This post was part of a series about managing how you spend your investments, if you aim to live long, live within your means, and, perhaps, leave some for others. The main ideas have been updated and moved to an article on Retirement Income, on the Retired Now menu. If you wish, you can still read the original series, using the links below.

- Retail strategies, using all-in-one mutual funds. Simple solutions that work for some.
- Insurance strategies, using Social Security and maybe some annuities
**.**Part of everyone's plan. - Endowment strategies, adapted from foundations and universities. Good for a reserve fund.
**Finance strategies, based on life-expectancy and future payments. This post.**- Smooth consumption, a comprehensive method that uses excellent, free software.

## Preview

The previous post in this series described three goals for retirement payouts, and proposed an endowment strategy for one of them:

**Holding emergency reserves**: An all-in-one fund with about 30-40% invested in U.S. and international stocks, and the rest in diversified bonds, including some short-term TIPS, is excellent for a reserve fund. Managing it somewhat like an endowment fund, you could withdraw or transfer 1% to 3% annually, with little risk of depleting your reserve holdings. For an ounce of additional safety, you could skip the withdrawal after a down year in the markets.

**Covering expenses**and**leaving a bequest.**These goals compete, because the more you spend on your own expenses, the less you will be able to leave as gifts, and vice versa. Complicating the matter, you cannot be sure how long your expenses will continue, because none of us can foretell the day when we will last greet the dawn. Actuaries have tables that estimate longevity, and these prove to be very helpful. Accountants have formulas to adjust for inflation, keep spending within a set range, and calibrate today's budget against future income and payments. These are helpful, too. While it may sound complex, the very best solutions had very simple calculations. No fancy spreadsheet or algorithm is required!

## Historical Analysis

As detailed in earlier posts, I developed a set of historical data on U.S. stock and bond markets since 1924, and ran simulations for hypothetical investors who retired at age 65 and lived to 95. The retirees were in 62 cohorts whose retirements started from 1924 to 1985, and ended between 1974 and 2015. To cover a range of investments, I simulated two portfolios, one with 35% invested in stocks; the other with 65%.

After exploring numerous strategies, I stress-tested the best methods by having the simulated retirees live 15 years longer than the 30 they had expected. I applied the longer, 45-year simulations to the least fortunate cohorts, the ones starting in 1929, 1937, 1966 and 1969. These were cohorts which, early in their golden years, faced the double-whammy of a sharp decline in the stock market and a multi-year period of high inflation.

Of many strategies I tested, the ones reported here are those that first proved sound for all 30-year simulations and then survived the selected 45-year stress-tests, for both 35% and 65% invested in stocks. The survivors included three methods from earlier posts, plus three more. All six methods had merit, two of them especially. The chart below illustrates the differences among the six payout methods, using a hypothetical portfolio invested 50% in U.S. stocks, and the rest in various U.S. treasuries. (As explained below, some of the methods calibrate their payouts to the composition of the portfolio; others don't.)

After exploring numerous strategies, I stress-tested the best methods by having the simulated retirees live 15 years longer than the 30 they had expected. I applied the longer, 45-year simulations to the least fortunate cohorts, the ones starting in 1929, 1937, 1966 and 1969. These were cohorts which, early in their golden years, faced the double-whammy of a sharp decline in the stock market and a multi-year period of high inflation.

Of many strategies I tested, the ones reported here are those that first proved sound for all 30-year simulations and then survived the selected 45-year stress-tests, for both 35% and 65% invested in stocks. The survivors included three methods from earlier posts, plus three more. All six methods had merit, two of them especially. The chart below illustrates the differences among the six payout methods, using a hypothetical portfolio invested 50% in U.S. stocks, and the rest in various U.S. treasuries. (As explained below, some of the methods calibrate their payouts to the composition of the portfolio; others don't.)

