Optimal asset allocation strategies for retirement & saving

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.
Post Reply
Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Sat Oct 26, 2019 8:34 am

A while ago I read about glidepaths on ERN's blog, a magical way to combat "sequence of returns risk". I don't understand sequence of returns risk, how is sequence of returns risk different from normal risk? How does a glidepath prevent sequence of returns risk? Why can nobody agree what the best glidepath is? Are glidepaths path dependent and therefore suboptimal by definition?

I set out to do better. A better asset allocation derived directly from your goal. I created a dynamic programming based optimizer that is able to calculate the optimal asset allocation for more or less arbitrary goals. The resulting asset allocation is optimal under the condition that market timing is impossible. With market timing being impossible, I mean that your assumed probability distribution of future market movements does not change over time.

Intuitive explanation of the idea behind my solution
Recall the trinity study:
The optimal asset allocation for a 4% withdraw rate with a 30 year horizon is 50% stocks.
The optimal asset allocation for a 6% withdraw rate with a 20 year horizon is 100% stocks.

Suppose that that you start your 30-year withdrawal with a 4% withdrawal rate and the suggested 50% stock allocation. Ten years down the road, you capital has shrunk and you are now effectively using a 6% withdrawal rate. What do you do? Do you keep your 50% stock allocation, or do you switch to a 100% stock allocation as suggested by the trinity study?

Of course you switch to the 100% asset allocation. If the markets go up, down or sideways, check the trinity study tables each year and switch to the new best asset allocation. This improves your success rate.

If you follow this method, there is a flaw in the trinity study: the trinity study tables give you the best asset allocation allocation under the assumption that the asset allocation is constant over the entire duration. But we have just said that switching improves your success rate. Can we build a better trinity study that gives us the best asset allocation under the assumption that you switch to the new-best asset allocation each year? That is what I have done.


simple scenario: saving a fixed amount by a fixed date
Let's start with a simple example: saving for retirement. Our model retiree saves $10k per year and wants to build $1 million at some fixed point in the future (the target date). We define success as an ending value >= $1m and failure otherwise. The following figure displays the optimal asset allocation for this goal, optimal means that the chance of success is maximized:

Image

There is a lot to unpack here. The Y axis displays your current net worth. The X axis displays the amount of years left until the target date. The color indicates the asset allocation, dark blue means 0% equity (all bonds) and dark red means 100% equity. The contour lines indicate 50%, 80%, 95% and 99% chance of success.

Suppose that our retiree wants to retire in 50 years and currently has 0 capital. We look at the graph above and see that our retiree is on the 95% confidence line, that indicates our retiree has 95% chance of succeeding at this particular goal. The optimal asset allocation at this point is 100% equity.

some years later, our retiree saved 400k and has 30 years left before his/her retirement date. Our retiree is in the white/light red zone, this indicates that an asset allocation of approximately 50% equity 50% bonds maximizes the chance of success. Our retiree should consult this figure each year to determine his asset allocation for the following year.

An important thing to note here is that the strategy is path independent. If person A want to retire in 50 years and B wants to retire in 40 years, then we don't need two separate solutions. The solution for the last 40 years are identical for both A and B.


Saving a fixed amount as soon as possible
The previous scenario had an odd utility function: our retiree wants to save $1m to retire at a fixed date. But what if the retiree reaches his goal 10 years earlier? Shouldn't he retire earlier? The following scenario minimizes the time required to reach $1m rather than maximizing the chance of reaching it.

Image

the asset allocation is relatively straightforward. During the majority of your savings period, you should have an asset allocation that is 100% equity. When you get close to your goal (around 85%), gradually switch over to an asset allocation of 100% bonds (In practice, I would advice gradually switching to the asset allocation used in retirement instead of 100% bonds). The contour lines indicate the average number of years left when the goal is reached.

How about leverage?

Image

Note the change in color scale. The results indicate that leverage investing early in your career is useful. My model makes several generous assumptions when it comes to leverage (yearly re-leverage, wealth can't go below zero, no margin calls), these results are an approximation.

The following figure shows a cross section of the (slightly modified) utility function at y=60, showing the average number of years left before reaching $1m.

Image

The difference between 100% stocks and optimal variable asset allocation is minor.



Trinity study like retirement
The following figure displays the optimal asset allocation for a classic trinity study style retirement. Success is defined as a positive ending balance. The retirement horizon is between 60 and 0 years. The color indicates the asset allocation to use for each year. The contour lines depict 50%, 80%, 95% and 99% chance of success.

Image

Look up your remaining lifespan in the X axis and your current spending in the Y axis. What we see here is that if the market drops, you should take a more aggressive asset allocation. And if the market does well, you should use a less aggressive asset allocation. Ironically, this is the opposite of what usually happens. Towards the end of this post are various figures with different return assumptions.

The following figure displays a cross section of the success rate with a remaining horizon of 25 years, for this strategy, versus various fixed allocations.

Image

As you can see, the success rate are substantially higher than with a fixed asset allocation.

How about glidepaths? The following figure shows the success chance from the dynamic asset allocation, the maximum success rate of all possible constant allocations (0-100%, 2.5% increments), and the maximum of all possible glidepaths (glidepath from X to Y over the entire duration, 10% increments):

Image

It is a bit difficult to see on this figure, but choosing the right glidepath increases your success chance over a static allocation by a very small amount, the maximum improvement is around ~ 0.6%. The optimal variable asset allocation performs much better. Note that that is impossible to choose a glidepath that performs as well as this figure shows over the entire range of withdraw rates, a withdrawal rate of 4% requires different glidepath than a withdrawal rate of 4.1%. You don't need to choose between different dynamic asset allocations, there is only one.


Trinity study like retirement, with ending capital bonus.
When the goal is already certain, the optimizer chooses the safest possible asset allocation. But if the goal is already certain, why not try to optimize for a higher terminal value? The utility function used is:

Code: Select all

if ruin:
	return 0
else:
	return 1 + C * ending_capital
Where C is a small constant, such as 10^-9. This causes the optimizer to maximize the terminal value when success is already 99.9999% certain (or similar).

Image


We also optimize for the same scenario with leverage (the color scale is different in the following figure):

Image

The 80% confidence threshold is quite a bit higher than without leverage, but the overall strategy does not change.


Trinity study with capital preservation.
The following scenario optimizes for a terminal value of 25 or more, i.e. capital preservation.

Image

There is not much to discuss here. I personally think this scenario is worthless because there is no convincing reason to specifically optimize for capital preservation (rather than, say, maximizing the terminal value according to some utility function). This is similar to optimizing for maximum drawdown, which does not result in sane strategies. Or to be more precise I think there is no sane reason to optimize for drawdown.



Model verification with monte carlo
My solver is very different from the usual monte carlo simulation. Due to implementation details (discussed later), the model slightly underestimates the utility function. How well does the model results match that of a simple monte carlo implementation?

Image

The dotted lines indicate the survival chance as calculated by the dynamic programming solver. The lines indicate the survival chance according to a simple monte carlo backtester. We see that the mode very slightly under-estimates the survival chance, as expected.


My solver only supports path-independent solutions. With the aid of this monte carlo simulator, we can test path independent strategies such as bond first and prime harvesting. I have mentioned that the variable asset allocation is optimal, but does it beat prime harvesting in practice?

Image

We see that the variable asset allocation convincingly beats prime harvesting. With some parameter tuning, we can get prime harvesting to beat any linear allocation, but not all of them at the same time. This highlights the difficulty of choosing the right initial parameters. Apart from the equity risk premium, the variable asset allocation has no parameters.


So far, we have ran monte carlo simulations with random numbers. Real stock returns are not normally distributed (more on that later). To make the test more realistic, we run the simulation with random blocks of 10 consecutive years from the history of US stock and bond returns since 1871. The entire history was normalized to match the return assumptions used by the solver.

Image

How about 30 years?

Image

The lines dance around a bit (there are only 100 samples available, and most of them are dependent), but the variable asset allocation still convincingly beats the competition. Prime harvesting hits 100% success chance at a 3.86% withdrawal rate and the optimal variable asset allocation hits it at 4.41%.


In case you concerned I only tested prime harvesting with one initial parameter, here is a figure showing prime harvesting with various initial parameters:
Image

And also constant allocation with various initial parameters:
Image


Model details.
My model assumes that real stock and bond returns are normally distributed, mean 7% / std 16% for stocks and 2% / 4% for bonds (correlation = 0). Please ask me if you would like to see simulations with different assumptions.

Are stock returns normally distributed? Yes... but with a caveat. Randomly drawn years are strongly normally distributed. Blocks of 2 consecutive years are still strongly normally distributed. 10 years, data is still normally distributed but a standard deviation of 15% gives the best fit. At 30 years the best fit has a standard deviation of just 9%.

This could indicate that the US is an outlier. But it could also indicate that the stock market exhibits mean-reverting behavior and that consecutive years are not independent. This means that market timing is not impossible.

Finally, the model that I use is not continuous, the solutions in this post were generated with an X resolution of 1 year and Y resolution somewhere around 2000 bins. That means (example) that the model can represent the values $1000 and $1004, but a value of $1003 is rounded down towards $1000. This causes the model to under-estimate the probability of success.

The time complexity of the solver is approximately O(X * Y^2) with X being the amount of time steps and Y being the amount of $$$ steps.


Different expected returns
The figures in this post were generated with return assumptions of mean 7% std 16% for stocks and mean 2% std 4% for bonds, for no particular reason. Here are some different assumptions.

7%, 2%
7%, 1%
7%, 0%
6%, 2%
6%, 1%
6%, 0%
5%, 2%
5%, 1%
5%, 0%


Really optimal?
In theory, we can do better by:
  • Using factor investing to juice up the returns or get the same returns with lower risk.
  • Varying the ERP (equity risk premium) according to some formula. Also known as market timing.
  • Taking into account that retirees will increase their spending if the market does well and decrease their spending if the market does poorly. (VPW).

Conclusions
There are two messages that I want to give here. The first message is that, when saving for as early as possible retirement, I strongly recommend an allocation of 100% (or more) in equities.

The second message that I want to give here is that in my opinion, path dependent strategies such as prime harvesting, buckets and to a lesser extent glidepaths are dead and should not be used. A variable asset allocation is able to provide a higher chance of archiving your financial goals with fewer assumptions.

Please ask me if you would like to see simulations for different goals or different assumptions.


