Valuation-based market timing with PE10 can improve returns?

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rustymutt
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Post by rustymutt »

I agree with Mel. These studies prove nothing. Having a investment approach rooted in buy and hold, while keeping taxes and cost to a minimum is the key. While some of you believe this report easy to understand, I don't. I'm not about to follow something as confusing as the report which I read through to the end. It's very ambiguous and open to different interpretations. No thanks you. By the way, if it boiled down to something as simple as PE, someone would have a software program out taking advantage of it, and running the advantage into the ground.
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Post by lazyday »

fredflinstone wrote:If you ask me, a few folks think the standard Boglehead approach (fixed asset allocation; stay the course regardless of valuation) is the Holy Grail of Investing.
Maybe the Bogleheads way is the holy grail of investing, for most people.

Valuations do matter. High valuations imply lower expected long term returns. But there is so much noise in markets, and such a terrible record of those who have attempted timing, that I don't think it's wise to use valuations data in investing--with possible exceptions:

- A very small % of people meet all of Bill Bernstein's list of requirements to invest on one's own. They might carefully alter AA based on valuations.

- Some people believe that small conservative changes to AA may be made based on valuation, without the above requirement, but usually with some caveats.

- I believe but have not demonstrated, that when valuations are so greatly extended in an extreme bubble, that a risk asset has expected returns the same as a riskless asset, it makes sense to significantly reduce holdings of the risk asset.

The study makes a case for something entirely different than any of the above. The evidence it offers should be weighed against opposing evidence. It would shock me if it turned out to be a good way to invest. Rather, it might turn out to be a valuable contribution along the way to understanding, and maybe even improving investing strategies.
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fredflinstone
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Post by fredflinstone »

rustymutt wrote:By the way, if it boiled down to something as simple as PE, someone would have a software program out taking advantage of it, and running the advantage into the ground.
One would think so, yet the data show otherwise. As recently as 2009, the PE10 indicator worked remarkably well.

I wonder if you might be conflating "simple" with "easy." The PE10 market timing approach is simple but that does not mean it would be easy to implement in real life. It would take nerves of steel to overbalance into stocks during the depths of a bear market. Conversely, it would require the patience of Job to continuously shift assets from stocks to bonds at a time when the stock market is marching upward.

My guess is that relatively few people can pull it off. Doesn't mean it isn't worth a try.
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Post by fredflinstone »

rustymutt wrote:These studies prove nothing. Having a investment approach rooted in buy and hold, while keeping taxes and cost to a minimum is the key.
Well we've been around and around on the taxes and cost issue. For investors who own no equities inside their tax-deferred accounts, taxes are an issue. For everyone else, this is just a big fat red herring. Just buy and sell your equities inside a tax-deferred account. Problem solved.

Costs were relevant in the past but they are hardly a big issue now, especially if you stick to no-load mutual funds. Maybe when people hear the term "market timing" they envision dozens or hundreds of trades per day. Those costs could add up. By contrast, what Professor Pfau is talking about in his study is long-term market-timing--no more than 1 trade per year, I believe.
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Post by jeffyscott »

letsgobobby wrote: first, Dow 11,700 (in 2000) or Dow 14,300 (in 2007) or heck even Dow 36,000 means nothing to the value-based-allocator, because of his belief in reversion to the mean.
Yes, but what if that belief were to have turned out to be wrong? What did happen did not have to happen, suppose things had gone differently after 2000 and stocks never went below P/E10 of, say, 18? Would it still have turned out okay to have 0% stocks?

As has been indicated by the author of this study, the idea of reversion to the mean could surely turn out to be wrong the other direction, the economy as we have known it could end. That seemed like a possibility in 2008-09 and I am sure it also seemed like a possibility in the 1930s.

I just think the same could happen in the opposite direction, that stocks are never a real bargain during one's lifetime, but they still provide better returns than the alternatives.

I think we agree that in reality these problems can be reduced by avoiding 'black or white thinking.'
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Post by richard »

Mel Lindauer wrote:Previous studies have shown lots of back-tested schemes that worked -- that is, until they stopped working.
Back-test schemes that stop working are usually data mined oddities that make little economic sense. Calendar anomalies, such as the January effect, are good examples of this. Things that continue to work are usually things that are supported by underlying economics, such as diversification.

