A glidepath formula

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A glidepath formula

Post by AnonJohn » Sat Feb 10, 2018 6:41 pm

I have an investment policy statement specifying that my asset allocation should be "100 - Age + 14". This no longer makes sense to me as I move through my 40's. Since my retirement horizon is over 20 years away, falling below 70% equities seems too conservative. I realize that a few percent doesn't really matter, but my temperament is such that I can best stick to a plan if it's explicit, like a formula. I'll readily stipulate that a formula is not as good as a reasoned analysis, year by year, of my need and ability to take risks. But it will help me avoid overthinking things and making behavioral mistakes.

I've studied the wiki page on glide paths, but am confounded by the variability in commercial options. The breakpoints and discontinuities seem artificial, though I liked the Morningstar curves. The log(100-age)-1 approach is interesting, but seems to roll off too quickly.

I'm also mildly troubled by the fact that the closed formula I've seen are not dimensionally correct. They have a hidden / assumed timescale. Using dimensional analysis to nondimensionalize the problem, and requiring that a formula transitions away from risk as you approach the time-scale of important fluctuations led me to the following:

X = asymptotic % stocks when very young (e.g. .8)

Y = asymptotic % stocks when very old (e.g. .2)

Tf = Target Retirement Age (e.g. 65)

T = Age

t = Characteristic time scale for market recovery (e.g. 10 years)

Pf = Target Retirement portfolio

P = Current portfolio size

R = Savings rate ($ / year)

% Stocks = (X+Y)/2 + (X-Y)/2 * tanh ( ( (Tf - T) + (Pf - P)/R ) / t )

Note that the quantities in the Tanh are dimensionless. Tanh is an analytic approximation to a step function, who's sharpness depends on t, the market fluctuation / recovery timescale that you want to guard against. The italicized term describing dependence on portfolio size is optional.

The underlying ideas are that (1) there is a time before retirement at which you should begin de-risking the portfolio; (2) Your ability to take risks depends on the time to retirement (human capital); (3) Your need to take risks depends on your savings rate and how far you are from your goal.

This approach makes you more aggressive when further from retirement (either by age or by ability to acquire the needed portfolio), transitions as you near retirement, and then glides out to an end state. It is mildly counter-cyclical. This is reminiscent of what Sam Gamgee posted here. This formula is a combination of an age based rule and "% of enough" (see WCI post here, BH discussion here).

The general shape looks like this. This example assumes you approach the target portfolio evenly as you approach the target retirement age.


For this timescale (t = 10 years), this is close to the Morningstar glidepaths but transitions more quickly.

Comments? Advice? Like I said, I'm conscious that I'm over thinking it and that using a formula implies a precision that is illusory. I'm not using an existing commercial glidepath because my portfolio is spread across providers, all of whom differ.

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Re: A glidepath formula

Post by FiveK » Sat Feb 10, 2018 7:06 pm

AnonJohn wrote:
Sat Feb 10, 2018 6:41 pm
Comments? Advice? Like I said, I'm conscious that I'm over thinking it and that using a formula implies a precision that is illusory. I'm not using an existing commercial glidepath because my portfolio is spread across providers, all of whom differ.
See Equity Glide Paths — Bogleheads versus Target Funds for more overthinking. :)

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Re: A glidepath formula

Post by livesoft » Sat Feb 10, 2018 7:09 pm

I really have to laugh at all this. If you like 70%, then make your "glidepath" a constant 70% (or 75% or 80%). No formula needed. Seriously.

Then post back here in 10 years.
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Re: A glidepath formula

Post by digarei » Sat Feb 10, 2018 7:42 pm

I like your formula but since the assumptions don’t fit my situation my mind was immediately employed in an attempt to apply it to some other useful purpose. I haven’t yet come up with any helpful applications but such elegance should not go to waste. Sweeping S curves are beautiful to contemplate.

I took this at face value, as an approach to dealing with the sequence of returns problem.
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Re: A glidepath formula

Post by KlangFool » Sat Feb 10, 2018 7:54 pm


I have a very simple formula that works very well.

A) If you are aiming for retirement, use forecasted retirement expense (R)

B) If you are aiming for financial independence, use current expense (C)

For (A), mapped out your AA based on your portfolio size as a multiple of (R).

For (B), mapped out your AA based on your portfolio size as a multiple of (C).

For example, with (A), portfolio size

Between 0 and 5R, 80/20
Between 5R and 10R, 75/25
Between 10R and 15R, 70/30
Between 15R and 20R, 65/35
Between 20R and 25R, 60/40
Above 25R, 60/40.

If your retirement expense is 40K,

Between 40K and 200K, 80/20
Between 200K and 400K, 75/25
Between 400K and 600K, 70/30
Between 600K and 800K, 65/35
Between 800K and 1 million, 60/40
Above 1 million, 60/40.

