Looking deeply into the SWR Equation

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JWR1945
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Looking deeply into the SWR Equation

Post by JWR1945 »

Looking deeply into the SWR Equation

Part 1 of 4 Parts

The Problem

I was working on one of KenM's questions when I ran into a brick wall.

KenM had asked what happens to portfolio safety versus time. In a hypothetical example, he starts with a retirement portfolio designed to last 40 years at a high level of safety (such as 95%). If a particular retiree has been unfortunate and his portfolio balance has dropped during the first ten years, is his portfolio as safe as it was originally? Phrased differently: assume that someone else starts out with a portfolio that has an initial balance equal to the actual balance that the first retiree ended up with. Assume that this new retiree withdraws the same dollar amount as the first. Does he have the same level of safety over the next 30 years that the first retiree had when he started out (in this case 95%)?

The answer is no, of course. The 95% probability applied at the start. After ten years, our actual retiree was found to be among the less fortunate. His chances of success, going forward, have fallen. Some of the others, starting out under similar circumstances but at different times, would be among the very fortunate who had seen their portfolios grow. Their odds of success, going forward, would have increased. If you combined the chances of all of the various groups, the overall safety would still be 95%. That single number covered everybody at the start. After ten years, each group of retirees has a new number based on its own experiences.

In any event gummy has worked out a basic withdrawal rate formula and a lot of statistical formulas to go along with it. The formula looks like this:

Your portfolio balance after N years/your initial portfolio balance = a Gain_Product_Term * (1 - w*gMS), where gummy had come up with a lot of elegant mathematics describing the statistical behavior of the second term (1 - w*gMS). The second term can be used to identify when a portfolio fails. It would be zero or negative. From gummy's formulas, I could convert the statistics on the (1 - w*gMS) term into statistics on whether the portfolio would succeed. (The Gain_Product_Term is always positive. It is never zero. It is never negative.)

Then I wanted to look at a particular balance after a specified period of time and see what would happen from there. Again, I had full knowledge about the (1 - w*gMS) term. But what would I use for the Gain_Product_Term? I could not make it up out of thin air since it is calculated from the same annual investment returns as gMS (gummy's Magic Sum formula). It is not arbitrary.

Thud. I had hit a brick wall.

Continued in the next post.

Have fun.

John R.
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Post by JWR1945 »

Looking deeply into the SWR Equation

Part 2 of 4 Parts

The Insight

Then peteyperson shed some light when he mentioned one of John Bogle's findings.

Peteyperson has been reading some good financial literature and he has been thinking about how to manage cash buffer accounts. A simple but effective approach to insure safety is to have enough cash on hand to cover all of your withdrawals for an extended period of time. This gets you away from having to sell stocks when prices are down. Owning stocks (or similar investments) will eventually provide the growth necessary to replenish your cash account and to extend the life of your retirement portfolio.

John Bogle cited a recent example of its taking over fifteen years for stocks to produce a positive real return (i.e., a positive return after adjusting for inflation). Remember the 1970s in the United States? Remember Whip Inflation Now buttons and Stagflation? That's when it happened.

Fifteen years with zero appreciation is a long time and peteyperson has come up with some interesting ideas about how to handle it. We will certainly want to continue looking into this particular problem.

But as for me and related to my own efforts, I saw some daylight. That Gain_Product_Term in gummy's equation is just as simple as what peteyperson was preparing for and what John Bogle had warned about. You only need two numbers to calculate the Gain_Product_Term. Divide the final value of your index by the initial value. It is that simple. For analytical purpose, you can examine a series of cases in which you look at the starting index value and the final value. Everything in between is part of the (1 - w*gMS) term. The exact sequence of good years and bad years does not affect the first term. It does affect the latter.

Continued in the next post.

Have fun.

John R.
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Post by JWR1945 »

Looking deeply into the SWR Equation

Part 3 of 4 Parts

The Mathematical Details

Gummy has set up the mathematical formulation to be simple enough and powerful enough at the same time.

These are the various terms:
a) The gain multipliers are one plus the investment return for a particular year. If you gain 12% in the third year, g3 = 1.12. If you lose 8% in the fourth year, g4 = 0.92.
b) The balance at year N is Bal(N), where N is an integer. The initial balance is Bal(0).
c) The withdrawal percentages (actually, the percentages divided by 100%) are w1, w2, w3 and so on for the first, second and third years (and so on). The withdrawal amounts are w1*Bal(0), w2*Bal(0), w3*Bal(0) and so on. When all withdrawal percentages are identical, we simply use w = w1 = w2 = ... = wN.
d) The Gain_Product_Term at year N = g1*g2*...*gN. As gummy points out, this is also the Final Balance/the Initial Balance (or cumulative total return) when there are no withdrawals. This also equals the Final Index Value/the Initial Index Value.
e) Gummy's Magic Sum at year N = gMS(N) = 1/(g1) + 1/(g1*g2) + 1/(g1*g2*g3) + ... + 1/(g1*g2*g3*...*gN).
f) For those with extensive technical backgrounds, I will write w = (w1, w2,..., wN) and gMS = ( 1/(g1), 1/(g1*g2), 1/(g1*g2*g3),..., 1/(g1*g2*g3*...*gN) ). I will write w*gMS for the inner product of these two vectors = (w1/[g1]) + (w2/[g1*g2]) + ... + (wN/[g1*g2*...*gN]). (In fact, gummy has set up the equations in a full matrix form.)

