The optimal mean variance problem with inflation

The risk of ination is looming under the current low interest rate environment. Assuming that the investment includes a fixed interest asset and n risky assets under ination, we consider two scenarios: ination rate can be observed directly or through a noisy observation. Since the ination rate is random, all assets become risky. Under this circumstance, we formulate the portfolio selection problem and derive the efficient frontier by solving the associated HJB equation. We find that for a given expected portfolio return, investment at time t  is linearly proportional to the price index level. Moreover, the risk for the real value of the portfolio is no longer minimal when all the wealth is put into the fixed interest asset. Finally, for the mutual fund theorem, two funds are needed now instead of the traditional single fund. If an ination linked bond can be included in the portfolio, the problem is reduced to the traditional mean variance problem with a risk-free and n + 1 risky assets with real returns.