Similar work and references
Forsyth and Vetzal, "Robust Asset Allocation for Long-Term Target-Based Investing". Looks at mv-optimal asset allocation for target wealth style retirement.
Last edited by Uncorrelated on Fri Nov 01, 2019 5:48 pm, edited 1 time in total.

dbr
Posts: 30842
Joined: Sun Mar 04, 2007 9:50 am

Re: Optimal asset allocation strategies for retirement & saving

Post by dbr » Sat Oct 26, 2019 9:51 am

Uncorrelated wrote:
Sat Oct 26, 2019 8:34 am
I don't understand sequence of returns risk,
Is this a question you want someone here to answer for you or are you saying that you have answered it in your long and interesting post? Note the Trinity Study is an example of an answer to what sequence of returns risk amounts to though they may not use that term and don't necessarily clarify a distinction between SoR and risk overall.

OK. I guess I will post an answer. There are some subtleties involved.

The scenario is that both while accumulating and while disaccumulating (or both over a period of time) one will receive random annual returns. One can compound those returns and compute a compound average return for that set of returns. If one were to invest money at the beginning of a period of years and leave it there until the end one could also compute the end point wealth for that CAGR. The order in which the annual returns occur would make no difference to the CAGR or the endpoint wealth. What the returns are is random and what final CAGR one gets is random. This is risk in the plain ordinary sense.

Now shift the scenario to a case where money is being contributed to or withdrawn from the portfolio. For the same set of annual returns the result in wealth accruing or remaining in the portfolio will depend on the order in which the returns occur for the same CAGR. When withdrawing poor returns bunched early on will deplete the portfolio more and when contributing poor returns bunched at the end will result in less accumulation. So this sequence of returns effect is an additional variability above and beyond the variability in possible CAGR or average return. Trinity was one of the first studies that reported the fact that a bad sequence of returns could be a worst case for the same assumption of average return that would reduce the safe withdrawal rate more than had been assumed at the time.

So, in summary, risk can take the overall form that one does not know what average return one will actually get and in addition one does not know what sequence the returns will have even for the same CAGR.

longinvest
Posts: 3983
Joined: Sat Aug 11, 2012 8:44 am

Re: Optimal asset allocation strategies for retirement & saving

Post by longinvest » Sat Oct 26, 2019 4:26 pm

The SWR model, where the retiree withdraws a constant inflation-adjusted amount from a portfolio of fluctuating assets regardless of market performance, is a broken and illogical portfolio withdrawal method.

I suggest that the original poster (OP) investigates, instead, the use of flexible variable withdrawals such as our wiki VPW method. In particular interest for the OP, there's a VPW Accumulation And Retirement Worksheet which covers both accumulation (flexible variable savings which adapt to market returns, asset allocation, retirement horizon, and salary) and retirement (flexible variable withdrawals which adapt to market returns, asset allocation, age, and pensions).

My personal opinion is that the use of a constant balanced allocation all lifelong, with an all-in-one globally-diversified fund such as Vanguard's LifeStrategy Moderate Growth Fund (VSMGX, 60/40 stocks/bonds), along with the VPW accumulation and retirement worksheet, is good enough.

See The One-Fund Portfolio as a default suggestion for more details about the portfolio suggestion.
Bogleheads investment philosophy | One-ETF global balanced index portfolio | VPW

D-Dog
Posts: 73
Joined: Sun Dec 02, 2007 12:44 am

Re: Optimal asset allocation strategies for retirement & saving

Post by D-Dog » Sat Oct 26, 2019 8:08 pm

Uncorrelated wrote:
Sat Oct 26, 2019 8:34 am


Please ask me if you would like to see simulations for different goals or different assumptions.

Very interesting work. You've obviously put a lot of time into this. I'm subscribing to this topic, and am interested to see other's reactions. I read your post through once and think I will need to read it through a couple more times to really take it all in. Since you offered, I would be interested in knowing a few things:
  • I'd like to see some examples of applying your variable asset allocation to some historical data using a 4% SWR and 30 year retirement. Take some of the worst starting retirement years (1929, 1966, etc.) and show how your strategy's asset allocation would have changed for each year of retirement, for each of the various starting years.
  • Applying your variable asset allocation strategy to a 30 year retirement, what is the SWR you calculate using actual historical data? Is the calculated SWR sensitive to the assumptions you use to calibrate your strategy? For example, if you had used 5% and 0% real returns to calibrate your model, how different would the historical SWR have been?

AlohaJoe
Posts: 4944
Joined: Mon Nov 26, 2007 2:00 pm
Location: Saigon, Vietnam

Re: Optimal asset allocation strategies for retirement & saving

Post by AlohaJoe » Sat Oct 26, 2019 9:27 pm

Uncorrelated wrote:
Sat Oct 26, 2019 8:34 am
Similar work and references
Forsyth and Vetzal, "Robust Asset Allocation for Long-Term Target-Based Investing". Looks at mv-optimal asset allocation for target wealth style retirement.
SInce you only listed one reference I figured I should point out the work of Gordon Irlam who was the first person I know of to take this approach.

See his papers:

Irlam, G. (2014). Portfolio Size Matters. The Journal of Personal Finance. 13(2), 9-16.
Irlam, G. & Tomlinson, J. (2014). Retirement Income Research: What Can We Learn from Economics?. The Journal of Retirement, 1(4), 118-128.

And for free on SSRN
Asset Allocation Confidence Intervals in Retirement: https://papers.ssrn.com/sol3/papers.cfm ... id=2675390
Human Capital, Social Security, and Asset Allocation: https://papers.ssrn.com/sol3/papers.cfm ... id=3016824

He also has a free asset allocation calculator that takes into account a wide variety of things (home equity, social security, pensions, annuities, etc)

https://www.aacalc.com/calculators/aa

It hasn't been updated in a few years but it is open source, hosted in github.

He's also a (very) occasional poster on Bogleheads under the username gordoni2.

Here's one of his posts that has similar results to what you talk about above

viewtopic.php?t=123342
SDP [ed: stochastic dynamic programming] suggests a relatively small portfolio should favor stocks. A medium sized portfolio should be balanced. And a relatively large portfolio should favor bonds if the owner is indifferent to leaving an inheritance, or stocks if the owner cares about leaving an inheritance.
I think this is an under-explored avenue for retirement planning because most people involved in retirement research don't really have the programming skills necessary to make it happen. If you look at retirement research overall, I mentally classify it into 3 "generations". The first generation, up to the late 1990s was the "calculator" generation. You plug some linear assumptions into a financial or scientific calculator and it would tell you a number. "If we assume 4% real growth for 26 years, how much money will I have?" That kind of thing.

The second generation, from Bengen onwards is the "spreadsheet" generation. Even most monte carlo analysis fits into this. Even today, if you look at the leading researchers like Pfau or Blanchett (or McClung) they do everything in an Excel spreadsheet. Excel is pretty powerful but it does have limits. It can't really do things like your approach.

So that leads us to the third generation, which is "data science" where they need some level of programming skills in R or python or whatever because they've reached the limits of Excel. But there just aren't very many people interested in retirement research that have the background for that right now. Which is why, half a decade after Irlam started publishing on stochastic dynamic programming there's been virtually zero follow up from anyone else.

AlohaJoe
Posts: 4944
Joined: Mon Nov 26, 2007 2:00 pm
Location: Saigon, Vietnam

Re: Optimal asset allocation strategies for retirement & saving

Post by AlohaJoe » Sat Oct 26, 2019 10:05 pm

Uncorrelated wrote:
Sat Oct 26, 2019 8:34 am
We define success as an ending value >= $1m and failure otherwise.

The following figure displays the optimal asset allocation for a classic trinity study style retirement. Success is defined as a positive ending balance.

When the goal is already certain, the optimizer chooses the safest possible asset allocation. But if the goal is already certain, why not try to optimize for a higher terminal value? The utility function used is:

Code: Select all

if ruin:
	return 0
else:
	return 1 + C * ending_capital
Over time I've come to believe that these kind of discrete or jump-step utility functions often obscure important information and having a continuous utility function usually isn't too hard to use instead.

Failure after 5 years isn't the same as failure after 29 years. Missing your retirement goal of $1m by $5 isn't the same as missing it by $500,000.

I've grown to like the approach that Estrada & Kritzman offer in their 2018 paper "Evaluating Retirement Strategies : A Utility-Based Approach".

The first step is to simply replace the pass-fail evaluation with a "coverage ratio": if a strategy only suppors 26 years of withdrawals (and the goal is 30 years), then the coverage ratio is 26/30 = 0.86. If the strategy lasted all 30 years and had a final portfolio balance equal to 3 years of withdrawals (e.g. if you were withdrawing $40,000 a year and the final portfolio balance is $120,000) then the coverage ratio is 33/30 = 1.1.

With the coverage ratio in hand, you can apply a typical power utility function:

Image

To get something like

Image

In and of itself, you might argue that the switch from discrete to continuous utility functions doesn't hugely change the results. But I think that is only true under the (ridiculously unrealistic) assumption of known, fixed planning horizons. Once you introduce stochastic planning periods, having a continuous utility function helps paint a truer picture of the differences (or lack of differences) in various strategies.

rossington
Posts: 327
Joined: Fri Jun 07, 2019 2:00 am
Location: Florida

Re: Optimal asset allocation strategies for retirement & saving

Post by rossington » Sun Oct 27, 2019 3:25 am

There is absolutely no reason at all to imply that one should consider this level of complexity prior to retirement or during retirement. The math is simple...you accumulate as much as you can, add all additional income and live within your means. Sequence of returns can be easily managed if carefully considered prior to retirement. 100% stocks could possibly produce the highest accumulation up to retirement...but then again?
"Success is going from failure to failure without loss of enthusiasm." Winston Churchill.

User avatar
JoeRetire
Posts: 3951
Joined: Tue Jan 16, 2018 2:44 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by JoeRetire » Sun Oct 27, 2019 5:37 am

rossington wrote:
Sun Oct 27, 2019 3:25 am
There is absolutely no reason at all to imply that one should consider this level of complexity prior to retirement or during retirement.
Well sure. But if optimizing, rather than satisficing, is your goal the fun possibilities are endless.
Don't be a lemming.