Buy, hold and rebalance is often supported by back-testing, often without adjusting for taxes and other costs. Are you suggesting it's a back-tested schemes that worked -- that is, until they stopped working? If not, why not? Should advocates stop citing unadjusted numbers?

What are your suggestions for adjusting historic data for taxes and costs? http://www.bogleheads.org/forum/viewtop ... 97#1067697

One of the fundamental tenets of finance is that higher risk results in higher expected returns. Wade's results, at bottom, are an illustration of that. His major problem, IMO, is using conventional risk metrics, which is hardly an unusual problem. See http://www.bogleheads.org/forum/viewtop ... 25#1067125
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Post by Mel Lindauer »

Something that seems to be missing in this study is the fact that "buy-and-hold" investors don't really just buy a portolio on a single day and hold it forever.

Rather, they're constantly DCAing into their tax-deferred plans at work, so when the market is down, they're buying equities at the lower prices. Therefore, in a way, they're participating and taking advantage of the down market. They're benefitting from the lower PE or whatever market-timing metric one chooses to use without following, or even paying attention to, an "all or nothing" bat signal.

Additionally, most investors don't hold a single 50/50 asset allocation for their investing liftime. Most are probably closer to 100% in equities in their early years, perhaps moving to 80/20, 70/30 and then 60/40 prior to, or at, retirement. So 50/50 isn't real world for most investors. And, we've seen that higher equity allocations do better.
Last edited by Mel Lindauer on Mon May 30, 2011 1:12 pm, edited 1 time in total.
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Post by dmcmahon »

jeffyscott wrote: Yes, but what if that belief were to have turned out to be wrong? What did happen did not have to happen, suppose things had gone differently after 2000 and stocks never went below P/E10 of, say, 18? Would it still have turned out okay to have 0% stocks?

As has been indicated by the author of this study, the idea of reversion to the mean could surely turn out to be wrong the other direction, the economy as we have known it could end. That seemed like a possibility in 2008-09 and I am sure it also seemed like a possibility in the 1930s.

I just think the same could happen in the opposite direction, that stocks are never a real bargain during one's lifetime, but they still provide better returns than the alternatives.

I think we agree that in reality these problems can be reduced by avoiding 'black or white thinking.'
Agreed completely re. black-and-white thinking. Your point about the nature of an overvaluation correction process is also well-taken, and one I've thought about constantly as I hold my nose while owning equities.

I like to read John Hussman's weekly assessments of the market. Even if one shares his grim assessment of valuations (and I do), it's still the case that equities look more attractive over the next 10 years than bonds do at current absurd interest rates. So what's a punter to do if both major asset classes are overvalued? If equities correct sharply to what Hussman thinks is an appropriate valuation, despite the dismal expected returns on bonds, then great, one can start ramping up the equity allocation. If, however, they correct via a slow process of underperforming historic averages, someone waiting it out will be left with less assets by virtue of having held bonds. Hussman's strategy of hedging his positions seems worse still, as he must pay dearly for protection as the market chums along sideways to slightly up.

I don't think a black-and-white approach is being put forth as an alternative, though, but a more nuanced process of shading AAs to take valuations into account.

I have to some extent been following just such an approach since 2000, and arguably it's failed miserably, except in one respect - I bailed out of equities completely over several months early in 2000 and avoided one of the worst crashes of the period. But I was slow to get back in as I never saw the slam-dunk low valuations I was hoping for, was still caught by the 2008 implosion with the limited part of my AA that had been put back into equities, and lacked the courage to do more than hold my positions through the bottom in 2009. Throughout this process I've had to endure increasingly lower risk-adjusted returns on my fixed income investments. Having a more mechanical approach would almost certainly have helped, and so I applaud the efforts to subject the idea to some academic rigour.
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Post by novastepp »

fredflinstone wrote:
novastepp wrote:Does the study take into account a strict rebalancing strategy that many Bogelheads use; say a 5% deviation or an annual effort?

I see that the study crosses its numbers with buy and hold, but I do not see mention as to the always-forgotten rebalancing strategy.
the study evaluates the following strategies:

- 100% equities buy-and-hold forever
- 50/50 balanced fund, rebalanced annually
- various market-timing strategies

The market-timing strategies produce much higher returns than the 50/50 balanced fund, with similar risk.

