, these companies provide small portfolios of ETFs for investment.  ETF selection and investing is clearly a relevant problem for individual investors today.

%The focus on ETF portfolio selection is motivated by a similar approach in \cite{puelz2015optimal}.  In this paper, the authors select subsets of ETFs to replicate the covariation in a set of financial anomalies widely recognized as representing fundamental dimensions of variation in U.S. financial markets, see for example \cite{FF3} and \cite{FF5}.  These anomalies are complex trading strategies so investing in them directly is impracticable. \cite{puelz2015optimal} derive a loss function from the log-likelihood of their statistical model based on arbitrage pricing theory \citep{Ross}.  Their loss function is then optimized over a block of the covariance matrix describing the relationship between the financial anomalies and ETFs, and the inference and selection of ETFs adapts the procedure of \cite{HahnCarvalho}.  

%The empirical investigation in this paper is focused on ETF investing. We develop a portfolio selection approach based on a new mean-variance criterion and procedure which separates learning (via inference) from the portfolio decision, and this methodology is designed for investors desiring both optimality and simplicity.


\subsection{Previous research}

The seminal work of \cite{markowitz1952portfolio} provides the foundation of utility design for portfolio optimization related to this paper. One area of relevant research highlighting Bayesian approaches to this problem may be found in \cite{zhou2014bayesian}.  In this paper, the authors consider portfolio construction with sparse dynamic latent factor models for asset returns. They show that dynamic sparsity improves forecasting and portfolio performance.  However, sparsity in their context is induced at the factor loading level, not the portfolio decision.  In contrast, our methodology seeks to sparsify the portfolio decision directly for \textit{any} generic dynamic model. 

Additional areas of research focus on the portfolio selection problem, particularly in stock investing and index tracking.  \cite{polson1999bayesian} consider the S\&P 500 index and develop a Bayesian approach for large-scale stock selection and portfolio optimization from the index's constituents.  Other insightful Bayesian approaches to optimal portfolio choice include \cite{johannes2014sequential}, \cite{irie2016bayesian}, \cite{zhao2016dynamic}, \cite{gron2012optimal}, \cite{jacquier2010simulation}, \cite{puelz2015optimal} and \cite{pettenuzzo2015optimal}.  Methodological papers exploring high dimensional dynamic models relevant to this work are \cite{carvalho2007dynamic} and \cite{wang2011dynamic}. 


%
%A second stream of research is dedicated to regularization techniques in portfolio optimization. Portfolio optimization is a cornerstone of finance where optimal weights are found by maximizing portfolio return for a fixed level of risk \citep{markowitz1952portfolio}.  Since this procedure relies on estimates of the first two moments of return, the optimal weights depend heavily on the modeling choice for these moments.  It is well documented that sample estimates for the mean and variance produce portfolios with extreme weights that perform poorly out-of-sample \citep{jobson1980estimation}, \citep{best1991sensitivity},  \citep{broadie1993computing}, \citep{britten1999sampling}, \citep{demiguel2009optimal}, \citep{frankfurter1971portfolio}, \citep{dickinson1974reliability}, and \citep{frost1988better}.  Estimation errors are the culprit, and it is widely acknowledged that errors in the expected return are much larger than errors in the covariance matrix, see for example \cite{merton1980estimating}.  As a result, researchers have focused on ways to stabilize, through regularization, the entire optimization process. For a detailed literature review of finance literature related to regularized portfolio optimization, we refer the reader to the appendix.

%Regularization methods for portfolio optimization follow two schools of thought.  The first attacks estimation errors directly by attempting to regularize statistical estimates.  This is often done in a Bayesian context by shrinking the mean estimates to a ``grand mean" \citep{james1961estimation}.  Similar applications of shrinkage used for regularization include \cite{jorion1985international} and \cite{jorion1986bayes} who set the mean return of the minimum variance portfolio to be the grand mean.  In addition to the mean, others have focused on shrinking the covariance matrix in the portfolio selection problem, including \cite{ledoit2003improved}, \cite{ledoit2003honey}, and \cite{garlappi2007portfolio}.  In \cite{garlappi2007portfolio}, they consider parameter and model uncertainty to improve portfolio selection and out-of-sample Sharpe ratio. The second stream of literature focuses on regularizing the portfolio weights directly.  This literature is distinct in that regularization is delegated to the portfolio optimization step in contrast to inference of the optimization inputs.  This is a popular approach since mean-variance optimization can be formulated in terms of a Least Absolute Shrinkage and Selection Operator (LASSO) objective function from \cite{Tib} with fast computation of the optimal solution.  In this setting, the weights in the objective function are penalized with an $l_1$ norm.  Among the first to investigate weight regularization in the context of portfolio optimization is \cite{brodie2009sparse}.  They document improved out-of-sample performance as measured by the Sharpe ratio and consistently beat a naive strategy of investing an equal amount in each asset.  \cite{demiguel2009generalized} is a separate study showing similar results.  These more recent studies have clarified and expanded on earlier work by \cite{jagannathan2002risk} who showed that constraining the weights is equivalent to shrinking sample estimation of the covariance matrix.  \cite{fan2012vast} show that $l_1$ regularization limits gross portfolio exposure, leads to no accumulation of estimation errors, and results in outperformance compared to standard mean-variance portfolios.  Recent work on regularization for portfolio selection includes \cite{yen2013note}, \cite{yen2014solving}, \cite{carrasco2011optimal}, \cite{fernandes2012regularized}, \cite{fastrich2013constructing} with applications to index tracking in \cite{fastrich2014cardinality}, \cite{takeda2013simultaneous}, \cite{Wu2}, and \cite{Wu}.
%
% Some motivate it statistically for overcoming estimation error and overfitting while others simply acknowledge that certain forms of regularization improve out-of-sample performance.

%Under a new decision-theoretic lens, this paper frames weight regularization in a portfolio selection methodology by viewing it as a necessary part of the investor's utility.  This is markedly different from previous literature where weights are regularized for statistical benefits -- thus mixing investor preferences with the modeling of the optimization inputs.  Our loss function regularizes the weights to mirror the investor's joint preference for portfolio \textit{simplicity} and high risk-adjusted return, and we are able to tailor the utility to many other desires of the investor (see section 4).  The paradigm of \textit{decoupling shrinkage and selection} used in this paper is intended to clearly separate statistical modeling from the investor's utility optimization problem.  In this way, we disentangle the motivation for using a penalized utility from everything statistical -- a contribution that does not currently exist in the literature.  Additionally, we show financial benefits of this new method in a practical out-of-sample exercise in section \ref{DLMsection}. 

%, and demonstrate uncertainty's role in portfolio selection by considering several models for the mean and variance of asset returns. 

%We use two well known Bayesian models (normal-inverse-Wishart and latent factor model) as well as a dynamic regression model to estimate the mean and variance and choose the amount of regularization using the ex ante uncertainty in the portfolio Sharpe ratio.  Our contribution to the literature is an approach that clearly separates challenging inference from the investor's preference for simplicity.


% The paper is structured as follows.  In section 2, we derive a loss function for a mean-variance investor to be used for portfolio choice.  In section 3, we demonstrate how this loss function may be used to construct portfolios of ETFs. 
%
%The first part of section 3 is an illustration of the procedure -- we consider two static models for asset returns (a latent factor model and normal-inverse-Wishart model) to emphasize how the methodology is agnostic to choice of prior and likelihood. Since the \textit{DSS} framework permits the separation of the \textit{learning stage} from the \textit{decision stage}, we show how two reasonable choices for learning the moments of asset returns propagate into the overall portfolio decision.
%
%The second part of section 3 considers a practical application of the methodology anchored in the ideas from the illustrative section.  We construct a dynamic model for asset returns and show how an investor may implement an out-of-sample portfolio strategy which performs well compared to several alternatives.
%

\section{Overview}

The focus will be loss functions of the following form:
\begin{equation} \label{overallloss}
	\begin{split}
		\textbf{L}_{\lambda_t}(w_t,\tilde{R}_t) = \mathcal{L}(w_t,\tilde{R}_t) + \Phi(\lambda_t,w_t)
	\end{split}
\end{equation}where $\mathcal{L}$ is the negative utility of an investor, $\tilde{R}_t$ is a vector of $N$ future asset returns, $w_t$ is a vector of portfolio weights, and $\lambda_t$ is a penalty parameter governing the degree to which the complexity function $\Phi$ is influential in the overall loss function. Let the future asset returns be generated from a model parameterized by $\Theta_t$ so that $\tilde{R_t} \sim \Pi(\Theta_{t})$ and $\Pi$ is a general probability distribution.

The time-varying preferences in (\ref{overallloss}) take into account an investor's negative utility as well as her desire for portfolio simplicity.  Optimization of (\ref{overallloss}) in practice poses an interesting challenge since there is uncertainty in model parameters $\Theta_t$ and future asset returns $\tilde{R}_t$.  Also, the penalty parameter $\lambda_t$ is not known by the investor a priori, making her risk preferences ambiguous in portfolio complexity. A further obvious complication is that all of these unknowns are varying in time.

We propose a three-step approach to constructing a sequence of sparse dynamic portfolios.  This procedure will be based on an investor's regret from investing in an alternative (target) portfolio defined by $w_t^*$.  The three general steps are:
\begin{enumerate}
	\item \textit{Model specification:} Model the future asset returns $\tilde{R_t} \sim \Pi(\Theta_{t})$.
\vspace{3mm} 
	\item \textit{Loss specification:} Specify $\mathcal{L}$ and $\Phi$ in Loss (\ref{overallloss}).  Then, the expected loss given by $\textbf{L}_{\lambda_t}(w_t) = \mathbb{E}[\textbf{L}_{\lambda_t}(w_t,\tilde{R}_t)]$ may be minimized for a sequence of $\lambda_t \hspace{1mm} \forall t$. Define the collection of optimal portfolios in the cross-section as $\{ w_{\lambda_t}^* \}$.
\vspace{3mm} 
	\item \textit{Regret-based summarization:} Compare regret-based summaries of the optimal portfolios versus a target portfolio $w_t^*$ by thresholding quantiles of a \textit{regret} probability distribution, where regret as a random quantity is given by $\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t)$.  This random variable is a function of a sparse portfolio decision $w_{\lambda_t}^*$, the target portfolio $w_{t}^*$, and future asset returns.
\end{enumerate}The expectation and probability are both taken over the joint distribution of unknowns $(\tilde{R}_t, \Theta_t \mid \textbf{R}_{t-1})$ where $\textbf{R}_{t-1}$ is observed asset return data.  Flexibly, the regret function $\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t)$ can be any metric that is a function of the portfolio weights and unknowns and may be constructed using any target portfolio $w_t^*$.   In this paper, we consider the difference in loss between the sparse portfolio decision and the target portfolio  $\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t) = \mathcal{L}(w_{\lambda_t}^{*},\tilde{R}_t) - \mathcal{L}(w_t^*,\tilde{R}_t)$ as our measure of regret in keeping with the usual decision theoretic definition. 

We can now see how portfolio sparsity appears in the dynamic setting. The dynamic model given by $\tilde{R}_t \sim \Pi(\Theta_t)$ interacts with the portfolio decision $w_t$ via the expected loss minimization in step 2.  Iterating step 3 over time gives a sequence of sparse portfolio summaries $\{ w_{\lambda_t^*}^* \}$ where $\lambda_t^*$ provides an index for the selected sparse decision. Ultimately, these sparse portfolio summaries select which subsets of assets are relevant for our ``simple portfolio".  

%In the language of \cite{HahnCarvalho}, the loss function distills the complex posterior distribution from our model $\Pi(\Theta_t)$ into a sequence of sparse, point summaries (portfolios) for each time $t$.  

%Without using ex-ante statistical uncertainty for sparsity selection from step 3, it is unclear how to induce portfolio sparsity in a dynamic setting -- traditional approaches such as cross-validation are not applicable in this context.

The details of this procedure will be fleshed out in the following subsection.

\subsection{Details of Regret-based summarization} \label{details}
 In following section, we discuss the specifics of the regret-based summarization procedure.  We focus on expanding on step 3 of the methodology which represents the main innovation of this paper. Model specification and fitting as well as loss specification comprising steps 1 and 2 are only highlighted; a detailed formulation of these first two steps is presented in the Application section. 

Suppose that we have inferred a model for future asset returns given observed data $\textbf{R}_{t-1}$ that is parameterized by the vector $\Theta_t$: $\tilde{R}_t \sim \Pi(\Theta_t \mid \textbf{R}_{t-1})$. Let the resulting posterior distribution of parameters and future asset returns be $p(\Theta_t, \tilde{R}_t \mid \textbf{R}_{t-1})$.  For example, $\Pi$ may be parameterized by dynamic mean and variance parameters $\Theta_t = (\mu_t,\Sigma_t)$.  Further, suppose we have specified the utility and complexity components of the investor's loss function: $\textbf{L}_{\lambda_t}(w_t,\tilde{R}_t) = \mathcal{L}(w_t,\tilde{R}_t) + \Phi(\lambda_t,w_t)$.

For each investing period $t$, we obtain a sequence of portfolio decisions indexed by $\lambda_t$ by optimizing the expected loss function $\mathbb{E}[\textbf{L}_{\lambda_t}(w_t,\tilde{R}_t)]$.  Regret-based summarization is an approach to select the appropriate optimal decision from the collection (i.e., select $\lambda_t$) for each time $t$, and this choice may be visualized by using sparsity-regret tradeoff plots.

Revisiting the regret, samples of the $\tilde{R}_t$ margin from posterior distribution $(\tilde{R}_t, \Theta_t \mid \textbf{R}_{t-1})$ define the distribution for the regret random variable $\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t)$ given by a difference in loss:
\begin{equation} \label{regretdefn}
	\begin{split}
		\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t) &= \mathcal{L}(w_{\lambda_t}^{*},\tilde{R}_t) - \mathcal{L}(w_t^*,\tilde{R}_t),
	\end{split}
\end{equation}where $w_{\lambda_t}^{*}$ are the optimal sparse portfolio weights for penalty parameter $\lambda_t$ and $w_t^*$ are the weights for a target portfolio -- any portfolio decision the investor desires to benchmark against.  Regret (\ref{regretdefn}) is a random variable whose uncertainty is induced from the joint posterior distribution $(\tilde{R}_t, \Theta_t \mid \textbf{R}_{t-1})$ from step 1 of the procedure.

We use the regret random variable as a tool for sparse portfolio selection.  Each portfolio decision indexed by $\lambda_t$ is assigned a number:
\begin{equation} \label{probdefn}
	\begin{split}
		\pi_{\lambda_t} = \mathbb{P}[\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t) < 0]
	\end{split}
\end{equation}which is the probability that the sparse portfolio is no worse than the dense (target) portfolio.  In other words, $\pi_{\lambda_t}$ is the probability that I will not ``regret" investing in the sparse $\lambda_t$-portfolio over the target portfolio.  This may also be called the \textit{satisfaction probability} for the sparse $\lambda_t$ portfolio decision.

In Figure (\ref{deltaexamplenew}), we provide an illustration of the connection bewteen the loss and regret random variables.  This figure is constructed using returns on 25 passive indices and a next period ``log cumulative wealth" utility function.  This is done for a snapshot in time with a focus on one sparse decision that is invested in 4 out of the 25 indices. The investor is considering this decision versus her target -- a portfolio optimized over all 25 indices.  The left figure displays the loss distributions of the sparse decision and target. The probability mass of the sparse loss is gathered at larger values compared with the target loss.  It is ``more costly" (higher loss potential) to neglect diversification benefits and invest in fewer assets.



\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centerline{
  \includegraphics[scale=.4]{lossexamplegraph-cumloss}}
  \caption{Loss (left) and regret (right) for an example using returns on 25 passive indices.  The loss is defined by the log cumulative wealth. The sparse decision is a portfolio invested in 4 indices and is represented by the light shaded gray region.  The target decision is a portfolio optimized over all 25 indices and is represented by the shaded black region. The regret distributions shown on the right represent the random variables constructed by subtracting the sparse decision loss from the target loss.  Additionally, the black shaded region on the right shows $\pi_{\text{sparse decision}}$: The probability that the sparse decision is no worse than the target decision. }
  \label{deltaexamplenew}
\end{figure}



The right plot in Figure (\ref{deltaexamplenew}) displays the regret distribution for the sparse decision.  This is constructed by taking the difference between the sparse and target losses, as given in Equation (\ref{regretdefn}), defining the regret random variable.  With the regret distribution in hand, we can compute the probability that the sparse decision is no worse than the target portfolio given by Equation (\ref{probdefn}) -- this may be referred to as the ``satisfaction probability" for the sparse decision.  This is shown by the black shaded area on the right in Figure (\ref{deltaexamplenew}).  The larger this probability, the ``closer" the sparse decision's loss is to the target loss.  By making a decision that satisfies a lower bound on this probability called $\kappa$, we are able to control our chance of being the same or better than a target portfolio.  A lower bound ($\kappa$) on the probability of satisfaction (no regret) implies an upper bound ($1-\kappa$) on the probability of regret.


The investor's portfolio decision boils down to answering the question first posed in the introduction: \textit{With what degree of certainty do I want my simple portfolio to be no worse than the target portfolio?}  As the investor moves through time, the loss and regret distributions will evolve and so will the probability associated with the sparse $\lambda_t$ portfolio decisions.  A dynamic sparse portfolio decision extends this probability thresholding approach to a time varying framework. The investor chooses the a portfolio decision satisfying $\pi_{\lambda_t} > \kappa \hspace{3mm} \forall t$, ensuring she holds a portfolio that satisfies her regret tolerance in every investing period.
\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centering
  \includegraphics[scale=.43]{deltapilambdagraph-examplesmoother2}
  \caption{Regret distributions (left vertical axis) and $\pi_{\lambda_t}$ (right vertical axis) for increasing $\pi_{\lambda_t}$ values from left to right on the horizontal axis. Displayed are 300 sparse portfolio decisions indexed by $\lambda_t$. As the satisfaction probability ($\pi_{\lambda_t}$) increases, the mean regret represented by the gray dots will will typically trend downwards.  Gray bands represent 20\% centered posterior credible intervals for the regret.}
  \label{deltaexample}
\end{figure}

In Figure (\ref{deltaexample}), we show an example sequence of regret distributions (right vertical axis and gray bars) indexed by $\lambda_t$ as well as the satisfaction probability $\pi_{\lambda_t}$ (left vertical axis and open circles).  Specifically, we show the regret distributions for 300 sparse decisions under the log cumulative wealth utility and versus a target portfolio optimized over all available assets.  The highest regret decisions (by satisfaction probability) are on the left, and they become less regretful as one moves to the right on the horizontal axis. These sparse portfolio decision are all fairly close in terms of loss to the target decision.  Therefore, the corresponding regret distributions hover around zero, and the $\pi_{\lambda_t}$'s hover around 0.5.  Exploratory plots like Figure (\ref{deltaexample}) aid in choosing a proper value for the $\kappa$ threshold.  In this case $\pi_{\lambda_t} > 0.5$ is quite high.

