3% SWR for 56 Years

Research on Safe Withdrawal Rates

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JWR1945
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3% SWR for 56 Years

Post by JWR1945 » Mon Oct 13, 2003 3:19 pm

3% SWR for 56 Years

Many people have been dismayed at stock returns since 2000 and what they mean in terms of retirement portfolios. For the investments in the traditional studies, the safe withdrawal rate in 2000 was between 2.0% and 2.5%, not the 4% that had been claimed. We have been successful in restoring the safety at a 4% withdrawal rate. It requires us to use long-term TIPS and to switch portfolio allocations in accordance with P/E10. The idea is to coast along using TIPS for income until stocks become attractive once again. But what if someone remains unconvinced? Some people talk about reducing their withdrawals. Some people talk in terms of a 2% withdrawal rate.

That is entirely unnecessary. Even if you reject the idea of switching outright, you can still withdraw 3% of your initial balance (plus inflation) for 56 years in today's market. That sets a floor on safe withdrawal rates for most of us.

Tables

I have made three tables that show how much principal remains if your portfolio is made up of TIPS alone. I have listed the number of years N and the corresponding percentages for withdrawal rates of 3.0%, 3.5% and 4.0% at the TIPS interest rates of 2.2%, 2.5% and 2.8%. Since we are withdrawing more than the interest rate in every case, we must sell some securities along the way.

For purposes of analysis, I have treated the principal as remaining constant (after adjusting for inflation). In reality, there would be price fluctuations for those TIPS that are sold early. The fraction of your income subject to such fluctuations is very small until you come close to maturity. This is similar to the build up in principal on a mortgage.

For purposes of analysis I have acted as if 30-year TIPS were still available. That is not quite true, but there are some available on the secondary market that will last until 2032. For purposes of analysis, I do not include the effect of taxes. Before applying these results, you should look at taxes very carefully. Changes in the principal of TIPS (caused by inflation) are taxed immediately. It applies to the full change in the principal, which can be equal to or higher than your withdrawals.

Percentage of principal remaining for a 3.0% withdrawal rate.

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TIPS    N=10   N=15    N=20     N=25    N=30
2.2%   91.2%   86.0%   80.2%   73.7%   66.5%
2.5%   94.4%   91.0%   87.3%   82.9%   78.1%
2.8%   97.7%   96.3%   94.7%   92.9%   90.8%



Percentage of principal remaining for a 3.5% withdrawal rate.


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TIPS    N=10   N=15    N=20     N=25    N=30
2.2%   85.6%   77.2%   67.8%   57.3%   45.6%
2.5%   88.8%   82.1%   74.5%   65.8%   56.1%
2.8%   92.0%   87.2%   81.6%   75.1%   67.8%



Percentage of principal remaining for a 4.0% withdrawal rate.


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TIPS    N=10   N=15    N=20     N=25    N=30
2.2%   80.1%   68.4%   55.4%   40.8%   24.7%
2.5%   83.2%   73.1%   61.7%   48.8%   34.2%
2.8%   86.4%   78.0%   68.4%   57.4%   44.7%

As you can see, we can coast along without buying any stocks for a long time.

After 30 Years

The remaining table tells you the number of years that you can continue to make your withdrawals after thirty years. For purposes of this analysis, I assume that your withdrawals remain the same (in terms of real dollars) and that your balance matches inflation exactly (i.e., zero percent real interest). Add thirty years to that number. That tells you how long you can make it even if you never buy stocks or any other growth vehicle.

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TIPS   3.0% rate    3.5% rate    4.0% rate
2.2%   22.2 years   13.0 years    6.2 years
2.5%   26.0 years   16.0 years    8.6 years
2.8%   30.3 years   19.4 years   11.2 years


We have seen long-term TIPS interest rates of 2.2% through 2.8% in the past few months. The last time that I checked, the rates were just above 2.6%.

If you are able to buy long-term TIPS at a 2.5% interest rate and you withdraw 3% of your initial balance (and make increases that match inflation), your total portfolio lifetime would be 56.0 years. That allows you a lot of time to sit on the sidelines. You have complete safely.

Have fun.

John R.

JWR1945
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Post by JWR1945 » Tue Oct 14, 2003 10:07 am

Here are the mathematical details. They are presented for completeness.

There may be some people who wish to apply these formulas to their own particular circumstances. It is not necessary to understand the derivations.

TIPS Equivalent Safe Withdrawal Rates (TESWR)

This is the formula. If you own TIPS (so that we do not have to make any special adjustments to account for inflation) with an interest rate r, you can withdraw the TIPS Equivalent Safe Withdrawal Rate TESWR for exactly N years. At the end of N years, your balance is exactly zero.