These are the equations for the portfolio balances.
a) Bal(1)/Bal(0) = g1 - w1 = g1*(1 - [w1/g1]). That is, the balance grows by g1*Bal(0) but an amount w1*Bal(0) is withdrawn. Remember that all gain multipliers are positive and notice the factorization.
b) Bal(2)/Bal(0) = g2*Bal(1)/Bal(0) - w2 = g2*(g1*[1 - (w1/g1)] ) - w2 = g2*g1*(1 - [w1/g1] ) - w2 = g2*g1*(1 - [w1/g1] ) - g2*g1*(1/g2*g1)*w2 = g2*g1*[(1 - [w1/g1] - (1/[g1*g2])*w2] = g2*g1*(1 - [w1/g1] - [w2/(g1*g2)] ) = Gain_Product_Term(2)*(1 - [w1/g1] - [w2/(g1*g2)] ). Notice the factorization and how gummy's Magic Sum formula comes into play when all withdrawals are the same w = w1 = w2.
c) You may have noticed the pattern. Bal(3)/Bal(0) = g3*Bal(2)/Bal(0) - w3 and when you factor out the Gain_Product_Term, you find that Bal(3)/Bal(0) = Gain_Product_Term(3)*(1 - [w1/g1] - [w2/(g1*g2)] - [w3/(g1*g2*g3)] ).
d) When all of the withdrawals are the same, Bal(N)/Bal(0) = Gain_Product_Term(N)*(1 - w*gMS(N)).
e) When the withdrawals vary, Bal(N)/Bal(0) = Gain_Product_Term(N)*(1 - w*gMS(N)).

It is very easy to adjust for inflation or decide not to. If all of the gain multipliers are adjusted for inflation, then the withdrawal rate is adjusted for inflation. If not, then the withdrawal rate is not adjusted to match inflation. If you model portfolio returns and inflation sequences separately (as gummy does, but as raddr does not do), you simply produce your gain multiplier sequences (by a random number generator or as determined from some theoretical requirement) and inflation sequences individually. You divide each gain multiplier by the corresponding inflation term for that year (e.g., if you have 2% inflation, you divide by 1.02). If you introduce mean reversion (as raddr does, but as gummy does not do), you limit yourself to using only those gain multiplier sequences that satisfy the appropriate restrictions. (I.e., the standard deviation decreases with the number of years N faster than 1/[the square root of N] and in accordance with the historical data.)

All of the gain multipliers are single year values. When given any two: the arithmetic mean and/or the annualized return and/or standard deviation, gummy's approximation allows you to calculate the third. Annualized (or geometric) mean return = arithmetic mean + (1/2)*(standard deviation)^2.

Continued in the next post.

Have fun.

John R.
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Post by JWR1945 »

Looking deeply into the SWR Equation

Page 4 of 4 Pages

Special Cases

We can learn a lot just by looking at gummy's formula.

Look at the two terms individually. The Gain_Product_Term does not depend upon the sequence of the gain multipliers. For example, g1*g2 = g2*g1. When you see a single number, this product is what someone is talking about. This product is reasonably well behaved statistically.

The other term depends heavily on exact sequences. That is, a calculation such as (1/g1) + (1/[g1*g2]) is usually quite different from (1/g2) + (1/[g1*g2]). This is what William Bernstein is focusing on when he mentions the dangers arising from the exact sequences of returns.

Notice that this introduces a second mechanism by which valuations affect portfolio safety. Normally, we think only about the Gain_Product_Term. Using the price at any specific date in the future, a lower starting price will produce a greater ratio of that future price to the starting price. Always. It is a mathematical certainty. That is all that is needed to calculate the Gain_Product_Term. It is the overall return when you make no withdrawals.

From the formula for gMS, we see that valuations also affect portfolio safety to the extent that they influence returns in the earliest years. Even if the overall return is the same in the absence of withdrawals, portfolios are the safest when the earliest returns include substantial gains. Portfolios are in great danger when the earliest returns include heavy losses. Higher initial valuations are more likely to be accompanied by lower returns in the earliest years. Higher initial valuations mean less safety during retirement, statistically, even when annualized returns are identical.

Notice that gummy's Magic Sum always increases with the number of years (since all of the gain multipliers are positive). Now consider the effects of having no lower limit on the gain multipliers. Specifically, consider the effects of volatile investments such as stocks, which can have deep losses. Eventually, you will have at least one year with a very deep loss. That means, that your gain multiplier will be low and 1/(the gain multiplier) will be very large. That means that gMS will be very large. That means that (1 - w*gMS) is likely to be negative (assuming that you are making withdrawals and that you are not putting money back into your portfolio). Even if one year's deep loss does not make gMS big enough to make (1 - w*gMS) negative, gMS always grows. There will be other bad years in the future. Under almost all circumstances, the term eventually becomes negative and your portfolio has been depleted. (You can create theoretical cases in which some occasional, spectacular returns isolate the effects of later years.)