% Similar to Figure (\ref{deltaexamplenew}), this plot display the tradeoff between regret and portfolio sparsity -- regret will approach zero as portfolios become less sparse.  Note that even for weak penalties, the regret distribution will have non-zero variance, so even if the mean regret is approximately zero, there will still probability mass above zero.  Thus, $\pi_{\lambda_t} > 0.5$ is high.

Once the time invariant threshold $\kappa$ is specified, a dynamic selection strategy is easily implementable. At each time $t$, we are presented with a set of decisions such as those displayed in Figure (\ref{deltaexample}) and choose the sparse portfolio decision such that its $\pi_{\lambda_t} > \kappa$.  If there are several sparse decisions satisfying the threshold, we may choose the one whose $\pi_{\lambda_t}$ is closest to $\kappa$. Of course, there is flexibility in the final step of selecting among admissible sparse decisions.  For example, one may select a sparse decision at time $t$ that is ``close" (in terms of a norm or assets held) to the previous sparse decision at time $t-1$ to reduce transaction costs related to the buying and selling of assets.  These features will be discussed in the Application section.

% The dynamic strategy is easily implemented by iterating this procedure over time. For each investing period, we evaluate an entire set of sparse portfolio decisions and select the sparsest decision that satisfies our regret threshold.  

%The selection of a decision (equivalently, a $\lambda_t$) is similar to the decoupling shrinkage and selection approach in \cite{HahnCarvalho} and \cite{puelz2016variable}.  In these papers, variables are selected by examining plots showing the predictive deterioration attributable to penalty-induced sparsification.  The current paper develops a framework for selection of dynamic decision with two innovative features.  First, we formulate the selection problem flexibly in terms of a probability of regret -- a desirable metric that is more intuitive than dealing with pure predictive uncertainty. Second, our approach is developed for sparsity in a \textit{dynamic} setting where traditional approaches for penalty parameter selection are unavailable.

%
%\subsubsection*{Step 2: Loss specification}
%We consider a loss function defined by the negative log cumulative return of a portfolio decision for $N$ assets.  Recalling general form of the loss function: $\textbf{L}_{\lambda_t}(w_t,\tilde{R}_t) = \mathcal{L}(w_t,\tilde{R}_t) + \Phi(\lambda_t,w_t)$, define:
%\begin{equation} \label{lossorig}
%	\begin{split}
%		\mathcal{L}(w_t,\tilde{R}_t) &= - \log \left( 1 + \sum_{k=1}^{N} w_t^k\tilde{R}_t^k \right) ,
%		\\
%		\Phi(\lambda_t,w_t) &=  \lambda_t \norm{ w_t }_{1}.
%	\end{split}
%\end{equation}This may be viewed as a version of the Kelly portfolio criterion \citep{kelly1956new} where the investor's preferences involve the portfolio growth rate.  
%
%Portfolio decisions $w_t$ may now be evaluated using the negative log cumulative return preferences given by $\mathcal{L}(w_t,\tilde{R}_t)$.  However, in order to find these portfolio decisions, we must first optimize the expectation of Loss (\ref{lossorig}).  We do this in two steps: First, we approximate the loss using a second order Taylor expansion, and second, we take the expectation over all unknowns and optimize for a range of penalty parameters $\lambda_t$.  
%
%Following the work of \cite{rising2012partial}, we consider a convenient second order Taylor approximation $\mathcal{L}(w_t,\tilde{R}_t) \approx \mathring{\mathcal{L}}(w_t,\tilde{R}_t) $ of the original Loss (\ref{lossorig}) expanded about $w_{0} = \vec{0}$:
%\begin{equation}
%	\begin{split}
%		\mathring{\mathcal{L}}(w_t,\tilde{R}_t) &=  \frac{1}{2}\sum_{k=1}^{N}\sum_{j=1}^{N} w_t^kw_t^j\tilde{R}_t^k\tilde{R}_t^j - \sum_{k=1}^{N} w_t^k\tilde{R}_t^k,
%		\\
%		\Phi(\lambda_t,w_t) &=  \lambda_t \norm{ w_t }_{1}.
%	\end{split}
%\end{equation}where we write the approximate loss including the penalty function as $\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t) = \mathring{\mathcal{L}}(w_t,\tilde{R}_t) + \Phi(\lambda_t,w_t)$.  The approximate expected loss $\mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)]$ is written as:
%\begin{equation}
%	\begin{split}
%		\mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)] = \mathbb{E}\left[  \frac{1}{2}\sum_{k=1}^{N}\sum_{j=1}^{N} w_t^kw_t^j\tilde{R}_t^k\tilde{R}_t^j - \sum_{k=1}^{N} w_t^k\tilde{R}_t^k + \lambda_t \norm{ w_t }_{1}  \right].
%	\end{split}
%\end{equation}
%
%With the posterior distribution $(\Theta_t, \tilde{R}_t \mid \textbf{R}_{t-1})$ in hand from step 1, we can take the expectation. We integrate over $(\tilde{R}_t \mid \Theta_t, \textbf{R}_{t-1})$ followed by $(\Theta_t \mid \textbf{R}_{t-1})$ to obtain the integrated approximate loss function: 
%%$\mathring{\textbf{L}}_{\lambda_t}(w_t)=\mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)]$:
%\begin{equation} \label{integratedloss}
%	\begin{split}
%		\mathring{\textbf{L}}_{\lambda_t}(w_t) = \mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)] & = \mathbb{E}_{\Theta_t}\left[\mathbb{E}_{\tilde{R}_t \mid \Theta_t}\left[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)\right]\right]
%		\\
%		  & = \mathbb{E}_{\Theta_t}\left[\frac{1}{2}w_t^{T}{\Sigma}_t^{\text{\tiny nc}} w_t - w_t^{T}\mu_t + \lambda_t \norm{ w_t }_{1}\right]
%		\\
%		& = \frac{1}{2}w_t^{T}\overline{\Sigma}_t^{\text{\tiny nc}} w_t - w_t^{T}\overline{\mu}_t + \lambda_t \norm{ w_t }_{1}
%	\end{split}
%\end{equation}where the overlines denote posterior means of the mean $\mu_t$ and  non-central second moment $\Sigma_t^{\text{\tiny nc}}$.  The non-central second moment is calculated from the variance as $\Sigma_t^{\text{\tiny nc}} = \Sigma_t + \mu_t\mu_t^T$.  Loss function (\ref{integratedloss}) may be minimized for a range of $\lambda_t$ values at each time $t$.  Specific details for the optimization may be found in Appendix (\ref{applossfunopt}). Since our motivation is forming simple portfolios for layman investors, we also constrain the weights to be positive: $w_t \geq 0$, i.e., short selling is prohibited.  Then, as detailed in step 3 below, the choice of a particular $\lambda_t$ for all penalized assets utilizes an intuitive probability of regret distribution.
%
%
%
%\subsubsection*{Step 3: Regret-based summarization}
%An important contribution of the paper is summarized in this section.  Here, we discuss how to choose the penalty parameter $\lambda_t$ governing portfolio complexity in our loss function.  This choice may be visualized by using sparsity-regret tradeoff plots.
%
%The selection of $\lambda_t$ is similar to the decoupling shrinkage and selection approach in \cite{HahnCarvalho} and \cite{puelz2016variable}.  In these papers, variables are selected by examining plots showing the predictive deterioration attributable to penalty-induced sparsification.  The current paper applies these ideas to portfolio selection and extends the existing framework in two ways.  First, we formulate the selection problem flexibly in terms of a probability of regret -- a desirable metric that is more intuitive than dealing with pure predictive uncertainty. Second, our approach is developed for sparsity in a \textit{dynamic} setting where traditional approaches for penalty parameter selection are unavailable.
%
%Revisiting the regret, samples of the $\tilde{R}_t$ margin from posterior distribution $(\tilde{R}_t, \Theta_t \mid \textbf{R}_{t-1})$ define the distribution for the regret random variable $\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t)$ given by a difference in loss:
%\begin{equation} \label{regretdefn0}
%	\begin{split}
%		\rho(w_{\lambda_t}^{*},w_t^*,\tilde{R}_t) &= \mathcal{L}(w_{\lambda_t}^{*},\tilde{R}_t) - \mathcal{L}(w_t^*,\tilde{R}_t),
%	\end{split}
%\end{equation}where $w_{\lambda_t}^{*}$ are the optimal sparse portfolio weights for penalty parameter $\lambda_t$ and $w_t^*$ are the weights for a target portfolio.  This target portfolio has yet to be specified; it can be any portfolio decision the investor desires. A natural choice for the target portfolio is the unpenalized ``Kelly optimal" decision from the original (approximated) loss:
%\begin{equation} \label{wtheta}
%	\begin{split}
%		w_t^*(\Theta_t) = \arg\min_{w_t \geq 0} \mathring{\mathcal{L}}(w_t,\Theta_t),
%	\end{split}
%\end{equation}where $\mathring{\mathcal{L}}(w_t,\Theta_t) =   \mathbb{E}_{\tilde{R} \mid \Theta_t} \left[ \mathring{\mathcal{L}}(w_t,\Theta_t) \right] = \frac{1}{2}w_t^{T}{\Sigma}_t^{\text{\tiny nc}} w_t - w_t^{T}\mu_t$.  The objective in (\ref{wtheta}) does not involve the complexity function and optimizes over the entire set of assets.  In contrast to the sparse portfolio decisions indexed by $\lambda_t$, we refer to this target decision as the ``dense portfolio".  Since the target portfolio is now defined, we may write out Regret (\ref{regretdefn0}) explicitly as:
%\begin{equation} \label{regretdefn}
%	\begin{split}
%		\rho(w_{\lambda_t}^{*},w_t^*(\Theta_t),\tilde{R}_t) &= \mathcal{L}(w_{\lambda_t}^{*},\tilde{R}_t) - \mathcal{L}(w_t^*(\Theta_t),\tilde{R}_t).
%	\end{split}
%\end{equation}Regret (\ref{regretdefn}) is a random variable whose uncertainty is induced from the entire joint posterior distribution $(\tilde{R}_t, \Theta_t \mid \textbf{R}_{t-1})$ from step 1 of the procedure -- posterior uncertainty now enters via the target (dense) portfolio weights.  These weights are uncertain insofar as they are defined by unknown mean and second moment parameters. 
%
%%\textcolor{red}{ Thus, we may refer to $w_{t}^{*}(\Theta_t)$ as the \textit{unknown} Kelly optimal decision. {\bf -- Not sure if I should say this.} }
%
%We use the regret random variable as a tool for sparse portfolio selection.  Each portfolio decision indexed by $\lambda_t$ is assigned a number:
%\begin{equation} \label{probdefn}
%	\begin{split}
%		\pi_{\lambda_t} = \mathbb{P}[\rho(w_{\lambda_t}^{*},w_t^*(\Theta_t),\tilde{R}_t) < 0],
%	\end{split}
%\end{equation}which is the probability that the sparse portfolio is no worse than the dense (target) portfolio.  In other words, $\pi_{\lambda_t}$ is the probability that I will not ``regret" investing in the sparse $\lambda_t$-portfolio over the dense portfolio.  This may also be called the sparse $\lambda_t$ portfolio decision.
%
%In Figure (\ref{deltaexamplenew}), we provide an illustration of the connection bewteen the loss and regret random variables.  This is done for a snapshot in time with a focus on two sparse decisions. The investor is considering these decisions versus a target: the dense portfolio.  The left figure displays the loss distribution of each decision and the target.  Decision 1 is a ``sparser" portfolio than decision 2 -- it corresponds to a decision with a larger penalty parameter governing portfolio complexity (i.e., $\lambda_{\text{decision 1}} > \lambda_{\text{decision 2}}$ and decision 1 invests in fewer assets than decision 2).  As a result, the probability mass of its loss is gathered at larger values compared with decision 2 and the target decision.  It is ``more costly" (higher loss potential) to neglect diversification benefits and invest in fewer assets.
%
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centerline{
%  \includegraphics[scale=.4]{lossexamplegraph-newMVloss}}
%  \caption{Loss (left) and regret (right) for a two decision example using returns on 25 passive indices.  \textbf{Decision 1} is a sparse portfolio decision, and \textbf{Decision 2} is a ``less sparse" portfolio decision (where more assets are included in the portfolio).  The shaded black region in the left figure is the loss for the target portfolio. The regret distributions shown on the right represent the random variables constructed by subtracting the sparse portfolio decisions from the target.  Additionally, the black shaded region on the right shows $\pi_{\text{decision 2}}$: The probability that sparse decision 2 is no worse than the target portfolio decision. }
%  \label{deltaexamplenew}
%\end{figure}
%
%The right plot in Figure (\ref{deltaexamplenew}) displays the regret distributions for the two decisions.  These are constructed by taking the difference between the target and sparse losses, as given in Equation (\ref{regretdefn}), defining the regret random variable.  Intuitively, the regret for the decision 1 is higher variance with more probability mass at larger regret values.  Decision 2's loss is much closer to the target loss, so its regret distribution has higher precision and mean closer to zero.  With the regret distribution in hand, we can compute the probability that decision 2 is no worse than the target portfolio given by Equation (\ref{probdefn}).  This is shown by the black shaded area on the right in Figure (\ref{deltaexamplenew}).  The larger this probability, the ``closer" the sparse decision's loss is to the target loss and the denser the portfolio.  By making a decision that satisfies a lower bound on this probability called $\kappa$, we are able to control our chance of being the same or better than a dense target portfolio.  A lower bound ($\kappa$) on the probability of ``no regret" implies an upper bound ($1-\kappa$) on the probability of regret.
%
%
%The investor's portfolio decision boils down to answering the question first posed in the introduction: \textit{With what degree of certainty do I want my simple portfolio to be no worse than the target portfolio?}  As the investor moves through time, the loss and regret distributions will evolve and so will the probability associated with the sparse $\lambda_t$ portfolio decision.  A dynamic sparse portfolio decision extends this probability thresholding approach to a time varying framework. The investor chooses the sparsest portfolio satisfying $\pi_{\lambda_t} > \kappa \hspace{3mm} \forall t$, ensuring she holds the simplest possible portfolio that satisfies her regret tolerance in every investing period.
%
%
%In Figure (\ref{deltaexample}), we show an example sequence of regret distributions (right vertical axis and gray bars) indexed by a decreasing penalty parameter complexity parameter $\lambda_t$ as well as $\pi_{\lambda_t}$ (left vertical axis and open circles).  The sparsest decisions are on the left, and they become less sparse as $\lambda_t$ decreases to the right.  Similar to Figure (\ref{deltaexamplenew}), this plot display the tradeoff between regret and portfolio sparsity -- regret will approach zero as portfolios become less sparse.  Note that even for weak penalties, the regret distribution will have non-zero variance, so even if the mean regret is approximately zero, there will still probability mass above zero.  Thus, $\pi_{\lambda_t} > 0.5$ is high.
%
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.42]{deltapilambdagraph-newMVloss4.pdf}
%  \caption{Regret distributions (left vertical axis) and $\pi_{\lambda_t}$ (right vertical axis) for decreasing penalties $\lambda_t$'s from left to right on the horizontal axis. As $\lambda_t$ becomes smaller, the sparse portfolio approaches the dense portfolio, the regret distribution will collapse around zero, and $\pi_{\lambda_t}$ will increase to a maximum.  Gray bands represent 60\% centered posterior credible intervals for the regret.}
%  \label{deltaexample}
%\end{figure}
%
%
%
%
%
%
%As previously noted, selection may be performed by specifying a time invariant threshold $\kappa$ and selecting the sparsest portfolio decision at each time $t$ such that its $\pi_{\lambda_t} > \kappa$. This is done by evaluating each decision from sparsest (far left) to densest (far right) and selecting the first sparse $\lambda_t$-portfolio that satisfies $\pi_{\lambda_t} > \kappa$.  The dynamic strategy is easily implemented by iterating this procedure over time. For each investing period, we evaluate an entire set of sparse portfolio decisions described by the solution path and select the sparsest decision that satisfies our regret threshold.  
%
%
%
%%Choice of the $\lambda$ parameter is an important practical issue.  To this end, we employ the Bayesian approach adapted from \cite{HahnCarvalho} by examining plots showing the predictive deterioration attributable to $\lambda$-induced sparsification.  These plots convey the posterior uncertainty in any posterior metric chosen.  For this paper, we consider the Sharpe ratio, a portfolio performance metric of return divided by risk and first studied in \cite{sharpe1966mutual}.  The intuition of this heuristic can be articulated as follows (for instance): choose $\lambda$ such that, with posterior probability 95\%, the predictive deterioration in the sparsified portfolio's Sharpe ratio is no more than 10\% worse than the unsparsified optimal portfolio's Sharpe ratio.  The Sharpe ratio is a natural metric to consider since its inputs, the portfolio mean and variance, are embedded in utility function \ref{optprob} which leads to our final loss function.  
%
%
%
%%\subsection{Connections to utility and decision theory}

\section{Application}

This section is divided into three parts.  First, we describe a dynamic linear model used to infer time-varying moments of asset returns and fulfill step 1 of the methodology. Second, we specify loss and resulting regret functions used for selection (steps 2 and 3).  Third, we demonstrate the methodology using a set of 25 passive funds.  In the demonstration sections, we consider a simple example and a more practical case study. 

\subsection{Model specification and data}
The general model we infer parameterizes the distribution of future asset returns with a mean and covariance indexed by time: $\tilde{R}_t \sim \Pi(\mu_t,\Sigma_t)$. An important feature of our proposed methodology is that any model providing estimations of these time varying moments may be used.  