## Covering Expenses

**Flat5%**. Each year, this method withdrew 5% of the then-current value of the portfolio, emulating how universities and foundations typically manage their endowments. Among commercially available funds with low fees, Vanguard's Managed Payout Fund is similar. It pays out 4% annually, computing the percentage against the fund's average value over the prior three years. My simulations found modest, somewhat inconsistent benefits for a three-year average, and implied that 5% without averaging offered safety from depletion. The bottom line for this method is that it is very easy to compute and may be an option worth considering if your objective is to leave a bequest whose real, inflation-adjusted value after 30 years is approximately equal to your portfolio's initial value at the time you retire.**4%-or-RMD**. This method withdrew 4% per year from ages 65 to 70, and then took annual payouts as prescribed by the Internal Revenue Service's rules for Required Minimum Distributions (RMD). Except for funds held in Roth accounts or a taxable account, this method may be attractive because once you hit 70.5 years of age, you can have your investment firm calculate the required withdrawal. The result will be cautious payouts initially, then rapid escalation starting at age 80. Because of the escalation, the amount left for bequests after 30 years tended to be smaller than for most other methods.**Collared Inflation**. This method was a variant of the oft-mentioned strategy of withdrawing 4% initially, then adjusting the dollar-amount of the payout annually to align with the change in consumer prices over the prior 12 months. As many others have reported, and as my own simulations found, blindly starting at 4% and inflating the payout each year sometimes depletes the portfolio. To rectify this problem, I modified it in two ways. First, the initial percentage was set by a formula that matches recommendations from Vanguard's extensive testing. Second, the annual adjustment was collared; it was not allowed to exceed a fixed percentage, even if inflation was worse than that percentage. A similar tactic is often used by endowments and is described, as one of many examples, by this actuary. After being modified, this method went from worst to best.*If your intent were to take minimal payouts and leave a large bequest, this was the preferred method, historically*. As shown by the line marked CI-1937 in the chart, which depicts results for the 1937 cohort, the payout varies from year to year (and from cohort to cohort), depending on actual changes in consumer prices. Simple steps for calculating this method's payout are in the "Summing Up" section at the end of this post.**Life+6**. Financial forecasts, such as life-insurance premiums and RMD, use population estimates of longevity, the years remaining from the current age until the age at which death is likely. After some testing, I found a very simple variant of longevity to be extremely effective. The trivially easy steps for computing it are in the "Summing Up" section below.*If you wanted to maximize your payouts and leave a modest bequest, this method would have been best, historically*. As shown by the nearly straight line for "Longevity+" in the chart, this method starts more conservatively than 4%-or-RMD, and increases in a steady fashion.**FlexPay1**. This method generated flexible payouts with a formula drawn from a well-known economic theory concerning the time-value of money. I examined numerous applications of this theory, including one proposed by the Bogelheads forum. The applications varied in their assumptions about how much the portfolio might earn in the future, how much should remain after the payout-period ended, and other factors. The settings for FlexPay1 were among the best I found in my simulation study. They may be most suitable for special circumstances where RMD is irrelevant, and leaving a bequest is important. An example might be a Roth or taxable account from which you want to make some withdrawals, while leaving at least half the inflation-adjusted value to charities or a trust fund or your estate, at some indeterminate point in the future. As the chart shows, this method steadily increases the payout, but at a gradual rate. Details of the settings for FlexPay1 are at the end of this post, in the "read more" section.**FlexPay2**. This method was also based on the time-value of money. It is linked to the same Vanguard recommendations as the Collared Inflation method, and has settings particularly well-suited for early retirement. It calibrates the initial payout according to both life expectancy and portfolio composition, and it manages the payout stream to prevent depletion and leave a bequest similar to that returned by the 4%-or-RMD method. Conceptually, the intent of this method is much like Betterment's excellent algorithm for retirement income (although the implementation details may differ). Where no RMD requirement exists, such as a Roth or taxable account at any age or a traditional IRA, 401(k) or 403(b) before age 70.5, the FlexPay2 method is a good choice. The growth rate of payouts for this method is intermediate between the lines marked FlexPay1 and Longevity+ in the chart. You'll find details by clicking the "read-more" link at the end of this post.

How did the six methods compare on covering expenses? For this question, the chart shown above is misleading. Over time, the inflation-adjusted value of your portfolio (its real buying power) will change as you take withdrawals, as your stocks and bonds gain or lose in the markets, and as consumer prices rise or fall. What matters is the buying power of your payouts over time, not the percentage of your portfolio that you spend. The aim is for your payouts' buying power to be steady or to rise, while your assets are protected from depletion.

