When the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space, it is a biased estimator for what can be expected on unseen data (out-of-sample). We derive (1) an unbiased estimator adjusting for both sources of bias: noise fit and estimation error. We then show...

When optimizing the Sharpe ratio over a k-dimensional parameter space the thus obtained in-sample Sharpe ratio tends to be higher than what will be captured out-of-sample. For two reasons: the estimated parameter will be skewed towards the noise in the in-sample data (noise fitting) and, second,...

The investment boundaries defined by Grenadier (2002) for an oligopoly investment game determine equilibria in open loop strategies. As closed loop strategies, they are not equilibria, because any firm by investing sooner can preempt the investments of other firms and expropriate the growth...

This paper presents a model in which (1) fiat money has strictly positive value in the unique trembling hand equilibrium. This holds as each bank note is both: (a) a witness for the existence of some agent in the economy with debt, backed by collateral, and (b) the only matter that allows the...