To demonstrate our methodology, we estimate a dynamic linear model (DLM) motivated by \cite{FF5} who detail five ``risk factors" with future returns $\tilde{R}_t^{F}$ as relevant to asset pricing. Specifically, we model the joint distribution of future asset returns and factor returns $p(\tilde{R}_{t},\tilde{R}_{t}^{F})$ compositionally by modeling $p(\tilde{R}_{t} \mid \tilde{R}_{t}^{F})$ and $p(\tilde{R}_{t}^{F})$.  Following the dynamic linear model setup from \cite{harrison1999bayesian}, the future return of asset $i$ is a linear combination of future factor returns ($\tilde{R}_{t}^{i} \mid \tilde{R}_{t}^{F}$):
\begin{equation}\label{DLM1}
	\begin{split}
		\tilde{R}_{t}^{i} &= (\beta_{t}^{i})^{T}\tilde{R}_{t}^{F} + \epsilon_{t}^{i}, \hspace{6mm} \epsilon_{t}^{i} \sim N(0,1/\phi_{t}^{i}),
		\\
		\beta_{t}^{i} &= \beta_{t-1}^{i} + w_{t}^{i}, \hspace{6mm} w_{t}^{i} \sim \text{T}_{n_{t-1}^{i}}(0,W_{t}^{i}),
		\\
		\vspace{0mm}
		\\
		\beta_{0}^{i} &\mid D_{0} \sim \text{T}_{n_{0}^{i}}(m_{0}^{i},C_{0}^{i}),
		\\
		\phi_{0}^{i} &\mid D_{0} \sim \text{Ga}(n_{0}^{i}/2,d_{0}^{i}/2),
		\\
		\vspace{0mm}
		\\
		\beta_{t}^{i} &\mid D_{t-1} \sim \text{T}_{n_{t-1}^{i}}(m_{t-1}^{i},R_{t}^{i}), \hspace{6mm} R_{t}^{i} = C_{t-1}^{i}/\delta_{\beta},
		\\
		\phi_{t}^{i} &\mid D_{t-1} \sim \text{Ga}(\delta_{\epsilon} n_{t-1}^{i}/2,\delta_{\epsilon} d_{t-1}^{i}/2),
%		W_{t}^{i} &= C_{t-1}^{i}\left( \frac{1-\delta_{\beta}}{\delta_{\beta}} \right),
%		\\
%		\phi_{t}^{i} &= \frac{\gamma_{t}^{i}\phi_{t-1}^{i}}{\delta_{\epsilon}},
%		\\
%		\text{with:} \hspace{3mm} \gamma_{t}^{i} &\sim \text{Beta}\left( \delta_{\epsilon}n_{t-1}^{i}/2, (1-\delta_{\epsilon})n_{t-1}^{i}/2 \right).
	\end{split}
\end{equation}where $W_{t}^{i} = \frac{1-\delta_{\beta}}{\delta_{\beta}}C_{t-1}^{i}$ and $D_{t}$ is all information up to time $t$.  This model permits the coefficients on the factors as well as the observation and state level variances to vary in time.  Pre-specified discount factors $\delta_{\epsilon}$ and $\delta_{\beta}$ $\in (0.8,1)$ accomplish this goal for the observation and state level variances, respectively.  Also, note that $C_{t}^{i}$ (the posterior variance of the state equation for $\beta_{t}^i$) is updated through moments of the prior $\beta_{t}^{i} \mid D_{t-1}$ and the one-step ahead forecast distribution $\tilde{R}_{t}^{i} \mid D_{t-1}$. Theorem 4.1 in \cite{harrison1999bayesian} provides the general updating equations for the univariate DLM.  Table 10.4 in the book summarizes the recurrence relationships in the special case of variance discounting, providing the moments of the posteriors of the parameters $\{m_t^i,C_t^i,n_t^i,d_t^i\}$ for all $t$ and each asset $i$.


We model the five factor future returns $\tilde{R}_{t}^{F}$ with a full residual covariance matrix using the following matrix normal dynamic linear model:
%%%%%%%%%%%%%
%% MEAN MODEL
%%%%%%%%%%%%%
\begin{equation}\label{DLM2}
	\begin{split}
		&\tilde{R}_{t}^{F} = \mu_{t}^{F} + \nu_{t} \hspace{10mm} \nu_{t} \sim \text{N}(0,\Sigma_{t}^{F}), 
		\\
		&\mu_{t}^{F} = \mu_{t-1}^{F} + \Omega_{t} \hspace{6mm} \Omega_{t} \sim \text{N}(0,W_{t},\Sigma_{t}^{F}),
		\\
		\vspace{0mm}
		\\
		&(\mu_{0}^{F},\Sigma_{0}^{F} \mid D_{0}) \sim \text{NW}_{n_{0}}^{-1}(m_{0},C_{0},S_{0}),
		\\
		\vspace{0mm}
		\\
		&( \mu_{t}^{F}, \Sigma_{t}^{F} \mid D_{t-1}) \sim \text{NW}_{\delta_{F}n_{t-1}}^{-1}(m_{t-1},R_{t},S_{t-1}),
		\\
		&R_{t} = C_{t-1}/\delta_{c},
%		\Sigma_{t}& \mid D_{t-1} \sim \text{W}_{\delta_{f}n_{t-1}}^{-1}(S_{t-1})
%		\\
%		W_{t} &= \frac{1-\delta_{c}}{\delta_{c}}C_{t-1}
	\end{split}
\end{equation}where $W_{t} = \frac{1-\delta_{c}}{\delta_{c}}C_{t-1}$.  Analogous to Model (\ref{DLM1}), the discount factors $\delta_{F}$ and $\delta_{c}$ in Model (\ref{DLM2}) serve the same purpose of permitting time variation in the observation and state level variances, respectively. An added benefit of (\ref{DLM2}) is that $\Sigma_{t}^{F}$ is a full residual covariance matrix.  

Elaborating on the intuition behind Models (\ref{DLM1}) and (\ref{DLM2}) and guided by \cite{harrison1999bayesian}, the purpose of variance discounting is to provide a natural way to evolve the variance from the posterior to the prior while maintaining conjugacy for sequential updating.  For example, consider the posterior of the precision in Model (\ref{DLM1}):
\begin{equation} \label{phiexample}
	\begin{split}
		\phi_{t-1}^i \mid D_{t-1} \sim \text{Ga}(n_{t-1}^{i}/2,d_{t-1}^{i}/2).
	\end{split}
\end{equation}To construct $p(\phi_{t}^i \mid D_{t-1})$, we wish to maintain a Gamma form so it is conjugate with the likelihood function for $\tilde{R}_t^i$ given by the one-step ahead forecast distribution. One reasonable approach is to preserve the mean of Distribution (\ref{phiexample}), but inflate the variance by discounting the degrees of freedom parameter $n_{t-1} \rightarrow \delta_{\epsilon}n_{t-1}$.  The prior distribution then becomes:
\begin{equation}\label{phiexample2}
	\begin{split}
		\phi_{t}^i \mid D_{t-1} \sim \text{Ga}(\delta_{\epsilon}n_{t-1}^{i}/2,\delta_{\epsilon}d_{t-1}^{i}/2).
	\end{split}
\end{equation}Moving from Distribution (\ref{phiexample}) to (\ref{phiexample2}) increases the dispersion of the prior to represent a ``loss of information" characteristic of moving forward to time $t$ with a lack of complete information in $D_{t-1}$.  The remaining discount factors $\delta_{\beta},\delta_{C}$, and $\delta_{F}$ in Models (\ref{DLM1}) and (\ref{DLM2}) serve the analogous purpose of \textit{variance inflation} for their respective stochastic processes.

The limiting behavior of variance-discounted learning corresponds to exponentially weighting historical data with decreasingly smaller weights given to values further in the past (see sections 10.8.2-3 in \cite{harrison1999bayesian}).  Larger discount factors correspond to slower decaying weights and suggest the time series of parameters is slower-fluctuating.  Smaller discount factors intrinsically mean we have less data with which to estimate the parameters because the sequence is believed to be  more rapidly fluctuating.  Thus, the choice of discount factors amounts to choosing decaying weights for previous data that are relevant for predicting the parameters today.  


Models (\ref{DLM1}) and (\ref{DLM2}) together constitute a time-varying model for the joint distribution of future asset and factor returns: $p(\tilde{R}_{t},\tilde{R}_{t}^{F}) = p(\tilde{R}_{t} \mid \tilde{R}_{t}^{F})p(\tilde{R}_{t}^{F})$. As detailed in \cite{harrison1999bayesian}, they are Bayesian models that have closed form posterior distributions of all parameters at each time $t$, and the absence of MCMC is convenient for fast updating and uncertainty characterization -- a necessary ingredient for our regret-based portfolio selection procedure.  Under these models, we obtain the following mean and covariance structure:
\begin{equation}\label{moments}
	\begin{split}
		\mu_{t} &= \beta_{t}^{T}\mu_{t}^{F},
		\\
		\Sigma_{t} &= \beta_{t} \Sigma_{t}^{F} \beta_{t}^{T} + \Psi_{t},
	\end{split}
\end{equation}where column $i$ of $\beta_{t}$ are the coefficients on the factors for asset $i$, $\beta_{t}^{i}$.  Also, $\Psi_{t}$ is a diagonal matrix with $i$th element $\Psi_{tii} = 1/\phi_{t}^{i}$.  The parameters $\Theta_t = (\mu_t,\Sigma_t)$ are inputs to step 2 of the procedure.
%
%Posterior means of moments \ref{moments} are obtained by filtering estimations of models \ref{DLM1} and \ref{DLM2} and may be substituted into optimization problem \ref{optprobfinaltimevary} to solve the portfolio selection problem at each time step.  Similar to the static portfolio case, we solve the optimization problem (now \textit{at each time $t$}) along the solution path and choose the simplest portfolio that remains within the uncertainty region for the Sharpe ratio of the unpenalized portfolio.  This procedure is done at each time $t$ to provide a simple ETF portfolio that updates over time.


\subsubsection{Data and choice of discount factors}

We use data on the 25 most highly traded (i.e., most liquid) equity funds from ETFdb.com as our investable assets. This is monthly data from the Center for Research in Security Prices (CRSP) database from February 1992 through May 2016 \citep{CRSP}.  If the fund's inception date is after 1992, we backfill its return data with the index it is mandated to track.

The fund names, tickers, and sample statistics are displayed in Table (\ref{tablepassivelist}).  The returns on the Fama-French five factors are obtained from the publicly available data library on Ken French's website.\footnote{http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/}  We start with 10 years of data to train the model and begin forming portfolios in February 2002.  
\begin{table}[H]
{\footnotesize  {\centering
\begin{tabular}{lccc}
\textbf{Fund name} & \textbf{Ticker} & \textbf{Return (\%)} & \textbf{St. Dev. (\%)} \\
  \midrule
SPDR Dow Jones Industrial Average & DIA & 8.06 & 14.15 \\ 
SPDR S\&P 500 ETF & SPY & 7.46 & 14.34 \\
SPDR Industrial Select Sector & XLI & 9.19 & 16.49 \\
Guggenheim S\&P 500 Equal Weight ETF & RSP & 9.37 & 15.92 \\
iShares MSCI Emerging Markets & EEM & 6.06 & 22.78 \\ 
iShares MSCI EAFE & EFA & 3.94 & 16.43 \\  
iShares MSCI Germany & EWG & 6.82 & 22.23 \\  
iShares MSCI Japan & EWJ & 0.22 & 19.28 \\ 
iShares MSCI United Kingdom & EWU & 4.63 & 15.88 \\ 
iShares MSCI South Korea Capped & EWY & 9.46 & 36.78 \\
iShares MSCI Eurozone & EZU & 6.00 & 19.85 \\
iShares S\&P 500 Value & IVE & 7.36 & 14.95 \\
iShares Core S\&P 500 & IVV & 7.48 & 14.34 \\
iShares Russell 1000 & IWB & 8.22 & 14.00 \\
iShares Russell 1000 Value & IWD & 8.10 & 14.26 \\
iShares Russell 1000 Growth  & IWF & 7.63 & 14.56 \\
iShares Russell 2000 & IWM & 8.00 & 18.76 \\
iShares Russell 2000 Value & IWN & 7.78 & 18.67 \\ 
iShares Russell 2000 Growth & IWO & 6.63 & 22.14 \\
iShares Russell Mid-Cap Growth & IWP & 8.52 & 19.98 \\
iShares Russell Mid-Cap & IWR & 9.52 & 15.96 \\
iShares Russell 3000 & IWV & 7.52 & 14.61 \\
iShares US Real Estate & IYR & 9.32 & 19.12 \\ 
iShares US Technology & IYW & 11.64 & 26.09 \\
iShares S\&P 100 & OEF & 7.32 & 14.61 \\
  \midrule
\end{tabular}
}}
  \vspace{3mm}
   \caption{List of exchange-traded funds (ETFs) used for the empirical study. Also displayed are the ticker symbols and realized return and standard deviation (annualized) over their sample period.}\label{tablepassivelist}
\end{table}

Step 1 in our procedure is specifying and inferring a model for asset returns.  For our empirical analysis, we use Models (\ref{DLM1}) and (\ref{DLM2}). The discount factors for the factor coefficient and factor mean processes are set to $\delta_{c} = \delta_{\beta} = 0.9925$, and we consider time varying residual variances $\delta_{F} =\delta_{\epsilon}=0.97$.  Evidence of time varying residual variance is well-documented in the finance literature (see for example, \cite{ng1991tests}).  The discount factors are chosen to incorporate an adequate amount of data in the exponentially weighted moving window.  When $\delta = 0.9925$ ($0.97$), data eight (three) years in the past receives half the weight of data today.  We require slower decay for the factor coefficients and mean processes because more data is needed to learn these parameters than residual volatility.



\subsection{Loss and regret specification}

%\textcolor{red}{add below paragraph in somewhere}
%

We consider a loss function defined by the negative log cumulative return of a portfolio decision for $N$ assets.  Recalling general form of the loss function: $\textbf{L}_{\lambda_t}(w_t,\tilde{R}_t) = \mathcal{L}(w_t,\tilde{R}_t) + \Phi(\lambda_t,w_t)$, define:
\begin{equation} \label{lossorig}
	\begin{split}
		\mathcal{L}(w_t,\tilde{R}_t) &= - \log \left( 1 + \sum_{k=1}^{N} w_t^k\tilde{R}_t^k \right),
%		\\
%		\Phi(\lambda_t,w_t) &= \norm{w_t}_0  \hspace{2mm} \text{and} \hspace{2mm} \lambda_t \norm{ w_t }_{1}, 
	\end{split}
\end{equation}The utility in Loss (\ref{lossorig}) may be viewed as a version of the Kelly portfolio criterion \citep{kelly1956new} where the investor's preferences involve the portfolio growth rate.  The complexity function $\Phi(\lambda_t,w_t)$ is separately specified in each of the two examples to follow. 


%Two choices for the complexity function in Loss (\ref{lossorig}) are the $\ell_0$-norm $\norm{w_t}_0$ and the $\ell_1$-norm $\lambda_t\norm{w_t}_1$ of the portfolio decision $w_t$.  In the $\ell_1$ case, $\lambda_t$ corresponds directly to a single portfolio decision by amplifying the penalty in the loss function.  This function measures the degree of ``denseness" of a decision by summing the absolute value of each position: $\norm{ w_t }_{1} = \Sigma_i | w_t^i |$.  Sparser decisions will generally evaluate to smaller $\ell_1$-norms. The $\ell_0$ case has an even easier interpretation. The investor cares about the \textit{number} of assets invested in for a given decision -- more assets corresponds to a larger evaluation of the penalty function. We consider both penalties in this paper and absorb $\lambda_t$  into the penalty for $\ell_0$ case, thereby allowing the \textit{cardinality} of a decision to enter directly into the overall loss function.  In the $\ell_0$ case for this paper, we enumerate optimal portfolio decisions of varying cardinalities, and $\lambda_t$ may be thought of as a convenient indexing of each possible portfolio decision.


Portfolio decisions $w_t$ may now be evaluated using the negative log cumulative return preferences given by $\mathcal{L}(w_t,\tilde{R}_t)$.  However, in order to find these portfolio decisions, we must first optimize the expectation of Loss (\ref{lossorig}).  We do this in two steps: First, we approximate the loss using a second order Taylor expansion, and second, we take the expectation over all unknowns and optimize for each $\lambda_t$.  

Following the work of \cite{rising2012partial}, we consider a convenient second order Taylor approximation $\mathcal{L}(w_t,\tilde{R}_t) \approx \mathring{\mathcal{L}}(w_t,\tilde{R}_t) $ of the original Loss (\ref{lossorig}) expanded about $w_{0} = \vec{0}$:
\begin{equation}
	\begin{split}
		\mathring{\mathcal{L}}(w_t,\tilde{R}_t) &=  \frac{1}{2}\sum_{k=1}^{N}\sum_{j=1}^{N} w_t^kw_t^j\tilde{R}_t^k\tilde{R}_t^j - \sum_{k=1}^{N} w_t^k\tilde{R}_t^k,
%		\\
%		\Phi(\lambda_t,w_t) &=  \lambda_t \norm{ w_t }_{1}.
	\end{split}
\end{equation}where we write the approximate loss including the penalty function as $\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t) = \mathring{\mathcal{L}}(w_t,\tilde{R}_t) + \Phi(\lambda_t,w_t)$.  The approximate expected loss $\mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)]$ is written as:
\begin{equation}
	\begin{split}
		\mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)] = \mathbb{E}\left[  \frac{1}{2}\sum_{k=1}^{N}\sum_{j=1}^{N} w_t^kw_t^j\tilde{R}_t^k\tilde{R}_t^j - \sum_{k=1}^{N} w_t^k\tilde{R}_t^k + \Phi(\lambda_t,w_t)  \right].
	\end{split}
\end{equation}

With the posterior distribution $(\Theta_t, \tilde{R}_t \mid \textbf{R}_{t-1})$ in hand from step 1, we can take the expectation. We integrate over $(\tilde{R}_t \mid \Theta_t, \textbf{R}_{t-1})$ followed by $(\Theta_t \mid \textbf{R}_{t-1})$ to obtain the integrated approximate loss function: 
%$\mathring{\textbf{L}}_{\lambda_t}(w_t)=\mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)]$:
\begin{equation} \label{integratedloss}
	\begin{split}
		\mathring{\textbf{L}}_{\lambda_t}(w_t) = \mathbb{E}[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)] & = \mathbb{E}_{\Theta_t}\left[\mathbb{E}_{\tilde{R}_t \mid \Theta_t}\left[\mathring{\textbf{L}}_{\lambda_t}(w_t,\tilde{R}_t)\right]\right]
		\\
		  & = \mathbb{E}_{\Theta_t}\left[\frac{1}{2}w_t^{T}{\Sigma}_t^{\text{\tiny nc}} w_t - w_t^{T}\mu_t + \Phi(\lambda_t,w_t)\right]
		\\
		& = \frac{1}{2}w_t^{T}\overline{\Sigma}_t^{\text{\tiny nc}} w_t - w_t^{T}\overline{\mu}_t + \Phi(\lambda_t,w_t),
	\end{split}
\end{equation}where the overlines denote posterior means of the mean $\mu_t$ and  non-central second moment $\Sigma_t^{\text{\tiny nc}}$.  The non-central second moment is calculated from the variance as $\Sigma_t^{\text{\tiny nc}} = \Sigma_t + \mu_t\mu_t^T$.  Loss function (\ref{integratedloss}) may be minimized for a range of $\lambda_t$ values at each time $t$.  



In the subsections to follow, we will present two analyses using this model and data.  First, we discuss a simple unsupervised example to demonstrate the regret-based selection procedure.  Second, we present an in depth practical case study. 


