Mean‐variance efficiency of the market portfolio and futures trading

We derived an intertemporal capital asset pricing model in which the mean‐variance efficiency of the market portfolio is neither a necessary nor a sufficient condition. We obtained this result by modeling a frictionless, continuously open financial market in which nonredundant futures contracts are available for trade, in addition to cash assets. Introducing such contracts modifies the way investors optimally allocate their wealth. Their portfolios then comprise the riskless asset, a perturbed mean‐variance‐efficient portfolio of cash assets, and a perturbed mean‐variance‐efficient portfolio of futures contracts. Furthermore, a (3 + K) mutual fund separation is obtained, with K being the number of economic state variables, in lieu of the usual (2 + K) fund separation. Mean‐variance efficiency of the market portfolio is a necessary condition only when cash assets are the sole traded assets. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:329–346, 2001     

Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation

The mean‐variance formulation by Markowitz in the 1950s paved a foundation for modern portfolio selection analysis in a single period. This paper considers an analytical optimal solution to the mean‐variance formulation in multiperiod portfolio selection. Specifically, analytical optimal portfolio policy and analytical expression of the mean‐variance efficient frontier are derived in this paper for the multiperiod mean‐variance formulation. An efficient algorithm is also proposed for finding an optimal portfolio policy to maximize a utility function of the expected value and the variance of the terminal wealth.     

Who should sell stocks?

Never selling stocks is optimal for investors with a long horizon and a realistic range of preference and market parameters, if relative risk aversion, investment opportunities, proportional transaction costs, and dividend yields are constant. Such investors should buy stocks when their portfolio weight is too low and otherwise hold them, letting dividends rebalance to cash over time rather than selling. With capital gains taxes, this policy outperforms both static buy‐and‐hold and dynamic rebalancing strategies that account for transaction costs. Selling stocks becomes optimal if either their target weight is low or intermediate consumption is substantial.     

Semimartingale theory of monotone mean–variance portfolio allocation

We study dynamic optimal portfolio allocation for monotone mean–variance preferences in a general semimartingale model. Armed with new results in this area, we revisit the work of Cui et al. and fully characterize the circumstances under which one can set aside a nonnegative cash flow while simultaneously improving the mean–variance efficiency of the left‐over wealth. The paper analyzes, for the first time, the monotone hull of the Sharpe ratio and highlights its relevance to the problem at hand.     

BUY‐LOW AND SELL‐HIGH INVESTMENT STRATEGIES

Buy‐low and sell‐high investment strategies are a recurrent theme in the considerations of many investors. In this paper, we consider an investor who aims at maximizing the expected discounted cash‐flow that can be generated by sequentially buying and selling one share of a given asset at fixed transaction costs. We model the underlying asset price by means of a general one‐dimensional Itô diffusion X , we solve the resulting stochastic control problem in a closed analytic form, and we completely characterize the optimal strategy. In particular, we show that, if 0 is a natural boundary point of X , e.g., if X is a geometric Brownian motion, then it is never optimal to sequentially buy and sell. On the other hand, we prove that, if 0 is an entrance point of X , e.g., if X is a mean‐reverting constant elasticity of variance (CEV) process, then it may be optimal to sequentially buy and sell, depending on the problem data.     

MEAN‐VARIANCE POLICY FOR DISCRETE‐TIME CONE‐CONSTRAINED MARKETS: TIME CONSISTENCY IN EFFICIENCY AND THE MINIMUM‐VARIANCE SIGNED SUPERMARTINGALE MEASURE

The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.     

OPTIMAL INVESTMENT STRATEGIES FOR CONTROLLING DRAWDOWNS

We analyze the optimal risky investment policy for an investor who, at each point in time, wants to lose no more than a fixed percentage of the maximum value his wealth has achieved up to that time. In particular, if M _t is the maximum level of wealth W attained on or before time t , then the constraint imposed on his portfolio choice is that W_tα M _t, where α is an exogenous number betweenα O and 1. We show that, for constant relative risk aversion utility functions, the optimal policy involves an investment in risky assets at time t in proportion to the "surplus" W _t - α M _t. the optimal policy may appear similar to the constant-proportion portfolio insurance policy analyzed in Black and Perold (1987) and Grossman and Vila (1989). However, in those papers, the investor keeps his wealth above a nonstochastic floor F instead of a stochastic floor α M _t. the stochastic character of the floor studied here has interesting effects on the investment policy in states of nature when wealth is at an all-time high; i.e., when Wt = M _t. It can be shown that at W _t= M _t, α M _t is expected to grow at a faster rate than W _t, and therefore the investment in the risky asset can be expected to fall. We also show that the investment in the risky asset can be expected to rise when W _t is close to α M _t. We conjecture that in an equilibrium model the stochastic character of the floor creates "resistance" levels as the market approaches an all-time high (because of the reluctance of investors to take more risk when W _t= M _t).     