Now look at what happens if your portfolio has a lower limit on the gain multiplier. A mixture of stocks and cash (or stocks and bonds) will do this. Even if your companies go bankrupt and your stock holdings become worthless, you will still have the fixed income component. With re-balancing, you can purchase new stocks (presumably at bargain rates from solid companies) to reintroduce a growth component. The term gMS still grows continually. However, those theoretical cases that would prevent the term (1 - w*gMS) from going negative suddenly become quite realistic, at least in a statistical sense. You need some occasional, good returns to isolate the effects of later years. You no longer need spectacular returns because you have never incurred the heavy losses that create huge values among the individual components of gMS, 1/(the gain multipliers).

Now let's get a little fancy. You could design a withdrawal scheme that matches the annual variation in stock prices each year for a specified number of years N. Starting with an initial value of w0, you make the first year's withdrawal equal to g1*w0. You make the second year's withdrawal equal to g2*w1. That is, you scale your withdrawal amount up or down strictly according to your previous year's withdrawal amount and its latest gain multiplier (which can be a decrease). We see that g2*w1 = g1*g2*w0. When we look at the form of (1 - w*gMS) suitable for varying withdrawal sequences, it becomes (1 - w1/g1 - w2/[g1*g2] - w3/[g1*g2*g3] - ...) = [1 - ( [w0*g1]/[g1] ) - ( [w0*g1*g2]/[g1*g2] ) - ( [w0*g1*g2*g3]/[g1*g2*g3] ) - ...] = (1 - w0 - w0 - w0 -...). If you select w0 so that w0 = 1/N, your portfolio will last exactly N years. Your portfolio balance after k years has the Gain_Product_Term(k) multiplied by (1- k*w0). The Gain_Product_Term(k) is simply the cumulative stock index increase at year k.

Now let us become a little bit more realistic. So far, I have talked about gain multipliers and index values in a casual manner. The proper terms to use assume total reinvestment of dividends and this can be difficult to construct. Actually, it is tedious. Seldom is there an easy way to do it. Dividends depend upon stock prices and your portfolio allocations affect their size.

Since we are making estimates, looking forward, and we are looking at various sensitivities, we can select our problems to ease our calculation burdens. For example, peteyperson's reference to John Bogle's example suggests an excellent special case. We can examine when the annualized return of stocks is zero, both in terms of stocks alone and when dividends are reinvested. (Heaven help us if we have to include taxes.)

If we are using the price index alone, we can select our withdrawal strategy as one that simply lives off the dividends. We can adjust for inflation very simply by using real prices and real dividends (scale all index values by the ratio of the consumers price index of the base year divided by the consumer price index of the current year). We can interpret our results later by considering small adjustments to the historical dividends. This strategy should work indefinitely even when the long-term price growth is zero.

I mention this example because of my recent finding that tracking S&P 500 index values during the first decade was quite successful (in a historical sense) for monitoring portfolio safety. Until recently, the safe withdrawal rate numbers came very close to this exact strategy. Safe withdrawal rates and dividend yields were almost identical. My guess is that my zero price growth assumption corresponds very closely to the actual situation among the worst case historical conditions. Reflecting upon today's lower dividend yields suggests that you should do one of two things: reduce your withdrawal rates so that your strategy matches dividend yields once again or select stocks with higher dividend yields. Then, make adjustments for buying and selling stocks according to your own projections of long-term price growth and/or price decreases. Take care to get reasonable prices.

We can estimate long-term stock returns for the next decade or so and then use that estimate for the Gain_Product_Term. (When we allocate among several asset classes, we combine the effects of all.) We can then examine the effects of volatility and exact sequences.

This is just the beginning of another way of looking at the Safe Withdrawal Rate issue. We have used the historical sequence method, along with raddr's related sensitivity studies. We have used two complementary Monte Carlo simulations (from raddr and gummy). Now we are designing withdrawal rate strategies. We can look at what has been typical historically, extract certain features directly, and then introduce them into equations that are always true. (The equations are always true. The inputs and assumptions are not necessarily true.) This should be of assistance in designing cash buffers and in tracking portfolio safety. By the nature of this approach, we would emphasize sensitivity studies, similar to using the Monte Carlo models. This emphasis will help make monitoring portfolio safety much easier and much more accurate.

Have fun.

John R.
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Post by peteyperson »

<cough> Err. sorry?

8)

Well I'll speak for myself when I say that almost all of the formulas went right over my head. The only possible way I might be able to understand them is if I see numbers in an example, with the example explained every single step along the way. Having said that, clearly four excellent well articulated posts.

I look at the issue simply. If you have assumed a certain return on your collection of assets and if over the long haul that delivers less, then you will have to reduce your w/d rate. A serious cash buffer will offer some insurance against selling low and allow you to ride out many of the 1-5 year dips in the market (turning 1929 - 1932 into a minor blip, for instance). Limiting your exposure on such asset classes with high standard deviations will help also. These steps in combination give you the best chance, along with perhaps having more modest estimates of future performance and building in a cushion in the budget where non-essentials can be trimmed for a time if things look bad in the medium term. I think together these are a powerful combination to the issue of volatility and uncertain future returns.

That's my simplistic take on it.