\subsection{Simple example: Portfolio decisions with limited gross exposure}

In this section, we present a simple example demonstrating the main components of regret-based selection.  We complete Loss function (\ref{overallloss}) by specifying the complexity function as the $\ell_1$ norm of the weight vector: $\Phi(\lambda_t,w_t) = \lambda_t \norm{ w_t }_1$. The complexity function measures \textit{gross exposure} of a decision by summing the absolute value of each position: $\norm{ w_t }_{1} = \Sigma_i | w_t^i |$.  Decisions with larger absolute value components will evaluate to larger $\ell_1$ norms. The penalty parameter $\lambda_t$ corresponds directly to a single portfolio decision by amplifying the penalty in the loss function.

The approximate Loss function (\ref{integratedloss}) is now convex and may be written as:
\begin{equation} \label{integratedlossconvex}
	\begin{split}
		\mathring{\textbf{L}}_{\lambda_t}(w_t) = \frac{1}{2}w_t^{T}\overline{\Sigma}_t^{\text{\tiny nc}} w_t - w_t^{T}\overline{\mu}_t + \lambda_t \norm{ w_t }_1.
	\end{split}
\end{equation}Loss function (\ref{integratedlossconvex}) is readily optimized by a variety of software packages -- please see Appendix (\ref{applossfunopt}) for details.  Given its computational convenience, it possesses a couple important features worth noting.

First, Loss (\ref{integratedlossconvex}) requires no enumeration of decisions; it can be minimized quickly for a range of $\lambda_t$.  In this way, it is an ``unsupervised" approach to the sparse portfolio selection problem. Second, $\lambda_t$ now has explicit meaning beyond indexing the decisions. Since it is multiplying the complexity function, larger (smaller) $\lambda_t$ will generally correspond to sparser (denser) portfolio decisions.  Conveniently, this displays the regret-based procedure's usefulness in selecting tuning parameters in penalized optimization problems with time-varying inputs.  The dynamic nature of asset return data renders traditional cross validation approaches using i.i.d. sampled testing and training splits inappropriate.


\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centering
  \includegraphics[scale=.43]{deltapilambdagraph-examplesmootherLASSO2}
  \caption{Regret distributions (left vertical axis) and $\pi_{\lambda_t}$ (right vertical axis) for increasing $\pi_{\lambda_t}$ values from left to right on the horizontal axis. Displayed are 300 of the sparse portfolio decisions indexed by $\lambda_t$ for March 2002. As the satisfaction probability ($\pi_{\lambda_t}$) increases, the mean regret represented by the gray dots will will typically trend downwards.  Gray bands represent 20\% centered posterior credible intervals for the regret.}
  \label{deltaexampleLASSO}
\end{figure}


%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.43]{pilambda-March2002-rawLASSO}
%  \caption{Histograms of the satisfaction probabilities ($\pi_{\lambda_t}$) for the $\lambda_t=0$, or dense, target decision.  These are shown for the March 2002 investing period and is the distribution across all 500 sparse decisions obtained from optimization of Loss (\ref{integratedlossconvex}) for 500 different $\lambda_t$ $\forall t$.}
%  \label{pilamexampleLASSO}
%\end{figure}



We optimize Loss function (\ref{integratedlossconvex}) for a range of $\lambda_t$ and for each time $t$.  Specifically, we consider 500 $\lambda_t$ values spanning the sparsest ``one asset decision" to the $\lambda_t=0$ decision. The latter decision is referred to as the unpenalized \textit{Kelly optimal} portfolio and puts nonzero weight in all available assets. This is used as the target decision since there is no limit on gross exposure and will also be called the \textit{dense portfolio}.  We also normalize all decisions to sum to one, and allow long \textit{and} short positions in assets.  We normalize all decisions so that the investor is neither borrowing nor investing in the risk free (cash) rate. Instead, she is fully invested in a risky asset portfolio.  In other words, denoting $w_{\text{cash}}$ and $w_{\text{risky}}$ as the percentage of wealth in cash and risky assets (the ETFs) respectively, $w_{\text{cash}}+w_{\text{risky}}=1$ will always hold, and this paper considers the case when $w_{\text{risky}}=1$.

At each point in time, the investor would like to choose among the 500 sparse decisions indexed by $\lambda_t$.  This may be done by first computing the corresponding satisfaction probabilities $\pi_{\lambda_t}$ for each of the 500 decisions under consideration and then choosing one that satisfies and a pre-specified threshold $\kappa$.  Recall that $\pi_{\lambda_t}$ is defined in (\ref{probdefn}) as the probability that the regret (versus the dense target) is less than zero.  The utility, as specified in Equation (\ref{lossorig}), is the next period log cumulative wealth.

Figure (\ref{deltaexampleLASSO}) displays the cross-sectional regret distributions (left vertical axis) and satisfaction probabilities $\pi_{\lambda_t}$ (right vertical axis) for 300 of the sparse decisions in March 2002.  As prescribed by the regret-based procedure, the investor uses this information to select a portfolio. The satisfaction probabilities span 0.4 to 0.5 indicating that the decisions in this investing period are all quite similar. Guided by this figure, we choose a $\kappa=42.5\%$ threshold to construct an example sequence of selected portfolios. 

Once the static threshold is selected, we can iterate the selection procedure through time.  At each time $t$, the investor is confronted with ex ante regret information provided by a cross-sectional plot like Figure (\ref{deltaexampleLASSO}) and selects a portfolio that satisfies the threshold $\kappa$.  Once the sequence of decisions is constructed, we can look at how the regret distribution varies \textit{over time}.


Figure (\ref{regretevolveLASSO}) shows precisely how the regret of the selected decisions evolve over time.  This example demonstrates how both the mean (black line) and variance (surrounding shaded black regions) of regret can vary substantially. Notice that the regret is close to zero with small variance for most periods of time.  However, surrounding the financial crisis in 2009, the mean increases and then drops below zero and the variance increases.  When regret is negative, the utility of sparse portfolio decision exceeds that of the dense portfolio.  During crisis periods shortly into 2009, sparse portfolio decisions appear to be preferred (as measured by ex ante investor utility) to the dense portfolio.  Nonetheless, this drop in mean is accompanied by increased variance which informs the investor to be wary of the precision of her regret assessment.



\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centerline{
  \includegraphics[scale=.43]{lossplot2JASAnewexampleLASSO}}
  \caption{The evolution of the ex ante regret distributions for the sparse long/short portfolio decision given by a $\kappa=42.5\%$ threshold and versus the unpenalized Kelly optimal target.  The mean regret is shown by the lines and the surrounding areas represent the evolving centered 60\% posterior credible intervals.}
  \label{regretevolveLASSO}
\end{figure}


%\hspace{3mm} 
%
%
%\begin{table}[H]
%
%\footnotesize
%\centerline{
%\begin{tabular}{K{0.8cm}|K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}}
%\textbf{Dates} & SPY & DIA & EZU & EWU & EWY & EWG & EWJ & OEF & IVV & EEM & IVE & EFA & IWP \\
%  \midrule
%2003 & -8.5 & 69 & 7.2 & 14 & 5.6 & -3 & 3.3 & - & -3.5 & 9.6 & -32 & 15 & -35 \\ 
%  2004 & - & 8.9 & - & - & 1 & - & - & - & - & - & - & - & - \\ 
%  2005 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2006 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2007 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2008 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2009 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2010 & - & - & - & - & - & - & - & - & - & 9.8 & - & - & -9.3 \\ 
%  2011 & - & - & - & - & - & - & -4.2 & -28 & - & - & -180 & - & -83 \\ 
%  2012 & - & 170 & -11 & 16 & 0.12 & -16 & -43 & -360 & - & 1.9 & -620 & -8.4 & -230 \\ 
%  2013 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2014 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2015 & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
%  2016 & - & - & - & - & - & - & - & - & - & - & - & - & - \\    \midrule
%\end{tabular}
%}
%
%\vspace{3mm}
%
%\centerline{
%\begin{tabular}{K{0.8cm}|K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}K{0.55cm}}
%\textbf{Dates}  & IWR & IWD & IWF & IWB & IWN & IWO & IWM & IWV & IYW & IYR & RSP & XLI \\ 
%  \midrule
%  2003 & 23 & 44 & -94 & 47 & 47 & -34 & 55 & -49 & -15 & 3.6 & 14 & 16 \\ 
%  2004 & - & 23 & - & 25 & 20 & - & 20 & - & -37 & 26 & 7.8 & 5.3 \\ 
%  2005 & - & - & - & - & - & - & - & - & - & - & 100 & - \\ 
%  2006 & - & - & - & - & - & - & - & - & - & 7.4 & 93 & - \\ 
%  2007 & - & - & - & - & - & - & - & - & - & - & 100 & - \\ 
%  2008 & - & 4 & - & - & - & - & - & - & - & 15 & 81 & - \\ 
%  2009 & - & - & - & - & - & - & - & - & -130 & 230 & - & - \\ 
%  2010 & 18 & - & - & 73 & - & - & 87 & - & -190 & 5 & 93 & 11 \\ 
%  2011 & 36 & - & - & 220 & - & - & 180 & - & -130 & - & 37 & 53 \\ 
%  2012 & 140 & - & 310 & 530 & -39 & -79 & 230 & - & -130 & -0.8 & 140 & 89 \\ 
%  2013 & - & - & - & - & - & - & - & - & - & 100 & - & - \\ 
%  2014 & - & - & - & - & - & - & 31 & - & - & - & - & 69 \\ 
%  2015 & - & - & - & 19 & - & - & - & - & - & - & 21 & 60 \\ 
%  2016 & - & - & - & - & - & - & - & - & - & - & 1.6 & 98 \\  
%    \midrule
%\end{tabular}
%}
%% using raggedright is an option here, but it gives me an error
%\caption{Sparse portfolio decision (in percent) for DLMs (\ref{DLM1}) and (\ref{DLM2}) with $\delta_{F}= \delta_{\epsilon}= 0.97$ and $\delta_{c}= \delta_{\beta}= 0.9925$.  The portfolio decision is obtained by optimization of Loss function (\ref{integratedlossconvex}) and selection of $\lambda_t$ such that the decision satisfies the $\kappa=45\%$ threshold.  The target decision is the unpenalized Kelly optimal portfolio when $\lambda_t=0$ -- also known as the dense portfolio.  Note that annual weights are shown for brevity although portfolios are updated monthly.} 
%\label{DLMweightsk45lasso}
%\end{table}

%
%For each time, these are decisions that are better (in terms of utility) than the dense portfolio with 45\% probability. Notice how the sparsity of the decision adjusts over time to accommodate the threshold.  In 2003, the portfolio is invested in 24 out of the 25 funds. In contrast, the chosen decision for 2016 only invests in two funds.  The long and short positions for these portfolios are occasionally quite extreme; in 2012, there is a 530\% allocation to IWB funded by large short positions in OEF and IVE, among others. In the following case study section, we provide a realistic construction of decisions that avoids these issues associated with an unsupervised approach.
%


\subsection{Case study: Selection among a large set of enumerated decisions}

%When $\Phi(\lambda_t,w_t) = \norm{ w_t }_0$, minimization of (\ref{integratedloss}) becomes a challenging nonconvex combinatorial optimization problem even for a relatively small number of assets.  To combat this, we structure our set of decisions using realistic constraints for a layman investor.  Details of the enumeration and optimization of decisions for the $\ell_0$ case are presented in the following Results section.

The purpose of the following case study is to demonstrate regret-based portfolio selection for a layman investor.  We assume our investor would like to hold a broad market fund (SPY) and a couple more diversifying positions in other funds.  Additionally, we consider a scenario where the investor cannot hold negative positions in funds; i.e., short selling is prohibited.  Therefore, we consider decisions of only positive weights: $w_t \geq 0$ $\forall t$.  We construct a set of portfolio decisions for a layman investor using the following rules for a sparse portfolio with $q < N$ funds:

\begin{enumerate}
	\item $\geq 25\%$ of the portfolio is invested in SPY, a broad market fund tracking an index comprised of the 500 largest US companies by market capitalization.
	\item $\geq 25\%$ of the portfolio is diversified across the $q-1$ non-market funds in the following way: The $q-1$ non-market funds each have weights $\geq \frac{25}{q-1}\%$.
\end{enumerate}We consider portfolios of two, three, four, and five funds, all of which include SPY.  Each of these sparse portfolios are optimized using the unpenalized \textit{Kelly optimal} loss as the objective (Loss (\ref{integratedloss}) without the complexity function) and constraints defined as above.  Since our data has 24 funds excluding SPY, enumeration of decisions in this way results in $\Sigma_{i=1}^4 {24 \choose i} = 12,950$ sparse portfolios to select among.  Enumeration of sparse decisions implies a complexity function that measures the \textit{cardinality} or number of funds included in a portfolio.  Since the complexity function is now implicit in the sparse enumeration, $\lambda_t$ may be thought of as a convenient indexing of each possible portfolio decision.


%Since our motivation is forming simple portfolios for layman investors, we also constrain the weights to be positive: $w_t \geq 0$, i.e., short selling is prohibited.  Then, as detailed in Subsection (\ref{details}), the choice of a particular decision $\lambda_t$ utilizes an intuitive probability of regret distribution.

As presented in the initial example, we must specify a target decision which then defines the regret random variable defined in Equation (\ref{regretdefn}).  We consider two targets at opposite ends of the sparsity spectrum for the empirical analysis.  

\begin{enumerate}
	\item \textbf{Dense target}: The unpenalized \textit{Kelly optimal} decision; a portfolio optimized over all available assets. Define the Kelly optimal decision as $w_t^* = \arg\min_{w_t \geq 0} \mathring{\mathcal{L}}(w_t)$ where $\mathring{\mathcal{L}}(w_t) = \mathbb{E}[\mathring{\mathcal{L}}(w_t,\tilde{R}_t)] = \frac{1}{2}w_t^{T}\overline{\Sigma}_t^{\text{\tiny nc}} w_t - w_t^{T}\overline{\mu}_t$; the optimal decision in absence of the penalty function. This is the same target used in the ``$\ell_1$ penalty" example presented above, now with a positivity constraint on the weights. 
	\item \textbf{Sparse target}: The \textit{Market}; a portfolio composed of one asset representing broad exposure to the financial market.  We choose SPY as the market fund.
\end{enumerate}

The choice of each target will give an investor vastly different perspectives on sparse portfolio selection. In the dense target case, the investor desires a sparse portfolio decision that is close (in terms of regret) to a potentially unattainable decision involving all possible funds. The sparse target turns this approach on its head.  In this case, the sparse target approach will inform the investor of the added benefit (if any) in diversifying away from a broad market fund.


\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centering
  \includegraphics[scale=.43]{pilambda-March2002.pdf}
  \caption{Histograms of the satisfaction probabilities ($\pi_{\lambda_t}$) for two target decisions: The dense portfolio (black) and SPY (i.e., sparse target, in gray).  These are shown for the March 2002 investing period and are distributions across all 12,950 enumerated sparse decisions.}
  \label{pilamexample}
\end{figure}


Each of the 12,950 sparse decisions has a probability of satisfaction versus a target ($\pi_{\lambda_t}$) which can be readily calculated via simulation at each point in time using Equations (\ref{regretdefn}) and (\ref{probdefn}) and the distribution of future returns given by Models (\ref{DLM1}) and (\ref{DLM2}). In Figure (\ref{pilamexample}), we show histograms of the satisfaction probabilities for March 2002 across all 12,950 sparse decisions.  It is related to Figure (\ref{deltaexample}) in that the satisfaction probabilities corresponding to the right vertical axis are shown in histogram form, now for various targets.  The probabilities versus the dense (SPY) target are shown in black (gray).  The dense target is the \textit{dense} portfolio optimized over all 25 funds. Satisfaction versus this dense portfolio decision are gathered at smaller probabilities when compared with the SPY portfolio decision.  Of course, the satisfaction rate versus a diversified dense portfolio will intuitively be lower than versus a sparse portfolio of a single fund.

Figure (\ref{pilamexample}) aids in the proper choice of the regret threshold $\kappa$.  When evaluated under the next period cumulative wealth utility, all long only portfolio decisions are somewhat similar.  Thus, the regret (difference in loss) will generally be gathered around values close to zero, and the satisfaction probabilities will be gathered around value close to 0.5.  As a next step, we select a $\kappa$ and present the resulting dynamic portfolio decision for the dense target and SPY target. 

\begin{table}[H]

\footnotesize
\centerline{
\centering
\begin{tabular}{K{0.8cm}|K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}}
\textbf{Dates} & \multicolumn{2}{c}{SPY} &  \multicolumn{2}{c}{EZU} &  \multicolumn{2}{c}{EWU} &  \multicolumn{2}{c}{EWY} &  \multicolumn{2}{c}{EWG} &  \multicolumn{2}{c}{EWJ} &  \multicolumn{2}{c}{OEF} &  \multicolumn{2}{c}{IVV} &  \multicolumn{2}{c}{IVE} &  \multicolumn{2}{c}{EFA} &  \multicolumn{2}{c}{IWP} \\
  \midrule
2003 & 25 & 75 & - & - & 58 & - & - & - & - & - & - & - & - & 8.3 & - & - & - & - & - & - & - & 8.3 \\ 
  2004 & 25 & 75 & - & - & 43 & - & - & - & - & - & 20 & - & - & - & 6.2 & - & - & - & - & - & - & - \\ 
  2005 & 25 & 75 & - & - & 25 & - & - & - & 6.2 & - & 13 & - & - & 8.3 & - & - & - & - & - & - & - & - \\ 
  2006 & 62 & 75 & - & - & - & - & - & - & 6.2 & - & 19 & - & - & 12 & - & - & - & - & - & - & - & - \\ 
  2007 & 75 & 75 & - & - & - & - & 25 & - & - & 8.3 & - & - & - & - & - & - & - & - & - & - & - & 8.3 \\ 
  2008 & 44 & 75 & - & - & - & - & - & 12 & 8.3 & 13 & 21 & - & - & - & - & - & - & - & 26 & - & - & - \\ 
  2009 & 30 & 45 & - & - & - & - & 6.2 & - & - & - & 41 & - & - & - & - & 34 & - & - & 17 & 21 & 6.3 & - \\ 
  2010 & 75 & 55 & - & - & - & - & 8.3 & - & - & - & - & - & - & - & - & 26 & - & - & - & 11 & 8.3 & - \\ 
  2011 & 58 & 57 & - & - & 25 & - & - & - & - & - & - & - & - & - & - & 26 & - & - & - & - & 8.3 & - \\ 
  2012 & 29 & 25 & 8.3 & - & - & - & - & - & - & - & 54 & - & - & 56 & - & 6.2 & - & - & - & - & - & - \\ 
  2013 & 34 & 25 & - & - & - & - & - & 6.2 & - & 6.2 & 49 & - & - & 56 & - & - & - & - & - & - & 8.3 & - \\ 
  2014 & 25 & 75 & - & - & - & - & - & - & - & - & 37 & - & 26 & - & - & - & - & - & 6.2 & - & - & - \\ 
  2015 & 45 & 25 & - & - & - & - & - & - & - & - & 39 & 36 & - & 27 & - & - & 8.3 & - & - & 6.2 & 8.3 & - \\ 
  2016 & 35 & 75 & - & - & - & - & - & - & - & - & 40 & - & - & - & 17 & - & - & 8.3 & - & 8.3 & 8.3 & 8.3 \\ 
    \midrule
\end{tabular}
}