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Sun Oct 27, 2019 6:03 am

longinvest wrote:
Sat Oct 26, 2019 4:26 pm
The SWR model, where the retiree withdraws a constant inflation-adjusted amount from a portfolio of fluctuating assets regardless of market performance, is a broken and illogical portfolio withdrawal method.
I'm glad someone mentioned this. I agree in general that trinity style utility functions are a broken measure of performance.

However, the research presented here provides the retiree with better answers to simple retirement questions. Such as estimating the required savings for retirement and determining the probability of success at a certain level of spending. Previous retirement research also provided answers to these questions, but did so by taking the illogical assumption of either a constant asset allocation or pre-determined market timing strategy.

I have often argued that McClung's withdrawal strategies should not be used because they are theoretically suboptimal, but many found that argument unconvincing. I didn't wrote this to encourage people to use my strategy, but rather to provide clearer evidence on the sub optimality of other withdrawal strategies.

I like VPW, but the underlying assumptions are a bit crude. I hope that we will be able to combine VPW with a dynamic asset allocation to maximize your personal utility function.
My personal opinion is that the use of a constant balanced allocation all lifelong, with an all-in-one globally-diversified fund such as Vanguard's LifeStrategy Moderate Growth Fund (VSMGX, 60/40 stocks/bonds), along with the VPW accumulation and retirement worksheet, is good enough.
I don't really agree with this. During retirement with VWP an asset allocation of 60/40 is good enough, although I suspect it is possible to do better with a variable asset allocation. During the savings phase a portfolio of 100% stocks appears to be very close to optimal.

longinvest
Posts: 3983
Joined: Sat Aug 11, 2012 8:44 am

Re: Optimal asset allocation strategies for retirement & saving

Post by longinvest » Sun Oct 27, 2019 6:39 am

Uncorrelated, it would be so simple if we knew the distribution of future returns; we don't.

In your analyses, you had to make assumptions. I'm pretty sure you chose parameters closer to historical US stock and bond returns than Japan ones. So, it isn't surprising to me to see that the output of your analysis is that an accumulation portfolio should be concentrated into the historical winning US asset. But, this is just another way to read history. It doesn't predict the future.

You're free to continue analysing illogical withdrawal methods and make all sorts of assumptions about future returns to discover what you think is an "optimal" asset allocation. I'm just expressing my opinion that the assumptions could be broken; that the future has a habit of surprising us with unanticipated outcomes.

I personally prefer to use an assumption-resilient good-enough globally-diversified balanced index portfolio all lifelong.

Good luck!

P.S. I repeat my suggestion to investigate the VPW accumulation and retirement approach which is based on the principle of successive approximations and doesn't aim for a specific portfolio size, during accumulation, nor a specific withdrawal amount during retirement.
Bogleheads investment philosophy | One-ETF global balanced index portfolio | VPW

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Sun Oct 27, 2019 9:08 am

D-Dog wrote:
Sat Oct 26, 2019 8:08 pm
I'd like to see some examples of applying your variable asset allocation to some historical data using a 4% SWR and 30 year retirement. Take some of the worst starting retirement years (1929, 1966, etc.) and show how your strategy's asset allocation would have changed for each year of retirement, for each of the various starting years.
1929 4%: https://imgur.com/1bF2FUT
1929 4.5%: https://imgur.com/8cdGBOc
1966 4%: https://imgur.com/ZT5P2VN
1966 4.5%: https://imgur.com/z5KlwAX

My implementation of prime harvesting starts at 60% stocks for no particular reason. The contour lines show 50%, 80%, 95% and 99% confidence thresholds (some labels are out of view).

The general strategy is to be as conservative as possible when we can, and as aggressive as possible when we must.
Applying your variable asset allocation strategy to a 30 year retirement, what is the SWR you calculate using actual historical data? Is the calculated SWR sensitive to the assumptions you use to calibrate your strategy? For example, if you had used 5% and 0% real returns to calibrate your model, how different would the historical SWR have been?
I don't think max SWR is an usable metric, but here are the results:

The max SWR for my strategy when the assumptions match the underlying data (7% and 2%) is 4.49%.
The max SWR for my strategy with the assumptions 5% and 0% is 4.11%.
The max SWR for prime harvesting if you choose the perfect initial parameters is approx 4.3%.

I repeat that these numbers are useless.

D-Dog
Posts: 73
Joined: Sun Dec 02, 2007 12:44 am

Re: Optimal asset allocation strategies for retirement & saving

Post by D-Dog » Sun Oct 27, 2019 11:24 am

Uncorrelated wrote:
Sun Oct 27, 2019 9:08 am
D-Dog wrote:
Sat Oct 26, 2019 8:08 pm
I'd like to see some examples of applying your variable asset allocation to some historical data using a 4% SWR and 30 year retirement. Take some of the worst starting retirement years (1929, 1966, etc.) and show how your strategy's asset allocation would have changed for each year of retirement, for each of the various starting years.
1929 4%: https://imgur.com/1bF2FUT
1929 4.5%: https://imgur.com/8cdGBOc
1966 4%: https://imgur.com/ZT5P2VN
1966 4.5%: https://imgur.com/z5KlwAX

My implementation of prime harvesting starts at 60% stocks for no particular reason. The contour lines show 50%, 80%, 95% and 99% confidence thresholds (some labels are out of view).

The general strategy is to be as conservative as possible when we can, and as aggressive as possible when we must.

The 1929/4.5% example is especially interesting. The starting stock percentage is 40%, but then after the market crashes the optimal stock percentage increases to 100% by 1932. At this point, the withdrawal rate is now a much larger proportion of the smaller portfolio and the probability of success is much lower.

Of course, in real life, rather than maintaining the constant withdrawal amount and letting the probability of success fall, someone might instead decide they should maintain a high probability of success and let the withdrawal rate fall. If the strategy was to maintain a constant probability of success, wouldn't your work imply a variable percentage withdrawal rate and a relatively constant asset allocation? For example, based on the chart below, if I were to follow the 80% success line, then it looks like my withdrawal percentage (of my current portfolio) would increase over time and my asset allocation would stay fixed at about 70% stocks. This sounds a lot like what longinvest is advocating.
Uncorrelated wrote:
Sat Oct 26, 2019 8:34 am

Trinity study like retirement
The following figure displays the optimal asset allocation for a classic trinity study style retirement. Success is defined as a positive ending balance. The retirement horizon is between 60 and 0 years. The color indicates the asset allocation to use for each year. The contour lines depict 50%, 80%, 95% and 99% chance of success.

Image

Look up your remaining lifespan in the X axis and your current spending in the Y axis. What we see here is that if the market drops, you should take a more aggressive asset allocation. And if the market does well, you should use a less aggressive asset allocation. Ironically, this is the opposite of what usually happens. Towards the end of this post are various figures with different return assumptions.
It's interesting that following the 95% or 99% lines seems to imply that a declining equity glidpath. I'm not sure if I'm interpreting this correctly, though.

dbr
Posts: 30842
Joined: Sun Mar 04, 2007 9:50 am

Re: Optimal asset allocation strategies for retirement & saving

Post by dbr » Sun Oct 27, 2019 12:00 pm

Uncorrelated wrote:
Sun Oct 27, 2019 9:08 am

The max SWR for my strategy when the assumptions match the underlying data (7% and 2%) is 4.49%.
The max SWR for my strategy with the assumptions 5% and 0% is 4.11%.
The max SWR for prime harvesting if you choose the perfect initial parameters is approx 4.3%.

I repeat that these numbers are useless.
A possible not useless conclusion is that a reasonable estimate for SWR is generally a little more than 4%. This would be distinct from telling people they can count on 7% with impunity or should be fearful of attempting anything more than 2%.

You are absolutely right that if someone is trading on the distinction between 4.49 and 4.3, then the numbers are useless. Then again who uses two decimal places to report a number that is likely to be subject to random error of half a percent or more?

I hope I understand your comment correctly.

withrye
Posts: 32
Joined: Mon Feb 29, 2016 1:48 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by withrye » Sun Oct 27, 2019 1:17 pm

I really enjoyed reading this work, and I am grateful for your efforts in generating these enlightening graphs.

How sensitive are some of the earlier graphs on accumulation (specifically, 1M goal with early exit, 10k/yr) to changes in their assumptions? For example, the 1M goal with 10k/yr amounts to an annual savings rate of 1% of the capital goal set aside in principal savings each year. Prodigious savers may be saving 4% or more of the capital goal each year (i.e. someone earning 80k who manages to save 40k/yr with a goal of 1M and a 4% withdrawal rate target). Is it still true that this saver should hold 100% until around 85% of target is achieved and then downshift to a (dynamic) retirement asset allocation? Or does this higher savings rate change when (X% of target) and how quickly the shift to bonds is enacted?

To further generalize your findings, is there a way to link the accumulation graph to the Trinity withdrawal with capital bonus? The retiree following the findings of your analysis dutifully holds 100% stocks until their portfolio is near its target size, and then downshifts in allocation. I imagine their desired stock allocation is related to how long they expect their retirement to be and what their target portfolio is (capital / yearly withdrawal), but is there an elegant way to see an actionable timetable for this shift as the retiree approaches target?

Again, thank you for your efforts. Absolutely fascinating, and exciting to see that dynamic programming can offer new insights into portfolio allocation and drawdowns beyond simple constant allocations or glidepaths.

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Sun Oct 27, 2019 1:40 pm

D-Dog wrote:
Sun Oct 27, 2019 11:24 am
The 1929/4.5% example is especially interesting. The starting stock percentage is 40%, but then after the market crashes the optimal stock percentage increases to 100% by 1932. At this point, the withdrawal rate is now a much larger proportion of the smaller portfolio and the probability of success is much lower.

Of course, in real life, rather than maintaining the constant withdrawal amount and letting the probability of success fall, someone might instead decide they should maintain a high probability of success and let the withdrawal rate fall. If the strategy was to maintain a constant probability of success, wouldn't your work imply a variable percentage withdrawal rate and a relatively constant asset allocation? For example, based on the chart below, if I were to follow the 80% success line, then it looks like my withdrawal percentage (of my current portfolio) would increase over time and my asset allocation would stay fixed at about 70% stocks. This sounds a lot like what longinvest is advocating.
I'm not sure how much of this work can be generalized to a variable withdrawal strategy. The position of the 80% confidence line changes between 100% stocks and 50% stocks depending on the underlying return assumptions and utility function. The theme of a decreasing glidepath allocation appears to be robust across many different utility functions.

