It is generally believed that stock markets, interest rates and investment returns move in periodic cycles. Unfortunately, these cycles are swamped with noise and only become evident with the benefit of hindsight. The current academic consensus is that they are extremely difficult to predict or measure in advance. Nevertheless, these cycles — once they do materialize — can have a profound impact on the sustainability of retirement income.

If you retire and start to withdraw money from a diversified investment portfolio just as the economy moves into a bear-market cycle your portfolio’s longevity can be at risk. Your nest egg will not last as long as it would under an equivalent spending plan which started during a bull-market cycle. This observation is often labeled by the term “sequence-of-returns” risk and is used by many in the insurance industry to explain the importance of downside protection during the so-called “retirement risk zone.”

I, too, have elaborated elsewhere on the role of guarantees early-on in retirement using analogies of spinning triangles, roulette wheels and Monte Carlo simulations. In this brief article I would like to take a slightly different approach to the issue — one that hopefully balances realism and accessibility.

My plan here is to illustrate exactly how a bull or bear market cycle can impact the sustainability of your portfolio by appealing to ideas from basic high school trigonometry, a.k.a. the mathematics of sine and cosine waves. And, although your skills might be rusty after these many years — and I’m not even sure they teach this anymore in high school — the main story should be accessible to all.

First, allow me to review the basic arithmetic of generating income, sustainability and ruin. Assume you start retirement with a nest egg of exactly $100 and you allocate this to an investment fund that earns a real (i.e., after inflation) 5 percent per annum during every year of your retirement. For simplicity, I take this 5 percent to be an annual percentage rate (APR) that compounds continuously over time. Remember, this implies that if inflation is 3 percent per annum, then your nominal return is (approximately) 5% + 3% = 8%. If inflation is 4 percent, then your nominal return is (approximately) 9 percent.

Now let’s spend some money. If you withdraw $6 (also inflation-adjusted) per year from this portfolio the nest egg will be exhausted — and your retirement ruined — in exactly 35.8 years. In contrast, if you withdraw $7 (inflation-adjusted) per year the funds will last for 25.1 years. At $8 of spending you will be ruined in 19.6 years. Remember, all of this assumes your portfolio earns the same consistent inflation-adjusted 5 percent APR for ever.

At this point I need you to suspend your disbelief and imagine a perfectly cyclical (sine wave) financial market; and I mean perfect with no randomness, noise or real-world uncertainty. I’m going to describe two symmetrically opposed scenarios. Look at Figure No. 1 for a picture that’s worth the next 200 words.

In that figure, the blue line represents how the portfolio starts out earning an APR of 5 percent on the first day of retirement. To be exact, this is 2 basis points during the first day of retirement, which is a 5 percent APR divided by 250 trading days. The market then moves into a bull-market cycle so that your annualized returns slowly increase until it peaks at 20 percent per annum (8 basis point per day) in approximately 19 months. In the language of sine waves, the market peaks after approximately ?/2 years. Remember the Greek letter ? is equal to approximately 3.14 (years), which is 37.68 months, so ?/2 is just shy of 19 months.

Then after hitting this peak financial markets start to decline so that approximately 19 additional months later the market is back to earning an APR of 5 percent. It has gone from 5 percent up to 20 percent and then down to 5 percent over approximately 3.14 years.

Bear with me here. Imagine that markets then continue to decline for another 19 months and your portfolio’s investment return hits -10 percent annualized (which is -4 basis points per day). Please see the blue line in Figure #1 for an illustration of the evolution of this entire cycle from start to finish over 2? (= 6.28) years. This sine wave I have constructed exhibits the amplitude (volatility) of plus or minus 15 percent, and ranges in value from negative 10 percent to positive 20 percent.

As a perfectly symmetric alternative, consider the scenario in which you retire and start to withdraw funds while the market is earning the same 5 percent per annum, but it immediately moves into a bear-market cycle so that 19 months into retirement you are earning -10 percent (i.e. losing money) per annum, and 19 months later you are back to 5 percent per annum and 19 months after that you are earning 20 percent, etc.