The next two charts use a baseline of $100 that 4% of a portfolio would have bought at the start of retirement. Payouts after the first one were adjusted for inflation, to represent a constant buying power of $100. Some methods optimistically started a bit higher than $100; some, pessimistically lower. Look at the first chart, which is for a portfolio consistently invested 65% in stocks, 20% in 10-year treasuries, and 15% in 2-year treasuries.

The next two charts use a baseline of $100 that 4% of a portfolio would have bought at the start of retirement. Payouts after the first one were adjusted for inflation, to represent a constant buying power of $100. Some methods optimistically started a bit higher than $100; some, pessimistically lower. Look at the first chart, which is for a portfolio consistently invested 65% in stocks, 20% in 10-year treasuries, and 15% in 2-year treasuries.

The chart shows the results for 90% of the cohorts (excluding the worst 5% and the best 5%). The blue box shows ones that were below the mid-point; the green box, those above the midpoint. A corresponding chart for portfolios with less invested in stocks (35%) is below.

Clearly, the main message here is that if you want payouts with consistent buying power, Collared Inflation is far and away the best.

To see the advantages of Collared Inflation and Life+6 (longevity), look at the next two charts, which show the same data, arranged this time by cohort. Here are the cohorts at 65% in stocks ...

**But**. It comes at a price. The payouts for Collared Inflation were consistently below the base level of $100. That's because they cautiously started a bit below 4%, and rarely rose above that mark. For payouts that varied more, but were highest overall, the longevity or Life+6 method was best. It had the highest highs and the highest medians; its lows, depending on the portfolio, were either best (35% stocks) or second best (65% stocks).To see the advantages of Collared Inflation and Life+6 (longevity), look at the next two charts, which show the same data, arranged this time by cohort. Here are the cohorts at 65% in stocks ...

... and here they are at 35% in stocks.

For both portfolios, almost every cohort got the highest payouts with Life+6, the longevity method, and the lowest with Collared Inflation. It's important to note that the longevity method, although it varied radically, fell below the $100 benchmark only once, in the worst-case scenario of a conservative 35% portfolio that started in the most challenging year, 1937.

## Leaving a Bequest

Before you leap ecstatically to the conclusion that Life+6 (longevity) is the payout method for all seasons, remember this. The more you pay yourself, the less will be left for bequests. As we turn to the amount left in a portfolio after 30 years, you may suspect what's coming.

There you have it. Both at 65% stocks (above) and 35% stocks (below), the tables are turned. Now it's Collared Inflation that has the highest results, coupled with the most variation, while Life+6, the longevity method, has the lowest results and the least variation.

For bequests, as for payouts, the top-ranked and bottom-ranked methods were remarkably consistent across portfolios and cohorts. Here are the cohorts' results for 65% in stocks ...

... and for 35% in stocks.

## Summing Up

The historical analysis of this study makes the following recommendations, although it bears repeating that past results and simulated portfolios are no guarantee of future returns.

- For the combination of stable, conservative payouts and possibly large bequests, Collared Inflation worked best. And you can calculate the payouts easily, like this. The first year, get your starting percentage either from Vanguard's tool or from our Safe Payout calculator. In subsequent years, calculate a number (let's call it the Ratio) as the current Consumer Price Index (CPI), divided by the CPI 12 months ago. If the Ratio is below zero (which should be rare), change the Ratio to zero. If the Ratio is above 1.067, change it to 1.067. Now multiply last year's dollar-payout by your Ratio. That's the payout for this year. (The number 1.067 provided the best protection against depletion, even at 45 years for the most challenging cohorts. Other planners may recommend numbers such as 1.05 or 1.10. If you take these other recommendations, check out the data analysis, if any, that's available to support them.)
- For the combination of high (and highly variable) payouts, plus a bequest that may be about 50% or less of your portfolio's original buying power, Life+6 works best. It's super-easy to calculate. Every year, use Social Security's Life Expectancy calculator to find your longevity. Add six years. Divide your portfolio's current dollar value by that number. The result is your payout for the current year.
- As compromise methods, the two that used the time value of money have much to recommend them, perhaps with a slight nod to FlexPay1. Its payouts were almost as good as Life+6, and its bequests tended to be predictably between 50% to 100% of the original portfolio's buying power. It offers a nice bargain: half goes to generate your annual payouts, with little risk of lost buying power and possibly some very splendid years, and half or more remains for your heirs. Yes, with historically normal market-returns, you and your heirs may both get more than half. Sweet!