\vspace{3mm}

\centerline{
\centering
\begin{tabular}{K{0.8cm}|K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}K{0.2cm}Q{0.2cm}}
\textbf{Dates} & \multicolumn{2}{c}{IWR} &  \multicolumn{2}{c}{IWF} &  \multicolumn{2}{c}{IWN} &  \multicolumn{2}{c}{IWM} &  \multicolumn{2}{c}{IYW} &  \multicolumn{2}{c}{IYR} &  \multicolumn{2}{c}{RSP} &  \multicolumn{2}{c}{EEM} &  \multicolumn{2}{c}{IWO} &  \multicolumn{2}{c}{IWV} \\
  \midrule
2003 & - & - & - & - & 8.3 & - & - & - & - & 8.3 & - & - & 8.3 & - & - & - & - & - & - & - \\ 
  2004 & - & - & - & 12 & - & - & - & - & - & 12 & - & - & 6.2 & - & - & - & - & - & - & - \\ 
  2005 & - & - & - & 8.3 & - & - & - & - & - & 8.3 & 30 & - & - & - & - & - & - & - & - & - \\ 
  2006 & - & - & 6.3 & - & - & - & 6.2 & - & - & 12 & - & - & - & - & - & - & - & - & - & - \\ 
  2007 & - & - & - & - & - & - & - & - & - & 8.3 & - & - & - & - & - & - & - & - & - & - \\ 
  2008 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
  2009 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\ 
  2010 & - & - & - & - & - & - & - & - & 8.3 & 8.3 & - & - & - & - & - & - & - & - & - & - \\ 
  2011 & - & - & - & - & - & - & - & - & 8.3 & 8.3 & - & - & - & - & - & - & - & - & - & 8.3 \\ 
  2012 & - & - & - & 6.2 & - & - & - & - & 8.3 & - & - & - & - & - & - & - & - & - & - & 6.3 \\ 
  2013 & - & - & - & - & - & - & - & - & 8.3 & - & - & - & - & - & - & 6.3 & - & - & - & - \\ 
  2014 & 6.2 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 12 & - & 12 & - & - \\ 
  2015 & - & - & - & - & - & - & - & - & - & 6.2 & - & - & - & - & - & - & - & - & - & - \\ 
  2016 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\    
  \midrule
\end{tabular}
}


\caption{Sparse portfolio decisions (in percent) for DLMs (\ref{DLM1}) and (\ref{DLM2}) with $\delta_{F}= \delta_{\epsilon}= 0.97$ and $\delta_{c}= \delta_{\beta}= 0.9925$.  Shown are the selected portfolio decisions for the two targets: dense portfolio (left column in white) and SPY (right column in gray).  Note that annual weights are shown for brevity although portfolios are updated monthly.  In this dynamic portfolio selection, the regret threshold is $\kappa = 45\%$ for both targets.}
\label{DLMweightsk45}
\end{table}



We show the selected sparse decisions with the $\kappa=45\%$ threshold for the dense (portfolio optimized over all funds in white columns) and sparse (portfolio of only SPY in gray columns) target decisions in Table (\ref{DLMweightsk45}).  Each of these decisions possesses the property that, at each point in time, the satisfaction probability of investing in this decision versus the respective target is at least 45\%.  The portfolios are updated on a monthly basis; the table displays annual weights for brevity.  There are many decisions that will satisfy this threshold for each of the target (see for example the probability mass above 0.45 in Figure (\ref{pilamexample})).  In this case, we have added flexibility in which sparse decision to choose. We construct the sparse decisions so that at most one fund is selected or removed from month to month. For example, if the current portfolio at time $t$ has SPY, OEF, and IVV, two admissible portfolios at $t+1$ could include the funds be SPY, OEF, IVV, and EWG \textit{or} SPY and OEF assuming they both also satisfy the $\kappa$ threshold.

In Table (\ref{DLMweightsk45}), the sparse decision for the SPY target has larger allocations to SPY over the trading period compared with the sparse decision for the dense portfolio target.  Also, it possesses a consistent allocation to the US technology sector fund IYW.  In contrast, the sparse decision for the dense target often possesses a significant allocation to the Japanese equity specific fund EWJ. 




Figure (\ref{regretevolve1}) displays the evolution of the regret distributions for the sparse decisions shown in Table (\ref{DLMweightsk45}).  The lines are the expected regret, and the surrounding areas correspond to the centered 60\% posterior credible intervals.  The expected regret for both decisions remains close to zero and for most investing periods is slightly above zero; this is by construction since we choose the sparse decision that satisfies the $\kappa=45\%$ threshold at each point in time.  Overall, these decisions do not result in much regret. Indeed, many of the enumerated long only decisions appear similar in terms of the next period log wealth utility. 


\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centerline{
  \includegraphics[scale=.43]{lossplot2JASA-BnewFULLandSPY}}
  \caption{The evolution of the ex ante regret distributions for the sparse decisions in Table (\ref{DLMweightsk45}) versus the two targets: dense portfolio (black) and SPY (gray).  The mean regret is shown by the lines and the surrounding areas represent the evolving centered 60\% posterior credible intervals.}
  \label{regretevolve1}
\end{figure}


The variance of the regret distributions in Figure (\ref{regretevolve1}) changes substantially over the investing period. The range of log cumulative wealth difference for the ``dense portfolio as target" at the beginning is large ($\sim0.98 \text{ to } 1.02$ on the cumulative wealth scale).  The sparse decision for the dense target collapses in variance around 2005 exactly when the sparse decision is very close to the dense portfolio. Notice also that the variance of regret for the sparse decision with SPY as the target is, in general, smaller than the dense target decision.  Since all of the enumerated decisions have at least 25\% of the portfolio allocated to SPY (the target itself) with other diversifying positions, it is intuitive that the uncertainty in regret should be smaller. 

The evolution of regret for the sparse decision with SPY as the target sheds light on another question: Are there diversification benefits of allocating to other funds in consideration?  Selecting among the 12,950 sparse decisions including up to four non-SPY funds, the expected regret appears to be essentially zero (see the gray line in Figure (\ref{regretevolve1})).  This analysis suggests that under the log cumulative wealth utility and considering the large set of enumerated decisions defined at the beginning of this section, the best sparse decision from an \textit{ex ante} perspective may be SPY itself!

The ex ante evolution of other metrics, such as the portfolio Sharpe ratio, may be studied for the sparse decisions displayed in Table (\ref{DLMweightsk45}).  The Sharpe ratio is not a utility since it is not a function of future returns $\tilde{R}_t$.  However, it is a function of our model parameters whose uncertainty is characterized by the posterior distribution.  Specifically, define the predictive portfolio Sharpe ratio:
	\begin{equation} \label{regretdefnSR}
	\begin{split}
		\mathcal{SR}(w_t,\Theta_t) &= w_t^{T}\mu_t / (w_t^{T}\Sigma_t w_t)^{1/2},
		\\
		\rho_{\mathcal{SR}}(w_{\lambda_t}^{*},w_t^*,\Theta_t) &= \mathcal{SR}(w_t^*,\Theta_t)-\mathcal{SR}(w_{\lambda_t}^{*},\Theta_t),
	\end{split}
\end{equation}where $\rho_{\mathcal{SR}}(\cdot)$ is predictive in the sense that future returns $\tilde{R}_t$ conditional on the model parameters are integrated out. This portfolio metric differs from the Kelly criterion loss in that it focuses on a ratio of the portfolio expected return and variance. It may be used as an exploratory tool to accompany selection from regret-based portfolio selection.  



\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centerline{
  \includegraphics[scale=.43]{lossplot2JASAnewFULLandSPYSR}}
  \caption{The evolution of the ex ante ``Difference in annualized Sharpe ratio" distributions for the sparse decisions in Table (\ref{DLMweightsk45}) versus the two targets: dense portfolio (black) and SPY (gray).  The mean regret is shown by the lines and the surrounding areas represent the evolving centered 60\% posterior credible intervals.}
  \label{SRevolve1}
\end{figure}


We utilize this ``difference in Sharpe ratio" distribution in an exploratory fashion in Figure (\ref{SRevolve1}) shown on an annualized scale.  The evolution of the difference in Sharpe ratio is similar to the regret in Figure (\ref{regretevolve1}). In this case, a larger positive difference in Sharpe ratio means the selected sparse decision possesses a smaller return-risk tradeoff compared to the target decision.  The sparse decision for the dense target is larger variance and trends around larger positive values compared with the sparse decision for the SPY target.  The rationale for these features is similar: The enumerated sparse decisions are constructed to contain SPY, so the Sharpe ratios (like the loss) of the sparse decision and the SPY target decision will often be close. Following the financial crisis around 2009, the difference in Sharpe ratio stabilizes at lower values for both sparse decisions.


\subsubsection{What happens when $\kappa$ is varied?}

The selection of dynamic portfolio decisions will change based on the regret threshold $\kappa$.  In Figure (\ref{expectedregret}), we show how expected regret and difference in Sharpe ratio (on an annualized scale) change for selected sparse decisions using the SPY target.  The evolution of these metrics is shown for sparse decisions constructed using three $\kappa$ thresholds: $\kappa=45\%$ (black), $50\%$ (dark gray), and $55\%$ (light gray).  The black lines in both figures correspond to the ``SPY as target" paths in Figures (\ref{regretevolve1}) and (\ref{SRevolve1}), now compared to other $\kappa$ choices. 

Since $\kappa$ is a lower bound on the satisfaction probability, increasing this lower bound should lead to dynamic sparse decisions with generally smaller regret.  In other words, if the investor would like to be satisfied with higher probability, a ``lower regret"-sequence of sparse decisions should be selected. Figure (\ref{expectedregret}) demonstrates this when SPY is the target.  Larger $\kappa$ generally lead to smaller expected regret and difference in Sharpe ratio paths.  The $\kappa=55\%$ sparse decision leads to expected regret and difference in Sharpe ratio that are mostly negative from 2002 through 2016, indicating that portfolios with SPY and diversifying funds may be preferred to just SPY alone at this high satisfaction threshold.  However, these differences in expectation are still close to zero and small, especially for the evolution of expected regret. 
\begin{figure}[H]
\renewcommand{\figurename}{Figure}
\centerline{
  \includegraphics[scale=.4]{lossplotJASAnewfullkappacompare}}
  \caption{Expected regret (left) and expected difference in Sharpe ratio (right) for sparse decisions with SPY as the target.  These metrics are shown for three regret thresholds (the lower bound on probability of satisfaction): $\kappa=45\%$ (black), $50\%$ (dark gray), and $55\%$ (light gray).  Note that as the lower bound on the probability of satisfaction increases, both the expected regret and difference in Sharpe ratio tend to decrease for the selected sparse decisions.}
  \label{expectedregret}
\end{figure} 

\subsubsection{Enumerated decisions without using the utility}

The enumerated decisions considered up to this point are constructed by optimizing the integrated approximate loss.  An investor might prefer to construct decisions without any consideration for utility and statistical model. An equal-weighted portfolio (where each of the $N$ assets is given weight $1/N$) is one such example of a ``utility agnostic" decision.  Financial practitioners often advocate for this decision because of its  out of sample performance and purity in not involving errors in the statistical model and optimization procedure \citep{demiguel2007optimal}.  The regret-based procedure can readily accommodate a set of decisions with these characteristics as well.

In the following analysis we consider the set of sparse enumerated equal-weighted portfolios with up to four funds. This amounts to $\Sigma_{i=1}^4 {25 \choose i} = 15,275$ decisions to choose among at each point in time. We choose the ``dense $1/N$" portfolio as the target decision.  This target has 4\% invested in each of the 25 funds. To remain consistent with the previous analysis, we consider selection when $\kappa=45\%$.


\hspace{3mm}
\begin{table}[H]

\footnotesize
\centerline{
\begin{tabular}{K{0.8cm}|K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}K{0.4cm}}
\textbf{Dates} & SPY & DIA & EZU & EWU & EWY & EWJ & OEF & EEM & IVE & EFA & IWP & IWR & IWF & IWO & IWM & IYW & XLI \\
  \midrule
2003 & - & - & - & - & - & - & - & - & - & 25 & 25 & - & - & - & 25 & 25 & - \\ 
  2004 & - & - & - & - & 33 & 33 & - & - & - & - & - & - & - & - & - & 33 & - \\ 
  2005 & - & - & - & - & 25 & - & 25 & - & - & - & - & - & - & - & - & 25 & 25 \\ 
  2006 & - & - & - & 33 & - & - & - & - & - & - & - & 33 & - & - & - & 33 & - \\ 
  2007 & - & - & - & 33 & - & - & - & - & - & - & 33 & - & 33 & - & - & - & - \\ 
  2008 & - & - & 25 & - & - & - & 25 & - & - & - & 25 & - & 25 & - & - & - & - \\ 
  2009 & - & 33 & - & - & - & 33 & - & - & - & - & - & - & - & - & - & 33 & - \\ 
  2010 & - & - & - & - & - & 33 & - & 33 & - & - & - & - & - & - & - & 33 & - \\ 
  2011 & - & - & - & - & - & - & - & 33 & - & 33 & 33 & - & - & - & - & - & - \\ 
  2012 & - & - & - & 33 & 33 & - & 33 & - & - & - & - & - & - & - & - & - & - \\ 
  2013 & - & - & - & - & - & - & 33 & - & 33 & - & - & - & - & 33 & - & - & - \\ 
  2014 & 33 & - & - & - & - & - & - & - & - & 33 & - & - & - & 33 & - & - & - \\ 
  2015 & - & - & 25 & - & - & 25 & - & - & 25 & - & - & - & - & - & 25 & - & - \\ 
  2016 & - & - & - & 33 & - & 33 & - & - & 33 & - & - & - & - & - & - & - & - \\  
  \midrule
\end{tabular}
}

% using raggedright is an option here, but it gives me an error
\caption{Sparse portfolio decision (in rounded percent) for DLMs (\ref{DLM1}) and (\ref{DLM2}) with $\delta_{F}= \delta_{\epsilon}= 0.97$ and $\delta_{c}= \delta_{\beta}= 0.9925$.  Each point in time represents an equal-weighted portfolio and corresponding $\lambda_t$ such that the decision satisfies the $\kappa=45\%$ threshold.  The target decision is the equal-weighted portfolio of all 25 funds -- also known as the dense $1/N$ portfolio.  Note that annual weights are shown for brevity although portfolios are updated monthly.} 
\label{DLMweightsk45EW}
\end{table}

The weights for the selected portfolio decision are shown in Table (\ref{DLMweightsk45EW}).  At the $\kappa=45\%$ threshold, all portfolios have either three or four funds included, and the portfolio decisions have sustained exposures to EWJ (Japanese equity) and IYW (technology) throughout the investing period.  

This approach to equal-weighted portfolio selection possesses innovative and important features that should be highlighted.  While traditional ``$1/N$" approaches avoid the investor's utility and model for future asset returns altogether, the regret-based procedure still accounts for both.  The decisions themselves may be constructed without a utility or model in mind, but the characterization of regret must involve the utility and model.  Regret is computed by the difference in \textit{utility} between the target and sparse decisions, and it is a random variable that may be simulated using the \textit{model} for future asset returns.  Regardless of the set of decisions considered by the investor, the utility and model will always play a crucial role in a regret-based selection procedure. 



\subsubsection{Ex post decision analysis}

In this section, we consider the realized performance of the three sparse portfolio decisions presented in Tables (\ref{DLMweightsk45}) and (\ref{DLMweightsk45EW}) relative to their target decisions. We present out of sample statistics for the six decisions in Table (\ref{OOStablenew}). Shown is the annualized Sharpe ratio, standard deviation of return, and mean return.  


%In this section, we consider the realized performance of the three sparse portfolio decisions presented in Tables (\ref{DLMweightsk45}) and (\ref{DLMweightsk45EW}) relative to their target decisions. In contrast to the \textit{ex ante} analysis presented in the previous section, the analysis to follow is \textit{ex post}. Figure (\ref{OOSgraph2new}) shows the growth of \$1 invested in the three sparse decisions (solid lines) as well as their respective targets (dotted lines). The dense target decision is shown in dotted black, the SPY target decision is shown in dotted dark gray, and the equal-weighted target decision is shown in dotted light gray and is labeled with ``EW".  One should compare all same colored decisions to each other. Additionally, we present out of sample statistics for the six decisions in Table (\ref{OOStablenew}). Shown is the annualized Sharpe ratio, standard deviation of return, and mean return.  