Petey




JWR1945 wrote: Looking deeply into the SWR Equation

Page 4 of 4 Pages

Special Cases

We can learn a lot just by looking at gummy's formula.

Look at the two terms individually. The Gain_Product_Term does not depend upon the sequence of the gain multipliers. For example, g1*g2 = g2*g1. When you see a single number, this product is what someone is talking about. This product is reasonably well behaved statistically.

<snip>

Have fun.

John R.
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Post by hocus »

Higher initial valuations mean less safety during retirement, statistically, even when annualized returns are identical.

This is counter-intuitive. I would like to understand better why it is so.

You are saying it is because a higher valuation increases the prospects of a bad return year turning up in the early years of a retirement. Is this due to a reversion -to-the-mean phenomenon?

I have a question that will seem basic to numbers-oriented people, but perhaps the answer will help me better understand the point you are making here. If you took all of the annual percentage gains and losses in S&P stocks that took place in the 30-year period from 1929 to 1959 and applied them in the same order to a portfolio for a retirement beginning in 2000 (while also making 4 percent withdrawals), I presume that the value of the latter-era retirement would go to zero at the end of the 30-year period but not before. It does not seem to me that it would change things that you started from a higher valuation level and that the large percentage losses in the early years were therefore in a sense "worse.(because the big fall was taking place in a more puffed-up construct)." Is that right?

Presuming that it is, then why is it that higher initial valuations mean less safety even when annualized returns are identical? Is there some greater volatility injected into the mix when you are starting from a higher valuation level?
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RE: Looking deeply into the SWR Equation

Post by BenSolar »

:shock:

Wow, John

You've been putting some extensive thought into this, eh? It will take me a read or two (or three or four) to absorb it and comment quasi-intelligently, but I wanted to say 'good work' before then!

Good work! :)
"Do not spoil what you have by desiring what you have not; remember that what you now have was once among the things only hoped for." - Epicurus
JWR1945
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Post by JWR1945 »

hocus:
Presuming that it is, then why is it that higher initial valuations mean less safety even when annualized returns are identical? Is there some greater volatility injected into the mix when you are starting from a higher valuation level?

I am assuming that higher initial valuations are more likely to result in lower returns in the first few years than if you start with lower valuations. That is, I am assuming that the initial valuation affects the order of future returns. My assumption is arguable because random effects can dominate in the near-term and mask this tendency.

If you were to take the exact returns from 1929 to 1959 and shift their order, the ending balances would vary all over the place provided that you were making withdrawals. Only if you withdrew nothing would the final balances all agree.

Annualized returns are returns that happen when don't make any withdrawals. If your stock holdings drop to half of their original value in the first year, but you withdraw your preplanned, fixed amount (with or without adjusting for inflation), you have to sell a lot of shares. If, instead, your stock holdings double in value in the first year, you only have to sell a few shares. In the first case, you end up with many fewer shares after the first year. In both circumstances, the dollar amounts that you have sold are identical.

Now look at the first example and assume that your stock returns double in value. If you hadn't sold any shares, you would be back to where you started. Since you have sold many shares, your recovery is limited.

Look at the second example and assume that your stock returns fall in half. Your fall would not be as bad because you would have sold very few shares. Once again, if you had not made any withdrawals at all, you would have ended up where you started.

For the first case, assume that you have 1000 shares of a $100 stock ($100 000 total) and you wish to withdraw $4000 per year (4%). In the first case, at the end of the first year, your stock is worth $50 per share. Your total holdings are down to $50 000. You withdraw your preplanned, fixed amount of $4000. That requires you to sell 80 shares. Your portfolio balance is $46 000 after making your withdrawal. Notice that this is 8% of your current balance even though it is still 4% of your original balance. At the end of the first year, you have 920 shares.

Your holdings double in the second year. Each share is now back to $100 per share. You holdings have grown to $92 000. This is twice the $46 000 previous balance after making withdrawals. It is also 920 shares at $100 per share. You now withdraw your preplanned, fixed amount of $4000. You get it by selling 40 shares (at the current $100 per share). You end up with 880 shares. Your balance, after making your withdrawal is $88 000.

Look at the second case. At the end of the first year, your stocks have doubled. You have 1000 shares selling at $200 per share. You sell 20 shares at $200 per share for $4000. Notice that this is only 2% of the current portfolio's value, but it is still 4% of your portfolio's initial value. You now have 980 shares remaining and your portfolio balance is $196 000 and you have withdrawn $4000.

In the second year, your stocks fall in half, back to $100 per share. You have 980 shares. They are worth $98 000. You must sell 40 shares to withdraw $4000. You end up with 940 shares. They are priced at $100 per share. Their value totals $94 000.

In the first case, the gain multipliers were 0.5 followed by 2.0. Absent withdrawals, their product (0.5)*(2.0) = 1 = the initial amount. In the second case, the gain multipliers were 2.0 followed by 0.5. Their product is (2.0)*(0.5) = 1, which is again equal to your initial amount provided that you have made no withdrawals.