I will need to define and optimize for an utility function that measures the success of a variable withdrawal strategy before we can draw sane conclusions.
dbr wrote:
Sun Oct 27, 2019 12:00 pm
Uncorrelated wrote:
Sun Oct 27, 2019 9:08 am

The max SWR for my strategy when the assumptions match the underlying data (7% and 2%) is 4.49%.
The max SWR for my strategy with the assumptions 5% and 0% is 4.11%.
The max SWR for prime harvesting if you choose the perfect initial parameters is approx 4.3%.

I repeat that these numbers are useless.
A possible not useless conclusion is that a reasonable estimate for SWR is generally a little more than 4%. This would be distinct from telling people they can count on 7% with impunity or should be fearful of attempting anything more than 2%.

You are absolutely right that if someone is trading on the distinction between 4.49 and 4.3, then the numbers are useless. Then again who uses two decimal places to report a number that is likely to be subject to random error of half a percent or more?

I hope I understand your comment correctly.
The problem with max SWR is that the metric is determined almost entirely by 2 or 3 unlucky sequences in the history of US stock returns. I much prefer to point to the chart with random data or blocksize 10 data as I believe those results are more robust indicators of the expected differences between the different withdrawal strategies.

I believe that the statement that the max SWR with a proper strategy is a little more than 4% on US data is accurate, but I would not depend on that going forward due to possible survivorship bias embedded in the US data.

D-Dog
Posts: 73
Joined: Sun Dec 02, 2007 12:44 am

Re: Optimal asset allocation strategies for retirement & saving

Post by D-Dog » Sun Oct 27, 2019 2:56 pm

Uncorrelated wrote:
Sun Oct 27, 2019 9:08 am
D-Dog wrote:
Sat Oct 26, 2019 8:08 pm
I'd like to see some examples of applying your variable asset allocation to some historical data using a 4% SWR and 30 year retirement. Take some of the worst starting retirement years (1929, 1966, etc.) and show how your strategy's asset allocation would have changed for each year of retirement, for each of the various starting years.
1929 4%: https://imgur.com/1bF2FUT
1929 4.5%: https://imgur.com/8cdGBOc
1966 4%: https://imgur.com/ZT5P2VN
1966 4.5%: https://imgur.com/z5KlwAX

My implementation of prime harvesting starts at 60% stocks for no particular reason. The contour lines show 50%, 80%, 95% and 99% confidence thresholds (some labels are out of view).

The general strategy is to be as conservative as possible when we can, and as aggressive as possible when we must.
First, thanks for doing all of this. It's one of the more interesting posts I've seen in a long time. Even though it's quite theoretical, I still think there is a lot that can be learned from studying and discussing things like this.

It would be interesting to see the four scenarios above, but with Prime Harvesting starting at the same stock percentage as your optimal strategy (i.e., 20% or 40%). I suspect that part of the reason your optimal strategy does well in situations like these is because it starts with a low stock percentage which mitigates the impact of the poor stock performance early in retirement.
Uncorrelated wrote:
Sun Oct 27, 2019 1:40 pm

I will need to define and optimize for an utility function that measures the success of a variable withdrawal strategy before we can draw sane conclusions.
I think looking at variable percentage withdrawal strategies would be very interesting.

Since variable percentage withdrawal strategies never fail, it seems like the utility function should be attempting to measure how much you are able to withdraw and how stable those withdrawals are from year to year.

Once you have the variable withdrawal utility function defined, it seems like you would be able to jointly optimize the variable withdrawal strategy and the asset allocation strategy.

For a variable percentage withdrawal strategy, the capital/spending ratio is fixed, by definition, for any given year (i.e., it doesn't vary based on prior market performance). Does this mean that the optimal asset allocation would vary only by the number of years left?

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Sun Oct 27, 2019 2:58 pm

longinvest wrote:
Sun Oct 27, 2019 6:39 am
Uncorrelated, it would be so simple if we knew the distribution of future returns; we don't.

In your analyses, you had to make assumptions. I'm pretty sure you chose parameters closer to historical US stock and bond returns than Japan ones. So, it isn't surprising to me to see that the output of your analysis is that an accumulation portfolio should be concentrated into the historical winning US asset. But, this is just another way to read history. It doesn't predict the future.

You're free to continue analysing illogical withdrawal methods and make all sorts of assumptions about future returns to discover what you think is an "optimal" asset allocation. I'm just expressing my opinion that the assumptions could be broken; that the future has a habit of surprising us with unanticipated outcomes.

I personally prefer to use an assumption-resilient good-enough globally-diversified balanced index portfolio all lifelong.

Good luck!

P.S. I repeat my suggestion to investigate the VPW accumulation and retirement approach which is based on the principle of successive approximations and doesn't aim for a specific portfolio size, during accumulation, nor a specific withdrawal amount during retirement.

Mixed feelings here. First, I absolutely agree that assumptions can change, be wrong or inaccurate. I strive to use as few assumptions as possible to choose my asset allocation. My calculation method allows you to determine your asset allocation from your goal and two relatively simple assumptions. In that sense, this is the purest form of asset allocation possible, it only uses assumptions that you gave it. The strategy cannot accidentally learn to exploit mean reversion, because it doesn't exist according to the model. The strategy cannot exploit market timing, because it doesn't exist according to the model.

Compare this with prime harvesting. Which assumptions does prime harvesting use? I don't know. It was designed for a particular use case and a particular set of data. How much market timing (overfitting) made it to the final version of the algorithm? We don't know. How effective is the CAPE initialization looking forward? We don't know. My strategy doesn't have that problem because all assumptions are clearly noted.

And that brings us to the subject of VPW. What assumptions does VPW use? It uses a mixture of stocks and bonds, from that we can infer that you made some assumptions about future stock and bond returns. Why did you choose an asset allocation of 60/40? Why not 50/50, or 80/20? Did you choose 60/40 because it represents a particular optimum, or are you just guessing? Why does it not have an allocation to gold? Why does it use a linear allocation instead of a glidepath? Was that shown to be optimal or are you just guessing? If so, for which assumptions is that optimal? If I disagree with your assumptions, what modifications should I make?

More to the point, I don't believe that VPW uses fewer or more robust assumptions than my solution. The assumptions are just hidden somewhere behind various formula's and backtests instead of being explicit. When I have more time I will read the thread on VPW in it's entirely to learn more about the development process.
Last edited by Uncorrelated on Mon Oct 28, 2019 3:51 am, edited 1 time in total.

longinvest
Posts: 3983
Joined: Sat Aug 11, 2012 8:44 am

Re: Optimal asset allocation strategies for retirement & saving

Post by longinvest » Sun Oct 27, 2019 3:50 pm

Uncorrelated, I suggest to try different goals. I think that the one's you've tried (specific portfolio target size during accumulation and specific withdrawal amount during retirement) are illogical.

As for the portfolio suggestion, I've provided a link to a post which explains it. Here are excerpts from that linked post:

The One-Fund Portfolio as a default suggestion
longinvest wrote:
Mon Aug 12, 2019 8:10 am
I think that the following could be a good default portfolio suggested in answer to many queries about portfolio construction:
  • Portfolio 1: Vanguard LifeStrategy Moderate Growth Fund (VSMGX) -- a globally-diversified balanced index portfolio with a moderate home bias, appropriate for investors of all ages and all wealth levels, or
  • Portfolio 2: [...]
In theory, the "ideal" default portfolio would be William Sharpe's Market Portfolio but it has various problems: (a) calculating asset weights is challenging, (b) the actual weightings are approximate because float-adjusted market capitalisations corresponding to Vanguard's total world stocks (VT) and bonds (BNDW) aren't all available*, and (c) there are good reasons for most investors (around the world) to keep a reasonable amount of home bias in their portfolios, as explained in Vanguard's paper "The role of home bias in global asset allocation decisions".

* They aren't available for free, if they're available.

As a consequence, I think that portfolio 1 is a very good default portfolio for investors of all ages and all wealth levels. This includes experienced investors who have finally realized the importance simplicity as well as the futility of trying to engineer a better portfolio, accumulating investors who want to spend their life doing other things than worrying about their portfolio, and even new investors who don't know how to choose an asset allocation. It has a fixed 60/40 stocks/bonds allocation. It's very broadly-diversified, currently holding over 25,000 securities. It's actually a very good practical proxy for Bill Sharpe's ideal Market Portfolio adapted for a U.S. investor with a moderate home bias.
[...]
My personal preference is for portfolio 1, representing a globally-diversified lifelong 60/40 stocks/bonds allocation because I consider that all investment assets are risky, but in different ways. I think that it's best to broadly diversify across them all lifelong***.

*** In retirement, combining variable portfolio withdrawals with Social Security (possibly delayed to age 70) and a pension (if any) often results into mild total income fluctuations. When necessary, Total Retirement Income fluctuations can be further dampened by using a small part of the portfolio to buy an inflation-indexed Single Premium Immediate Annuity (SPIA) instead of increasing the bond allocation above 40%.
One could argue whether the float-adjusted Market Portfolio is closer to 60/40, 50/50, or 40/60. But, the differences between these three allocations are small enough to ignore. The Market portfolio is definitely not concentrated into stocks (e.g. 80/20).

Good luck!
Bogleheads investment philosophy | One-ETF global balanced index portfolio | VPW

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Sun Oct 27, 2019 3:54 pm

AlohaJoe wrote:
Sat Oct 26, 2019 10:05 pm
I've grown to like the approach that Estrada & Kritzman offer in their 2018 paper "Evaluating Retirement Strategies : A Utility-Based Approach".
Image
reference image with trinity study
reference image with trinity study + bonus

It appears that the 0 value of this utility function is approximately equal to the 85% confidence line with the trinity study utility function. During the last 10 years the asset allocation recommended by either utility function (below the 0 line) is almost identical. Further away from the end, the estrada-kritzman utility function seems to prefer asset allocations in the 60-80% range. The trinity study utility function is more eager to take a position of 100% stocks or less than 50% stocks.

I think this new utility function is better because it accurately penalizes early failures. The bonus points that the estrada-kritzman utility function awarded for having funds leftover at the end is less interesting. If you look closely, you'll see that the strategy always enters a 100% stocks position in the last year of retirement.

Another observation is that if you create a VPW withdrawal strategy using the 0 contour line of this utility function, the asset allocation resembles a decreasing glidepath.