DYNAMIC PORTFOLIO OPTIMIZATION WITH A DEFAULTABLE SECURITY AND REGIME‐SWITCHING

We consider a portfolio optimization problem in a defaultable market with finitely‐many economical regimes, where the investor can dynamically allocate her wealth among a defaultable bond, a stock, and a money market account. The market coefficients are assumed to depend on the market regime in place, which is modeled by a finite state continuous time Markov process. By separating the utility maximization problem into a predefault and postdefault component, we deduce two coupled Hamilton–Jacobi–Bellman equations for the post‐ and predefault optimal value functions, and show a novel verification theorem for their solutions. We obtain explicit constructions of value functions and investment strategies for investors with logarithmic and Constant Relative Risk Aversion utilities, and provide a precise characterization of the directionality of the bond investment strategies in terms of corporate returns, forward rates, and expected recovery at default. We illustrate the dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a detailed economic and numerical analysis.     

DISUTILITY, OPTIMAL RETIREMENT, AND PORTFOLIO SELECTION

We study the optimal retirement and consumption/investment choice of an infinitely-lived economic agent with a time-separable von Neumann–Morgenstern utility. A particular aspect of our problem is that the agent has a retirement option. Before retirement the agent receives labor income but suffers a utility loss from labor. By retiring, he avoids the utility loss but gives up labor income. We show that the agent retires optimally if his wealth exceeds a certain critical level. We also show that the agent consumes less and invests more in risky assets when he has an option to retire than he would in the absence of such an option.
An explicit solution can be provided by solving a free boundary value problem. In particular, the critical wealth level and the optimal consumption and portfolio policy are provided in explicit forms.     

The Asymmetric Commodity Inventory Effect on the Optimal Hedge Ratio

Hedging strategies for commodity prices largely rely on dynamic models to compute optimal hedge ratios. This study illustrates the importance of considering the commodity inventory effect (effect by which the commodity price volatility increases more after a positive shock than after a negative shock of the same magnitude) in modeling the variance–covariance dynamics. We show by in‐sample and out‐of‐sample forecasts that a commodity price index portfolio optimized by an asymmetric BEKK–GARCH model outperforms the symmetric BEKK, static (OLS), or naïve models. Robustness checks on a set of commodities and by an alternative mean‐variance optimization framework confirm the relevance of taking into account the inventory effect in commodity hedging strategies.     

STATIC FUND SEPARATION OF LONG‐TERM INVESTMENTS

This paper proves a class of static fund separation theorems, valid for investors with a long horizon and constant relative risk aversion, and with stochastic investment opportunities. An optimal portfolio decomposes as a constant mix of a few preference‐free funds, which are common to all investors. The weight in each fund is a constant that may depend on an investor's risk aversion, but not on the state variable, which changes over time. Vice versa, the composition of each fund may depend on the state, but not on the risk aversion, since a fund appears in the portfolios of different investors. We prove these results for two classes of models with a single state variable, and several assets with constant correlations with the state. In the linear class, the state is an Ornstein–Uhlenbeck process, risk premia are affine in the state, while volatilities and the interest rate are constant. In the square root class, the state follows a square root diffusion, expected returns and the interest rate are affine in the state, while volatilities are linear in the square root of the state.     

Continuous‐time mean–variance portfolio selection: A reinforcement learning framework

We approach the continuous‐time mean–variance portfolio selection with reinforcement learning (RL). The problem is to achieve the best trade‐off between exploration and exploitation, and is formulated as an entropy‐regularized, relaxed stochastic control problem. We prove that the optimal feedback policy for this problem must be Gaussian, with time‐decaying variance. We then prove a policy improvement theorem, based on which we devise an implementable RL algorithm. We find that our algorithm and its variant outperform both traditional and deep neural network based algorithms in our simulation and empirical studies.     