The second portion of the equation (1 - w*gMS) has w = 4% (of the initial balance) and gMS = [ 1/(g1) ] + [ 1/(g1*g2) ]. This portion is sensitive to which gain multiplier comes first. In the first example, the stock dropped in the first year so that g1=0.5. It doubled in the second year, so that g2=2.0. Calculating gMS, we find that gMS = [ 1/(0.5) ] + [ 1/(0.5*2.0) ] = 2 + 1 = 3.

In the second example, the gain multiplier is 2.0 in the first year (so that g1=2.0) and it is 0.5 in the second year (so that g2=0.5). The calculation is a little bit different this time. gMS = [ 1/(2.0) ] + [ 1/(2.0*0.5) ] = 0.5 + 1 = 1.5.

In our formula, w = 4% so that (1 - w*gMS) = (1 - 0.04*3) = (1 - 0.12) = 0.88 in the first example and (1 - w*gMS) = (1 - 0.04*1.5) = (1 - 0.06) = 0.94 in the second example. In both cases, g1*g2 = 1 (or 0.5*2.0 = 2.0*0.5 = 1). The full formula gives the right answers. In the first example the balance at the end of the second year is 1*0.88*(the $100 000 initial balance) or $88 000. In the second example the balance at the end of the second year is 1*0.94*(the $100 000 initial balance) or $94 000. (I used the form Gain_Product_Term*(1 - w*gMS)*Bal(0), with the Gain_Product_Term = 1 in both cases, the (1 - w*gMS) terms differ because the sequences were different in the two cases and the Bal(0) term = $100 000 in both cases.)

I hope that this helps.

Have fun.

John R.
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Post by JWR1945 »

To Mr. Peteyperson, Sir!

I hope that you will understand why I have been so slow getting back to your posts. You have made a significant advance which I want to help develop. I think that your ideas can be generalized and that they will lead to a host of useful retirement portfolio withdrawal strategies. As it happens your earlier comments have led to this distraction. I think that these results will apply to your ideas and be helpful.

Have fun.

John R.
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Post by JWR1945 »

peteyperson
I look at the issue simply. If you have assumed a certain return on your collection of assets and if over the long haul that delivers less, then you will have to reduce your w/d rate. A serious cash buffer will offer some insurance against selling low and allow you to ride out many of the 1-5 year dips in the market (turning 1929 - 1932 into a minor blip, for instance). Limiting your exposure on such asset classes with high standard deviations will help also....

I couldn't let this one go by. Around that time, President Herbert Hoover told our nation that the economy was doing well. We were not entering on of those horrible panics of earlier eras. We were only entering a depression. Or, as you would say, a minor blip!

Have fun.

John R.
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Post by peteyperson »

This is patently wrong and the opposite of what is true. (Explanation below)
JWR1945 wrote: Higher initial valuations mean less safety during retirement, statistically, even when annualized returns are identical.


If a high valuation reduces down to a more rational valuation over time, that doesn't necessarily create a situation where you are getting a sub-par return. You have the room in the initial high valuation to let some of the air out of the valuation over time and still receive reasonable levels of growth. It might not be immediately obvious because the value of your portfolio (uncorrected for high P/E) is dropping and that would throw most people.
JWR1945 wrote: I am assuming that higher initial valuations are more likely to result in lower returns in the first few years than if you start with lower valuations. That is, I am assuming that the initial valuation affects the order of future returns. My assumption is arguable because random effects can dominate in the near-term and mask this tendency.



The valuation level at the point of retirement is an interesting issue. I feel it is true that a high valuation gives you some wriggle room. This works because your prices are pumped up and you have to sell less shares to deliver your $4,000. But only if valuation is taken into account at the time. If people blindly retire thinking they can take 4% from a market investment priced at 28.2 P/E (2x historical norm), then when the market reverts to the mean (and sometimes lower still like in the 1970s) they're basically screwed with their pants on as they withdrawal $4,000 but take out 8% a year..

I'll run thru the possibilities:-

If you retire at the point of low valuation, perhaps half the historical P/E of 14.1, down to 7.05.. you will need to ensure $4,000 is 4% of your assets at that valuation at that start point.

If you face a sudden increase shortly after retiring with a high valuation, then you have even more wriggle room still.

If you face a sudden increase shortly after retiring with a low valuation, you'll be in a good position there too. You'll be taking $4,000, 2% of the assets out. Originally it was $4,000, 4% of the assets.

If you face a sudden decrease shortly after retiring with a high valuation, you should be fine. Your 2% w/d totalling $4,000 will now be a 4% withdrawal which still totals $4,000.

If you face a sudden decrease shortly after retiring with a low valuation (P/E of 7 was the example shown), then you're in deep shit (John's worst nightmare scenario). P/E will be at incredibly low levels, the confidence in stock market investing will have dramatically reduced at that time. This happened in the 1970s and confidence returned. Indeed, we've seen bumper performance since that time. Here you are the most vulnerable you're likely to be. If you calculated your $4,000 withdrawal based on your low P/E valuation at the time, then your withdrawal is still $4,000 but it is now 8%. Bleeding a low valuation portfolio having suffered more falls since, it cannot withstand depletion of the asset base with cost of living withdrawals. You're like The Enterprise with shields down and one more torpedo on target will finish you off. This is where starting FIRE with a multi-year cash fund is a critical strategy!