This assumes that you can sell some TIPS at the same price at which you bought them. That is unlikely to be true. But very few TIPS have to be sold until you get close to their maturity. It causes very little fluctuation in your total income until then.

Let g = 1+r (for convenience in presenting the formula). Then:
TESWR = r * [ 1 / (1 - [ 1 / g^N] ) ].

If r = 2.5% = 0.025, then g = 1.025. If N = 10 years, then g^N = (1.025)^10 = 1.2800845 and [ 1 / g^N ] = 0.7811984 and (1 - [ 1 / g^N] ) = 1 - 0.7811984 = 0.2188016 and [ 1 / (1 - [ 1 / g^N ] ) ] = 4.5703506.

Finally, TESWR = r*[ 4.5703506 ] = 2.5% * [4.5703506] = 11.425876%, which I round to 11.426%.

Here is a table of TIPS Equivalent Safe Withdrawal Rates for interest rates of 2.2%, 2.5% and 2.8% and N = 10, 15, 20, 25 and 30 years.

Code: Select all

TIPS     N=10     N=15     N=20     N=25      N=30
2.2%   11.249%   7.899%   6.234%   5.243%   4.589%
2.5%   11.426%   8.077%   6.415%   5.428%   4.778%
2.8%   11.604%   8.256%   6.598%   5.616%   4.971%



Derivation of the TESWR Formula

The TESWR formula is very similar those involving mortgages.

From this post about Rocket Science:
http://nofeeboards.com/boards/viewtopic.php?t=1206

A stream of annual payments P beginning at year N = 1 invested at an interest rate r for N years will total T according to this formula:
T = P*[ ( g^N - 1 ) / r ], where g = 1+r.

This formula was derived using the proof of the formula for the geometric mean (with a finite number of terms). Accountant are likely to prefer using n = N-1 so that the first term is the present value at n = 0.

Rearranging the formula, the payment amount is:
P = T*[ r / (g^N - 1) ] = T * r * [ 1 / (g^N - 1) ]

The withdrawal amount WA is equivalent to adding this payment to an initial interest of r*T. That is, it consists of principal starting at P and interest starting at r*T.
WA = T * r * [ 1 / (g^N - 1) ] + r * T = [r*T] * ( [ 1 / (g^N - 1) ] + 1) = [r*T] * ( [1 + (g^N - 1)] / [g^N - 1] ) = [r*T] * (g^N / [g^N - 1]).
The withdrawal rate is the withdrawal amount divided by T or TESWR = WA / T = r * [g^N / (g^N - 1)] = r * [1 / (1 - [1 / g^N] ) ].

Intermediate Balances

Once you know the TIPS Equivalent Safe Withdrawal Rate for N years, you can calculate the percentage of your principal that remains at the end of N years. This is the formula: the remaining fraction RF of your initial balance when you withdraw at a rate of WR from TIPS with an interest rate of r is:

RF = [ (TESWR - WR) / (TESWR - r) ]

For example, if the interest rate r is 2.5%, the TESWR for N = 10 years is 11.426% (as shown in the first example). If the withdrawal rate WR is 3%, the remaining fraction is RF = [ (11.426% - 3%) / (11.426% - 2.5%) ] = [ 8.426 / 8.926 ] = 0.9439839, which I have rounded off to 94.4% in the 3% withdrawal rate table in the main text.

That is where the tables in the main text came from.

Derivation of the Intermediate Balance Formula

The withdrawal amount can be broken into two components. One starts with an initial balance of A. It is withdrawn at the TESWR for exactly N years. It has a final balance of exactly zero after N years. The other component's principal B is left entirely untouched. Its interest of r*B is added to the withdrawal amount. The initial balance equals A+B. The final balance after N years is B.

Equating withdrawal amounts:
WR * (A+B) = TESWR*A + r*B or
WR = TESWR*x + r*y
where x = A / (A+B) and y = B / (A+B) and x+y = 1.

We are interested in solving for the remaining fraction RF, which is the same as y. Since RF = y and x = 1-y = 1 - RF:
WR = TESWR*(1 - RF) + r*RF

Expanding and then grouping terms:
WR = TESWR + RF*(-TESWR + r) or
RF*(r - TESWR) = WR - TESWR or
RF = [ (WR - TESWR) / (r - TESWR) ] = [ (TESWR - WR) / (TESWR - r) ]

Years Remaining

Long-term TIPS will mature just before 30 years ended. To a good approximation, just divide the remaining fraction after 30 years by the withdrawal rate WR. That is the number of additional years that you can continue to make withdrawals, assuming no interest (i.e., the real interest rate equals zero).

