The objective of this paper is to study the mean-variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in [2] where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time inconsistent control developed in [4], we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.Key words: Mean-variance, time inconsistency, time inconsistent control, dynamic programming, stochastic control, Hamilton-Jacobi-Bellman equation * The authors are greatly indebted to Ivar Ekeland, Ali Lazrak, Traian Pirvu, and Suleyman Basak for very helpful discussions. We are also very grateful to two anonymous referees for a number of comments, which have improved the paper considerably.
In this paper, which is a continuation of the discrete-time paper (Björk and Murgoci in Finance Stoch. 18:545-592, 2004), we study a class of continuoustime stochastic control problems which, in various ways, are time-inconsistent in the sense that they do not admit a Bellman optimality principle. We study these problems within a game-theoretic framework, and we look for Nash subgame perfect equilibrium points. For a general controlled continuous-time Markov process and a fairly general objective functional, we derive an extension of the standard HamiltonJacobi-Bellman equation, in the form of a system of nonlinear equations, for the determination of the equilibrium strategy as well as the equilibrium value function. The main theoretical result is a verification theorem. As an application of the general theory, we study a time-inconsistent linear-quadratic regulator. We also present a study of time-inconsistency within the framework of a general equilibrium production economy of Cox-Ingersoll-Ross type (Cox et al. in Econometrica 53:363-384, 1985
We show that CDS premiums of sovereigns are significantly affected by the foreign exposures of their domestic banks. Our analysis uses a simple risk-weighted exposure measure which aggregates detailed data on the composition and risk of banks' foreign exposures. A 1 basis point change in our risk weighted exposure measure corresponds to an average change of approximately 0.4 bp in sovereign CDS spreads. Extensive robustness checks confirm that the explanatory power of our measure is not due to common factors in CDS premiums. We also measure the size and riskiness of the sovereign's implicit and explicit guarantees extended to its domestic banking system.
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