John, as you can see, you only included some of the possible outcomes, I've tried to demonstrate them all above. High valuation gives you plenty of room for market correction if your withdrawal percentage was based around mean valuation levels. If you don't do that, if you say I'm taking 4% of this inflated valuation, then when the market reverts to mean, you'll be taking 8% and realising you did your math all wrong.

So I agree that retiring at different valuation levels does change the level of risk & the risk factors involved. A 50/50 stock to cash type investment portfolio balances those risks even if they are the low valuation (using that as your baseline valuation from the start point) and you drop another 50% a year into FIRE. It limits the slide of your total assets as a whole and removes the need to sell at stupidly low levels (I'm thinking here that you retire at half average P/E, then it halfs again a year into FIRE as an extreme case). The cash fund allows you to weather even this. Non-essential budget items that can be cut back or cut out in these times allows further flexibility.

P.S. Just to throw a spanner in the works, one could even argue that a low valuation when FIREing, is a safer position to start from. You've only go so much further you can drop down to... The standard deviation would also indicate from that low point, you're statistically much more likely to go back up than drop still further (I'm thinking FIREd at P/E of 7). It does have the penalty that your investments haven't probably done all that well and are held down by that low P/E valuation. This could delay the FIRE date as you save more to plug the gap left by all your stock investments having a low valuation perhaps a year before you planned to FIRE. This was one of the factors why hocus sold out of the market when he got near the time he wanted to FIRE. Reducing your market exposure gradually down to 50% in the last 5-10 years before FIRE date would help there, though it would limit your growth just at the point where you have the most invested to do some good! The double edged sword.

Now... everyone still with me? Class dismissed! :roll:

Petey

JWR1945 wrote: Annualized returns are returns that happen when don't make any withdrawals. If your stock holdings drop to half of their original value in the first year, but you withdraw your preplanned, fixed amount (with or without adjusting for inflation), you have to sell a lot of shares. If, instead, your stock holdings double in value in the first year, you only have to sell a few shares. In the first case, you end up with many fewer shares after the first year. In both circumstances, the dollar amounts that you have sold are identical.

Now look at the first example and assume that your stock returns double in value. If you hadn't sold any shares, you would be back to where you started. Since you have sold many shares, your recovery is limited.

Look at the second example and assume that your stock returns fall in half. Your fall would not be as bad because you would have sold very few shares. Once again, if you had not made any withdrawals at all, you would have ended up where you started.

For the first case, assume that you have 1000 shares of a $100 stock ($100 000 total) and you wish to withdraw $4000 per year (4%). In the first case, at the end of the first year, your stock is worth $50 per share. Your total holdings are down to $50 000. You withdraw your preplanned, fixed amount of $4000. That requires you to sell 80 shares. Your portfolio balance is $46 000 after making your withdrawal. Notice that this is 8% of your current balance even though it is still 4% of your original balance. At the end of the first year, you have 920 shares.

Your holdings double in the second year. Each share is now back to $100 per share. You holdings have grown to $92 000. This is twice the $46 000 previous balance after making withdrawals. It is also 920 shares at $100 per share. You now withdraw your preplanned, fixed amount of $4000. You get it by selling 40 shares (at the current $100 per share). You end up with 880 shares. Your balance, after making your withdrawal is $88 000.

Look at the second case. At the end of the first year, your stocks have doubled. You have 1000 shares selling at $200 per share. You sell 20 shares at $200 per share for $4000. Notice that this is only 2% of the current portfolio's value, but it is still 4% of your portfolio's initial value. You now have 980 shares remaining and your portfolio balance is $196 000 and you have withdrawn $4000.

In the second year, your stocks fall in half, back to $100 per share. You have 980 shares. They are worth $98 000. You must sell 40 shares to withdraw $4000. You end up with 940 shares. They are priced at $100 per share. Their value totals $94 000.
Last edited by peteyperson on Wed Jul 16, 2003 2:18 pm, edited 8 times in total.
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Post by peteyperson »

I thank you and I concur.

I have just finished writing a reply which extends out your example of different valuations points on FIRE and major rises or drops after that. I found that you didn't include all possible scenarios and your analysis was getting blindsided as it was incomplete. I hope I was correct in reaching this assumption.

I said, okay we start with low or we start with high. We then go lower or higher relative to where we began. What is the net result, where are the risks there. That was my approach, a continuation of yours. It made clear where the risks were you were concerned about.

Thanks for your comment regarding slow reply. I think we have made more progress in the last few days than I have seen previously all in a flurry, so I am encouraged. This even though ataloss feels were are only discussing valuation and no new advances are being made. I believe major advances are happening. Now I understand how w/d rate moves in hocus's approach and why a high P/E creates a low w/d percentage but same $ amount, I have been able to move my thinking (and hopefully advance it) in applying this new understanding onto the various possible valuation scenarios at FIRE and movement after FIRE. I'm repeating myself a bit tonight, sorry, been a long day here.

Petey



JWR1945 wrote: To Mr. Peteyperson, Sir!

I hope that you will understand why I have been so slow getting back to your posts. You have made a significant advance which I want to help develop. I think that your ideas can be generalized and that they will lead to a host of useful retirement portfolio withdrawal strategies. As it happens your earlier comments have led to this distraction. I think that these results will apply to your ideas and be helpful.

