One of the most hotly contested debates in personal finance is over the safe withdrawal rate for retirement. How much can you expect to spend sustainably from your investments during retirement? This question can drive you to madness, perhaps because it brings up a fundamental reality about our existence: We cannot predict the future, especially when it comes to financial market returns. What do we need to do to be safe?

To better understand what is at stake, let’s take a step back and spend some time to understand the PMT and NPER functions available on business calculators and Microsoft Excel. All of our analysis about systematic withdrawals will fundamentally tie into these formulas. First, the PMT formula is defined as:

=PMT(rate,nper,pv,fv,type)

Rate is the assumed investment return, nper is the number of periods that the nest egg should last, pv is the amount you have saved at retirement, and fv is the amount you wish to be able to take out for a legacy at the end of retirement (counterintuitively, it needs to have negative sign if you want to have money left over at the end). It is more conservative to set type to a value of 1 to reflect that you will take your distributions at the start of each time period. For instance,

=PMT(0.0133,30,1000000,0,1)

will provide you with an answer of $40,109, meaning if you assume an inflation-adjusted compounding investment return of 1.33%, want your money to last thirty years, start with $1 million, want to end up with $0, and you take your distributions at the start of each year, then you can sustainably spend $40,109, with subsequent inflation adjustments for your yearly spending amount. As a percentage of initial assets, that is just slightly above a 4% spending rate.

With all else being equal, if you increase the value of *rate* (your investment return), then you can increase the sustainable spending amount. For instance, replace 1.33% with 7% in the function, and the spending answer jumps to $75,314. This represents a 7.53% withdrawal percentage from initial assets. Perhaps surprisingly as well, with a 7% return assumption, if you wanted to fully preserve the purchasing power of your initial retirement portfolio after thirty years (which requires putting in a value of -1000000 for *fv*), the sustainable spending rate falls only to 6.54%.

NPER is a variation on this theme. It works the same way as PMT, but in this case you input how much you wish to spend and the formula tells you how long your money will last. Moshe Milevsky recently suggested in the* Financial Analysts Journal* that this formula will be a good starting point for understanding how our spending assumptions impact portfolio sustainability. NPER is expressed as:

=NPER(rate,**pmt**,pv,fv,type)

So, reflecting the first problem we investigated above,

=NPER(0.0133,-40109,1000000,0,1)

will give us an answer of thirty years. With a 1.33% return, we can spend $40,109 from a $1 million portfolio sustainably for thirty years and be left with nothing at the end of that period.

If we wish to spend $80,000 with these other assumptions, NPER would inform us that our portfolio would only last for 13.6 years before depletion. Milevsky likes to start with this formula because he believes it provides a more useful reality check about our assumptions.

Incidentally, the RATE function in Excel works in the same basic way as these two formulas, instead telling us the rate of return we would require to meet our *pmt* spending objective for *nper*years.

These formulas provide useful starting points to understand retirement spending because they clearly link spending amounts, rates of return, and time horizons. This is the heart of the sustainable spending problem: spend more and your money will not last as long. But these fundamentally important links are easy to lose sight of later when we enter the world of the “4% rule” for retirement spending.

The question then turns to determining appropriate values for the underlying variables. How much do we want to be able to spend? How long should our spending strategy work? What is a reasonable rate of return assumption for our portfolio? How much should we try to leave at the end of the time horizon? The problem gets more complicated as we think about how to consider sequence of returns risk, and whether we have the capacity to adjust our spending along with the market’s performance.

Returning to an earlier example, we saw that a 7% rate of return would allow for a 7.53% sustainable spending rate over thirty years. That was the state of analysis in the mid-1990s when William Bengen came onto the scene and his research led him to coin the 4% spending rule. If the long-term average real compounding return from the stock market is 7%, does that really mean one can safely use a 7.5% withdrawal rate from a 100% stocks portfolio without worrying about running out of wealth for thirty years? No, it does not.

## Basics for choosing a portfolio return assumption

For a lifetime financial plan, the most intuitive way to express a portfolio return assumption is as an inflation-adjusted compounding return. Unfortunately, this is generally not the most common way returns are expressed. A quick review of the steps needed to arrive at a real compounded return is in order.

I will illustrate this by focusing on the compounded real returns generated historically by a 50/50 asset allocation to the S&P 500 and intermediate-term US government bonds.

For the period from 1926 through 2015, Morningstar data reveals that the S&P 500 enjoyed an average (arithmetic) return of 12%, while intermediate-term government bonds earned 5.3%. Removing inflation so the numbers allow for a better understanding of purchasing power growth, the real arithmetic returns fell to 9% and 2.3%, respectively.

For those simulating long-term financial plans, we also have to account for volatility and the lack of symmetry in outcomes for positive and negative returns. We calculate the compounded returns over time to account for this volatility. The S&P 500 compounded real return fell to 6.9%, while the compounded real return for the less volatile bonds fell only slightly to 2.2%.

I will simulate each asset class separately and combine them into a 50/50 portfolio rebalanced annually. For 100,000 Monte Carlo simulations over thirty-year periods, the estimated arithmetic real return from the 50/50 portfolio was 5.6% and the standard deviation for returns was 10.8%. The compounded real return was 5.1%.

A number of further adjustments could and should be made, such as incorporating taxes, accounting for the fact that today’s interest rates are much lower than historical averages, removing the impacts of advisory and investment fees, and considering the possibility of outperformance or underperformance with respect to the underlying market indices.

Also of utmost importance is a downward adjustment to increase the probability that the assumed return will be met. For one’s best guess about the forward-looking compounded real return, there is still a 50% chance that the realized return will be lower.

This could derail a plan that requires a higher return in order to succeed. Monte Carlo simulations generally focus on building a plan with a high probability of success, which implicitly means a low assumed investment return. How should we adjust our return assumptions to account for this aspect of risk?

Stocks do not earn their average real return each year. Some years they go up and some years they go down, as recent investors know all too well. For a retiree who is taking distributions from their savings, the sequence of market returns matters. If a retiree’s portfolio drops in value during the early retirement period, portfolio withdrawals will dig a further hole. Climbing out of this hole becomes increasingly difficult, even if a subsequent market recovery arrives. This is sequence of returns risk. Sequence risk amplifies the impact of traditional investment volatility, meaning that the *rate* assumption has to be further reduced to improve the odds that the plan will work.

The debate today is over whether Bengen took the analysis far enough to provide a sufficiently conservative projection for the safe withdrawal rate. Is a thirty-year planning horizon appropriate for today’s retirees? Is it appropriate to assume retirees can annually rebalance and maintain a rather aggressive stock allocation and precisely earn the underlying market returns net of fees and taxes?

While a 4% spending rate represents the worst case outcome with United States historical data, how do we reflect that we have very little experience with knowing what happens when people retire at times when interest rates are at such low levels as today, while stock market valuations are also well above their historical averages? On the other hand, inflation is also quite low now. What implications does this have? How does spending flexibility impact all of this analysis?

**To find out more about investing in retirement, read our eBook 8 Tips to Becoming a Retirement Income Investor.**