You could do the same with C aka current annual expense.


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Re: A glidepath formula

Post by asset_chaos » Sat Feb 10, 2018 8:29 pm

Anything is good enough that keeps you comfortable with your plan and that you can follow. Where are you now, where do you want to be a specific number of years from now, draw a straight line, and there's your first order approximation to a "perfect" glide path.
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Re: A glidepath formula

Post by 2015 » Sat Feb 10, 2018 10:36 pm

Makes me wanna hang myself.

The more complexity the less time for L.I.F.E., the less fun, the more dull boy imbalanced Johnny becomes. I don't find AA decisions before or after retirement to be rocket science.

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Re: A glidepath formula

Post by Tyler Aspect » Sat Feb 10, 2018 10:46 pm

This is the one that I usually post.

Past result does not predict future performance. Mentioned investments may lose money. Contents are presented "AS IS" and any implied suitability for a particular purpose are disclaimed.

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Re: A glidepath formula

Post by skierrex » Sat Feb 10, 2018 10:59 pm


You crack me up!

I'm going with a step function.

< 25X annual expenses, 100/0

> 25X annual expenses, 50/50

It's slightly simpler, probably similarly effective.

Yours is a super-nice curve, though!

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Re: A glidepath formula

Post by AlohaJoe » Sat Feb 10, 2018 11:14 pm

AnonJohn wrote:
Sat Feb 10, 2018 6:41 pm
Without any testing whatsoever it is hard to offer much meaningful feedback. So my feedback would be to test it. The internet is full of financial forum posts from people who have ideas that they've never tested.

I mean, what's the definition of success here? The curve looks better when eyeballed?

What actual real world differences does it make?

These are two papers that I like that show the kind of research people do when investigating glidepaths to determine whether one glidepath is better or worse than another glidepath.

"Dynamic Allocation Strategies for Distribution Portfolios: Determining the Optimal Distribution Glidepath", David Blanchett (2007)

The focus is on retirement glidepaths but he looks at 43 different kinds of glidepaths, 21 different time periods, and 51 different withdrawal rates.

"The Glidepath Illusion: An International Perspective", Javier Estrada (2014)

Looks at glidepaths during the accumulation phase and considers 15 different kinds of glidepaths across 19 countries (plus "European", "Global", and "Cross-Sectional" results).

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Re: A glidepath formula

Post by AnonJohn » Sun Feb 11, 2018 11:47 am

Thanks for all the thoughts!

@FiveK - Fascinating. If I had read that thread ...

@Livesoft - You're right, of course. But I've had it set between 70 and 80% for over 20 years. Now that my portfolio is larger, I've found myself wondering about my willingness to take risks. So I'm trying to be more deterministic about when I will start shifting. Of course, the plot I propose follows your advice for 10 years ...

@diagrei - Wikipedia tells me that "The logistic function is an offset and scaled hyperbolic tangent function" ... so I'm apparently chasing the same pretty sigmoids as in the link FiveK posted ...

@Klang - I like your ratios as a sanity check. The principle is similar to the italicized term (target portfolio size). You're following "% of enough" but with a clear mapping, which I like, to asset allocations. Of course the mathematical discontinuities bother me, but that's a personal hangup. :)

@Asset_Chaos - Yes, I've been following the first order approximation for a while, but have found it wanting, hence looking for higher-order approximation.

@2015 - There's definitely wisdom in removing these deliberations and enjoying life. But, for me, complexity doesn't always map to extra time. I've been wasting too much much time worrying about a simple approach. But then I find AA decisions to be very hard. I know little about my need and ability to take risks on a 20+ year time scale. Yes, it's not rocket science, but there are unknowables.

@Tyler - Thanks. Your approach seems similar to the Vanguard target date funds. I've found the linear ramp down in the middle to steep, but I could of course start it later.

@skierrex - Well, glad I gave you a chuckle. I have enough self-awareness that (a) I agree (b) I also laugh (c) My laugh gets slightly nervous as I worry about whether I'm crazy. :)

@AlohaJoe - I really appreciate you challenging my reasoning! Great questions. To answer them: My definition of success is that (a) the function is analytic; (b) It transitions between two end points; (c) It passes through 50% equities at retirement; (c) It approximates other, professionally developed glidepaths; and (d) the balance between "conservative" and "aggressive" is determined by a natural parameter. So a more than "eyeballed" (not much), though (c) is just eyeballing.

The reason for this is my definition of success is my answer to your second question: I don't think it makes a real-world difference, or at any rate not a difference I could determine. That was the lesson I took from the lack of agreement among professionally-constructed glidepaths. But the papers you shared help me evaluate that - thanks! The Blanchett one is excellent, though it only looks at distribution glidepaths.

I took from them the idea that "becoming too conservative just when your portfolio gets large" can be a real mistake depending on life expectancy. One thing I conclude is that an approach that includes some "% of enough" thinking is useful.