Have fun.

John R.
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Post by peteyperson »

John,

I actually have a book out from the library called " Rainbow's End " all about the crash and the reasons for it. It will no doubt talk about unchecked margin borrowing, bankruptcy of individuals unable to make the margin calls, nor sell property as many were in the same position. This of course was also a time when those with money to invest did stupidly well because there were bargains to be had. This links nicely into the Ben Stein book that hocus is reading about.

If you hadn't needed to sell stocks and had not invested on margin as a FIREd person, then 1929 was a complete irrelevancy. If it had lasted over 10 years then it would have taxed a good cash buffer however but most major drops don't last longer than 3 years (Siegel data).

P.S. The economy probably was doing well. People were just overleveraged and a major drop in valuations screws with stocks bought on margin. It multiplies the risk factor many times over when that happens, which has nothing to do with the economy per se. You could argue the economy was overcooked because of the rampant speculation, investments in the market on borrowed money. This perhaps is similar to Japan and how their economy took a tumble because it was financed with too much debt which wasn't substainable long term. But like I say, I haven't read the book yet <smirk>.

Petey


JWR1945 wrote: peteyperson
I look at the issue simply. If you have assumed a certain return on your collection of assets and if over the long haul that delivers less, then you will have to reduce your w/d rate. A serious cash buffer will offer some insurance against selling low and allow you to ride out many of the 1-5 year dips in the market (turning 1929 - 1932 into a minor blip, for instance). Limiting your exposure on such asset classes with high standard deviations will help also....

I couldn't let this one go by. Around that time, President Herbert Hoover told our nation that the economy was doing well. We were not entering on of those horrible panics of earlier eras. We were only entering a depression. Or, as you would say, a minor blip!

Have fun.

John R.
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Re: RE: Looking deeply into the SWR Equation

Post by peteyperson »

Ben, ben..

You're missing all the fun!

Petey



BenSolar wrote: :shock:

Wow, John

You've been putting some extensive thought into this, eh? It will take me a read or two (or three or four) to absorb it and comment quasi-intelligently, but I wanted to say 'good work' before then!

Good work! :)
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Post by galagan »

John,

I've enjoyed the analysis, although I fear that it may be difficult for new board members to follow discussions here, given the speed with which new ideas flow from old ones. If you weren't part of all the conversations that came before, it's hard to catch up!

I'll throw a few questions in to muddy the waters a bit:

1. I find it interesting to consider the hypothesis that high valuations do not affect the product of all future annual returns but merely changes the order in which those annual returns come. But I'm not sure how to reconcile that with your statement in part 4 that the product of all future annual returns must mathematically vary depending on the starting valuation. Am I missing a distinction, or are you talking about two separate hypotheses?

2. I also find the initial discussion in part 1 interesting, in which you suggest the possibility of dynamic adjustment of investment and withdrawal strategies in mid-retirement to accommodate changing conditions. I may have missed longer discussions of this, but it appears that you are presently focusing on adjusting withdrawal strategies while leaving investment strategies constant. Is this a deliberate decision - are we trying to consider each variable separately? It seems to me that a retiree who suffered unfortunate initial results would be highly tempted to make adjustments to investing strategies, and in many cases those changes would prove to be counterproductive. To use a simple example, one might be willing to bet initially that a coin won't come up heads 10 times in a row. But the coin comes up 9 times in a row, that person might choose to accept an opportunity to get out of the bet. This might be rational, but if you introduce some bias toward mean reversion, there's a good possibility that second-guessing yourself mid-stream would lead to suboptimal results.

Apologies if this isn't the appropriate thread to be raising these questions. Having spent a decent amount of time going through your analysis, I wanted to make sure I wasn't missing your main point.

thanks,
dan
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Post by ataloss »

My guess is that my zero price growth assumption corresponds very closely to the actual situation among the worst case historical conditions.


which is probaby why chips "naval analogy" approaches have been popular historically (preserving manuvering room, keeping powder dry)
Have fun.

Ataloss
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Post by hocus »

In the first case, you end up with many fewer shares after the first year. In both circumstances, the dollar amounts that you have sold are identical.

I understand this and what follows from it.

I am assuming that higher initial valuations are more likely to result in lower returns in the first few years than if you start with lower valuations. That is, I am assuming that the initial valuation affects the order of future returns.

This is the part I am struggling with.

My assumption is arguable because random effects can dominate in the near-term and mask this tendency.

This point is the source of my confusion.

I have been working from the premise that random effects dominate in the near-term. I am not wedded to the idea, and I am not aiming to challenge the assumption that initial valuation affects the order of returns. I am trying to understand the basis of the assumption. I'm a little uneasy with it.

How important is this assumption to the remainder of the analysis? Am I focusing on something of relatively little consequence to the points you are ultimately driving at?
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Post by peteyperson »

I have yet to read or see any data following a high valuation that indicates that the return was lower from that point on and by how much. I would like to see this in respect of the changing valuation levels too, not just one figure changing to another figure, because we both agree that valuation changes things considerably.

Petey



hocus wrote: In the first case, you end up with many fewer shares after the first year. In both circumstances, the dollar amounts that you have sold are identical.