Although a mixture of bonds, stocks, and guaranteed benefits may be safer than investing exclusively in one class of assets, diversification cannot guarantee a positive return. Losses are always possible with any investment strategy. Nothing here is intended as an endorsement, offer, or solicitation for any particular investment, security, or type of insurance.

## Is 4% a Magic Number?

Well, no, it's not. If payouts are never adjusted for inflation and instead a flat percentage is taken each year, all my simulations showed 5% to be safe from depletion. You can readily find lots of additional recommendations about setting percentages other than 4%. I find the most convincing case to be made in Vanguard's explanation of their online calculator for setting the initial payout percentage. The explanation points to a thorough, mathematically sound, and highly informative exposition of Vanguard's Global Capital Markets Model. Vanguard gets kudos for being both rigorous and transparent, well beyond what is typical of most investment firms.

With some reverse engineering, I was able to generate a simple formula that almost exactly reproduces the percentages given by Vanguard's calculator. Expressed in the format used by Excel and Google Sheets, the formula is: PMT( 0.3 * Expected_Return, Retirement_Period, -1, 0.15 ), where Expected_Return is the expected real (inflation adjusted) return of the investor's portfolio, and Retirement_Period is in years. I have no theoretical justification for this formula. It's just the result of parameter-estimation, with Vanguard's percentages as the to-be-fitted variable.

One benefit of the formula is that it allows interpolation to values not allowed by Vanguard's calculator. The calculator is limited to portfolios invested 20%, 50%, or 80% in stocks, and to retirement periods between 10 and 40 years in increments of five. The formula is not.

I use the formula to get a retiree's starting point for the Collared Inflation method, because Vanguard's calculator was designed with the explicit assumption that payouts after the first would be adjusted for inflation.

With some reverse engineering, I was able to generate a simple formula that almost exactly reproduces the percentages given by Vanguard's calculator. Expressed in the format used by Excel and Google Sheets, the formula is: PMT( 0.3 * Expected_Return, Retirement_Period, -1, 0.15 ), where Expected_Return is the expected real (inflation adjusted) return of the investor's portfolio, and Retirement_Period is in years. I have no theoretical justification for this formula. It's just the result of parameter-estimation, with Vanguard's percentages as the to-be-fitted variable.

One benefit of the formula is that it allows interpolation to values not allowed by Vanguard's calculator. The calculator is limited to portfolios invested 20%, 50%, or 80% in stocks, and to retirement periods between 10 and 40 years in increments of five. The formula is not.

I use the formula to get a retiree's starting point for the Collared Inflation method, because Vanguard's calculator was designed with the explicit assumption that payouts after the first would be adjusted for inflation.

## Payouts Based on Time Value of Money

- Life+6 is a degenerate case of the time value of money. It's nothing more than PMT( 0, Longevity + 6, -1, 0 ). In short, the method calculates a payout as if the portfolio earns no interest, and is to be fully paid in equal allotments over a period that lasts 6 years longer than the retiree's current life expectancy. In effect, adding 6 to the payout period ensures that even beyond the age of 100, payouts can't exceed one-sixth of the portfolio, thus reducing the risk of depletion. It remains mathematically possible, however, that the portfolio could become vanishingly small.
- FlexPay1 is PMT( 0.5 * Expected_Return, Longevity, -1, 0.5 ). By using 0.5 * Expected_Return, the method assumes that the future may hold sub-normal returns for the investor; this pessimism tends to favor larger than normal payouts in good markets. Protection of the bequest is accomplished by setting Longevity as the number of periods to be paid, and 0.5 as the expectation that the future value (after the last payout) is at least half the initial value.
- FlexPay2 is almost identical to the formula for initial percentage under Collared Inflation. It's PMT( 0.3 * Expected_Return, Longevity + 6, -1, 0.15). The idea here is that if the formula works well for an initial payout, it may also work for subsequent payouts, provided that the period is updated for the investor's current longevity. With the future value set a bit low at 0.15, risk of depletion is reduced by adding 6 to longevity.