The market-timing strategies produce similar returns to the 100% equities buy-and-hold forever approach, but with much less risk.
In regards to rebalancing, does the study look at each year and whether the investor needed to invest into bonds or stocks? Or does it simply move all monies back into a 50/50 split?

Just curious. Thanks
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Post by 555 »

Fixed your post. :roll:
fredflinstone wrote:It would take nerves of silicon to overbalance into stocks during the depths of a bear market.
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Post by Bongleur »

Maybe we can frame a question & explore whether the PE data can help answer it. From a lifetime investment goals perspective, the investor has a certain number of years to reach his goal. He needs a given rate of return per year.

When PE is low he can reach his goal with less risk (less equity exposure, at baseline rate of return). Unless the low PE is persistant too long, at which time he must raise his baseline to "catch up" to his timeline.

OR

he can remain fully invested and reduce the number of years it takes to reach his goal (higher rate of return) -- which also reduces his lifetime risk because once the goal is reached he is out of the risky market entirely. The trick is to "get out" prior to losing _all_ his excess gains above his baseline, once the PE has risen and then reverses.

And vice versa.

So is there an AA variation strategy that helps make these decisions to over or under weight vs the lifetime baseline?
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Post by jeffyscott »

Bongleur wrote:When PE is low he can reach his goal with less risk (less equity exposure, at baseline rate of return). Unless the low PE is persistant too long, at which time he must raise his baseline to "catch up" to his timeline.
This option seems backwards to me. Expected returns from equities are high, so why put less in them? A young person starting out, should put everything in stocks while they can do so at low prices because you never know how quickly the low prices are going to last. If low stock prices become the new normal there would still be plenty of time to buy bonds later on, but if p/e return to normal or above they will have lost the opportunity to get as much of that gain as possible.

An older person has to also consider when they will need the money, so they should have some constraints so that money that is needed in, for example, 10 years is not put in stocks no matter how cheap they may appear to be.
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Post by 555 »

Bongleur wrote:"When PE is low he can reach his goal with less risk (less equity exposure, at baseline rate of return). Unless the low PE is persistant too long, at which time he must raise his baseline to "catch up" to his timeline."
Maintaining constant risk would then mean underrebalancing instead of overrebalancing. So maybe mere rebalancing is the sensible compromise. :wink:
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Valuations and timing

Post by Lauren Vignec »

Hello Everyone,

At the beginning of this year I told everyone I work with, "Last year turned out to be great. But valuations on stocks are high now, and interest rates on bonds are low. Don't expect good returns anywhere in the near future."

It's not just that "valuations matter". If we talk about investing to anyone, we have a responsibility to explain to people when expected returns are low, as they still are now for both stocks and bonds.

As far as timing goes? Well, the all-or-nothing approach will never work. It's too stressful to change allocations that dramatically--no one is going to do it in real life. And in order to follow the strategy outlined in Pfau's paper, an investor would have had to stay out of stocks for the entire decade of the 1990's. And the whole time, they would have had to have the willingness and emotional fortitude to push all their money into the stock market as soon as Great Depression II hit. No one is like that.

However, the advice to change allocations in smaller increments based on valuations is more plausible. The problem here is that people will lower their stock allocations when markets get more richly valued, but they won't increase their allocations when markets drop. Then again, this is the same problem people have with buy-hold-rebalance. In general, investors have a tougher time buying stocks after market drops than they do selling them when markets go up.

And in any case, the current expected returns of both stocks and bonds are low by historical standards. Normally that may not have been an issue, but it is certainly an issue now! So I'm left thinking along exactly the same lines as Dick Purcell and a few of the other posters--yes valuations matter, and no, market timing isn't going to help people.

L
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Mel Lindauer
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Re: Valuations and timing

Post by Mel Lindauer »

Lauren Vignec wrote:Hello Everyone,

At the beginning of this year I told everyone I work with, "Last year turned out to be great. But valuations on stocks are high now, and interest rates on bonds are low. Don't expect good returns anywhere in the near future."

It's not just that "valuations matter". If we talk about investing to anyone, we have a responsibility to explain to people when expected returns are low, as they still are now for both stocks and bonds.

