Multi-stage Optimization for Long-term Investors

Multi-stage simulation and optimization models are effective for solving long-term financial planning problems. Prominent examples include: asset-liability management for pension plans, integrated risk management for insurance companies, and long-term planning for individuals. Several applications will be briefly mentioned. A multi-stage framework provides advantages over single-period myopic approaches. First, the investor gains an understanding of the risks that a long-term goal will be unfulfilled, such as retiring with adequate wealth. A multi-stage model can be more realistic than a single period model. Thus, assets such as equity, which reduce long-term risks while increasing short-term volatility, can be evaluated in a temporal setting. The tradeoff between long and short-term gains becomes apparent in a multi-period context. As a second advantage, enhanced returns are possible with dynamic investment strategies. For instance, the traditional approach of rebalancing assets to a fixed strategic benchmark generates higher returns when assets possess increased volatility. This “volatility pumping” is dampened by transaction and market impact costs. Only by solving a multi-stage optimization model can we discover the optimal rebalancing rules. Likewise, moving a large portfolio to a new strategic benchmark can be optimized. As a third example, individuals often hold assets with large embedded gains. Selling these assets triggers a capital gains tax. Again, these decisions can be evaluated by means of a multi-stage model. A real-world example from pension planning illustrates the concepts. Three distinct approaches are available for solving the multi-stage optimization model: 1) dynamic stochastic control, 2) stochastic programming, and 3) optimizing a stochastic simulation model. We briefly review the pros and cons of these approaches; it seems unlikely that a single approach will dominate the others. We conclude with some topics for future research.