I will come back to your other post once I have read all the references.

AlohaJoe
Posts: 4944
Joined: Mon Nov 26, 2007 2:00 pm
Location: Saigon, Vietnam

Re: Optimal asset allocation strategies for retirement & saving

Post by AlohaJoe » Sun Oct 27, 2019 9:33 pm

Uncorrelated wrote:
Sun Oct 27, 2019 3:54 pm
The trinity study utility function is more eager to take a position of 100% stocks or less than 50% stocks.
Yep, it is exactly those big jumps that make me dislike the discrete utility functions. They are a bit too much black or white, leading to black or white asset allocations. "If the plan only lasted 359 out of 360 months, then you should switch from 100% stocks to 30% stocks and use that instead"...just doesn't seem like a rational conclusion to me.

You should also try running things against other countries. A good source of data is the Jordà-Schularick-Taylor Macrohistory Database, which includes 17 economies since 1870. They provide both Stata and Excel files here: http://www.macrohistory.net/data/#DownloadData. If you're using something like R or pandas, it is easy to import Stata data directly. Interpreting the results isn't easy -- it includes WW1 & WW2 which seems unlikely to repeat. It assumes unrealistic levels of home bias. (A Belgian investor in 1900 would not have been 100% invested in domestic equities.) And so on. But it at least offers a different lens on the problem and some semblance of out-of-series testing.

I used it to generate charts showing the SWR for each retirement year in each country: https://medium.com/@justusjp/safe-withd ... 0b2215e138. It is amazing to see how terrible equity investing has been in France, for instance, a country that gets much less press than Japan, even though France's woes lasted from the 1890s (from the Panama scandals at the beginning of the Third Republic) until the 1970s (after the heavy-handed, patriarchal Gaullists lost power)....nearly 80 years.

Image
The bonus points that the estrada-kritzman utility function awarded for having funds leftover at the end is less interesting.
The good/bad part about more complicated utility functions that they handle this. The bad part is you have to make decisions that other people will always disagree with. Lots of people claim that extra money is totally worthless to them. (I think they are kidding themselves; after all, I don't see them throwing away that extra money or even donating it to a charity.) But the good part is that at least the utility function can directly model things like that instead of leaving it implicit and un-examined.

arthur450
Posts: 1
Joined: Mon Oct 28, 2019 5:28 am

Re: Optimal asset allocation strategies for retirement & saving

Post by arthur450 » Mon Oct 28, 2019 5:55 am

I've just created an account in order to post an answer here. Great work Uncorrelated!

I've been experimenting with different withdrawal strategies myself and I also came to the conclusion that a dynamic asset allocation was the way to go. Increasing your stock allocation if the market goes down definitely makes sense. What I've also realized is that cash and gold are worthy assets when it comes to optimizing the SWR. Some of my work is summarized here, for the curious: https://old.reddit.com/r/financialindep ... rategy_an/

My request: could you add cash and gold to your algorithm? Instead of computing the optimal stock/bond allocation each year, you would instead compute the optimal stock/bond/cash/gold allocation each year. I'm sure that it would yield interesting results. By the way: did you use annual returns or monthly returns? It might make sense to use monthly returns, even if you only rebalance to the optimal asset allocation once a year.

Thank you for your hard work, it is much appreciated!

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Mon Oct 28, 2019 8:01 am

withrye wrote:
Sun Oct 27, 2019 1:17 pm
I really enjoyed reading this work, and I am grateful for your efforts in generating these enlightening graphs.

How sensitive are some of the earlier graphs on accumulation (specifically, 1M goal with early exit, 10k/yr) to changes in their assumptions? For example, the 1M goal with 10k/yr amounts to an annual savings rate of 1% of the capital goal set aside in principal savings each year. Prodigious savers may be saving 4% or more of the capital goal each year (i.e. someone earning 80k who manages to save 40k/yr with a goal of 1M and a 4% withdrawal rate target). Is it still true that this saver should hold 100% until around 85% of target is achieved and then downshift to a (dynamic) retirement asset allocation? Or does this higher savings rate change when (X% of target) and how quickly the shift to bonds is enacted?

To further generalize your findings, is there a way to link the accumulation graph to the Trinity withdrawal with capital bonus? The retiree following the findings of your analysis dutifully holds 100% stocks until their portfolio is near its target size, and then downshifts in allocation. I imagine their desired stock allocation is related to how long they expect their retirement to be and what their target portfolio is (capital / yearly withdrawal), but is there an elegant way to see an actionable timetable for this shift as the retiree approaches target?

Again, thank you for your efforts. Absolutely fascinating, and exciting to see that dynamic programming can offer new insights into portfolio allocation and drawdowns beyond simple constant allocations or glidepaths.

I generated solutions for this scenario with different savings rates:

save 5k / year
save 10k / year
save 20k / year
save 40k / year
save 80k / year

The annotations indicate the middle of the 50% stocks allocation, and the point where the optimizer suggests switching over from 100% stocks to 90% stocks. I think that the differences between the figures have something to do with the order of operations. The overall conclusion (100% stocks until you are 85% along to way to your goal) appears robust across different savings rates.

I also tried a logarithmic utility function (not shown) and got similar results.


Various return assumptions:
7%, 2%
7%, 0%
5%, 2%
5%, 0%
3%, 2% (unstable)
3%, 0%
3%, 3% (unstable. Not sure what's going on with the horizontal lines)
I think a better utility function is needed to make some of the unstable solutions stable. All solutions that are stable give similar results.


Linking the accumulation graph to any of the de-accumulation graphs would require a solver that supports path dependent problem specifications, mine doesn't do that. But we can create an approximate solution by varying desired end capital. The following figure displays a the strategy for capital accumulation where the goal number is equal to the 90% confidence threshold in the trinity study:
Image

It is a bit hard to see, but the steepness of the glidepath increases as you move towards the right side of the image. The X resolution (years) of the simulation is a bit too low to give an accurate answer. I generated a chart in a higher resolution which confirms that the transition zone grows thinner towards the right side of the image. I will post the higher-resolution image later once I have the kinks ironed out.

Edit: monthly simulation
This higher-resolution simulation suggests a steeper glidepath, allocating 100% to stocks until 95% of your goal number is reached. The 95% number stays constant over the entire duration. I wonder if the resolution issue also affects the other accumulation scenario's.

Note that the contour lines are quite different from the 1-year simulation. This is because I increased the X (time) resolution without increasing the Y (dollar amount) resolution, this exacerbates the 'always rounds down' issue with my solver.

withrye
Posts: 32
Joined: Mon Feb 29, 2016 1:48 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by withrye » Thu Oct 31, 2019 9:46 pm

Uncorrelated wrote:
Mon Oct 28, 2019 8:01 am

I generated solutions for this scenario with different savings rates:
...
The overall conclusion (100% stocks until you are 85% along to way to your goal) appears robust across different savings rates.
Thank you so much for running these numbers, they are illuminating.

Would the following approach be the correct interpretation for a young saver looking at a 40+ year retirement?

Save aggressively at 100% stocks, targeting some historically conservative SWR (3.5%, or a 28.5x capital portfolio). At approximately 85% of target, the saver shifts toward the optimal AA for a 40-year retirement at a 28.5x portfolio, which is approximately 40% stocks according to the "Trinity with capital bonus" graph.

At this point, the saver-turned-retiree follows the Trinity with capital bonus graph each year of retirement. If the portfolio is successful, the retiree holds an increasingly conservative portfolio until perhaps 20-ish years out from "completed retirement." If the portfolio experiences distress early in retirement, the retiree holds pat or shifts toward increasingly aggressive allocations if things are particularly bad. Is this a fair hypothetical?

What do you think is the best way for a retiree to use a tool like this? Given the solution needs to only be found once for a given set of inputs, could the output graph (e.g. Trinity with capital bonus) be converted to a large lookup table, where each cell is the intersection of [years left] and [capital/yearly spend] with the cell value being the %stock to hold?

Thanks again for your work and for continuing to engage with us by creating new graphs and running new solutions.

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Fri Nov 01, 2019 9:33 am

withrye wrote:
Thu Oct 31, 2019 9:46 pm
Uncorrelated wrote:
Mon Oct 28, 2019 8:01 am

I generated solutions for this scenario with different savings rates:
...
The overall conclusion (100% stocks until you are 85% along to way to your goal) appears robust across different savings rates.
Thank you so much for running these numbers, they are illuminating.

Would the following approach be the correct interpretation for a young saver looking at a 40+ year retirement?
I believe that interpretation is accurate. But I'm not sure about the asset allocation to use in retirement. because:
  1. The simulation assumes years are independent. If stocks exhibit long term mean-reverting behavior, then stocks are less risky over the long term which should result in more aggressive stock allocations for very early retirees.
  2. The asset allocation is moderately sensitive to the exact utility function used. The asset allocation with Estrada-Kritzman utility function a few posts back is considerably more aggressive. Although this function is not perfect I think it's slightly better than trinity with capital bonus.
  3. I have not explored VPW yet. Gordon Irlam states that the optimal variable withdrawal asset allocation is in the range of 70-100% stocks (factoring in guaranteed social security income).
The images can be used as a look up table. An actual table would be very large.

GAAP
Posts: 966
Joined: Fri Apr 08, 2016 12:41 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by GAAP » Fri Nov 01, 2019 11:16 am

AlohaJoe wrote:
Sat Oct 26, 2019 9:27 pm
I think this is an under-explored avenue for retirement planning because most people involved in retirement research don't really have the programming skills necessary to make it happen. If you look at retirement research overall, I mentally classify it into 3 "generations". The first generation, up to the late 1990s was the "calculator" generation. You plug some linear assumptions into a financial or scientific calculator and it would tell you a number. "If we assume 4% real growth for 26 years, how much money will I have?" That kind of thing.

The second generation, from Bengen onwards is the "spreadsheet" generation. Even most monte carlo analysis fits into this. Even today, if you look at the leading researchers like Pfau or Blanchett (or McClung) they do everything in an Excel spreadsheet. Excel is pretty powerful but it does have limits. It can't really do things like your approach.