As far as timing goes? Well, the all-or-nothing approach will never work. It's too stressful to change allocations that dramatically--no one is going to do it in real life. And in order to follow the strategy outlined in Pfau's paper, an investor would have had to stay out of stocks for the entire decade of the 1990's. And the whole time, they would have had to have the willingness and emotional fortitude to push all their money into the stock market as soon as Great Depression II hit. No one is like that.

However, the advice to change allocations in smaller increments based on valuations is more plausible. The problem here is that people will lower their stock allocations when markets get more richly valued, but they won't increase their allocations when markets drop. Then again, this is the same problem people have with buy-hold-rebalance. In general, investors have a tougher time buying stocks after market drops than they do selling them when markets go up.

And in any case, the current expected returns of both stocks and bonds are low by historical standards. Normally that may not have been an issue, but it is certainly an issue now! So I'm left thinking along exactly the same lines as Dick Purcell and a few of the other posters--yes valuations matter, and no, market timing isn't going to help people.

L
Well said, Lauren.
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Post by Bongleur »

Apparently there exist passive funds which use value weighting. So could one be constructed to follow changes in value 1) beneficially (alpha) and 2) without excess costs vs a traditional benchmarked index fund? :

http://moneywatch.bnet.com/investing/bl ... blog-river

Q: Where would you draw the line between passive asset class investment and active management? I'm thinking, for example, of Joel Greenblatt's value-weighted indexing

A: I would view Joel's funds as passive as there is no attempt at stock selection or market timing.

His funds IMO are clearly passive asset class funds that just don't have a public index to benchmark against.

snip
I don't really know what actively passive means...To me the answer if someone is passive or active is do they do any fundamental analysis to pick stocks and do they shift their allocations around based on judgments about valuations--so not always fully invested

So using screens, even if it is a momentum screen, to me would be passive. Not active.

As to rebalancing, it is an activity, controlling risk, but not active management. NOT rebalancing to me would be active---you are actively deciding to OVERRIDE your investment plan.
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Post by wade »

There are so many good comments here. And even some LOL comedy from grayfox
grayfox wrote:And I believe that the IRS has ruled that hypothetical investors don't have to pay taxes.
I am kind of busy this week, but I will definitely be studying this thread very closely in the coming weeks.

Fred made a nice chart:
fredflinstone wrote:just for fun, I came up with an equity allocation formula that takes into account three factors: the investor's age, the PE10 ratio, and the 5-year yield on TIPS.

Voila:

=MIN(95,(MAX(10,(152.5-(AGE*1.25)-(2.5*PE10)-(20*TIPS))))),

where AGE is the investor's age, PE10 is the Shiller PE10 ratio, and TIPS is the yield on 5-year TIPS bonds.

The maximum percentage to equities is 95 and the minimum is 10. Note that although I incorporate TIPS yields into the model, the investor is not restricted to buying TIPS in his or her fixed-income holdings.

Here are the resultant equity percentages under a few different scenarios:

Code: Select all

AGE	PE10	TIPS yield	% equities
20	   15	   0	          90
30	   15	   0	          77.5
40	   15	   0	          65
50	   15	   0	          52.5
60	   15	   0	          40
			
20	   25	   0             65
30	   25	   0	          52.5
40	   25	   0	          40
50	   25	   0	          27.5
60	   25	   0	          15
			
20	   35	   0	          40
30	   35	   0	          27.5
40	   35	   0	          15
50	   35	   0	          10
60	   35	   0	          10
These numbers look about right for me. Of course, others would probably want to modify it depending on their risk aversion, need for growth, and degree of tolerance for market-timing.
Anyone ultimately interested in this will have to make such a chart for themselves based on their own return and risk objectives. That's exactly what someone will need!

And then what I am able to offer is to test such allocation paths against some alternative that doesn't consider valuations and see how the two strategies performed over rolling historical periods. Of course, past performance does not guarantee future performance, and that always needs to remain as a background caveat! As well, there are some artificial "academic" assumptions that need to be made in order to be able to test anything, and people need to understand these limitations. My paper about using this for individuals will not be released until I am satisfied that I have fully explained all these limitations, and that is why these threads are so helpful for me.