## Technical Notes

An important technical note is that the payout methods had significant constraints. First, they could not deplete the portfolio within 45 years, even under inflation, and second, they had to generate payments that did not drift lower than about half of a baseline set at 4% of the portfolio's initial value. The depletion constraint meant that either longevity (number of periods) had to be incremented or the final portfolio (future value) had to be generous, or a bit of both. Additionally, the payout constraint meant that the expected return (rate) had to be reduced to ensure reasonable payouts in bad markets. The tactic used here to reduce payouts was to multiply the true expected return by a fraction, thus "cuffing" it. Other tactics are possible. For example, Betterment's payout algorithm appears to use a low-percentile estimate of the investor's likely future returns, given Betterment's model of the investor's portfolio.

Another technical issue is that the simulations assumed a planner would map out the full series of future payments before the first payout was taken, and present the series to a client to be used until they died. That, in essence, is what the IRS does when it presents the RMD table as the tax-payer's payment plan. Having to specify all the payments in advance creates a complication for the future value parameter. For example, if the future value were set at 25% of the portfolio's original value, when the client's longevity is 15 years, what should the future value be 10 years hence, when longevity is 9 years, not 5? If it were 5, standard applications of a spreadsheet's PMT formula would work. In the simulations, to make the payment series adaptive to changes in longevity, the future value at time j+1 was adjusted according to the future value at time j and the change in longevity between times j and j+1. Specifically, where future value is "fv" and longevity is "long":

fv(j+1) = fv(j) * long(j) / [ long(j+1) * (1-fv(j)) + long(j) * fv(j) ].

Then the j + 1 payout was PMT( i, long(j+1), -1, fv(j+1) ) where the rate i remained constant for the series. Note that in this approach, if long(j+1) = long(j), then fv(j+1) = fv(j).

Evidence that this adjustment was reasonable came from the FlexPay1 method. It had a target to preserve 50% of the portfolio's original value, while using straight longevity rather than longevity plus 6 years. In the simulations, after 30 years, FlexPay1 had a minimum final value of 51% and a median of 86% in the portfolio with 65% in stocks. For the portfolio with 35% in stocks, the minimum was 43% and the median 57%. Thus the future value setting in the FlexPay1 method can reasonably be viewed as a target to preserve at least that much of the original principal, over a period of 30 years.

Another technical issue is that the simulations assumed a planner would map out the full series of future payments before the first payout was taken, and present the series to a client to be used until they died. That, in essence, is what the IRS does when it presents the RMD table as the tax-payer's payment plan. Having to specify all the payments in advance creates a complication for the future value parameter. For example, if the future value were set at 25% of the portfolio's original value, when the client's longevity is 15 years, what should the future value be 10 years hence, when longevity is 9 years, not 5? If it were 5, standard applications of a spreadsheet's PMT formula would work. In the simulations, to make the payment series adaptive to changes in longevity, the future value at time j+1 was adjusted according to the future value at time j and the change in longevity between times j and j+1. Specifically, where future value is "fv" and longevity is "long":

fv(j+1) = fv(j) * long(j) / [ long(j+1) * (1-fv(j)) + long(j) * fv(j) ].

Then the j + 1 payout was PMT( i, long(j+1), -1, fv(j+1) ) where the rate i remained constant for the series. Note that in this approach, if long(j+1) = long(j), then fv(j+1) = fv(j).

Evidence that this adjustment was reasonable came from the FlexPay1 method. It had a target to preserve 50% of the portfolio's original value, while using straight longevity rather than longevity plus 6 years. In the simulations, after 30 years, FlexPay1 had a minimum final value of 51% and a median of 86% in the portfolio with 65% in stocks. For the portfolio with 35% in stocks, the minimum was 43% and the median 57%. Thus the future value setting in the FlexPay1 method can reasonably be viewed as a target to preserve at least that much of the original principal, over a period of 30 years.