So that leads us to the third generation, which is "data science" where they need some level of programming skills in R or python or whatever because they've reached the limits of Excel. But there just aren't very many people interested in retirement research that have the background for that right now. Which is why, half a decade after Irlam started publishing on stochastic dynamic programming there's been virtually zero follow up from anyone else.
You are correct, but the flip side of this is that it is probably much easier for the typical person to understand and implement 1st generation than 2nd generation -- let alone 3rd generation. It is also far easier for an "adviser" to sell 2nd generation solutions since they are sufficiently advanced to scare/impress the general public while being simple enough for the salesperson to understand. The financial industry rewards sales/profits from customers and has little incentive outside of academia to pursue 3rd generation research. These tendencies play out every week in this forum...
“Adapt what is useful, reject what is useless, and add what is specifically your own.” ― Bruce Lee

Amadis_of_Gaul
Posts: 121
Joined: Tue Dec 18, 2018 5:57 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Amadis_of_Gaul » Fri Nov 01, 2019 2:29 pm

Uncorrelated, let me add my thanks for all your work and for sparking a fascinating thread! I also appreciate AlohaJoe's comments about using a continuous rather than discrete utility function. The former makes more intuitive sense to me. If it looks like I'm going to have a shortfall in retirement, I might prefer the shortfall (after all, I might not live to 95!) to going double-or-nothing with 100 percent equities and _really_ wiping out.

I thought your graph of the Estrada-Kritzman utility function was fascinating. It seems more commonsensical to me than your other results. People who have comfortably made it (30x expenses) should essentially employ an ascending glidepath. People who haven't saved much should throw it into the stock market and hope. People who are on the edge should get very conservative (in real life, this might look like buying an annuity).

However, as someone who is still in the accumulation phase, I would like to know what the accumulation counterpart to this graph looks like. In some ways, it ought to be symmetrical. If the guy with 30x expenses ought to be in 60/40 the day after retirement, he ought to be in 60/40 the day before retirement too. In other ways, though, it probably won't be. If you would be willing to enlighten me, I'd greatly appreciate it! Thanks!

User avatar
siamond
Posts: 5048
Joined: Mon May 28, 2012 5:50 am

Re: Optimal asset allocation strategies for retirement & saving

Post by siamond » Fri Nov 01, 2019 3:55 pm

I concur with the comments that this approach really needs be generalized to more realistic withdrawal methods (e.g. variable) and thoroughly tested out-of-sample (e.g. Jorda and al). Also, life has a habit of not stopping when planned, so I'd be careful to check what happens when varying the 'planned' retirement period. Otherwise, none of it is realistic in real life. Easier said than done, admittedly.

But that's ok, one has to start somewhere and the OP certainly has displayed impressive skills, so I am quite curious (while keeping a big dose of healthy skepticism) where this will go next. I am part of the Excel generation, the 'R' generation undoubtedly has things to add...

As a side note, having played with withdrawal methods for years, it is indeed non-trivial to evaluate & compare them. Utility functions never mapped to anything clear and intuitive in my mind, nothing truly palatable in real life. What is clear to me is that focusing on the portfolio balance is the wrong approach. You don't eat and enjoy life based on your portfolio balance, you eat and enjoy life based on your annual income (i.e. spending budget), this is WAY different. McClung's motto about loss-of-income vs. loss-of-value as a much improved characterization of risk is right on, when applied over entire retirement periods.

Personally, for a given retirement period tested with a given test of parameters, I like to look at average annual income (fixed income + variable withdrawals) minus its standard-deviation. This simple metric conveys powerful and very practical meaning. Optimizing on this metric proved quite helpful in my (Excel!) endeavors. But that is in no way a silver bullet, risk is way too multi-faceted to be summarized in one single number...

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Fri Nov 01, 2019 7:19 pm

Amadis_of_Gaul wrote:
Fri Nov 01, 2019 2:29 pm
However, as someone who is still in the accumulation phase, I would like to know what the accumulation counterpart to this graph looks like. In some ways, it ought to be symmetrical. If the guy with 30x expenses ought to be in 60/40 the day after retirement, he ought to be in 60/40 the day before retirement too. In other ways, though, it probably won't be. If you would be willing to enlighten me, I'd greatly appreciate it! Thanks!
My previous scenario's looked at target wealth retirement. i.e. you save and retire when you hit your number, this works best for early retirement. We can also look at more traditional target date retirement, my solver can do that easily. The following scenario uses the trinity (binary) utility function with spending of 1 in the last 30 years of retirement and a spending of -0.2 (i.e. accumulation) before that. Essentially simulating a retiree that plans to retire at a fixed date in the future.

Image

The odd-shaped zone at Y = 30 is an artifact and can be ignored.

The right half of the image represents retirement (completely identical to the trinity chart in the first post). The left half of the image represents the asset allocation to be used during accumulation in order to maximize the final utility. The contour lines represent success probability.

The following scenario is the same, but uses the Estrada-Kritzman utility function.

Image

The result makes intuitive sense, but the utility function is much harder to interpret than the trinity utility function.


The following image overlays both utility functions, indicating that the (red) Estrada-Kritzman utility value of 0 approximately equals a success rate of 85% (blue). Ignore the color scale.

Image
Last edited by Uncorrelated on Sat Nov 02, 2019 3:16 am, edited 1 time in total.

Amadis_of_Gaul
Posts: 121
Joined: Tue Dec 18, 2018 5:57 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Amadis_of_Gaul » Fri Nov 01, 2019 8:37 pm

Uncorrelated wrote:
Fri Nov 01, 2019 7:19 pm
My previous scenario's looked at target wealth retirement. i.e. you save and retire when you hit your number, this works best for early retirement. We can also look at more traditional target date retirement, my solver can do that easily. The following scenario uses the trinity (binary) utility function with spending of 1 in the last 30 years of retirement and a spending of -0.2 (i.e. accumulation) before that. Essentially simulating a retiree that plans to retire at a fixed date in the future.
Many thanks! I have no idea how common my situation is, but I am in the position of not wanting to retire at all (I love my job!) yet potentially having to retire early. I've chosen to plan for the latter eventuality. It's very interesting to see how sequence-of-returns risk affects asset allocation. Much appreciated!

StillGoing
Posts: 10
Joined: Mon Nov 04, 2019 4:43 am

Re: Optimal asset allocation strategies for retirement & saving

Post by StillGoing » Sat Nov 23, 2019 1:24 pm

This is a very interesting piece of work - so many thanks to uncorrelated for posting this. I have followed your approach for producing the graph of initial capital/yearly spending as a function of years left in retirement but using historical data instead (I'm struggling to include the figures inline - it is clearly possible, so help would be much appreciated). Not too much of surprise, the basic form of the graph is the same although the AA is more weighted towards stocks (as you stated, it ought to be if reversion to the mean is present). I also obtain a better performance with variable AA than with fixed, but not to the same extent as for your model (this may be due to the crudity in which I have implemented my model, e.g. 1 or 2 year steps and steps of 2 in spending or because of the difference in the underlying database).

As others have already noted, maintaining a fixed spend can result in significant changes in the AA - personally I might struggle to change from, say, 20% to 100% stocks in a single year even though the rationale is entirely sensible. However, it strikes me that:
1) the AA could remain fixed, but the withdrawal rate could be varied
2) or a hybrid of variable AA and variable withdrawal could be made (e.g. by limiting the amount that the AA could change and then taking the appropriate withdrawal)

I think both of these could provide optimal solutions in terms of failure. At some point, I shall implement them and have a look. As others have suggested, following a line of constant success rate might also be interesting.

Finally, I was interested to note that using variable AA for historical cases, the legacy in 'good' retirement periods was generally a lot less than using fixed AA - I think this is because relatively low stock fractions are used when conditions are good. My own circumstances (I'm recently retired and lucky enough to have a large fraction of my income in the form of a pension) mean that I'm particularly interested in leaving a legacy from my investments, but not too much of one (while I have run my code for this, I have yet to look at the outcomes).

cheers

StillGoing

Lazareth
Posts: 127
Joined: Tue Nov 01, 2016 9:21 am
Location: New England

Re: Optimal asset allocation strategies for retirement & saving

Post by Lazareth » Sun Nov 24, 2019 8:22 pm

I found this helpful for me, a more simplistic illustration on the impact of various asset allocation strategies.

Image
"I want to light the lights of patriotism." - Lech Walesa

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Tue Nov 26, 2019 4:02 pm

StillGoing wrote:
Sat Nov 23, 2019 1:24 pm
I use imgur.com to upload images.

How did you manage to simulate actual stock market data with dependent years? Are you simulating with blocks of multiple years at once or do you have a different solution?

StillGoing
Posts: 10
Joined: Mon Nov 04, 2019 4:43 am

Re: Optimal asset allocation strategies for retirement & saving

Post by StillGoing » Wed Nov 27, 2019 5:19 am

Uncorrelated - thanks for the help (I finally actually found the board help for uploading figures too - shows that you should always read more before you post for the first time!)

Here is the asset allocation versus time to go (i.e. similar to your Trinity study graph)

Image
https://i.postimg.cc/PJwq86cX/aavstime-20191127.png

You can see that the resolution is relatively low (1 year for years 1-10, 2 years after that, a step of 2 in the initial capital/yearly spending ratio. My calculation is a brute force, loop around the months (this allows the software to happily calculate path dependent strategies, but means that there is little vectorisation in my matlab code and it runs quite slowly for this sort of study).

Here is the probability of survival with various approaches

Image
https://i.postimg.cc/gkrFzM5Q/survivalv ... 191127.png

This shows my code for two fixed values of AA (80% stocks and 40% stocks) as well as the same calculation for ERN's spreadsheet (https://earlyretirementnow.com/2018/08/ ... s-part-28/) and cFireSim (http://www.cfiresim.com) just to show my code gives similar results to publicly available simulators. As mentioned before, the variable AA approach does improve survival, but not so much at high survival rates.
How did you manage to simulate actual stock market data with dependent years? Are you simulating with blocks of multiple years at once or do you have a different solution?
If I understand your question correctly (let me know if I haven't), I just used the vanilla historical sequence with no randomisation of blocks and therefore the method suffers from having little independence in the data and a limited number of cases (the failure rates are quantised). There are advantages to this approach since it makes no assumptions about the distribution of returns (except that the future will be broadly similar to the past and that the historical worst cases are indicative of the likely future worst cases - both of which can be argued with).

cheers
StillGoing

User avatar
heart_in_san_francisco
Posts: 86
Joined: Sun Oct 09, 2011 3:43 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by heart_in_san_francisco » Thu Nov 28, 2019 12:09 am

Very interesting results. Thank you for all the work you put in.
My model makes several generous assumptions when it comes to leverage (yearly re-leverage, wealth can't go below zero, no margin calls), these results are an approximation.
These are generous assumptions indeed. I wouldn't recommend anyone go above 100% equities unless they are a day-trader. Leveraged ETFs are not suitable for buy-and-hold investors, it even says so in bold print in their prospectuses that they are meant for day-traders only. So I must ask, do your simulations take into volatility drag and fees for leveraged ETFs? It doesn't feel right that a leveraged ETF is the best "retire early" strategy given their many downsides. I guess if your goal was "early retirement or bust" then I can see the case for leveraged ETFs (easy way to beat the market: take on more risk and get lucky), but otherwise I'm a bit worried some readers will have the wrong takeaway.