Additional Comments

The TIPS Equivalent Safe Withdrawal Rate should be obtainable with mortgage calculators. The mathematics is the same. With a mortgage you build up your equity (or principal) as you pay down the loan. You return principal to the mortgage holder. With TIPS, you take the position of the mortgage holder (using real dollars instead of nominal dollars). It is likely that such calculators provide amortization (schedule) information as well.

What I have presented are the formulas that I actually use.

Have fun.

John R.

JanSz
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Post by JanSz » Thu Oct 16, 2003 10:11 am

You have made many wonderfull calculations, JWR1945
There is one I would like to see, possibly it exists and I have overlooked it.

Whole port resides at a broker.
It invests in stocks (S&P500 for simplicity) and cash.
Cash earns interest as paid by broker's MM
Stocks / cash portfolio allocations is in accordance with P/E10 (please provide your own details)
Withdrawal is based on formula
(2% of current value+half gains over 2% if any), averaged over past 5 years

Wonder how long that kind of port would last, and what kind of withdrawal rate would have been possible.

gummy's (sensible withdrawals.xls) calculator is able to deal with similar problem but it uses fixed stocks/cash ratio, hovewer it already shows advantage of waiting for actual gains before spending the cash.

Mike
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Post by Mike » Thu Oct 16, 2003 1:56 pm

Money market funds don't work nearly as well as TIPS do when employing a P/E switching strategy. If you have Excel, you can download a money market/S&P switching calculator at:

http://rehphome.tripod.com/re60.html

Even if you don't have Excel, the article shows that there is little advantage to switching back and forth between money markets and the S&P based solely on P/E. TIPS work much better with a switching strategy, at least if your money is in an IRA. In a taxable account you could wind up paying more in taxes than the TIPS generate in interest, depending upon your tax bracket and the inflation rate.

JWR1945
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Post by JWR1945 » Fri Oct 17, 2003 2:32 pm

JanSz wrote:Wonder how long that kind of port would last, and what kind of withdrawal rate would have been possible.

Indefinitely, still growing after 35 years.

Mike wrote:Even if you don't have Excel, the article shows that there is little advantage to switching back and forth between money markets and the S&P based solely on P/E.

That is what I thought. I ran some numbers. They surprised me.

Have fun.

John R.

JWR1945
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Post by JWR1945 » Fri Oct 17, 2003 4:12 pm

Mike wrote:Even if you don't have Excel, the article shows that there is little advantage to switching back and forth between money markets and the S&P based solely on P/E.

I just checked out the Retire Early article that you referenced.

intercst got switching wrong!

He is in error. He incorrectly lists three switching levels in that article. I think that he has corrected that information somewhere, sometime, but not at his own website and not now.

These are my results taken with the Retire Early Safe Withdrawal Calculator, Version 1.61, November 7, 2002, with no changes by me.

I checked switching between stocks and commercial paper, with 80% stocks when P/E10 is below 12.0 and 20% stocks when P/E is above 12.0. The thirty year Historical Database Rate was 4.7%! With 4.8% withdrawals, there was a single failure in 1911. It failed in 1941.

Without switching and with an 80% stock allocation, there was a single failure in 30 years at 4.0%. It was 1967. It failed in 1997. That is consistent with a reported 4.12% Historical Database Rate with a 60% stock allocation (without switching).

Have fun.

John R.

P.S. Thanks to JanSz! Without your question, we would never have discovered this.

P.P.S. Switching has worked all along. The reported results were in error!

JWR1945
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Post by JWR1945 » Fri Oct 17, 2003 5:25 pm

I have reconstructed what is shown at the Retire Early Home Page website that Mike referenced. Visit http://rehphome.tripod.com/re60.html

The only P/E10 threshold that works is 10.0. The other thresholds are meaningless. The center column with 60% stock allocations is meaningless. Only the first and the third column are used in the calculator. Otherwise, the results are accurate.

Here is a revised table.

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Low      High      Rate
50%      70%      3.85%
55%      65%      3.98%
60%      60%      4.12%
65%      55%      4.03%
70%      50%      3.91%
80%      40%      3.64%
100%      0%      2.31%


With allocations of 80% (below threshold) and 20% (above threshold), the Historical Database Rate is 3.05% for a threshold of 10.0. It is 4.79% for a threshold of 12.0.

Have fun.

John R.

JWR1945
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Post by JWR1945 » Sat Oct 18, 2003 8:06 am

Additional Information

This table lists the Historical Database Rates when switching allocations of stocks and commercial paper according to the switching threshold. Data were all taken from the Retire Early Safe Withdrawal [Rate] Calculator, version 1.61, November 7, 2002, as is and with none of my own modifications. Visit the Retire Early Home Page for more information.
http://rehphome.tripod.com/re60.html

Although the calculator has provisions for two thresholds (and three allocations), it implements only one. Data inputs at F19, B20 and I20 work. Inputs for I19 and F20 do not. That is, the Low PE to Mid PE threshold works and the stock allocations for the Low PE Years and the High PE Years work.