Then you just have to commit yourself to sticking to it no matter what, which means you should take some time and explore the possibilities and make sure you won't be scared away from following through with your plan.

One possible amendment though, is that as valuations change, your return objectives may also change. Increasing valuations may be creating wealth for you more quickly than expected, which means you could lower your stock allocation regardless of valuations in order to lock-in your goals with less risk.

Dick: the predictive power 5 years out is going to be so weak that your proposal for using it to feed Monte Carlo assumptions will hardly make any difference, I'm afraid. In order to meet investor's goals, there has got to be a more dynamic component that I haven't really yet considered, but hinted at in the previous paragraph.

Grayfox: yes, Dr. Markowitz uses 50/50 to minimize his overall regret (you would wish you had 100% stocks when markets go up, and 0% stocks when markets go down).

And to just hit a few key points again:

-In the real world, this doesn't have to be 0 stocks or 100 percent stocks. There are many in-between points

-It is not so easy for a mutual fund to do this, because it will require trailing the benchmarks for years at a time, which will cause all the customers to flee

-I am considering dollar-costs averaging in my other paper, which help reduce costs from rebalancing and can even provide some tax-loss harvesting for allocation changes.

- I will eventually incorporate modern taxes. I have a system planned out to do this, but it will probably take several weeks to program and debug.

- Trying to have someone in 1920 build their own index fund to implement these strategies is not practical or possible. How different would our historical data have been in the counterfactual situation that low-cost index funds have always been available? It's a good question, but I have no clue about the answer.

Again, thank you for all the comments, both good and bad. I'm a Boglehead (DRiP Guy said so, and I hope he doesn't rescind it) and very grateful for low cost index funds. I'm not a market timer, and this topic can only be considered as market timing in the very loosest sense of the term. I only called it market timing in this paper because I was following Fisher and Statman. It's not really tactical asset allocation either, because tactical asset allocation implies making short term changes to your strategic asset allocation due to short-term forecasts for capital markets. But it may take years for the changes to happen.
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Post by novastepp »

What about the idea of continually buying into the market at all times? I buy into my 403b and 457 with every paycheck (bimonthly). So, if the market is at the tippy top, I'm buying, if it is bottomed out, I'm buying. How would research into any timing approach take into account that kind of buy/hold/rebalance strategy?
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Post by wade »

novastepp wrote:What about the idea of continually buying into the market at all times? I buy into my 403b and 457 with every paycheck (bimonthly). So, if the market is at the tippy top, I'm buying, if it is bottomed out, I'm buying. How would research into any timing approach take into account that kind of buy/hold/rebalance strategy?
When you are buying, your buying is done to move yourself toward some particular overall asset allocation. The only difference is whether that target asset allocation has changed because of valuations. This figure is based on someone contributing new funds every year over 30 years:

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Post by FNK »

I must say this piques my interest, but I would make a much smaller change to AA. Say, adjust the allocation to equity funds by the difference between PE10 and PE at the day of rebalancing, with bonds picking up the balance.

The PEs are posted right on the site. But where can I find PE10 for TSM and TISM?

Also, what would be a similar strategy for TBM/TIPS?
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Post by Dick Purcell »

Wade –

1. For adjusting future return-rate assumptions based on present PE10, I didn’t mean adjust the rate for just year 5 ahead, I meant adjust the average arithmetic mean for all the next five years. Maybe next ten years is better. Look at Bob90245’s related graph at the link in my message near the top of this page (3). That looks like a significant slope to me: (a) next ten years stocks-minus-bonds v. (b) current PE10. And I’d expect the relationship I’m describing to have at least that steep a slope.

2. That’s the only way I can conceive for incorporating your paper’s message in making allocation decisions for the investment purpose – result probabilities for the investor’s particular dollar plan and goals. If I can’t see that an allocation switch appears better for the investment purpose, I ain’t agonna do it. Or recommend it. I'll recommend against it.

3. In the paragraph above the one addressed to me, you suggested that after a big upsurge to very high PE, maybe an investor should reduce her stock allocation because the upsurge has brought her portfolio value closer to her goal. Makes sense. But that decision does not require any special new PE-based decision rule. If you just do a Monte of your plan once a year, that year’s Monte will reveal that you now have better prospects for meeting your goal with a reduced-stocks portfilio.