StillGoing
Posts: 10
Joined: Mon Nov 04, 2019 4:43 am

Re: Optimal asset allocation strategies for retirement & saving

Post by StillGoing » Wed Dec 04, 2019 10:27 am

Here are some further (historical) results for dynamically adjusting the asset allocation (i.e. as uncorrelated has suggested, but using the optimal results arising from historical data) - I'll upload some results for dynamic withdrawal in a follow up post (the code is done and the results interesting)

Firstly, for a 30 year retirement, the real (i.e. adjusted for inflation) fraction of the starting portfolio balance at the end date and the minimum balance through the 30 years as a function of start date. This is for a fixed, inflation-indexed, withdrawal of 3.6% (chosen because it leads to zero historical failures). Except in the 1870s and dates since the 1970s the portfolio is pretty well spent up.

Image
https://i.postimg.cc/DZ2ns9sC/dynaa-wit ... 191204.png

The next graph is the maximum, mean, and minimum stock fraction found during the 30 year retirement as a function of start date. As expected (from uncorrelated's trinity plot and my reproduction of it), the stock fraction is pretty low except in periods where conditions are challenging.

Image
https://i.postimg.cc/mkW942Fd/dynaa-wit ... 191204.png

The last few graphs are some examples (those chosen produce some of the lowest end balances - the year is in the title of the graph. Note that I have used monthly data, withdrawals, and rebalancing throughout) of the stock fraction as a function of time for individual retirements. The optimal stock fraction can vary quite a lot in some cases and tends to decrease significantly towards the end of the retirement period (expected from uncorrelated's trinity plot).

Image
https://i.postimg.cc/nzgycNHS/dynaa-wit ... 191204.png

Image
https://i.postimg.cc/J0tzSfpT/dynaa-wit ... 191204.png

Image
https://i.postimg.cc/HnSBky4Y/dynaa-wit ... 191204.png

I note that applying this approach on a monthly basis would probably be unwise and not for the fainthearted!

cheers
StillGoing

StillGoing
Posts: 10
Joined: Mon Nov 04, 2019 4:43 am

Re: Optimal asset allocation strategies for retirement & saving

Post by StillGoing » Wed Dec 04, 2019 1:19 pm

Several approaches can be adopted in using the graph of capital/spending vs time left in retirement to follow an optimal solution (my earlier diagram reproduced here, uncorrelated's version is in the original post):

Image

1) fix the withdrawal and vary the AA (see uncorrelated's original post and my previous one)
2) fix the AA and allow the withdrawal to vary (the subject of this post)
3) a combination of 1 and 2, or
4) follow a path of fixed failure rate

If the AA is fixed such that we have a 60/40 portfolio (chosen as an example solely because it is a popular choice), the final and minimum real balances (as a fraction of the original balance) for a 30 year retirement as a function of start year is as follows.
Image
https://i.postimg.cc/2SmN2mvF/aa060-dyn ... 191204.png

Compared to fixing the withdrawal and allowing the AA to vary (previous post), the remaining capital is much lower (it is also much lower than the often large fortunes left with a fixed AA and fixed withdrawal). Of course, this would present a longevity risk in the worst cases where only about 9% of the original portfolio remains, i.e. two or three years worth of expenses - I recently read a comment in another thread, sorry cannot find the link, that said that the last few years of retirement expenditure is under-researched).

The following graph shows the mean real expenditure (i.e. the fraction of the original portfolio value) as a function of start date.
Image
https://i.postimg.cc/7hjwxQBH/aa060-dyn ... 191204.png

This approach leads to a minimum mean expenditure of 4% (with no historical failures - so better than the ~5% failure rate with fixed AA and fixed spend) and for good starting years (early 1980s) results in a mean real expenditure of 14%. Of course, the graph doesn't tell us when the withdrawals took place (as it happens, mostly around years 15-25 of the 30 year retirement).

Here are a few example retirements focussing on some of the worst cases for remaining balance (1890 and 1902) and the worst case for income (1965) (the top panel shows the nominal and real balance, the lower panel the real withdrawal rate)

Image
https://i.postimg.cc/DyMDWXdN/aa060-dyn ... 191204.png

Image
https://i.postimg.cc/L63twKh3/aa060-dyn ... 191204.png

Image
https://i.postimg.cc/gkgw3YnQ/aa060-dyn ... 191204.png

While I have yet to systematically study the effect of changing the AA on this approach, a larger share fraction tends to lead to higher withdrawals and vice versa (although I note 100% share fraction leads to a minimum mean withdrawal of 3.6% instead of 4%).

I still need to explore options 3 and 4 (see above) and to look at leaving a legacy (rather than attempting to spend the portfolio down). It would be useful if someone else could see if they at least get broadly similar outcomes to me (I'm fairly confident that my code is correct, but I'm not quite ready to change my preferred withdrawal/AA approach to this one).

cheers
StillGoing

pdavi21
Posts: 1292
Joined: Sat Jan 30, 2016 4:04 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by pdavi21 » Wed Dec 04, 2019 3:44 pm

I disagree.
"We spend a great deal of time studying history, which, let's face it, is mostly the history of stupidity." -Stephen Hawking

Topic Author
Uncorrelated
Posts: 165
Joined: Sun Oct 13, 2019 3:16 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Uncorrelated » Wed Dec 04, 2019 6:12 pm

heart_in_san_francisco wrote:
Thu Nov 28, 2019 12:09 am
Very interesting results. Thank you for all the work you put in.
My model makes several generous assumptions when it comes to leverage (yearly re-leverage, wealth can't go below zero, no margin calls), these results are an approximation.
These are generous assumptions indeed. I wouldn't recommend anyone go above 100% equities unless they are a day-trader. Leveraged ETFs are not suitable for buy-and-hold investors, it even says so in bold print in their prospectuses that they are meant for day-traders only. So I must ask, do your simulations take into volatility drag and fees for leveraged ETFs? It doesn't feel right that a leveraged ETF is the best "retire early" strategy given their many downsides. I guess if your goal was "early retirement or bust" then I can see the case for leveraged ETFs (easy way to beat the market: take on more risk and get lucky), but otherwise I'm a bit worried some readers will have the wrong takeaway.
I spent some time to research leverage recently. It appears that if you use options to get access to leverage, you will exactly mimic the 'yearly leverage' and 'no margin calls' properties of my optimizer. My optimizer assumes that the borrow rate is equal to the return on bonds (2% real), however, one can obtain leverage with options for far lower rates. Assuming I got the math right one can get 2x leverage for about 0.8% over the fed fund rate and 1.5x leverage for 0.2% over the fed fund rate by purchasing S&P 500 LEAP calls with an expiration time 2 years into the future and waiting until expiry. This appears to be quite close to the assumptions in my optimizer.

Of course there is a chance your options will expire worthless, but that chance is low enough that the optimizer (apparently) thinks it's worth it.

For readers that want to know more about leveraged investing I recommend to read the about lifecycle investing, there is a topic on this board about that: viewtopic.php?f=10&t=274390. The topic contains an exact mathematical approach to determine the best variable allocation over time. I have verified that my optimizer gives similar results. In part because I was able to use my optimizer to validate it, lifecycle investing has become my go-to recommendation for the accumulation phase.
StillGoing wrote:
Wed Nov 27, 2019 5:19 am

If I understand your question correctly (let me know if I haven't), I just used the vanilla historical sequence with no randomisation of blocks and therefore the method suffers from having little independence in the data and a limited number of cases (the failure rates are quantised). There are advantages to this approach since it makes no assumptions about the distribution of returns (except that the future will be broadly similar to the past and that the historical worst cases are indicative of the likely future worst cases - both of which can be argued with).

cheers
StillGoing
Thanks for validating my approach. There are some interesting artifacts in your results. In particular, with 23 capital left and 23 years left, the optimizer appears to prefer a more aggressive asset allocation than with 23 capital left and 25 years left. Maybe that's just the color coding?


Your approach makes hidden assumptions about the distribution of returns, namely that the distribution and sequence of returns is exactly identical to the past. This avoids the problem of requiring a model specification, but it also means that the strategy only has ~100 years of data to work with. I fear this presents a very large danger of over fitting.

As you mention there are advantages and disadvantages to this model, but I prefer to make it explicit which assumptions are used.
StillGoing wrote:
Wed Dec 04, 2019 10:27 am
I note that applying this approach on a monthly basis would probably be unwise and not for the fainthearted!
Yes that is definitely a huge problem. The current solutions are frankly unimplementable in practice. In order to make these strategies implementable, we need to come up with utility functions that make sense and good rules of thumb that allows novices to determine their strategy and stay the course.
StillGoing wrote:
Wed Dec 04, 2019 1:19 pm
The following graph shows the mean real expenditure (i.e. the fraction of the original portfolio value) as a function of start date.
Image
https://i.postimg.cc/7hjwxQBH/aa060-dyn ... 191204.png

This approach leads to a minimum mean expenditure of 4% (with no historical failures - so better than the ~5% failure rate with fixed AA and fixed spend) and for good starting years (early 1980s) results in a mean real expenditure of 14%. Of course, the graph doesn't tell us when the withdrawals took place (as it happens, mostly around years 15-25 of the 30 year retirement).
It's not clear what you're doing here. Could you detail how you determine the optimal withdrawal amount (also bruteforce?) and which utility function is being maximized? Are you maximizing the minimum spending or the mean spending, or a variation thereof?

In your scenario's, the withdrawal in the final year appears to be quite bad. Is that true or is that a display artifact?