% This is shown by the black line and labeled as \textit{sparse}.  Also shown are two other strategies: (\textit{i}) dense (gray) and (\textit{ii}) SPY (dotted gray). 
%
%The dense portfolio is the $\lambda_t=0$ solution for the integrated loss function: $w_{t\text{ dense}}^*= \arg\min_{w_t \geq 0} \mathring{\textbf{L}}_{0}(w_t)$ where the objective is defined in Equations (\ref{integratedloss}).  It is the strategy that optimizes over all 25 ETFs at each point in time using the posterior means of the first and second moments of returns.  Notably, the sparse portfolio outperforms the dense portfolio over this sample period.  Although the sparse portfolio decision is constructed to be \textit{ex ante} equivalent, or ``close", to the dense portfolio in terms of loss, it appears to exceed this expectation \textit{ex post}.  It also outperforms the sparse alternative of just investing in the market fund, SPY.
%
%
%
%Figure (\ref{OOSgraph2}) compares the sparse portfolio strategies to another reasonable alternative: a portfolio recommended by ETF investment advisor Wealthfront.  Its positions are: 50\% SPY (market), 15\% EEM (emerging markets), 30\% EZU (eurozone), and 5\% IYW (tech).  The portfolio is rebalanced to these strategic weights at each point in time.  The Wealthfront portfolio underperforms the sparse portfolio whose level of sparsity is allowed to adjust over time based on statistical uncertainty from our dynamic learning procedure.   The underperformance of this portfolio relative to the sparse portfolio decision highlights the benefit of selecting  \textit{different numbers} of ETFs at each time to satisfy the regret threshold. There appear to be out-of-sample gains from permitting flexibility in both the number and type of ETFs for investment.
%


%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.43]{cumretJASAnewALL}
%  \caption{Comparison of cumulative returns for the six portfolio strategies considered over the investing period February 2002 to May 2016.  The three solid lines correspond to the sparse portfolio decisions presented in Tables (\ref{DLMweightsk45}) and (\ref{DLMweightsk45EW}).  The three dotted lines correspond to the target decisions used for the regret-based selection procedure. ``EW" refers to the equal-weighted portfolio decision.}
%  \label{OOSgraph2new}
%\end{figure}
%

The sparse enumerated decision for the dense target performs similarly to the dense target.  This is comforting -- this sparse decision is a dynamic portfolio that is allocated to the market (SPY) and at most four other diversifying positions, and its out of sample performance is comparable to the dense portfolio optimized over all 25 funds at each time.  


\begin{table}[H]
\begin{center}
\footnotesize

\begin{tabular}{l*3{>{\centering\arraybackslash}m{0.4in}} @{}m{0pt}@{}}
 & \multicolumn{3}{c}{out-of-sample statistics} \\[1mm]
 \cline{2-4}
 &  Sharpe ratio & s.d. & mean return\\[.1ex] 
  \hline
sparse enumerated - dense as target & 0.40 & 14.98 & 6.02  \\ 
dense & 0.45 & 14.41 & 6.47  \\ 
  \hline
    sparse enumerated - SPY as target & 0.43 & 14.65 & 6.28 \\ 
    SPY & 0.43 & 14.63 & 6.28  \\ 
  \hline
  sparse EW enumerated - dense $1/N$ as target & 0.49 & 16.71 & 8.15  \\ 
 dense $1/N$ & 0.44 & 16.47 & 7.32  \\ 
  \hline
\end{tabular}

\end{center}
\caption{Comparison of out of sample statistics for the six portfolio strategies considered over the investing period February 2002 to May 2016. All statistics are presented on an annualized scale. ``EW" refers to the equal-weighted portfolio decision.}
\label{OOStablenew}
\end{table}



The sparse enumerated decision for the SPY target is equally interesting.  Since the SPY target is a sparse decision itself, comparison of it with selected sparse decisions provides insight into whether or not one should diversify away from investing in just the \textit{Market}. The out of sample performance shown in rows three and four of Table (\ref{OOStablenew}) display similar performance of the sparse enumerated decision and SPY.  Even after considering 12,950 sparse decisions containing up to four funds other than SPY -- the diversification benefits of exposure beyond SPY are negligible. The decisions that are \textit{ex ante} better than SPY with $45\%$ probability turn out to help out little \textit{ex post}.  Note that this conclusion is with respect to the next period cumulative wealth utility.  Future work will involve consideration of other utilities and compare how their selection and ex ante/post analyses differ.

While the sparse optimal strategies both underperform their targets, the sparse equal-weighted strategy slightly outperforms its dense $1/N$ target.  This is shown in rows five and six of Table (\ref{OOStablenew}). Interestingly, its out of sample performance even exceeds its sparse optimal counterparts shown in rows one and three in the Table. 



