A Mean-Variance Approach to Capital Investment Optimization

We develop an improved model of capital investment under uncertainty that incorporates the variance of the capital stock in the payoff functional to manage risk. Our model results in a mean field type control problem that cannot be solved by classical stochastic control methods. We solve our problem using techniques presented in Bensoussan, Frehse, and Yam [Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013]. The explicit solution is a feedback depending on the initial condition. Moreover, our model can be reduced to Abel's [Amer. Econ. Rev., 73 (1983), pp. 228-233]. Numerical results suggest that the risk reduction optimally exceeds the cost incurred. Following Bjork, Khapko, and Murgoci [Finance Stoch., 21 (2017), pp. 331-360], we solve for a time-consistent solution, i.e., the best possible feedback independent of the initial condition. The time-consistent policy discards our risk specification, with the resultant loss of value to the firm.