I updated my optimizer with the ability to determine the optimal spending level using an CRRA utility function at each time step (this paper details the math on page 15), this should allow for optimizing the asset allocation and withdrawal at the same time. But I'm having some difficulty in visualizing the results because the resulting solution is quite sensitive to input parameters (social security income and risk aversion), and has too dimensions to plot properly. When there is no social security income, the optimal asset allocation appears to be a constant. A lower risk aversion results in a more aggressive asset allocation, which puzzlingly results in the selection of lower withdrawal rates.

StillGoing
Posts: 10
Joined: Mon Nov 04, 2019 4:43 am

Re: Optimal asset allocation strategies for retirement & saving

Post by StillGoing » Thu Dec 05, 2019 4:37 am

Thanks for validating my approach. There are some interesting artifacts in your results. In particular, with 23 capital left and 23 years left, the optimizer appears to prefer a more aggressive asset allocation than with 23 capital left and 25 years left. Maybe that's just the color coding?
This is an artefact (not the colour coding) and, as far as I can tell, arises out of the limited historical data set available (i.e. changing the duration by a year or two can remove a 'recovery' that optimises on one particular AA than another. While, I didn't follow up the example you mentioned, that was the case for the reduction in aggressiveness that occurs at a capital/spend of 5 with about 10 years to go.
Your approach makes hidden assumptions about the distribution of returns, namely that the distribution and sequence of returns is exactly identical to the past. This avoids the problem of requiring a model specification, but it also means that the strategy only has ~100 years of data to work with. I fear this presents a very large danger of over fitting.

As you mention there are advantages and disadvantages to this model, but I prefer to make it explicit which assumptions are used.
I completely agree with this - this is why both historical and monte-carlo simulations have their place. Having spent part of my career looking at bit error rate calculations in communication systems in the presence of unusual channels (i.e. highly non-Gaussian), I worry about under specified tails, the shape of the distribution (while there may be better sources than this one, it is at least open source), https://sixfigureinvesting.com/2016/03/ ... of-normal/, and reversion to the mean (if that exists). In terms of over-fitting, retaining a realistic (whatever that actually means) approach to the precision of inputs and outputs will undoubtably be useful in addressing this.

cheers
StillGoing

Derpalator
Posts: 19
Joined: Sun Nov 15, 2015 2:52 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by Derpalator » Thu Dec 05, 2019 6:47 am

GAAP wrote:
Fri Nov 01, 2019 11:16 am
AlohaJoe wrote:
Sat Oct 26, 2019 9:27 pm
I think this is an under-explored avenue for retirement planning because most people involved in retirement research don't really have the programming skills necessary to make it happen. If you look at retirement research overall, I mentally classify it into 3 "generations". The first generation, up to the late 1990s was the "calculator" generation. You plug some linear assumptions into a financial or scientific calculator and it would tell you a number. "If we assume 4% real growth for 26 years, how much money will I have?" That kind of thing.

The second generation, from Bengen onwards is the "spreadsheet" generation. Even most monte carlo analysis fits into this. Even today, if you look at the leading researchers like Pfau or Blanchett (or McClung) they do everything in an Excel spreadsheet. Excel is pretty powerful but it does have limits. It can't really do things like your approach.

So that leads us to the third generation, which is "data science" where they need some level of programming skills in R or python or whatever because they've reached the limits of Excel. But there just aren't very many people interested in retirement research that have the background for that right now. Which is why, half a decade after Irlam started publishing on stochastic dynamic programming there's been virtually zero follow up from anyone else.
You are correct, but the flip side of this is that it is probably much easier for the typical person to understand and implement 1st generation than 2nd generation -- let alone 3rd generation. It is also far easier for an "adviser" to sell 2nd generation solutions since they are sufficiently advanced to scare/impress the general public while being simple enough for the salesperson to understand. The financial industry rewards sales/profits from customers and has little incentive outside of academia to pursue 3rd generation research. These tendencies play out every week in this forum...
To quote Sandtrap: "actionably", one cannot stop progress. These calculations have progressed according to the technology available. Once more robust artificial intelligence is available, and being able to combine with quantum computers, such investigations/calculations may approach being truly optimal. Such circumstances would leave us with the ability to pursue more interesting studies, like just what it IS to BE human.

StillGoing
Posts: 10
Joined: Mon Nov 04, 2019 4:43 am

Re: Optimal asset allocation strategies for retirement & saving

Post by StillGoing » Thu Dec 05, 2019 7:37 am

It's not clear what you're doing here. Could you detail how you determine the optimal withdrawal amount (also bruteforce?) and which utility function is being maximized? Are you maximizing the minimum spending or the mean spending, or a variation thereof?
I've ignored more than 30 years experience as a scientist/engineer and omitted a description of the method - my apologies.

For the fixed withdrawal approach:
1) Calculate the required spend as a fraction of the current portfolio (so, if the initial requirement was for 4%, then index-link the initial amount (in $) and then convert to the capital/spend ratio using the current capital). As the portfolio grows (in real terms) the capital/spend ratio increases and vice versa.
2) Using the value calculated in 1 and the years remaining, read off the required AA from the graph of capital/spend vs years left graph.
3) Loop round all months in a given historical sequence

For the variable withdrawal approach (fixed AA)
1) For a given value of years left, find the capital/spend ratio that corresponds to the required AA.
2) Because of the limited historical data set, there may be a range of values (e.g. with 100% stocks there are many capital/spend ratios): for values on the low boundary (i.e. 100% stocks) use the highest capital/spend ratio that satisfies the required AA (e.g. with 30 years left and 100% stocks, the range of capital/spend ratios lies from 5 to 24, 24 is then chosen). For values on the high boundary (i.e. with few years remaining, a large part of the graph indicates 0% stocks) choose the lowest value of capital/spend that satisfies the required AA.
3) Loop round all months in a given historical sequence

In other words, I am not actually optimising for spend etc., what I am doing is using the graph (which is optimal for failure rate, with some limitations) to explore actionable approaches - does this then remain optimal? - probably not*. Does it potentially improve on other spend/AA strategies - maybe (preliminary results would indicate yes - which is why your approach here is so interesting).

* I think I want to optimise for many things simultaneously
a) minimal portfolio failure (i.e. 'success' is not running out of money or leaving a certain amount - not too much and not too little)
b) maximising income (dependent on a) - but not falling below a certain level (that level may even be time dependent)

There may be other constraints such as a range of 'acceptable' AA, etc. Some of these constraints may lead to less than optimal outcomes.
In your scenario's, the withdrawal in the final year appears to be quite bad. Is that true or is that a display artifact?
I think this is a result of the portfolio being spent down heavily in the last couple of years (effectively the second method I've described above goes from being a percentage of initial portfolio to a percentage of current portfolio) - towards the end of retirement, the capital/spend ratio is likely to become 5 (in the way I've implemented this) and hence 20% of the portfolio is spent in each of the last few years. I note that for some cases this may be a problem for other variable withdrawal methods that aim to spend down the portfolio.

cheers
StillGoing

stocksurfer
Posts: 13
Joined: Tue Oct 09, 2018 11:30 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by stocksurfer » Thu Dec 05, 2019 6:34 pm

Whoa! This thread blows my mind! :-) Thanks so much for all the data / simulations / thoughts!

Uncorrelated, I'm not sure I fully understand the assumptions underlying your simulations. I'm particularly interested in the withdrawal with a E-K utility function. Are the assumptions really just:
  • annual stock return is 7% normally distributed with a 16% stddev
  • annual bond return is 2% normally distributed with a 4% stddev
  • stock and bond returns have zero correlation
  • year-to-year returns are independent
It seems you ran simulations that don't assume year-to-year independence, but I'm not sure?

What strikes me overall is that the graphs show large regions of almost all-stock or all-bonds and relatively little in-between. But when I look at your simulations that vary the mean stock/bond returns, that in-between regions moves (as one would expect). Doesn't that mean that for someone in the interesting region one would have to predict the mean return to make any use of these charts? If you believe that the future will be similar to the past then that may be fine, but if you put value into predictions that future returns will be lower then what these charts really highlight is how much of a crap shoot the AA is??

stocksurfer
Posts: 13
Joined: Tue Oct 09, 2018 11:30 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by stocksurfer » Thu Dec 05, 2019 9:41 pm

I just read the details of the E-K utility function. The one thing that struck me is that they (primarily) use the mean of the coverage ratio as utility function. I would think that using the minimum of the coverage ratio might be interesting as well as it captures better how bad it could get. Hmm, I wish I could run some of these simulations... :shock:

AlohaJoe
Posts: 4944
Joined: Mon Nov 26, 2007 2:00 pm
Location: Saigon, Vietnam

Re: Optimal asset allocation strategies for retirement & saving

Post by AlohaJoe » Thu Dec 05, 2019 10:34 pm

stocksurfer wrote:
Thu Dec 05, 2019 9:41 pm
I just read the details of the E-K utility function. The one thing that struck me is that they (primarily) use the mean of the coverage ratio as utility function. I would think that using the minimum of the coverage ratio might be interesting as well as it captures better how bad it could get. Hmm, I wish I could run some of these simulations... :shock:
It is a good question but I don't think just looking at the minimum is right, either. That would be optimising for the worst case without regard for how much it costs you. In real life, when you're buying insurance for something you don't buy the most expensive policy no matter what, do you? If you're rich, sure you can do that. You can afford to pay any likely cost. But when it comes to generating relatively universal investing guidelines? Telling people "eh, just be rich and make so much money that you can build a 50-year TIPS ladder and have $300,000 on top of that for long-term care and assorted spending shocks...all without compromising your quality of life in the 40 years before retirement" isn't exactly useful.

Figuring out the utility for a single retirement run is easy enough. What is trickier is how you come up with meaningful aggregate statistics. I think that, too often, people try to narrow it down to a single number hoping it will explain everything. But I'm not convinced that's really possible.

I often look at: the minimum, the 1st percentile (1% worst cases), 5th percentile, 10th percentile, 25th percentile, 50th percentile, etc.

stocksurfer
Posts: 13
Joined: Tue Oct 09, 2018 11:30 pm

Re: Optimal asset allocation strategies for retirement & saving

Post by stocksurfer » Fri Dec 06, 2019 12:10 am

Agreed that minimum may be too pessimistic. Hard to tell without seeing what it is compared to avg. I guess what I'm saying--like you--is that it would be nice to get a feel for the landscape across a couple of percentiles.

Post Reply