A Martingale Characterization of Consumption Choices and Hedging Costs with Margin Requirements

This paper examines optimal consumption and investment choices and the cost of hedging contingent claims in the presence of margin requirements or, more generally, of nonlinear wealth dynamics and constraints on the portfolio policies. Existence of optimal policies is established using martingale and duality techniques under general assumptions on the securities' price process and the investor's preferences. As an illustration, explicit solutions are provided for an agent with ‘logarithmic’ utility. A PDE characterization of the cost of hedging a nonnegative path‐independent European contingent claim is also provided.     

PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS

We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest,   r > 0  , and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark–Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical "delta hedging" approach.     

ROBUST ASSET ALLOCATION WITH BENCHMARKED OBJECTIVES

In this paper, we introduce a new approach for finding robust portfolios when there is model uncertainty. It differs from the usual worst‐case approach in that a (dynamic) portfolio is evaluated not only by its performance when there is an adversarial opponent (“nature”), but also by its performance relative to a stochastic benchmark. The benchmark corresponds to the wealth of a fictitious benchmark investor who invests optimally given knowledge of the model chosen by nature, so in this regard, our objective has the flavor of min–max regret. This relative performance approach has several important properties: (i) optimal portfolios seek to perform well over the entire range of models and not just the worst case, and hence are less pessimistic than those obtained from the usual worst‐case approach; (ii) the dynamic problem reduces to a convex static optimization problem under reasonable choices of the benchmark portfolio for important classes of models including ambiguous jump‐diffusions; and (iii) this static problem is dual to a Bayesian version of a single period asset allocation problem where the prior on the unknown parameters (for the dual problem) correspond to the Lagrange multipliers in this duality relationship. This dual static problem can be interpreted as a less pessimistic alternative to the single period worst‐case Markowitz problem. More generally, this duality suggests that learning and robustness are closely related when benchmarked objectives are used.     

THE NUMÉRAIRE PROPERTY AND LONG‐TERM GROWTH OPTIMALITY FOR DRAWDOWN‐CONSTRAINED INVESTMENTS

We consider the portfolio choice problem for a long‐run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude toward risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numéraire property through the notion of expected relative return and prove that drawdown‐constrained numéraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time‐horizon becomes distant, the drawdown‐constrained numéraire portfolio is given explicitly through a model‐independent transformation of the unconstrained numéraire portfolio. The asymptotically growth‐optimal strategy is obtained as limit of numéraire strategies on finite horizons.     

OPTIMAL INVESTMENT IN CREDIT DERIVATIVES PORTFOLIO UNDER CONTAGION RISK

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.     

EXPLICIT SOLUTIONS OF CONSUMPTION-INVESTMENT PROBLEMS IN FINANCIAL MARKETS WITH REGIME SWITCHING

We consider a consumption and investment problem where the market presents different regimes. An investor taking decisions continuously in time selects a consumption–investment policy to maximize his expected total discounted utility of consumption. The market coefficients and the investor's utility of consumption are dependent on the regime of the financial market, which is modeled by an observable finite-state continuous-time Markov chain. We obtain explicit optimal consumption and investment policies for specific HARA utility functions. We show that the optimal policy depends on the regime. We also make an economic analysis of the solutions, and show that for every investor the optimal proportion to allocate in the risky asset is greater in a "bull market" than in a "bear market." This behavior is not affected by the investor's risk preferences. On the other hand, the optimal consumption to wealth ratio depends not only on the regime, but also on the investor's risk tolerance: high risk-averse investors will consume relatively more in a "bull market" than in a "bear market," and the opposite is true for low risk-averse investors.     

基于确定性偏好的投资组合选择

文章在对马科维茨证券投资组合模型简要评述的基础上，针对投资者可选标的证券信息集非对称的现实，依据确定性偏好原理，将投资者对可选标的证券信息的确定性程度转换成偏好次序关系，同时结合行为金融学中的前景理论，依确定性偏好次序规则来确定权重函数，并在价值函数-风险的框架下探讨了证券投资组合模型的构建及其最优解，从而在行为金融理论下扩展了马氏证券投资组合模型。实证分析表明，我国证券市场投资者基本是采用线性赋权方式来处理非对称信息集下的投资组合选择的。     

POWER UTILITY MAXIMIZATION IN CONSTRAINED EXPONENTIAL LÉVY MODELS

We study power utility maximization for exponential Lévy models with portfolio constraints, where utility is obtained from consumption and/or terminal wealth. For convex constraints, an explicit solution in terms of the Lévy triplet is constructed under minimal assumptions by solving the Bellman equation. We use a novel transformation of the model to avoid technical conditions. The consequences for q‐optimal martingale measures are discussed as well as extensions to nonconvex constraints.