I discovered this when I modified a version to include switching between stocks and TIPS. The relevant code appears in the formula bar (at the top) when you highlight any cell in row 182 Stock Switch by P/E. Before then, I had assumed that all of the thresholds and allocations were active. So had others.

The data shows how Historical Database Rates vary with the P/E10 threshold. The stock allocation is 80% below the threshold and 20% above the threshold. The second column shows the 30-year HDBR results using the full set of data. For each threshold there was at least one failure when the withdrawal rate was increased by 0.01%. The third column shows the 30-year HDBR results using data from 1921-2002. For each threshold there was at least one failure when the withdrawal rate was increased 0.1%. The limited precision for the 1921-2002 entries is because it was necessary to reduce the data manually.

80% or 20% Stock Allocations, Switching, Stocks and Commercial Paper

Code: Select all

Threshold   1871-2002   1921-2002
10   3.05%   3.0%
11   4.15%   4.1%
12   4.79%   4.8%
13   4.85%   4.8%
14   4.46%   4.4%
15   4.43%   4.4%
16   4.49%   4.4%
17   4.45%   4.6%
18   4.48%   4.8%
19   4.34%   4.3%
20   4.34%   4.3%


In the modern era, early failures at thresholds of 10 and 11 occurred in (retirements started in) 1938. At thresholds of 12 and 13, early failures occurred in 1957. At higher thresholds, early failures occurred in the 1960s. (At thresholds of 14, 15, 19 and 20, early failures occurred in 1966 and 1967. At a threshold of 16, the first failure occurred in 1966. At a threshold of 17, the first failures occurred in 1963, 1966 and 1967. At a threshold of 18, the first failures occurred in 1961-1963 and 1965-1967.)

As noted earlier, switching improves performance considerably, even when the portfolio has commercial paper as its alternative to stocks.

Special Observation

These results surprise me because the 1871-2002 and 1921-2002 data are reasonably close to each other. Whether that holds up in general is yet to be seen. Without switching, the Historical Database Rates versus P/E10 data behaved differently in earlier times. There may be a common factor that switching suppresses.

Have fun.

John R.

Mike
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Post by Mike » Sat Oct 25, 2003 3:50 pm

I have verified that the above listed REHP calculator inputs do not work. The spreadsheet doesn't recalculate regardless of what values I place in the fields. Even nonsense numbers like 800% have no effect. This makes me wonder if this failure invalidates the entire REHP spreadsheet by causing it to mishandle crucial information for the data points that the inputs were designed to cover. It may simply ignore all years in the fields that were designated as "mid", or assign a pre programmed equity percentage for those years. Thank you for pointing this out John.

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Post by JWR1945 » Sat Oct 25, 2003 5:20 pm

Thanks, Mike.

I think that the problem is limited simply to the fact that the two inputs are ignored.

This goes back a while, but you may remember it. It was from a very early post of mine about The Modified RE Switching Calculator dated Sunday, Sep 07, 2003 at 12:22 pm CDT.
http://nofeeboards.com/boards/viewtopic.php?t=1365

I had provided some very detailed (spreadsheet) programming information and then wrote:

Notice that the formula shows $F$19, $I$20 and $B$20. Those correspond to cells F19, I20 and B20 respectively. Notice that there is no mention of cells I19 (or $I$19) and F20 ($F$20). Those are two of the cells in which you enter your switching criteria. Neither the "Switch to High PE"￾ threshold nor the allocation for "Mid PE Years"￾ have any influence on the results. The calculator does use the "Low PE to Mid PE"￾Â￾ entry to change allocations. The calculator uses the "Stock Allocation for Low PE"￾ and the "High PE Years"￾ entries for allocations.
If my memory serves me correctly, intercst (the creator of the Retire Early Safe Withdrawal Rate Calculator) originally wrote a program that used two levels for switching and it worked. He tried later to upgrade the calculator for three levels, but it had bugs. He has now returned to his original formula (in version 1.61).


I am not yet aware of any additional problems that would have affected us. There are some hidden surprises for someone who simply looks at the summary tables. For example, there are many years without treasury bond rates (because there were no bonds!). Understand that we are working with software that is only partially validated and there may be other bugs of consequence.

There are other, similar calculators. It is likely that all of the standard features are correct. Switching is a new feature. I believe that it is unique.

Have fun.

John R.

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