Dynamic portfolio choice without cash

We consider the dynamic mean-variance portfolio choice without cash under a game theoretic framework. The mean-variance criterion is investigated in the situation where an investor allocates the wealth among risky assets while keeping no cash in a bank account. The problem is solved explicitly up to solutions of ordinary differential equations by applying the extended Hamilton-Jacobi-Bellman equation system. Given a constant risk aversion coefficient, the optimal allocation without a risk-free asset depends linearly on the current wealth, while that with a risk-free asset turns out to be independent of the current wealth. We also study the minimum-variance criterion, which can be viewed as an extension of the mean-variance model when the risk aversion coefficient tends to infinity. Calibration exercises demonstrate that for large investments, the mean-variance model without cash yields the highest certainty equivalent return for both short-term and long-term investments. Furthermore, the mean-variance portfolio choices with and without cash yield almost the same Sharpe ratio for an investment with large initial wealth.