4. Here’s something interesting: Bongleur pointed out that if the PE is very low instead of high, an investor might want to adjust her allocation the same way as just above for a PE rise – reduce the % stocks, because with stocks now expected to probably do better, not as much of them is needed for good prospects of meeting the goal.

Think about that. Bongleur’s message and your #3 above are calling for the same reduce-the-stocks response to opposite PE extremes. And they both are reasonable possibilities. This is a great illustration of why the only responsible basis for the decision is dollar-result probabilities for that investor’s plan and goals, viewed by that investor for a decision that fits her priorities. Decision rules based on investment-theory abstractions are not adequate.

5. I understand your point about limitations in general-application academic studies, but I have a suggestion that I think can fit that limitation yet bring your individual-investor tests closer to real people’s plans. Instead of hypothesizing $ X invested for Y years, test three realistic triangular cash flow plans: an upslope triangle with cash put in each year, for a pre-retirement plan; a downslope triangle with withdrawal each year, for a retired person; and an isosceles triangle of up-and-down for a full pre and post retirement plan. These more realistic cash flow shapes will make sharp switches between conservative and aggressive allocations more dangerous, because there will be more risk that in the few years when the portfolio is richest, the allocations are far above or below the desired average allocation, throwing the investor way off her desired average allocation.

6. Back on page 2, Fredflinstone said this about you:
I wish everyone responded to criticism in such a calm, clear, rational manner. I know I don't always do so. Professor Pfau, you are a class act.
I agree, doubly so. Wade, you are absolutely marvelous in absorbing, thinking about, and responding to comments of all sorts on your work.

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Post by JW-Retired »

Wade wrote: JW Nearly Retired: I had an implicit assumption that choice of dates (January 1, etc.) and choice of valuations (PE6,...PE16, etc.) would not have any significant impacts on the results. As well, I was thinking in terms of something that would be easy to follow: only do a checkup at the end of each year rather than constant monitoring, and using PE10 which is readily available on the Internet. But you are right that I should check more of these possibilities as a form of sensitivity analysis. If I did accidentally cherry pick the only scenario that works, then of course it is a serious problem. I will check more about this. Thank you. -- The paper is not published yet, and the whole point of releasing the unpublished version is to get valuable feedback like this.
Apologies for my rudeness. Hope I'm wrong because I'm interested in timing schemes, but I bet you will find large impacts to making the trade on other days. M. Faber's monthly timing traded the last day of the month. Turns out if you test it doing it on different sets of 1-month period trading days you get widely different results, persumably because of market noise. The noise obviously won't repeat the same going forward.
See the 16th page in the 200-day market timing thread, 7th post down dated Jan 17, 2009.
http://www.bogleheads.org/forum/viewtop ... &start=750

Again, sorry for my over the top remarks. For what little it might be worth, I've deleted them.
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William Million
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Post by William Million »

I am skeptical this approach would beat ordinary re-balancing over time. Successful market-timers, and it's a small group, don't use a single measure, such as PE10.

However, even if this market-timing scheme works, I don't many people would be capable of implementing it. The Boglehead approach is hard enough. How many people re-balanced by purchasing stocks in 2008-09? When I did so, I really had to discuss at length with my wife and thought I might be hopping off a cliff.

This scheme is far more difficult because the percentages themselves change and based only on a single, somewhat limited metric.

Anyone who considers this approach should, as I mentioned above, decide at what point they jump off the sinking ship. If you adopt this approach and fall significantly behind buy-and-hold after 5 years, do you abandon? How about 10 years? Even if it works, you cannot guarantee that it'll work immediately. Can you handle the wait?

With the Boglehead approach, by contrast, you are guaranteed to buy low and sell high within a modest band through re-balancing.
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Post by letsgobobby »

William Million wrote:
With the Boglehead approach, by contrast, you are guaranteed to buy low and sell high within a modest band through re-balancing.
if this is true, how can tactical AA NOT be true? it's the same... only more so.
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Post by Rodc »

Sorry to jump in late, been off on a long weekend vacation.

