Optimal Strategies for Prudent Investors

I discuss some theoretical results with a view to motivate some practical choices in portfolio optimization. Even though the setting is not completely general (for example, the covariance matrix is assumed to be non-singular), I attempt to highlight the features that have practical relevance. The mathematical setting is Stochastic Portfolio Theory, which is flexible enough to describe most realistic assets, and it has been successfully employed for managing equity portfolios ...

We derive formulas for the performance of capital assets in continuous time from an efficient market hypothesis, with no stochastic assumptions and no assumptions about the beliefs or preferences of investors. Our efficient market hypothesis says that a speculator with limited means cannot beat a particular index by a substantial factor. Our results include a formula that resembles the classical CAPM formula for the expected simple return of a security or portfolio. This ve...

We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussi...

We formulate an infinite-horizon optimal investment and consumption problem, in which an individual forms a habit based on the exponentially weighted average of her past consumption rate, and in which she invests in a Black-Scholes market. The individual is constrained to consume at a rate higher than a certain proportion $\alpha$ of her consumption habit. Our habit-formation model allows for both addictive ($\alpha=1$) and nonaddictive ($0<\alpha<1$) habits. The optimal inve...

In simulations of some economic gas-like models, the asymptotic regime shows an exponential wealth distribution, independently of the initial wealth distribution given to the system. The appearance of this statistical equilibrium for this type of gas-like models is explained in a rigorous analytical way.

This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depend on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our...

We consider a stochastic game-theoretic model of a discrete-time asset market with short-lived assets and endogenous asset prices. We prove that the strategy which invests in the assets proportionally to their expected relative payoffs asymptotically minimizes the expected time needed to reach a large wealth level. The result is obtained under the assumption that the relative asset payoffs and the growth rate of the total payoff during each time period are independent and ide...

We study long-term growth-optimal strategies on a simple market with linear proportional transaction costs. We show that several problems of this sort can be solved in closed form, and explicit the non-analytic dependance of optimal strategies and expected frictional losses of the friction parameter. We present one derivation in terms of invariant measures of drift-diffusion processes (Fokker- Planck approach), and one derivation using the Hamilton-Jacobi-Bellman equation of ...

Two major financial market complexities are transaction costs and uncertain volatility, and we analyze their joint impact on the problem of portfolio optimization. When volatility is constant, the transaction costs optimal investment problem has a long history, especially in the use of asymptotic approximations when the cost is small. Under stochastic volatility, but with no transaction costs, the Merton problem under general utility functions can also be analyzed with asympt...

We consider an investor, whose portfolio consists of a single risky asset and a risk free asset, who wants to maximize his expected utility of the portfolio subject to managing the Value at Risk (VaR) assuming a heavy tailed distribution of the stock prices return. We use a stochastic maximum principle to formulate the dynamic optimisation problem. The equations which we obtain does not have any explicit analytical solution, so we look for accurate approximations to estimate ...