FINANCIAL PLANNING

A common concern of financial planners is determining a sustainable withdrawal rate that is, the percentage of a retirees assets (at the time of retirement) that can be withdrawn each year without incurring an unacceptable risk of running out of money while still alive. Obviously, this calculation will depend on the retirees life expectancy, the definition of unacceptable risk, and the expected returns on the assets remaining in the portfolio. These figures often include an allowance for inflation adjustments, but we will work in constant dollars.

Suppose a particular retiree (your client) wants to be 95% confident that his assets will last for 30 years. Simulate this by assuming that a fixed amount (expressed as a percentage of the beginning portfolio, which can be any figure you choose) that will be taken out at the beginning of each year. If the balance in the portfolio to start the year is less than this amount, then the withdrawal will be whatever will empty the account and it will be zero from that point on. Suppose further that the assets are invested in a bundle of equities whose price movements over any given year (based on the balance after the withdrawal) are expected to occur randomly with the following probability distribution:

Value Change (%)	-20	-10	-5	0	+5	+10	+15	+20
Probability	0.02	0.03	0.1	0.2	0.25	0.25	0.1	0.05

Use simulation (1,000 repetitions as usual) to determine the maximum withdrawal percentage that will give a less than 5% chance of running out of money in a 30-year period. What you will need to do is simulate a 30-year period as a dynamic simulation, then repeat it 1,000 times. The easiest way to do that is to keep your random numbers live and use a data table to get the 1,000 recalculations. Collect the value of the last withdrawal if it is zero, you ran out of money, if it isnt, you didnt. The disadvantage of this approach is that your results will also be live and will bounce around a fair bit, making it difficult to get an exact answer. You can re-calculate a few times to get a sense of the real answer, or increase the number of repetitions if you prefer.

Explore other confidence levels that you think might be acceptable and construct a graph of withdrawal rate versus confidence level. As stated above, values will most likely be live, so these numbers will have to be approximate, but you can get a pretty good handle on them by recalculating a few times. Your client has some additional concerns, for example, at the optimal withdrawal amount that you found above, what is the expected value of the estate at the end of 30 years (the inheritance he will leave to his children)? Further, in the case that his money doesnt last (5% of the time) what is the expected value of the total shortfall (the amount he will have to borrow from his children in later years)? All amounts will be expressed as percentages of the initial portfolio.