%
%
%
%{\noindent \bf \textcolor{red}{========================================}} \\
%{\noindent \bf \textcolor{red}{========================================}}
%
%
%%We minimize expected Loss function (\ref{integratedloss}) for a range of penalty parameters $\lambda_t$ to construct an entire collection of portfolio decisions which we may compare in terms of regret.  This optimization is done at each time $t$ by plugging in posterior means $\overline{\mu}_t$ and $\overline{\Sigma}_t^{\text{\tiny nc}}$ directly into (\ref{integratedloss}).  For this empirical analysis, we consider an investing scenario where only long (positive) positions in the assets are allowed. This is done since layman investors desiring to hold a small number of ETFs are typically constrained to long only portfolio positions.  Specifically, we optimize (\ref{integratedloss}) with a positivity constraint on the portfolio weights. A final technical note: After optimizing (\ref{integratedloss}) for selection, we reoptimize the portfolio of selected assets numerically for good measure.  This is done to ensure weights are not artificially ``overshrunk" due to the $\ell_1$ norm penalization. 
%%
%%A key component of the methodology is the specification of a static regret tolerance level $\kappa \in (0,1)$ for the investor -- a lower bound on the probability of being satisfied with a sparse portfolio decision (or equivalently, an upper bound on the probability of regret).  Recall that the investor would like to be at least as good as the target portfolio $\kappa\%$ of the time.  Once $\kappa$ is known, we are able to iterate our dynamic portfolio selection procedure over time to produce a sequence of sparse optimal portfolios.  In the analysis below, we examine the portfolio decision for the $\kappa = 55\%$ threshold and compare it to a much less stringent $\kappa = 10\%$ threshold.  
%%
%%A quick note: Asymmetry in the regret distributions imply that we will get reasonable sparse decisions even for $\kappa=55\%$, which appears to be high threshold.  We will display these features of the regret distributions below.
%%
%%%We include results for a very sparse portfolio decision $\kappa = 10\%$ in Appendix (\ref{appsparse}).
%
%%\subsubsection{Ex ante portfolio decision}
%%
%%Table (\ref{DLMweightsk4}) shows the weights for each year of the dynamic portfolio strategy for the regret tolerance level of $\kappa=55\%$.  Note that the portfolios are updated monthly as new data is observed -- annual weights are only shown for brevity.  This may be interpreted as the sparse portfolio decision that does ``no worse than the target portfolio" with 55\% probability during each investing period.  
%%
%%
%%%\begin{table}[H]
%%%\begin{center}
%%%\footnotesize
%%%
%%%\begin{tabular}{l|cccccccc}
%%% \textbf{Date}  & IWD & IWB & IWM & IYR & RSP & XLI \\
%%%  \midrule
%%%200202 & 50 & 50 & - & - & - & - \\ 
%%%  200302 & 49.9 & 50.1 & - & - & - & - \\ 
%%%  200402 & 35 & - & - & 65 & - & - \\ 
%%%  200502 & - & - & - & 44.6 & 27.8 & 27.5 \\ 
%%%  200602 & 71.3 & - & - & 28.7 & - & - \\ 
%%%  200702 & 41.6 & 38.4 & - & 20.1 & - & - \\ 
%%%  200802 & 81.9 & - & - & 18.1 & - & - \\ 
%%%  200902 & - & - & - & 100 & - & - \\ 
%%%  201002 & - & - & - & 100 & - & - \\ 
%%%  201102 & - & - & - & 100 & - & - \\ 
%%%  201202 & - & - & 16.5 & 5.69 & - & 77.9 \\ 
%%%  201302 & - & - & - & 13 & - & 87 \\ 
%%%  201402 & - & - & 100 & - & - & - \\ 
%%%  201502 & - & - & - & - & - & 100 \\ 
%%%  201602 & - & - & - & - & - & 100 \\    
%%%  \midrule
%%%\end{tabular}
%%%
%%%\end{center}
%%%\caption{Sparse ETF portfolio weights (in percent) for the dynamic linear model with $\delta_{F}= \delta_{\epsilon}= 0.999$.  Note that annual weights are shown for brevity although portfolios are updated monthly.  In this dynamic portfolio selection, the regret tolerance level $\kappa = 20\%$. Infrequent positions in DIA and IWN are also part of this portfolio decision but are not shown in the above summary table.}
%%%\label{DLMweightsk2}
%%%\end{table}
%%
%%%\begin{table}[H]
%%%\begin{center}
%%%\footnotesize
%%%
%%%\begin{tabular}{l|cccccccccccc}
%%% \textbf{Date} & IWD & IWB & IWN & IWM & IYR & RSP & XLI \\
%%%  \midrule
%%%200202  & 21.6 & 23.7 & 2.55 & 2.83 & 49.3 & - & - \\ 
%%%  200302 & 18.4 & 18.2 & - & - & 63.3 & - & - \\ 
%%%  200402 & 14.1 & 22.3 & - & - & 63.6 & - & - \\ 
%%%  200502  & 31.2 & 35.2 & - & - & 33.6 & - & - \\ 
%%%  200602  & 32.7 & 40.6 & - & - & 26.7 & - & - \\ 
%%%  200702  & - & - & - & - & 33 & 67 & - \\ 
%%%  200802  & 43.8 & 39.3 & - & - & 16.9 & - & - \\ 
%%%  200902  & - & 100 & - & - & - & - & - \\ 
%%%  201002  & - & - & - & - & - & 100 & - \\ 
%%%  201102  & - & - & - & 96.9 & 3.1 & - & - \\ 
%%%  201202  & - & - & - & 17.5 & 4.62 & - & 77.9 \\ 
%%%  201302  & - & 100 & - & - & - & - & - \\ 
%%%  201402  & - & 100 & - & - & - & - & - \\ 
%%%  201502  & - & - & - & - & - & - & 100 \\ 
%%%  201602  & - & 96.5 & - & - & 3.46 & - & - \\   
%%%  \midrule
%%%\end{tabular}
%%%
%%%\end{center}
%%%\caption{Sparse ETF portfolio weights (in percent) for DLMs (\ref{DLM1}) and (\ref{DLM2}) with $\delta_{F}= \delta_{\epsilon}= 0.97$ and $\delta_{c}= \delta_{\beta}= 0.9925$.  Note that annual weights are shown for brevity although portfolios are updated monthly.  In this dynamic portfolio selection, the regret tolerance level $\kappa = 55\%$.}
%%%\label{DLMweightsk4}
%%%\end{table}
%%
%%
%%
%%
%%In Table (\ref{DLMweightsk4}), allocations to large companies (IWB), large value companies (IWD) and real estate (IYR) are accompanied by modest allocations to small companies (IWN and IWM) to begin the investing period.  An allocation to real estate persists up until the financial crisis through 2009 along with an increasing allocation to large value companies in lieu of the broader large cap index.  Following the financial crisis, there is an allocation to large companies and the disappearance of the allocation to real estate.  In the final investing periods, there is a small reallocation to real estate and Dow Jones Industrial Average stocks (DIA). 
%%
%%Figure (\ref{weightgraphk45}) displays the evolution of the monthly weights for the $\kappa=55\%$ decision from February 2002 to February 2008.  The bear market resulting in the financial crisis began in October 2007.  The allocation to the US real estate ETF IYR is displayed in black, and allocations to the remaining funds are displayed in gray. As mentioned previously, the portfolio decision initially invests a large amount in real estate with reasonable sized positions in the remaining broader market funds.  As the bear market approaches, the position in IYR (black) is gradually curtailed and eventually disappears.
%%
%%The financial crisis is an inflection point -- immediately after the crash, all funds experience upward movements in their expected returns with IWB possessing the most favorable loss.  This fund comprises the entire portfolio through the end of the investing period when there is a small reallocation to IYR and DIA. 
%%
%%
%%\begin{figure}[H]
%%\renewcommand{\figurename}{Figure}
%%\centering
%%  \includegraphics[scale=.43]{weightplotJASANEW}
%%  \caption{Portfolio weights from February 2002 to February 2008 for the sparse $\kappa=55\%$ portfolio decision displayed in Table (\ref{DLMweightsk4}).  The allocation to the real estate fund IYR is shown in black, and the allocations to the remaining funds IWD, IWB, IWN, IWM and DIA are shown in gray.}
%%\label{weightgraphk45}
%%\end{figure}
%%
%%
%%
%%%In both portfolio decisions, SPY is always an included asset since it is unpenalized in the optimization. Note that for the denser portfolio in Table \ref{DLMweightsk4}, the positivity constrained numerical optimization zeros it out in certain periods since other market ETF ``comparables" exist. In Table \ref{DLMweightsk2}, the market is underweighted from 2003 through 2008 and is fully allocated to after the financial crisis (2009 to 2015).  Diversifying positions include emerging market stocks at the beginning and large and value stocks and real estate in the middle of the trading period.  The procedure's selection of the real estate ETF and significant allocation leading up to the financial crisis contribute to its performance over the trading period. 
%%
%%%	\begin{equation} \label{regretdefnSR}
%%%	\begin{split}
%%%		\mathcal{SR}(w_t,\Theta_t) &= w_t^{'}\mu_t / w_t^{'}\Sigma_t w_t,
%%%		\\
%%%		\mathcal{R}(w_{\lambda_t}^{*},w_t^*,\Theta_t,\tilde{R}_t) &= \mathcal{SR}(w_t^*,\Theta_t)-\mathcal{SR}(w_{\lambda_t}^{*},\Theta_t),
%%%	\end{split}
%%%\end{equation}
%
%\subsubsection{The regret distribution's evolution}
%
%Samples of the posterior distribution $(\Theta_t, \tilde{R}_t \mid \textbf{R}_{t-1})$ from step 1 allow the investor to visualize the sparse portfolio decision from Table (\ref{DLMweightsk4}) in several \textit{ex ante} contexts. In Figure (\ref{lossgraphk45}), we show how the regret distribution changes over time.  The vertical axis displays the regret distributions' mean (black line) and 60\% centered posterior credible intervals (shaded black area) for each investing period.  The units of this axis are the difference in monthly log cumulative wealth.  The target decision is also called the dense portfolio or unpenalized Kelly optimal portfolio; one which is allowed to optimize over the entire set of ETFs.  The sparse portfolio decision has regret that is smaller than zero with 55\% probability.
%
%Notice how the regret distribution's variance evolves over time.  At the beginning, the range of log cumulative wealth difference in small ($\sim0.985 \text{ to } 1.015$ on the cumulative wealth scale).  The regret threshold $\kappa=55\%$ produces a sparse portfolio decision that is \textit{close} to the dense decision (similar to ``decision 2" for the example displayed in Figure (\ref{deltaexamplenew})).  Thus, the loss distributions for the sparse portfolio decisions and dense portfolios will have significant probability mass that overlaps.  The mean regret of investing in the sparse portfolio over the target (black line) remains slightly above zero, and there are periods when it jumps to marginally higher levels.  An upward jump in the mean and variance of the regret distribution can be seen around the financial crisis in 2009.  The regret distributions preceding market bottom have small mean and variance, and regret becomes significantly less precise during the market crash. However, as the market began to rebound, the mean of regret stabilizes and its variance begins to decrease towards pre-crisis levels. 
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.43]{lossplot2JASADATEST055}
%  \caption{Comparison of the regret distributions evolution over time versus the dense target portfolio (left vertical axis) and the probability of regret being less than zero (right vertical axis). The regret is measured as a difference in the loss between the portfolio decisions given by the $\kappa=55\%$ threshold and the target portfolio.  Smaller levels of regret are more favorable.  The black line displays the mean regret, and the shaded black area displays the centered 60\% posterior credible interval.  The shaded gray region represents the probabilistic constraint on the regret distribution given by $\pi_{\lambda_t} > \kappa$ and used for selection.}
%\label{lossgraphk45}
%\end{figure}
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centerline{
%  \includegraphics[scale=.4]{lossexamplegraph-newMVloss}}
%  \caption{Loss (left) and regret (right) for a two decision example using returns on 25 passive indices.  \textbf{Decision 1} is a sparse portfolio decision, and \textbf{Decision 2} is a ``less sparse" portfolio decision (where more assets are included in the portfolio).  The shaded black region in the left figure is the loss for the target portfolio. The regret distributions shown on the right represent the random variables constructed by subtracting the sparse portfolio decisions from the target.  Additionally, the black shaded region on the right shows $\pi_{\text{decision 2}}$: The probability that sparse decision 2 is no worse than the target portfolio decision. }
%  \label{deltaexamplenew2}
%\end{figure}
%
%
%%\begin{figure}[H]
%%\renewcommand{\figurename}{Figure}
%%\centering
%%  \includegraphics[scale=.43]{lossplot2JASADATEST001}
%%  \caption{Comparison of the regret distributions evolution over time versus the dense target portfolio (left vertical axis) and the probability of regret being less than zero (right vertical axis). The regret is measured as a difference in the loss between the portfolio decisions given by the $\kappa=55\%$ threshold and the target portfolio.  Smaller levels of regret are more favorable.  The black line displays the mean regret, and the shaded black area displays the centered 60\% posterior credible interval.  The shaded gray region represents the probabilistic constraint on the regret distribution given by $\pi_{\lambda_t} > \kappa$ and used for selection.}
%%\label{lossgraphk45}
%%\end{figure}
%
%
%
%%Compare the evolution of regret given in Figure (\ref{lossgraphk45}), which is tightly about zero, with the regret distribution under a less stringent threshold of $\kappa = 20\%$ shown in Figure (\ref{lossgraphk2}).  The latter figure corresponds to a weaker regret threshold, so the resulting regret distributions for this sparse decision have means further above zero and larger variance. In other words, sparser decisions and \textit{more} regret is allowed when $\kappa$ is smaller.  The right vertical axis corresponding to the probability of regret less than zero shows probabilities that hover around the threshold (0.2) until the financial crisis in 2008.  During this time in which many funds struggled, the sparse and target decisions had similar loss distributions for the entire set of complexity parameters $\lambda_t$ (this set is discrete as shown in Figure (\ref{deltaexample})).  Thus, the resulting probability of regret jumped to larger values for the selected sparse decision.  This is an interesting feature of defining regret in terms of two losses (sparse and target), \textit{both} of which are random variables.
%%
%%
%%
%%\begin{figure}[H]
%%\renewcommand{\figurename}{Figure}
%%\centering
%%  \includegraphics[scale=.43]{lossplotJASAalpha02newmodel}
%%  \caption{Comparison of the regret distributions evolution over time versus the dense target portfolio (left vertical axis) and the probability of regret being less than zero (right vertical axis). The regret is measured as a difference in the loss between the portfolio decisions given by the $\kappa=20\%$ threshold and the target portfolio.  Smaller levels of regret are more favorable.  The black line displays the mean regret, and the shaded black area displays the centered 60\% posterior credible interval.  The shaded gray region represents the probabilistic constraint on the regret distribution given by $\pi_{\lambda_t} > \kappa$ and used for selection.}
%%\label{lossgraphk2}
%%\end{figure}
%
%
%\subsubsection{Alternative ex ante analyses}
%
%
%This analysis is not confined to only one specification of regret.  We can also define regret in terms of a difference in predictive portfolio Sharpe ratio:
%	\begin{equation} \label{regretdefnSR}
%	\begin{split}
%		\mathcal{SR}(w_t,\Theta_t) &= w_t^{T}\mu_t / (w_t^{T}\Sigma_t w_t)^{1/2},
%		\\
%		\rho_{\mathcal{SR}}(w_{\lambda_t}^{*},w_t^*(\Theta_t),\Theta_t) &= \mathcal{SR}(w_t^*(\Theta_t),\Theta_t)-\mathcal{SR}(w_{\lambda_t}^{*},\Theta_t),
%	\end{split}
%\end{equation}where $\rho_{\mathcal{SR}}(\cdot)$ is predictive in the sense that future returns $\tilde{R}_t$ conditional on the model parameters are integrated out. This portfolio metric differs from the Kelly criterion loss in that it focuses on a ratio of the portfolio mean and variance.  Thus, the distributions of the random variable $\mathcal{SR}(w_{\lambda_t}^{*},\cdot)$ and our original loss $\mathcal{L}(w_{\lambda_t}^{*},\cdot)$ are, in general, not the same.  Regret (\ref{regretdefnSR}) may be used directly in the selection step to produce a sparse portfolio decision like the one defined in Table (\ref{DLMweightsk4}) by thresholding quantiles of $\rho_{\mathcal{SR}}(\cdot)$ instead of $\rho(\cdot)$.  It may also be used as an exploratory tool to accompany selection from any other regret criteria (such as the difference in loss).  We utilize this ``difference in Sharpe ratio" regret distribution in an exploratory fashion in Figure (\ref{regretgraphSR1}).
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.43]{deltaplotJASADATEST055}
%  \caption{Comparison of the ``difference in Sharpe ratio" regret distribution's evolution over time versus two target portfolios: the dense portfolio and the fund SPY. SPY is an ETF that tracks an index comprised of the 500 largest companies by market capitalization. The regret is measured as a difference in predictive Sharpe ratio between the portfolio decisions given in Table (\ref{DLMweightsk4}) and the target portfolios.  Smaller levels of regret are more favorable.  The lines display the mean regret, and the regions display the centered 60\% posterior credible intervals.}
%\label{regretgraphSR1}
%\end{figure}
%
%Specifically, we consider the $\kappa=55\%$ portfolio decision displayed Table (\ref{DLMweightsk4}) versus two targets: The dense portfolio (the target defining the original regret, shown in black) and the fund SPY (gray).  SPY is an ETF that tracks an index comprised of the 500 largest US companies by market capitalization.  The lines show the mean annualized Sharpe ratio regret over time, and the regions show centered 60\% posterior credible intervals of the regret. 
%
%Similar to the difference in loss regret, the Sharpe ratio regret versus the dense portfolio (black region in Figure (\ref{regretgraphSR1})) hovers slightly above zero. It is intuitive that this regret is positive since the sparse portfolio will have a smaller Sharpe ratio than the dense (unpenalized Kelly optimal) -- there are fewer asset positions to provide diversification and a higher return-to-risk ratio.  The important feature is that the sparse portfolio must remain \textit{close}, in terms of regret, to the dense portfolio (as measured by posterior probability). 
%%Note that there are periods during bull markets (after the tech crash in the early 2000s and during the financial crisis) in which the variance of the Sharpe ratio regret increases. This behavior is similar to the difference in loss regret -- during ``up markets", the diversified dense portfolio more clearly separates from the sparse portfolio decisions; especially in terms of Sharpe ratio.  Thus, Sharpe ratio regret appears to have a larger mean and variance compared to ``down markets" when all decisions behave equally poorly.
%
%The gray region in Figure (\ref{regretgraphSR1}) displays the regret of the sparse portfolio decision versus investing in SPY.  This is calculated by replacing the Sharpe ratio for the target portfolio in Equations (\ref{regretdefnSR}) with the Sharpe ratio of SPY alone which also has a distribution induced by the posterior.  This regret is almost always negative, which means that it is ex ante \textit{beneficial} to invest in the sparse portfolio instead of just SPY.  Note the positive jumps in the regret versus SPY around 2002 and 2009.  These occurred when the market, and thus SPY, rallied before and after the tech bubble and after the financial crises, respectively.  Although the jumps were drastic, none were able to push a significant amount of the probability mass of regret into positive territory, indicating that the sparse portfolio decision is preferred to investing in only SPY (up to statistical uncertainty).
%
%
%
%%
%%\begin{figure}[H]
%%\renewcommand{\figurename}{Figure}
%%\centering
%%  \includegraphics[scale=.43]{deltaplotJASAalpha02SPYinFALSE}
%%  \caption{Comparison of the regret distributions evolution over time versus two target portfolios: The dense portfolio and the fund SPY. SPY is an ETF that tracks an index comprised of the 500 largest companies by market capitalization. The regret is measured as a difference in predictive Sharpe ratio between the portfolio decisions given in Table (\ref{DLMweightsk2}) and the target portfolios.  Smaller levels of regret are more favorable.  The lines display the mean regret, and the regions display the centered 60\% posterior credible interval.}
%%\label{regretgraph}
%%\end{figure}
%%
%%
%%\begin{figure}[H]
%%\renewcommand{\figurename}{Figure}
%%\centering
%%  \includegraphics[scale=.43]{probplotJASAalpha02SPYinFALSE}
%%  \caption{Comparison of the ``probability of \textbf{no} regret" evolution over time versus two target portfolios: The dense portfolio and the fund SPY.  Recall that for the sparse portfolio shown in Table (\ref{DLMweightsk2}) and represented by weights $w_{\lambda_{t}}^*$, we require $\pi_{\lambda_t} = \mathbb{P}[\mathcal{R}(w_{\lambda_t}^{*},w_t^*,\Theta_t,\tilde{R}_t) < 0] > 20\%$ where $w_t^*$ are the weights for the dense portfolio. Hence, the black line sits right against the edge of this probabilistic constraint.   }
%%\label{probregretgraph}
%%\end{figure}
%%
%%In Figure (\ref{probregretgraph}), we display the probability that the sparse portfolio decision is no worse than a target portfolio (also referred to as the probability of no regret) over time.  Similar to Figure (\ref{regretgraph}), these probabilities are displayed for the dense portfolio (black) and SPY (gray) as the targets.  This probability is given in Equation (\ref{probdefn}), and we select the sparse portfolio as the simplest portfolio whose $\pi_{\lambda_t}$ is bounded below by a regret tolerance threshold.  Since this threshold requires $\pi_{\lambda_t} > 20\%$, we see the black line as hovering right above the 0.2 probability region.  Note that the probabilistic constraint doesn't strictly bind (equal 0.2 exactly) at each time point because the collection of penalty parameters (sparse portfolio decisions) to select from at each point are discrete.  The gray line shows the probability that the sparse portfolio decision is the same or better than investing in SPY.  As expected, this probability is high and for certain periods is close to one. Also, the probability notably drops during periods when the market performs well (1990, 2000, and 2009) reflecting the regret distributions incorporation of target portfolio performance into its dynamics.
%
%As a final note, Figures (\ref{lossgraphk45}) and (\ref{regretgraphSR1}) are just two examples of exploratory plots readily available from this paper's statistical methodology.  There are benefits to constructing point summaries of the posterior to find simple portfolios while \textit{simultaneously} taking ex ante statistical uncertainty into account.  We know of no such portfolio selection methodology that is first-principled, dynamic, and admits these informative plots.
%
%
%\subsection{Ex post portfolio analysis}
%
%
%In this section, we consider the realized performance of the sparse portfolio decision presented in Table (\ref{DLMweightsk4}). In contrast to the \textit{ex ante} analysis presented in the previous section, the analysis to follow is \textit{ex post}. Figure (\ref{OOSgraph1}) shows the growth of \$1 invested in the sparse portfolio (also know as the cumulative return).  This is shown by the black line and labeled as \textit{sparse}.  Also shown are two other strategies: (\textit{i}) dense (gray) and (\textit{ii}) SPY (dotted gray). 
%
%The dense portfolio is the $\lambda_t=0$ solution for the integrated loss function: $w_{t\text{ dense}}^*= \arg\min_{w_t \geq 0} \mathring{\textbf{L}}_{0}(w_t)$ where the objective is defined in Equations (\ref{integratedloss}).  It is the strategy that optimizes over all 25 ETFs at each point in time using the posterior means of the first and second moments of returns.  Notably, the sparse portfolio outperforms the dense portfolio over this sample period.  Although the sparse portfolio decision is constructed to be \textit{ex ante} equivalent, or ``close", to the dense portfolio in terms of loss, it appears to exceed this expectation \textit{ex post}.  It also outperforms the sparse alternative of just investing in the market fund, SPY.
%
%
%
%Figure (\ref{OOSgraph2}) compares the sparse portfolio strategies to another reasonable alternative: a portfolio recommended by ETF investment advisor Wealthfront.  Its positions are: 50\% SPY (market), 15\% EEM (emerging markets), 30\% EZU (eurozone), and 5\% IYW (tech).  The portfolio is rebalanced to these strategic weights at each point in time.  The Wealthfront portfolio underperforms the sparse portfolio whose level of sparsity is allowed to adjust over time based on statistical uncertainty from our dynamic learning procedure.   The underperformance of this portfolio relative to the sparse portfolio decision highlights the benefit of selecting  \textit{different numbers} of ETFs at each time to satisfy the regret threshold. There appear to be out-of-sample gains from permitting flexibility in both the number and type of ETFs for investment.
%
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.43]{cumretJASAfig1NEWalpha055}
%  \caption{Comparison of the sparse portfolio decision's out-of-sample performance to the dense (or target) portfolio and SPY.}
%\label{OOSgraph1}
%\end{figure}
%
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.43]{cumretJASAfig2NEWalpha055}
%  \caption{Comparison of the sparse portfolio decision's out-of-sample performance to a model portfolio from the investment manager Wealthfront.com.}
%  \label{OOSgraph2}
%\end{figure}
%
%
%
%\begin{table}[H]
%\begin{center}
%\footnotesize
%
%\begin{tabular}{l*2{>{\centering\arraybackslash}m{0.7in}} @{}m{0pt}@{}}
% & \multicolumn{2}{c}{out-of-sample statistics} \\[1mm]
% \cline{2-3}
% &  Sharpe ratio & s.d. &\\[1ex] 
%  \hline
%\textbf{sparse} & 0.58 & 16.68 \\ 
%%\textbf{sparse min-var} & 0.51 & 17.82 \\ 
%%  \hline
%  \textbf{dense} & 0.45 & 14.41 \\ 
%%  \textbf{full min-var} & 0.44 & 14.46 \\ 
%   \hline
%%  \textbf{pick 5} & 0.54 & 15.00 \\ 
%  \textbf{Wealthfront} & 0.38 & 17.31  \\ 
%  \textbf{SPY} & 0.42 & 14.61 \\ 
%  \hline
%\end{tabular}
%
%\end{center}
%\caption{Comparison of out-of-sample portfolio statistics for the four portfolio strategies considered over the investing period February 2002 to May 2016.  The sparse ETF portfolio notably outperforms the dense (or full) portfolio as well as the two practical alternatives: (i) A Wealthfront.com static portfolio and (ii) The market ETF, SPY.}
%\label{OOStable}
%\end{table}
%
%
%Table (\ref{OOStable}) compares the out-of-sample Sharpe ratio and standard deviation of returns for all four portfolio decisions considered.  The sparse portfolio shown at the top should be contrasted with the dense portfolio (optimizing over the entire set of available ETFs) and other practical strategies shown in the last two rows.  The realized Sharpe ratios for the sparse portfolio outperforms each of the other three.  Also, the standard deviation for the sparse strategies is slightly higher than the others since these portfolios generally have fewer positions.  However, this added out-of-sample risk is made up for by increased return.
%
%
%
%%\subsection{Extension: Long-short decisions}
%%
%%\begin{figure}[H]
%%\renewcommand{\figurename}{Figure}
%%\centering
%%  \includegraphics[scale=.43]{wbadplotJASA-unconnew}
%%  \caption{Comparison of two portfolio strategies: Dense (gray) and sparse (black) \textbf{when short selling is allowed}.  The plot is zoomed into an unstable investing period during the financial crisis: October 2008 to September 2009. The regret threshold used to construct the sparse strategy is $\kappa = 20\%$.  The left vertical axis corresponds to the dense portfolio weights, and the right vertical axis corresponds to the sparse portfolio weights.  Note the difference in scales between the two vertical axes.}
%%  \label{unconweights}
%%\end{figure}
%%
%%
%%\begin{table}[H]
%%\begin{center}
%%\footnotesize
%%
%%\begin{tabular}{l*3{>{\centering\arraybackslash}m{0.7in}} @{}m{0pt}@{}}
%% & \multicolumn{3}{c}{out-of-sample statistics} \\[1mm]
%% \cline{2-4}
%% & cumulative return &  Sharpe ratio & s.d. &\\[1ex] 
%%  \hline
%%\textbf{sparse} & 2.40 & 0.43 & 18.66 \\ 
%%%\textbf{sparse min-var} & 0.51 & 17.82 \\ 
%%%  \hline
%%  \textbf{dense} & 23.59 & 0.38 & 116.34 \\ 
%%  \hline
%%\end{tabular}
%%
%%\end{center}
%%\caption{Comparison of out-of-sample portfolio statistics for the two portfolio strategies: Dense (gray) and sparse (black) \textbf{when short selling is allowed}.  The investing period is February 2002 to May 2016, and the regret threshold used to construct the sparse strategy is $\kappa = 20\%$.  }
%%\label{OOStable2}
%%\end{table}
%
%
%%
%%The model for the market portfolio return $R_{t}^{m}$ is:
%%
%%\begin{equation}
%%	\begin{split}
%%		R_{t}^{m} &= \mu_{t} + v_{t}, \hspace{14.5mm} v_{t} \sim N(0,\sigma^{2}),
%%		\\
%%		\mu_{t} &= \mu_{t-1} + w_{t}^{m}, \hspace{6mm} w_{t}^{m} \sim \text{T}_{n_{t-1}^{m}}(0,W_{t}^{m}),
%%		\\
%%		\vspace{0mm}
%%		\\
%%		W_{t}^{m} &= C_{t-1}^{m}\left( \frac{1-\delta_{\mu}}{\delta_{\mu}} \right).
%%	\end{split}
%%\end{equation}Defining the posterior at time $t-1$ as $\mu_{t-1} \mid D_{t-1} \sim \text{T}_{n_{t-1}^{m}}(m_{t-1}^{m},C_{t-1}^{m})$ and $\sigma^{-2} \mid D_{t-1} \sim \text{Ga}(n_{t-1}^{m}/2,n_{t-1}^{m}S_{t-1}/2)$, the model is completed by specifying the following priors at time t:
%%
%%\begin{equation}
%%	\begin{split}
%%		\mu_{t} \mid D_{t-1} &\sim \text{T}_{n_{t-1}^{m}}(m_{t-1}^{m},R_{t}^{m}), \hspace{6mm} R_{t}^{m} = C_{t-1}^{m}/\delta_{\mu}
%%	\end{split}
%%\end{equation}
%
%
%%\subsection{Future directions}
%%
%%Our analysis up to this point has assumed stationarity of returns and variances.  Although this stationarity assumption is a pillar of asset pricing theory, it is an empirical challenging to demonstrate.  As a natural next step, we investigate whether the selected portfolios change in different time periods.  We estimate the mean and covariance in overlapping 10 year windows from March 1995 to February 2015 and select a long/short portfolio using the Sharpe ratio heuristic in each.  The latent factor model is chosen for estimation. 
%%
%%Figures \ref{rollgraphs1}, \ref{rollgraphs2}, and \ref{rollgraphs3} show the three rolling window solution paths.  Note that the range of the Sharpe ratios are different for each, indicating the return to risk ratio \textit{does} depend on the time period.  First, we focus on the March 1995 through February 2005 portfolio shown in table \ref{ETFtabroll1}.  This optimal portfolio short sells emerging market equities, Japan, and small growth stocks.  The long positions include value, small, and mid-cap stocks, South Korea, and real estate and technology sector equities.  The largest long position is in small-cap stocks through IJR with the added exposure to large value stocks through IWD. This is similar to the ``small" and ``value" financial anomalies explored in Eugene Fama and Kenneth French's ``three factor model" paper \citep{FF3}.  In short, Fama and French argue that small and value stocks systematically outperform their large and growth stock counterparts, respectively.  Adding evidence to the Fama and French theory, growth stocks are indeed a short position in our selected portfolio through IWO. Real estate through IYR is a large long position and adds diversification to increase the Sharpe ratio.  The technology sector only has a modest allocation through IYW despite the explosive growth of tech companies during this time period.  These companies could increase their stock price by simply adding a ``.com" to the end of the company name\footnote{The tech market reached its peaked shortly after this time period on March 10, 2000.}.  This emphasizes the ability of our methodology to robustly choose systematic risk factors (such as small and value stocks) over idiosyncratic outperformance (such as the tech market) that may not last.     
%%
%%The portfolio selected for March 2000 through February 2010 is shown in table \ref{ETFtabroll2}.  This time period experienced a great deal of stock market volatility.  The tech market crashed in 2000, and the Great Recession began in 2008.  As a result, the selected portfolio has relatively large short and long positions.  Japanese, growth, and German equities are all short positions with the largest short position in the technology sector. Real estate, value, South Korean, and Brazilian equities comprise the long positions.  The Brazilian stock market performed well during this period which results in the largest long position.  Emerging market equities were coveted during this time and the acronym BRICs (Brazil, Russia, India, and China) was coined by Goldman Sachs in 2001 to identify these rapidly growing economies.
%%
%%\begin{figure}[H] 
%%\centering
%%\subfigure[March 1995 - February 2005]{\includegraphics[width=3.75in]{199503to200502ETFlongLossGraph}\label{rollgraphs1}}  
%%\subfigure[March 2000 - February 2010]{\includegraphics[width=3.75in]{200003to201002ETFlongLossGraph}\label{rollgraphs2}}
%%\subfigure[March 2005 - February 2015]{\includegraphics[width=3.75in]{200503to201502ETFlongLossGraph}\label{rollgraphs3}} 
%%\caption{Sharpe ratios of optimal portfolios formed on ETFs (short selling allowed).  March 1995 to February 2015. Gray band is the 99\% posterior uncertainty region. Parameters are estimated via the latent factor model.}
%%\end{figure}
%%
%%
%%% unconstrained tables
%%
%%\begin{table}[H]
%%\begin{center}
%%\footnotesize
%%\begin{tabular}{|m{.95cm}|c|m{.9cm}|m{.9cm}|m{.9cm}|m{.9cm}|m{.9cm}|m{.9cm}|m{.9cm}|c|c|}
%%\hline
%%\textbf{ETF}  &    EEM  &    IWD  &    IJR  &    EWJ   &    DIA   &     IWR   &     IWO   &     IYR   &     EWY   &     IYW  \\ \hline
%%\textbf{weight} &    -5.1 &    10.8 &    54.4 &    -32.4 &    8.4   &     44.7  &     -36.6 &     45.8  &     9     &     1 \\ \hline
%%\textbf{style} & \textit{EM} & \textit{large value} & \textit{small-cap} & \textit{Japan} & \textit{large value} &\textit{mid-cap} & \textit{small growth} & \textit{real estate} & \textit{SK} & \textit{tech}  \\ \hline
%%\end{tabular}
%%\end{center}
%%\caption{March 1995 - February 2005. Short selling allowed. ``EM" stands for emerging markets, and ``SK" stands for South Korea.}
%%\label{ETFtabroll1}
%%\end{table}
%%
%%
%%
%%
%%\begin{table}[H]
%%\begin{center}
%%\footnotesize
%%\begin{tabular}{|m{.95cm}|c|c|c|c|c|c|c|c|c|c|c|}
%%\hline
%%\textbf{ETF} &    EWJ    &    XLK    &    IWO  &    EWG  &    IYR   &     EWZ   &     EWY   &     IJJ  \\ \hline
%%\textbf{weight} &    -337.6 &    -403.6 &    -243 &    -46  &    201.9 &     502.6 &     261.4 &     164.4 \\ \hline
%%\textbf{style} & \textit{Japan} & \textit{tech} & \textit{small growth} & \textit{Germany} & \textit{real estate} & \textit{Brazil} & \textit{South Korea} & \textit{mid-value}  \\ \hline
%%\end{tabular}
%%\end{center}
%%\caption{March 2000 - February 2010. Short selling allowed.}
%%\label{ETFtabroll2}
%%\end{table}
%%
%%
%%\begin{table}[H]
%%\begin{center}
%%\footnotesize
%%\begin{tabular}{|m{.95cm}|c|c|c|c|c|c|c|c|c|c|c|}
%%\hline
%%\textbf{ETF} &    QQQ  &    XLF   &    XLP  &    IBB   \\ \hline
%%\textbf{weight} &    5.3  &    -24.5 &    63.2 &    56 \\ \hline
%%\textbf{style} & \textit{large growth} & \textit{financials} & \textit{con-staples} & \textit{biotech}  \\ \hline
%%\end{tabular}
%%\end{center}
%%\caption{March 2005 - February 2015. Short selling allowed.}
%%\label{ETFtabroll3}
%%\end{table}
%%
%%
%%
%%The portfolio selected for March 2005 through February 2015 is shown in table \ref{ETFtabroll3}.  This time period includes the Great Recession of 2008 and its recovery.  Financial institutions suffered the most, and this is reflected in the shorting of financials equities through XLF.  The representative broad market ETF is QQQ and added diversification comes from the consumer staples (XLP) and biotechnology (IBB) sectors. 
%%
%%
%%
%%These results indicate that a generic model of the mean and variance may be insufficient.  It is clear that over these shorter rolling time periods, the selection algorithm prefers well performing ETFs over poor performers.  In this case, statistical estimation of the mean in crucial.    Although beyond the scope of this paper, one may prefer a more carefully designed mean model that takes into account meaningful economic factors, such as those proposed in \cite{FF3} or \cite{FF5}.  Alternatively, one could carefully control the mean estimation by adjusting priors.  These are avenues for future work that apply the proposed methodology to specific financial and economic models beyond the scope of this paper. The time-varying selected portfolios derived here can be though of as a decision-theoretic dimension reduction tool for exploratory data analysis.

