Optimal Consumption, Investment, Insurance and Annuitisation Over the Life Cycle

Significant Achievements, Anticipated Outcomes and Future Work

We have determined the optimal values of consumption, investment, life insurance and annuitisation over our individual's life cycle. We have also been able to determine the expected paths of these control variables as well as the expected path of the state variables (wealth and income). Thus we have achieved our stated objectives.

Our results show that expected optimal consumption in this model is hump shaped during the individual's working life, then rises following retirement. We explain this using the buffer stock and life cycle theories of consumption. We find investment is expected to be initially lower than in an environment with safe income; it then rises to exceed the safe case, then trends down to the safe level at retirement. Following retirement investment is at Merton ratio. Life insurance and annuitisation follow similar patterns to the case of safe income. Risky income does, however, lead to later and less optimal annuitisation - thus it contributes to the observed thinness of annuity markets.

These are novel and interesting results and the result of techniques up to now little used in economics. Future work would involve extensions to the model. Of particular interest is the introduction of jump processes to model periods of unemployment as well as major market events. Correlation between the income and risky asset return process is also of interest.

Computational Techniques Used

The Hamilton-Jacobi-Bellman equation that gives the solution to our problem is a parabolic partial differential equation. This is solved using a Markov chain approximation technique developed by Kushner & Dupuis (2001). Here the method was implemented in a recursive fashion on a two-dimensional grid of state variables, which is an explicit scheme. Such an approach not only gives the values of the optimal controls, but as transition probabilities are available from the Markov chain we can also determine the expected values of the optimal controls. Also, using an adaptive procedure developed by Fitzpatrick & Fleming (1991), we are able to vary the backwards recursive step depending on the smoothness of the value function. This reduces computation time.

The solution method was coded by the principal investigator in C and parallelised by the High Performance Support Unit at the University of New South Wales. For the parameterisation of the model adopted we found a grid of 1000 x 1000 points gave good results.

Publications, Awards and External Funding

External Funding and Awards

Future work, to be undertaken under Australia Research Council Discovery-Projects Grant DP0209429, will use the results and methods developed of this current work.

Publications

T.S. Purcal, Growing old gracefully: optimal financial behaviour over a stochastic life cycle, PhD Thesis, University of New South Wales, Sydney, Australia, 2003, xiv+186pp.
J. Piggott, S. Purcal, M. Williams, Retirement provision: accumulation, security and insurance, in T. Tachibanaki (ed.), The economics of social security in Japan, Edward Elgar Publishing, 2004, chapter 7, ISBN 1-84376-682-5.