As a contribution to the discussion on the risks of stocks in the long run the present paper analyses the shortfall risks of stocks using the risk measures shortfall probability, mean excess loss and shortfall expectation for various deterministic as well as a stochastic benchmark. As a main...

Retirees must draw down their accumulated assets in an orderly fashion so as not to exhaust their funds too soon. We derive the optimal retirement portfolio from a menu that includes payout annuities as well as an investment allocation and a withdrawal strategy, assuming risk aversion,...