\section{Discussion}

This paper presents a new approach to portfolio selection based on an investor-specified regret tolerance.  A loss function defined by the expected portfolio growth rate is used in tandem with a new procedure that separates statistical inference from selection of a sparse dynamic portfolio.  We illustrate the procedure using a set of exchange-traded funds.  After analyzing two target decisions: (\textit{i}) A dense portfolio of all available assets, and (\textit{ii}) A portfolio comprised of a single market fund, we find that selected sparse decisions differ little from their targets in terms of utility; especially after taking into account uncertainty. This finding persists ex post, and a variety of sparse decisions perform similarly to their target decisions on a risk adjusted return (or Sharpe ratio) basis.

The procedure offers a fresh approach to portfolio selection.  While traditional approaches typically focus on either the careful modeling of parameters or the optimization procedure used to calculate portfolio weights, regret-based selection combines both through analysis of the regret random variable.  Portfolio decisions that are \textit{parsimonious} in nature are then evaluated in a framework that incorporates \textit{uncertainty} in the investor's utility. 

Areas of future research include alternative utility specifications.  Two relevant examples are: (\textit{i}) incorporation of fees and (\textit{ii}) minimization of transaction costs.  In each case, a variant of Loss function (\ref{lossorig}) may be considered.  Fees of the funds can be incorporated directly into the vector of future returns.  For example, suppose a vector of expense ratios (percentage fee charged of total assets managed) of all funds were given by $\tau$.  The vector of future returns within the loss function may be adjusted by $\tau$ to reflect an investor's sensitivity to fees:
\begin{equation}
	\begin{split} \label{optprobconclusion}
		\mathcal{L}(w_t,\tilde{R}_t) &= - \log \left( 1 + \sum_{k=1}^{N} w_t^k(\tilde{R}_t^k - \tau_{k}) \right) ,
%		\\
%		\Phi(\lambda_t,w_t) &=  \lambda_t \norm{ w_t }_{1}.
	\end{split}
\end{equation}where $\tilde{R}_t^k - \tau_{k}$ is the net return on investment in for fund $k$. 
%The expense ratio can also be incorporated at various time frequencies depending the units of $\overline{\mu}$.  For example, if $\overline{\mu}$ represents an expected monthly fund return, then $\tau$ (typically quoted annually), can be expressed on a monthly scale, too.


Sensitivity to transaction costs can be similarly accounted for by modifying the complexity (penalty) function $\Phi$.  This can be accomplished by penalizing the difference in consecutive weight vectors through time, $w_{t} - w_{t-1}$.  An example penalty function would look like:
\begin{equation}
	\begin{split} \label{optprobconclusion2}
	\Phi(\lambda_t^1,\lambda_t^2,w_t) &=  \lambda_t^1 \norm{ w_t }_{1} + \lambda_t^2 \norm{ w_t - w_{t-1} }_{1}.	
\end{split}
\end{equation}This penalty is designed to encourage sparsity as well as slow movement in portfolio positions over time so as to avoid frequent buying and selling of assets.  It poses an interesting challenge since there are two penalty parameters ($\lambda_t^1 \text{ and } \lambda_t^2$) that must be chosen.  This is precisely where the regret-based framework has merit. Portfolio decisions indexed by these two penalties can be mapped to a digestible single probability of regret.  Then, selection of an appropriate $\{\lambda_t^1,\lambda_t^2\}$ pair can be done in this intuitive ``regret space".


The remainder of this section takes a step back and discusses the modularity and important features of regret-based portfolio selection. The methodology is intended to be general -- the particular loss, model and dataset used for the empirical analysis are only chosen to demonstrate how the procedure works in practice.  The primitive components are: (\textit{i}) a utility function characterizing investor preferences, (\textit{ii}) a complexity function measuring how ``simple" a portfolio decision is, (\textit{iii}) a statistical model, and (\textit{iv}) the investor's regret tolerance; where regret is defined as a difference in utility.  The regret tolerance stitches together the first two primitives by answering the question: How does the investor view the \textit{tradeoff} between her utility and portfolio complexity?  Using the utility and posterior distribution defined by the statistical model (primitive three), one can construct a mapping between a set of penalty parameters $\{ \lambda_t \}$ and probabilities of regret (as displayed by the right vertical axis in Figure (\ref{deltaexample})).  However, this is not enough.  The collection of portfolio decisions indexed by $\lambda_t$ must be distilled down to one.  The fourth primitive accomplishes this by placing an upper bound on the probability of regret; a portfolio that satisfies this upper bound is selected.  By incorporating the four primitives, the main (and surprising) feature of this methodology is that a \textit{static} regret threshold produces a sequence of \textit{dynamic} portfolio decisions, one for each investing period.

%\subsection{Alternative utility specifications}
%
%We end this paper by briefly discussing extensions of the presented methodology.  The flexibility of the portfolio selection procedure to different utility specifications is an important characteristic of this paper. Two relevant examples were mentioned by the referees: (\textit{i}) incorporation of fees and (\textit{ii}) minimization of transaction costs.  In each case, a variant of loss function \ref{optprobfinal2} may be considered.  Fees of the funds can be incorporated directly into the first moment of returns.  For example, suppose a vector of expense ratios (percentage fee charged of total assets managed) of all funds were given by $\tau$.  The first moment of returns within the loss function may be adjusted by $\tau$ to reflect an investor's sensitivity to fees:  
%
%\begin{equation}
%	\begin{split} \label{optprobconclusion}
%		 \mathcal{L}(w) = \frac{1}{2} \norm{ L^{T}w-L^{-1}(\overline{\mu} -\tau) }_{2}^{2} + \lambda \norm{ w }_{1},
%	\end{split}
%\end{equation}where $\overline{\mu} -\tau$ is the net return on investment in each fund. The expense ratio can also be incorporated at various time frequencies depending the units of $\overline{\mu}$.  For example, if $\overline{\mu}$ represents an expected monthly fund return, then $\tau$ (typically quoted annually), can be expressed on a monthly scale, too. 
%
%Sensitivity to transaction costs can be similarly accounted for by modifying the dynamic version of the mean-variance loss function (the objective in \ref{optprobfinaltimevary}).  This can be accomplished by penalizing the difference in consecutive weight vectors through time, $w_{t+1} - w_{t}$.  An example loss function would look like:
%\begin{equation}
%	\begin{split} \label{optprobconclusion2}
%		 \mathcal{L}(w_{t+1}) = \frac{1}{2} \norm{ L_{t}^{T}w_{t+1}-L_{t}^{-1}\overline{\mu_{t}} }_{2}^{2} + \lambda \norm{ w_{t+1} - w_{t} }_{1}.
%	\end{split}
%\end{equation}This penalty is designed to encourage slow movement in portfolio positions (given by the weight vector) over time so as to avoid frequent buying and selling of assets.  In our empirical results, we focus on the original loss function in optimization \ref{optprobfinaltimevary} for two reasons.  First, ETFs are low-cost investments and $\tau$ is small relative to $\overline{\mu}$.  Second, our sparse dynamic portfolio weights (shown in table \ref{DLMweights}) under a variety of modeling choices vary gradually without a penalty on the weight difference.  
%
%
%
%
%


\appendix

%\section{Portfolio optimization review}\label{appregportopt}
%
%Regularization methods for portfolio optimization follow two schools of thought.  The first attacks estimation errors directly by attempting to regularize statistical estimates.  This is often done in a Bayesian context by shrinking the mean estimates to a ``grand mean" \citep{james1961estimation}.  Similar applications of shrinkage used for regularization include \cite{jorion1985international} and \cite{jorion1986bayes} who set the mean return of the minimum variance portfolio to be the grand mean.  In addition to the mean, others have focused on shrinking the covariance matrix in the portfolio selection problem, including \cite{ledoit2003improved}, \cite{ledoit2003honey}, and \cite{garlappi2007portfolio}.  In \cite{garlappi2007portfolio}, they consider parameter and model uncertainty to improve portfolio selection and out-of-sample Sharpe ratio. The second stream of literature focuses on regularizing the portfolio weights directly.  This literature is distinct in that regularization is delegated to the portfolio optimization step in contrast to inference of the optimization inputs.  This is a popular approach since mean-variance optimization can be formulated in terms of a Least Absolute Shrinkage and Selection Operator (LASSO) objective function from \cite{Tib} with fast computation of the optimal solution.  In this setting, the weights in the objective function are penalized with an $l_1$ norm.  Among the first to investigate weight regularization in the context of portfolio optimization is \cite{brodie2009sparse}.  They document improved out-of-sample performance as measured by the Sharpe ratio and consistently beat a naive strategy of investing an equal amount in each asset.  \cite{demiguel2009generalized} is a separate study showing similar results.  These more recent studies have clarified and expanded on earlier work by \cite{jagannathan2002risk} who showed that constraining the weights is equivalent to shrinking sample estimation of the covariance matrix.  \cite{fan2012vast} show that $l_1$ regularization limits gross portfolio exposure, leads to no accumulation of estimation errors, and results in outperformance compared to standard mean-variance portfolios.  Specifically, \cite{fan2012vast} and references therein discuss how to use a constraint on portfolio size to ``bridge the gap" between a no short sale optimization and unconstrained optimization. This paper represents useful theoretical treatment of the ideas presented in \cite{jagannathan2002risk} and their connections with the canonical mean-variance problem.   Recent work on regularization for portfolio selection includes \cite{yen2013note}, \cite{yen2014solving}, \cite{carrasco2011optimal}, \cite{fernandes2012regularized}, \cite{fastrich2013constructing} with applications to index tracking in \cite{fastrich2014cardinality}, \cite{takeda2013simultaneous}, \cite{Wu2}, and \cite{Wu}.


\section{Penalized optimization} \label{applossfunopt}
In this section, we show how to minimize a penalized loss function of the following form:
\begin{equation}
	\begin{split}
		\textbf{L}_{\lambda_t}(w_t) =  \frac{1}{2}w_t^{T}\overline{\Sigma}_t w_t - w_t^{T}\overline{\mu}_t + \lambda_t \norm{ w_t }_{1}.
	\end{split}
\end{equation}


 Completing the square in $w_t$, dropping terms that do not involve the portfolio weights, and defining $\overline{\Sigma}_t = L_tL_t^{T}$, we rewrite the first two terms of the loss function as an $l_{2}$-norm:
 \begin{equation}
	\begin{split}
	\textbf{L}_{\lambda_t}(w_t) &=\frac{1}{2}w_t^{T}\overline{\Sigma}_t w_t - w_t^{T}\overline{\mu}_t + \lambda_t \norm{ w_t }_{1}
	 \\
	 &\propto \frac{1}{2}(w_t-\overline{\Sigma}_t^{-1}\overline{\mu}_t)^{T}\overline{\Sigma}_t(w_t-\overline{\Sigma}_t^{-1}\overline{\mu}_t) + \lambda_t \norm{ w_t }_{1}
	 \\
	 &= \frac{1}{2}(w_t-L_t^{-T}L_t^{-1}\overline{\mu}_t)^{T}L_tL_t^{T}(w_t-L_t^{-T}L_t^{-1}\overline{\mu}_t) + \lambda_t \norm{ w_t }_{1}
	 \\
	 &= \frac{1}{2} \left[ L_t^{T}(w_t-L_t^{-T}L_t^{-1}\overline{\mu}_t) \right]^{T} \left[ L_t^{T}(w_t-L_t^{-T}L_t^{-1}\overline{\mu}_t) \right] + \lambda_t \norm{ w_t }_{1}
	 \\
	 &= \frac{1}{2} \norm{ L_t^{T}w_t-L_t^{-1}\overline{\mu}_t }_{2}^{2} + \lambda_t \norm{ w_t }_{1}.
	\end{split}
\end{equation} The optimization of interest is written as:
\begin{equation}
	\begin{split} \label{optprobfinal}
		 \min_{w_t \geq 0} \hspace{3mm} \frac{1}{2} \norm{ L_t^{T}w_t-L_t^{-1}\overline{\mu}_t }_{2}^{2} + \lambda_t \norm{ w_t }_{1}.
	\end{split}
\end{equation}
%After integrating the loss function over the predictive distribution, notice how portfolio optimality and simplicity appear in optimization \ref{optprobfinal}.  The $l_{2}$ norm includes the term, $L_t^{-1}\overline{\mu}_t$ and the weights are scaled by $L_t^{T}$, the right Cholesky factor of the posterior mean of the covariance.  Optimizing the first term is similar to minimizing the distance between the chosen and maximum Sharpe ratio portfolios.\footnote{In the unconstrained optimization, the maximum Sharpe ratio portfolio weights are given by $w_t^{*} \propto \overline{\Sigma}_t^{-1}\overline{\mu}_t$.}  The simplicity component appears in the $l_1$ norm on the weights.  As previously mentioned, this penalty zeros out asset weights for larger value of the tuning parameter, $\lambda_t$.  
Optimization (\ref{optprobfinal}) is now in the form of standard sparse regression loss functions, \citep{Tib}, with covariates, $L_t^{T}$, data, $L_t^{-1}\overline{\mu}_t$, and regression coefficients, $w_t$.  We may optimize (\ref{optprobfinal}) conveniently using existing software, such as the \texttt{glmnet} package of \cite{friedman2010regularization}.

% Some motivate it statistically for overcoming estimation error and overfitting while others simply acknowledge that certain forms of regularization improve out-of-sample performance.

%
%\section{Additional sparse portfolio decisions} \label{appsparse}
%
%
%
%\begin{table}[H]
%\begin{center}
%\footnotesize
%
%\begin{tabular}{l|ccccccccccc}
% \textbf{Date} & IWD & IWB & IWM & IYR & RSP & XLI \\ 
%  \midrule
% 200202 & 49.7 & 50.3 & - & - & - & - \\ 
%  200302 & 50.5 & 49.5 & - & - & - & - \\ 
%  200402 & - & - & - & - & 100 & - \\ 
%  200502 & - & - & - & - & 100 & - \\ 
%  200602 & - & - & - & - & 100 & - \\ 
%  200702 & - & - & - & - & 100 & - \\ 
%  200802 & - & - & - & 28.6 & 71.4 & - \\ 
%  200902 & - & - & - & 100 & - & - \\ 
%  201002 & - & - & - & 100 & - & - \\ 
%  201102 & - & - & - & 100 & - & - \\ 
%  201202 & - & - & - & 9.57 & - & 90.4 \\ 
%  201302 & - & - & - & 13.4 & - & 86.6 \\ 
%  201402 & - & - & 100 & - & - & - \\ 
%  201502 & - & - & - & - & - & 100 \\ 
%  201602 & - & - & - & - & - & 100 \\  
%  \midrule
%\end{tabular}
%
%\end{center}
%\caption{Sparse ETF portfolio weights (in percent) for DLMs (\ref{DLM1}) and (\ref{DLM2}) with $\delta_{F}= \delta_{\epsilon}= 0.97$ and $\delta_{c}= \delta_{\beta}= 0.9925$.  Note that annual weights are shown for brevity although portfolios are updated monthly.  In this dynamic portfolio selection, the regret tolerance level $\kappa = 10\%$. Infrequent positions in XLI and DIA are also part of this portfolio decision but are not shown in the above summary table.}
%\label{DLMweightsk4}
%\end{table}
%
%
%
%\begin{figure}[H]
%\renewcommand{\figurename}{Figure}
%\centering
%  \includegraphics[scale=.43]{lossplotJASAalpha01newmodel}
%  \caption{Comparison of the regret distributions evolution over time versus the dense target portfolio (left vertical axis) and the probability of regret being less than zero (right vertical axis). The regret is measured as a difference in the loss between the portfolio decisions given by the $\kappa=10\%$ threshold and the target portfolio.  Smaller levels of regret are more favorable.  The black line displays the mean regret, and the shaded black area displays the centered 60\% posterior credible interval.  The shaded gray region represents the probabilistic constraint on the regret distribution given by $\pi_{\lambda_t} > \kappa$ and used for selection.}
%\label{lossgraphk45}
%\end{figure}
%



\newpage
%\nocite{*}
\bibliographystyle{imsart-nameyear.bst}
\bibliography{DSSPortfolioOptimization.bib}


\end{document}



DSSOpt-BA.bbl
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