Peter Ponzo, a retired faculty member in Applied Mathematics, at the University
of Waterloo, passed away in 2020, at the age of 85.
During his retirement, Peter blogged about his misadventures in investing his retirement savings. He
was especially interested in * decumulation* strategies, i.e.
how to invest and spend down your
money during retirement.

Nobel laureate William Sharpe
has referred
to
decumulating savings
during retirement as *``the nastiest, hardest problem in finance''*.
Why is this so hard? Basically, you have to deal with uncertain longevity, volatility in the financial markets,
and deciding how much to withdraw each year. Although traditionally annuities have been suggested as
a way around the longevity issue, annuities are very
unpopular
with most retirees, due to meager payouts
in the current low-interest rate environment, lack of true inflation protection, poor pricing, and no access to cash in
the event of an emergency.

So, the retiree has to devise a strategy by him/her self. What about financial planners? Unfortunately, this is not a popular area for planners, who typically get paid a percentage of assets under management. A retiree's portfolio will be decreasing as withdrawals take place during retirement, hence the planners fees get smaller as the retiree gets older. So, just when the strategy becomes very important (smaller portfolios, uncertain number of years remaining), the planner gets the smallest fee. Good luck finding someone who wants to manage this.

One of the interesting strategies devised by Professor Ponzo, was what I'll call the Canasta strategy

*''If we have a good year,
we take a trip to China,...if we have a bad year, we stay home and play canasta.''*

Is this a good strategy? Well, the decumulation problem can be posed as an exercise in optimal stochastic control. Given an initial pot of money, we assume that a 65-year old male wants to maximize his total withdrawals over 30 years, and minimize the risk of running out of cash.

The probability of a 65-year old Canadian male attaining the age of 95 is about 0.13. So, we are being conservative here in terms of longevity, although there is still a chance that our retiree will attain his 96'th birthday.

Risk is measured in terms of the Expected Shortfall at the 5% level. What is Expected Shortfall? This is simply the mean of the worst 5% of the outcomes in terms of the amount of cash left in the portfolio at age 95.

There are two decision variables in this problem (which are also termed *controls*).

- The amount of cash withdrawn each year.
- It makes sense to impose minimum and maximum withdrawal amounts. The minimum amount is based on funding necessary expenses. We also impose a maximum withdrawal amount, since we assume the retiree has little use for a huge cash infusion in any one year.

- The asset allocation strategy.
- The retiree's portfolio consists of a stock index and a bond index, so the allocation strategy is simply the fraction of the portfolio invested in stocks. Use of Leverage or short selling is not permitted.

Being a retiree myself, I was very interested in solving the stochastic control problem. The results are summarized in the working paper (to appear in the North American Actuarial Journal in 2021).

Perhaps the most intriguing outcome of this work concerns the withdrawal controls,
i.e. the amount taken out of the portfolio each year. It turns out
that, to a good approximation, the withdrawal controls are * bang-bang*. This means that the optimal strategy is
to withdraw only at either the minimum or maximum amount each year.

Basically, if the portfolio increases in value (i.e. the stock market goes up), you should withdraw at the maximum amount. If the stock market drops, you should withdraw at the smallest possible rate.