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Re: A glidepath formula

Post by iceport » Sun Feb 11, 2018 2:18 pm

AnonJohn wrote:
Sun Feb 11, 2018 11:47 am
I took from them the idea that "becoming too conservative just when your portfolio gets large" can be a real mistake depending on life expectancy.
My only contribution to this interesting thread is along the lines of the observation above. Your analysis does not challenge the decision to drop equity exposures much below 50%, a decision that seems flawed, based on numerous portfolio withdrawal rate analyses.

It's confusing to me why the professionally developed funds drop their equity exposure so low, while at the same time providing guidance to their customers on sustainable withdrawal rates (SWRs) that rely on far higher equity allocations. This seems a basic contradiction. Thus (leaving aside the proposed rising-equity glidepaths, post-retirement), it seems a far narrower range of equity exposure is warranted than we see in the retirement funds.

It follows that if an investor elects to maintain a "terminal" equity allocation of 50% to 60%, with the intent of maximizing a SWR, then the shape of the glidepath becomes less critical. And it starts to resemble Tyler Aspect's proposal.
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Re: A glidepath formula

Post by PaulF » Sun Feb 11, 2018 9:48 pm

Well, if it makes you feel any better, I like it a lot. I, too, like a deterministic approach to de-risking. I had decided on a linear approach, and was happy with that. However, I later realized that I more or less had won the game about 10 year earlier than my linear glidepath assumed, and wanted to de-risk faster than my linear approach. I settled on an exp(-t/T) approach, which is approximately the same as yours in my regime.

Put another way, if I had had your approach 15 years ago, I probably would have coded into my excel spreadsheet and adopted it! :sharebeer

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Re: A glidepath formula

Post by Morse Code » Mon Feb 12, 2018 1:11 pm

AnonJohn wrote:
Sat Feb 10, 2018 6:41 pm
I have an investment policy statement specifying that my asset allocation should be "100 - Age + 14". This no longer makes sense to me as I move through my 40's...
You lost me after the third paragraph, so this may not be helpful, but it is a simple formulaic glide path that makes more sense to me than using age.

I use my target balance at retirement and target bond % at retirement.

I think I can retire when my portfolio reaches about 1.5 M. I've accumulated 34.3% of that. I plan to have 70% allocated to bonds when I retire. 70 X 34.3% = 24%, therefore, my current allocation is 76% equities, 24% bonds. On the day I hit "my number", regardless of my age, I will have exactly 30% equities, 70% bonds.
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Re: A glidepath formula

Post by Patzer » Mon Feb 12, 2018 1:50 pm

I use IF statements to choose when the glide path starts downwards and when it stops moving downwards.
This allows me to maintain maximum equity allocation until a set number of years before retirement, and move towards the retirement allocation in equal percents each year.
In the example below, 90% equity is held until 42, then equity allocation reduces by 2% per year until 62, when it stops at 50%.

IF(Current Age<(Retirment Age-Glide Path Years),Max Equity Allocation,IF(Current Age>Retirment Age,Retirement Equity Allocation,((Max Equity Allocation-Retirement Equity Allocation)*(Retirment Age-Current Age)/Glide Path Years)+Retirement Equity Allocation))

An example, in excel would be setting your variables in A1:B5
Retirement Age 62
Current Age 45
Max Equity Allocation 0.9
Retirement Equity Allocation 0.5
Glide Path Years 20
Then, putting the calculation in another cell:
=IF(B2<(B1-B5),B3,IF(B2>B1,B4,((B3-B4)*(B1-B2)/B5)+B4))=.84 Current Equity Allocation

*Note this is not my actual variables, just an example of how I use my formula.*

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Re: A glidepath formula

Post by AnonJohn » Mon Feb 12, 2018 9:49 pm

Thanks all for the stimulating discussion!

@Iceport: I agree. There's little to no justification for distribution stage shape of the curve. I was following the "wisdom" of the professional curves, which you and AlohaJoe have quite reasonably (IMHO) questioned.

@Paul: :sharebeer ; I have coded it in my spreadsheet. But I have 8-10 years before it changes appreciably from "75% stocks". :)

@Morse Code: I like your formula. Elegant/simple (and dimensionally correct!). Surprised I hadn't seen that posted before, but perhaps I'm just not well read enough. I think it's a continuous version of Klang's approach. ("% of Enough"). The main weaknesses I see are in the extremes: If you retire very young, risk may not match long-term needs; if you try but fail to reach a portfolio size, you could carry too much risk into a forced retirement. But these could be addressed by revisiting target allocation and portfolio size periodically.

The more I read and think on this, the more I appreciate White Coat Investor's post for capturing the relevant considerations.
https://www.whitecoatinvestor.com/desig ... lide-path/

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