In a world where it seems every client has a goal of having a portfolio that holds up through at least 35 years of retirement, it would be helpful to have some principles and mathematical measures to use as a guide in accumulating and dispersing retirement funds.

There is one principle that practically everyone knows: A retirement portfolio is likely to last longer if the withdrawal rate is lower. In other words, if the initial withdrawal rate is 4% from your retirement portfolio, the probability is lower that the portfolio will last at least 35 years than if the initial withdrawal rate is 3%.

But there are several other aspects of retirement portfolio survival that may not be quite as obvious. For example, how much is needed in the portfolio to produce a sufficient level of retirement income? Equally compelling is the issue of retirement portfolio design: How have different portfolio models behaved in the past?

Let’s explore those questions and a few more.

**THE RETIREMENT ACCOUNT MULTIPLE**

The first key principle in the mathematics of retirement portfolios is the retirement account multiple. RAM is the size of the retirement account balance as a multiple of the retiree’s final annual salary. So if the final salary was $100,000 and the new retiree has a retirement account balance of $700,000, he has a RAM of 7.

Once we know the RAM, calculating the income replacement ratio is straightforward: RAM x Withdrawal Rate = Income Replacement Ratio.

So let’s go back to the retiree who earned $100,000 in his final year of work and has a retirement account balance of $700,000, which means his RAM is 7.

Multiplying that RAM by an initial withdrawal rate of 5% produces the following equation: 7 x 5 = 35% income replacement. In other words, the retiree withdraws 5% of $700,000, which is $35,000; $35,000 is 35% of his final working salary of $100,000.

Income replacement (as a percentage of final working salary) is simply a mathematical relationship between RAM and the withdrawal rate percentage.

Next, we will consider the performance of the retirement portfolio. This is, understandably, a somewhat more complicated process.

Two retirement portfolios will be evaluated for illustrative purposes. The first is a 25% stock/75% fixed-income portfolio; this portfolio consists of 15% large-cap U.S. stocks, 10% small-cap U.S. stocks, 55% U.S. bonds and 20% cash.

**PORTFOLIO SURVIVAL**

To measure the performance of these various elements, some commonplace indexes were used. Large-cap U.S. stocks were represented by the S&P 500; small-cap U.S. stocks by the Ibbotson Small Stock Index in 1926-1978 and the Russell 2000 Index in 1979-2015; U.S. bonds by the Ibbotson U.S. Intermediate Government Bond Index in 1926-1975 and the Barclays Capital Aggregate Bond Index in 1976-2015; and U.S. cash was represented by 90-day Treasury bills.

The second retirement model was a 65% stock/35% fixed-income portfolio; this consists of 40% large-cap U.S. stocks, 25% small-cap U.S. stocks, 25% U.S. bonds and 10% cash. In both cases, the portfolios were rebalanced annually.

**A 90-YEAR ANALYSIS**

The time frame of this retirement portfolio survival analysis was the 90-year period from January 1926 through December 2015. Over this time, there were 56 rolling 35-year periods to analyze.

Success is defined as having the portfolio survive (not run out of money) for at least 35 years (which simulates a retiree from age 65 to 100).

The historical success rates of the two retirement portfolios are shown in the chart “Retirement Portfolio Math.”

This particular analysis was based on a retiree who selected about 50% income replacement during retirement. For example, if the retiree has a RAM of 7, he will need to invoke a 7% withdrawal rate to create a 49% income replacement (7 x 7% = 49% income replacement ratio).

At a 7% withdrawal rate and a 3% annual cost-of-living adjustment, the 25/75 portfolio lasted at least 35 years in only 21% of the 56 rolling 35-year periods between 1926 and 2015. The more growth-oriented 65/35 retirement portfolio model had a historical success rate of 71%.

If, however, the retiree had a RAM of 10, he needed only a 5% withdrawal rate to produce the 50% income replacement ratio. At the lower withdrawal rate of 5%, the 25/75 portfolio had a success rate of 59%, whereas the 65/35 portfolio had a survival success rate of 91%.

**A LOWER WITHDRAWAL RATE**

Here, we clearly see the relationship between a lower withdrawal rate and enhanced retirement portfolio survival. The luxury of using a lower withdrawal rate is created by having a larger RAM.

The table below shows that the goal of every retiree should be to have a high enough RAM to permit a withdrawal rate of 4% or lower.

Of course, that will not always be possible. Life happens. Many retirees may have experienced events that impaired their ability to save enough for retirement. Medical situations, divorce, job loss and many other challenging situations can, unfortunately, ravage a retirement account.

**BEYOND THE TABLE**

The information in the table on the previous page provides a template for understanding the math of retirement portfolios.

But we can assess other retirement scenarios as well by using the basic information in the table.

For example, perhaps a retiree wants a 70% income replacement during retirement and has a RAM of 10. We know that this particular scenario will require a 7% withdrawal rate (inasmuch as a RAM of 10 requires a 7% withdrawal rate to produce a 70% income replacement).

The table shows that a 7% withdrawal rate is associated with a 21% success rate if employing a 25/75 portfolio, and a 71% success rate with a 65/35 portfolio (where success rate is defined as the retirement portfolio lasting at least 35 years).

Alternatively, if this same retiree had a RAM of 14 and wanted a 70% income replacement, he would implement a 5% withdrawal rate (70% income replacement divided by 14 RAM = 5% withdrawal rate).

This is a much more encouraging situation inasmuch as a 5% withdrawal rate is associated with a much higher success rate for the two retirement portfolio models (59% for the 25/75 model and 91% for the 65/35 model).

The basic relationships between RAM, withdrawal rate and portfolio success rate that are illustrated in the table can be interpolated to provide guidance for a variety of retirement scenarios.

The retirement income replacement ratios used in this analysis assume that the needed retirement income is coming solely from the investment portfolio. But often this will not be the case for any individual retiree.

Clearly, there will be situations where the retiree will have other retirement income sources. For example, if the retiree is hoping for 75% income replacement during retirement and has 25% coming from other sources, then the investment portfolio has to produce only a 50% income replacement.

This will, naturally, take some pressure off the investment portfolio, which will allow for a slightly lower RAM or a lower withdrawal rate.

Of course, we want to encourage retirees to have a large RAM, because the ultimate goal is to lower the portfolio withdrawal rate as much as possible.

**DIVERSIFICATION**

This analysis also illustrates the importance of staying diversified during retirement. Portfolio diversification is a lifelong strategy, not just during the pre-retirement years.

Whether a 65/35 portfolio or a 25/75 portfolio (or some other asset allocation model) is appropriate for a retiree is largely determined by the RAM.

For instance, if a person at the moment of retirement has a RAM of 18 and needs a 50% income replacement from his investment portfolio, the needed withdrawal rate is only 3%. In this case, 25/75 might be the appropriate asset allocation model based on its 100% success rate over the past nine decades.

In short, there is no reason to take on more portfolio risk than is needed during retirement. A large RAM combined with a modest income replacement goal generally leads to a lower required withdrawal rate; this, in turn, permits retirees to have a portfolio design that can be a bit more on the conservative side.

The freedom to have a conservative retirement portfolio is, for many, earned by being good savers during their careers. But, as noted previously, I recognize that many people have been good savers and were then blindsided by unfortunate life events that impacted their retirement savings balance.

In the end, we can only try to do our very best in life. And in order to get the best results for our clients in their retirement years, it is important that we all know — and then do — the math.

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B. Jepson