The Great SWR Investigation-Part 2

Financial Independence/Retire Early -- Learn How!
therealchips
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Post by therealchips »

Thanks, Gummy. It's a great chart. Now, I hestitate to ask, is there a readable derivation of that fancy equation and its assumptions and interpretation? I don't mean to push. I'm happy to see your work in progress.

Readable by whom?, gummy may sensibly ask.
He who has lived obscurely and quietly has lived well. [Latin: Bene qui latuit, bene vixit.]

Chips
JWR1945
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Post by JWR1945 »

This is a technical sketch of what gummy has presented.

The geometric sum formula (finite form) is this:
SUM(x,N) = 1 + x + x^2 + ... + x^N
Then (1-x)*SUM(x,N) = 1 - x^(N+1) since all of the intermediate terms cancel out.

Looking at SUM(x,N-1) = 1 + x + x^2 + ... + x^(N-1), then
(1-x)*SUM(x,N-1) = 1 - x^N or
SUM(x,N-1) = [1-x^N]/[1-x]

Multiplying by x, we find that
x*SUM(x,N-1) = x + x^2 + x^3 + ... x^N and also that
x*SUM(x,N-1) = (x*[1-x^N])/(1-x)

In his derivations, gummy presents a magic sum term gMS. It is gMS = (1/g1) + (1/[g1*g2]) + (1/[g1*g2*g3]) + ... + (1/[g1*g2*...gN]) where the terms g1, g2 and so forth are multipliers from one year to the next. For example, if you gain 6% in the first year, g1 = 1.06. If you gain 15% in the second year, g2 = 1.15. If you lose 3% in the third year, g3 = 0.97.

If all of the multipliers are the same, g = g1 = g2 = g3 = ... = gN, then gummy's magic sum gMS has the form of x*SUM(x,N-1) where x = 1/g. That is, gMS = (1/g)*SUM([1/g],N-1) = (1/g)*[1-(1/g)^N]/(1-[1/g]).

It is easier to work in terms of x, until the very last step.

gummy's formula for final portfolio balance of zero after N years corresponds to having the withdrawal fraction W satisfy (1-W*gMS) = 0. That is, W = 1/gMS = g*(1-[1/g])/(1-(1/g^N)) = (g-1)/(1-[g^(-N)]).

For a lognormal distribution, gummy has shown that the annualized (or geometric) return is M-(S^2)/2 (approximately), where M is the arithmetic return (the arithmetic average of annual returns) and SD is the standard deviation of annual returns.

After one year, a lognormal distribution has a return of exp^(M-[(S^2)/2]). That is the term g in the numerator. After N years, the lognormal distribution has a return of exp^(N*(M-([S^2]/2)). This is the term in the denominator.

What about the inflation term i? gummy has replaced the multiplier g (before inflation) with g/i (after inflation). He should also have a term i^(-N) in the denominator, replacing g^(-N) with (g/i)^(-N) or [g^(-N)]*[i^N]. At least, that is what I show.

Summary:
1) gummy's formula for safe withdrawals with a constant annualized gain (or 1 + the percentage increase expressed as a decimal) of g is (1-W*gMS) = 0. (When this term is not zero, it is multiplied by another. The product equals the current balance.)
2) gummy's formula is based upon having a lognormal distribution. That is, the gain multipliers g1, g2, g3 and so forth have a normal distribution of 1+M with a standard deviation of S. That is, the annual percentage increases are normally distributed about a mean of M with a standard deviation of S.
3) The annualized return of a lognormal distribution is M-([S^2]/2) to a very good approximation. It is less than the average return M. If the standard deviation is large, the annualized return can be much less than the average return.
4) If you solve the formula (1-W*gMS) = 0 for a constant gain multiplier g and replace g by its equivalent form, assuming a lognormal distribution, you end up with his formula.

That's my sketch. Depending upon how familiar that you are with gummy's previously supplied links, this may help guide you through or it may cause you to give up or it may cause you to learn how to play canasta.

Have fun.

John R.
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Post by gummy »

John:
Thanks for finding the error !!!

In a nutshell:
The terms in gMS are products of ratios like InflationFactor/GainFactor (as you've pointed out).
For lognormal distributions the Expected Value of such a ratio is: M = I / exp(m-s^2/2) (InflationFactor/AnnualizedReturnFactor)
That gives: gMS = M + M^2 + M^3 + ... + M^n = M(1 - M^n)/(1-M) (summing the powers)
and, substituting M = I / exp(m-s^2/2), gives 1/gMS as

with the error you found corrected :^).

Alas, knowing Mean(gMS) and SD(gMS) doesn't give me the distribution of gMS ... tho' it looks familiar when I plot samples!
... it may cause you to learn how to play canasta.