I understand this and what follows from it.

I am assuming that higher initial valuations are more likely to result in lower returns in the first few years than if you start with lower valuations. That is, I am assuming that the initial valuation affects the order of future returns.

This is the part I am struggling with.

My assumption is arguable because random effects can dominate in the near-term and mask this tendency.

This point is the source of my confusion.

I have been working from the premise that random effects dominate in the near-term. I am not wedded to the idea, and I am not aiming to challenge the assumption that initial valuation affects the order of returns. I am trying to understand the basis of the assumption. I'm a little uneasy with it.

How important is this assumption to the remainder of the analysis? Am I focusing on something of relatively little consequence to the points you are ultimately driving at?
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Post by JWR1945 »

hocus quoting me (JWR1945):
I am assuming that higher initial valuations are more likely to result in lower returns in the first few years than if you start with lower valuations. That is, I am assuming that the initial valuation affects the order of future returns.

This is the part I am struggling with.

I fault William Bernstein for this confusion. He cites the relationship between risk and reward too loosely. Stated properly, you should never invest in anything risky unless you are likely to receive an adequate reward. Under that constraint, risk does equal reward.

But without that restrain, the adage can be turned on its head. If prices are high, then it is risky. Since risk equals reward, you will get better returns on your investment. Of course, it does not work that way. You do not get rewarded for taking reckless, unnecessary risks. That is foolhardy. You only get rewarded for necessary risks and only if you demand adequate compensation for taking those risks.

The proper way to look at this is to evaluate the upside potential and compare it to the downside risk. We are all familiar with this concept. It you start at a high valuation, your upside potential is much worse than if you start at a much lower price. Starting from a high valuation, your downside is much worse than if you start at bargain prices. Implicit in this discussion is an idea that there is a reasonable range of prices, both high and low, that exists even in the short term.

You are familiar with a such comparison. It is called the risk/reward ratio. It makes sense to invest when the rewards outweigh the risks. You should not invest when the risks exceed your potential rewards.

Even in the short term, the risk/reward ratio should have some influence on the likelihood of returns. It goes back to the notion that there are bounds as to what the market will do, at least in a statistical sense. As time increases this influence increases. The bubble corresponded to a case in which the market moved well outside of the normal range of variation.

In terms of my initial posts, I have looked at special cases for insights. I hypothesized a case in which the final prices (or index value) of stocks was the same after an extended period of time. peteyperson quoted John Bogle's example of such an event, an investor's nightmare. You want to be able to handle such a possibility during retirement because it has already occurred once. It is not typical. It is just something that you want to be able to handle.

You can find many cases over short time periods in which the final price equals the starting price. Look at a graph of an appropriate index. Prices should fluctuate quite a bit over time with an overall rising trend. Just draw a straight line parallel with the x-axis (i.e., your line shows price remaining constant versus time). Identify all points for which the index and your line cross each other. Read the dates (i.e., the times) at which they occurred. Between any two of those specific dates, the overall return has stayed the same. The final price equals the initial price. They have fluctuated greatly in between.

Similar examples can be found for specified ratios of final price to initial price. In those cases, however, the annualized rate of return varies because the time intervals vary. Only if the final price equals the initial price does the time interval become unimportant.

The reason for looking at a case in which the final price equal the initial price was for the insights that it provides. It is not typical. But something similar is, I suspect, typical of worst case situations ending in (financial) failure.

Keep this in mind as well. I am most concerned about the first decade of returns. There is a lot of noise in the returns during the first few years, taken individually. But taken as a composite, I suspect that following high valuations, there are more bad cases during that first decade than following lower valuations. Or, it could be that some especially bad result is more likely to occur once, sometime during that decade. You would not be able to say with confidence that it will happen in year number one or year number two and so forth. But you could say that it will happen at least once during those years.

Have fun.

John R.
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Post by JWR1945 »

galagan
I may have missed longer discussions of this, but it appears that you are presently focusing on adjusting withdrawal strategies while leaving investment strategies constant.

Not really. I am pointing out how much useful information that we can extract just by looking closely at gummy's equations. We can consider a host of special cases and reflect upon the insights that they provide.

The equations are quite general. In terms of investment strategies, per se, they are hidden in the gain multiplier terms g1, g2, etc. The withdrawal strategies are in the withdrawal terms w1, w2 etc. Some overall constraints can be introduced in the portfolio balance terms Bal(N).

If you wish to look at investment strategies, you can use gummy's equation to identify desirable properties of the gain multiplier terms (other than bigger is always better). In the simple case of investing in stocks and bonds, having the bonds introduces a lower bound on the gain multiplier terms. In this case the gain multiplier corresponds to a loss. That is, your overall portfolio ends up higher than the lower bound, which is bigger than zero but less than one. Then you see what you have to do to get your strategy to deliver those desirable properties.

Looking deeply into gummy's formula gives us another tool for analyzing safe withdrawal rate strategies. It is powerful.

Quite a few strategies have been developed over the years. But as you have indicated, not all of them work for real people. gummy has called such strategies cyber-fiction. He has developed an alternative for determining Sensible Withdrawal Rates.

Have fun.

John R.
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