Luca Ghezzi

This study focuses on efficient asset allocations that properly include T-bills, T-bonds, and the S&P 500 stock index. It checks that their annual real rates of linear return are both normal and almost lognormal. It reexamines how efficient portfolios based on the rates of linear return may turn into efficient portfolios based on the rates of logarithmic return. It finds that each efficient asset allocation has the lowest possible standard deviation as well as the highest possible arithmetic and geometric means. It eventually reconsiders the relationship between the confidence interval of a geometric mean and an expected long-run capital accumulation. As a consequence, it bridges a gap in the scientific literature by enabling financial advisors to trade off the mean rate of return on a portfolio more rigorously against the value at risk.

This study makes use of a very long time series of the S&P Composite Index, checking once more that the rates of return benefit from aggregational normality. It performs unit root tests as well as elementary statistical tests that take advantage of normality. It finds that mean blur is not consistent with the hypothesis of random walk with constant parameters, because the means of the annual real rates of linear return can be estimated as usual. It gives further evidence that the rates of return on the S&P Composite Index are mean‐reverting.

This statistical study refines and updates Sharpe’s empirical paper (1975, Financial Analysts Journal) on switching between US common stocks and cash equivalents. According to the original conclusion, profitable market timing relies on a representative portfolio manager who can correctly forecast the next year at least 7 times out of 10. Four changes are made to the original setting. The new data set begins and ends with similar price-earnings ratios; a more accurate approximation of commissions is given; the rationality of assumptions is examined; a prospective and basic Monte Carlo analysis is carried out so as to consider the heterogeneous performance of a number of portfolio managers with the same forecasting accuracy. Although the first three changes improve retrospectively the odds of profitable market timing, the original conclusion is corroborated once more.

The stage of strategic asset allocation is the most important one in a process of portfolio management: asset classes are selected and target weights are set. Careful decision-making benefits from the computation of an efficient frontier. In this work, weights are nonnegative and rebalanced once a year; portfolio returns are time uncorrelated and lognormal. A novel sufficient condition is obtained, whereby efficient portfolios based on linear returns may turn into efficient portfolios based on logarithmic returns. If that is met, the efficient frontier based on logarithmic returns is upward sloping, stretching from a corner portfolio with global minimum-variance to a corner portfolio with global maximum-variance. Such a complementary efficient frontier allows a decision maker to forecast the long-term portfolio value. The null hypothesis of lognormal portfolio returns is also tested by using two different data sets. It is always rejected in the latter; it is either accepted or rejected in the former, depending on the specific efficient portfolio.

An insightful problem of passive management is considered, where an aggregate portfolio is rebalanced annually to restore the percent weights of its strategic asset allocation. As its annual total returns are assumed to be time uncorrelated and lognormally distributed, multi-period optimization boils down to single period optimization. Expanding on previous theoretical results, it is shown how a minimum-variance set based on linear returns turns into a minimum-variance set based on logarithmic returns. More precisely, it is found that there can be two different qualitative patterns, one of which is unprecedented and striking. Both patterns are tentatively portrayed by using historical data. The resulting efficient frontier is readily complemented by a dynamic shortfall constraint. Each threshold return can be turned into a threshold accumulation that has the same shortfall probability; coeteris paribus, the more distant the time horizon, the smaller the shortfall probability. As our procedure is analytically tractable, it might be operationally useful, especially to financial advisors and individual investors.

An immunization problem is considered in which a bond portfolio is to be periodically rebalanced. Max–min optimal control is applied to the problem. The target is to maximize the final portfolio value under the worst possible evolution of interest rates. The optimal control law, obtained by means of dynamic programming, turns out to be different from any duration-based immunization policy.

This paper is devoted to the control of a nonlinear sampled system that can exhibit chaotic behaviour. The system is derived from a classical epidemiological model in which the vaccination rate is the control variable. It is shown that chaos can be removed by using a constant and suitably large vaccination rate. Nonetheless, reducing rather than suppressing chaos seems to be a more appropriate goal owing to both general and case-specific reasons. PID control laws, for the first time applied to this purpose, prove effective as well as robust, since they make the control system fairly insensitive to parameter misspecification. Bifurcation analysis and simulation play a chief role in the work.

Optimal control is applied to a chaotic system. Reference is made to a well-known one-dimensional map. Firstly, attention is devoted to the stabilization of a fixed point. An optimal controller is obtained and compared with other controllers which are popular in the control of chaos. Secondly, allowance is made for uncertainty and emphasis is placed on the reduction rather than the suppression of chaos. The aim becomes that of confining a chaotic attractor within a prescribed region of the state space. A controller fulfilling this task is obtained as the solution of a min-max optimal control problem.