Amen. I might go back to that meself!

therealchips:
The derivation of the gMS stuff is here:
http://home.golden.net/~pjponzo/sensibl ... tm#FORMULA

Takin a page from John's book (!), I've added some explanatory stuff at the end, here:
http://home.golden.net/~pjponzo/distributions-stuff.htm
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Post by therealchips »

Thanks, John and gummy. I will have to derive each line of the proof myself, by hand with paper and pencil and following your derivations, to know what it means and whether I understand it. My homework for this forum is mounting up. Meanwhile, I'm reading Four Pillars and, of course, enjoying the current rally in US stocks. (S&P touching 1000 again. Imagine that.) I turned more cautious this month; I dropped my expected real rate of return from 4% (which I used for more than ten years) to 2.5%. My new-found level of caution is probably just the contrary indicator the market needed to really take off. If the market does persistently better than my projection, the consequences for me will be a rising standard of living. I can endure that. :wink:
He who has lived obscurely and quietly has lived well. [Latin: Bene qui latuit, bene vixit.]

Chips
JWR1945
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Post by JWR1945 »

I just looked at gummy's write up. I was just about to extend his work a little bit and I noticed that he is far ahead of me.

If I am not mistaken (IINM), gummy has continued to add to his explanations. Maybe not. I had only glanced over it briefly previously.

In any event I will examine several questions in light of his current write up. I find this very interesting, especially since gummy has been able to come up with an excellent closed form approximation.

Have fun.

John R.
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Post by ataloss »

incredible
John and Mary are both 50 years old. Both would like to retire as early as possible. Both plan to take 4 percent withdrawals from their portfolios and to live on $40,000 per year.

As of July 1, 2003, John has saved $1,000,000, and Mary has saved $500,000.

John retires on July 2, 2003. The Dow is at 8,000.

Over the course of the next 12 months, the DOW skyrockets to 16,000. Mary's Portfolio doubles in value to a value of $1,000,000 on July 1, 2004. She retires on July 2, 2004.

Which retirement is more safe, or are they equally safe?

Intercst says that both retirements are 100 percent safe (let's call it 95 percent if you count in the chance of the future being unlike the past). He says the two retirements are equally safe.


hocus 5/30
http://nofeeboards.com/boards/viewtopic ... safe#p7109

Today TMF:
Intercst says that a 4% withdrawal survived all 30-year payout periods starting on Jan. 1st from 1871-2002.

John's portfolio "skyrocketed" along with Mary's and he now has about $2 million. If he is still taking about a $40,000/yr. withdrawal, his retirement is obviously "more safe" since that's only 2% of $2 million.
Have fun.

Ataloss
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Post by JWR1945 »

Intercst says that a 4% withdrawal survived all 30-year payout periods starting on Jan. 1st from 1871-2002.

John's portfolio "skyrocketed" along with Mary's and he now has about $2 million. If he is still taking about a $40,000/yr. withdrawal, his retirement is obviously "more safe" since that's only 2% of $2 million.

This is new and different. Previously, 100% safe always meant 100% safe so that both would be 100% safe. (hocus's parenthetical 95% was unfortunate.)

That led to an obvious paradox when portfolios fall. Whenever hocus mentioned that paradox, the answer was always the same: 100% safe is 100% safe.

Have fun.

John R.
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Post by hocus »

This is new and different. Previously, 100% safe always meant 100% safe....

The point you are making here is true enough, but it is not the point that I was trying to convey with the John and Mary scenario.

The SWR is calculated on the day of retirement. My point is that, on the day of retirement, both the John retirement and the Mary retirement are said to be 100 percent safe (or 95 percent safe, if you believe that there is a 5 percent chance that the future will be worse than the past).

It is absurd to say that the John and Mary retirements had equal propsects of safety on the day of retirement. John's portfolio value presumably possessed something close to $1,000,000 of real value. Mary's obviously did not. About $500,000 of her porfolio value was "real," the rest was "fluff," or in Bernstein's quoting of Bogle, "noise" rather than signal.

John's portfolio is generally comprised of non-fluff, signal type assets. Bernstein is saying that that sort of stuff will last you 30 years.

Mary's stuff is largely comprised of fluff, noise type assets. Bernstein is saying that that stuff looks good in the short term, but that it doesn't last.

I am saying that, if you are going to assess the safety of a portfolio, you must do something to assess how much of your portfolio is fluff. Mary's $1,000,000 in assets is comprised of a lot more fluff than John's, but the conventional methodology treats the two as if they are the same. There is no adjustment in the conventional methodology for the fluff factor. There is no adjustment for changes in valuation levels.
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Post by JWR1945 »

hocus
I am saying that, if you are going to assess the safety of a portfolio, you must do something to assess how much of your portfolio is fluff. Mary's $1,000,000 in assets is comprised of a lot more fluff than John's, but the conventional methodology treats the two as if they are the same. There is no adjustment in the conventional methodology for the fluff factor. There is no adjustment for changes in valuation levels.

This is a very good point.

The reason for my post was that someone might have concluded that you had misrepresented intercst's position previously. The fact is that you represented his position accurately. He has now changed his position.

Have fun.

John R.
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