Personally I think this is a more realistic look at the issues:

http://home.comcast.net/~rodec/finance/ ... mingV2.pdf

Note in that paper q-timing worked up until the 70s or so (ie on the data used to develop q) and largely failed after on independent data.

As noted above all or nothing is totally unrealistic.

To see a related interesting thread see the 200 day moving average thread. Lots of folks found that compelling in the beginning and we got to watch that fail in real time.

I only glanced through the paper, but it seems very dangerous to look at stock valuations in isolation from bonds and other assets (a short coming of my work as well).

Anything based on a threshold computed and tested on the same dataset is highly unlikely to be robust. How about this for a test. Compute your algorithm based on data up to 1950 and then test going forward?

And last, I know is this something of a ad homin attack, but really, anything that takes anything Stein says as serious work can't be considered serious.

Valuations matter for expected LONG term returns, but it is highly unlikely you can market time on signals with a time scale of decades.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.
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Post by 555 »

Is paper going to be peer reviewed and published in an academic journal?

I'll say one thing. Many academic works go unnoticed, so the author is actually quite lucky to have his work discussed and criticised on the internet like this.
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Post by Lumpr »

I took a quick look at the paper and plan to delve into it a bit more carefully, but two questions jumped to mind:

1 - How are dividends accounted for in the model? I searched the paper, but there is no mention of how dividends are accounted for. I searched this thread as well and there seems to be no mention of it. I assume they are included (i.e. reinvested in the buy and hold scenario), but would like clarification.

2 - How exactly is PE10 calculated on the date of the determination? The paper indicates that the market timing decisions are annual (page 5); however it is not clear precisely how the decision model works. Is the author basing this annual decision on the Q3 earnings information (which is what would be available on January 1st of any given year) or is the author using Q4 earnings information (which would not be available until some time in March).

Maybe the author would considered making his spreadsheet available so that interested people can take a more in depth look at the model?
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Post by fredflinstone »

William Million wrote:even if this market-timing scheme works, I don't many people would be capable of implementing it. The Boglehead approach is hard enough. How many people re-balanced by purchasing stocks in 2008-09? When I did so, I really had to discuss at length with my wife and thought I might be hopping off a cliff.

This scheme is far more difficult because the percentages themselves change and based only on a single, somewhat limited metric.

Anyone who considers this approach should, as I mentioned above, decide at what point they jump off the sinking ship. If you adopt this approach and fall significantly behind buy-and-hold after 5 years, do you abandon? How about 10 years? Even if it works, you cannot guarantee that it'll work immediately. Can you handle the wait?
Fair questions. May I play devil's advocate for a moment? Just for the sake of argument, I would like to propose that the market-timing approach may be EASIER for risk-averse investors to implement than the standard rebalancing approach. Sound crazy? Let me explain.

Hypothetically, let's say you are a 40-year old investor with less-than-average risk tolerance. Standard Boglehead advice probably would suggest something along the lines a 40/60 equity/bond split. Probably that would work very well for this investor as long as valuations are reasonable. Now let us suppose PE10 climbs above 40. My guess is that this investor is going to be very uncomfortable holding 40% in equities. Even "stay the course" Jack Bogle reduced his equity allocation when PE10 was above 40 back in 1999-2000.

But our investor has been reading lots of posts decrying market-timing at Bogleheads.org. Against every risk-averse fiber in his being, he stays the course. Now let's the housing bubble bursts and the stock market suddenly falls by 50 percent. This investor has almost instantly lost almost 20 percent of his assets. After the market crash, PE10 is below 20. The investor should be increasing his exposure to stocks, but he is so shell-shocked by his losses that he can't stomach taking on even more risk. So rather than rebalance, he decides to shift some of his equities to bonds.

If this same investor had been using a valuation-based market-timing approach, he would have been reducing his equity exposure as PE10 climbed. For awhile our risk-averse investor lags his friends but he is OK with that because he knows the market is getting frothy (as indicated by high PE10).

Once PE10 climbs above 40, our hypothetical investor is now down to an equity allocation of only 10%. The investor is risk-averse, but he knows stocks have excellent long-run returns and would welcome the opportunity to allocate more than 10% of his assets to stocks. In short, he is chomping at the bit for a major decline so that he will have an opportunity to plow more of his money into stocks. When the market declines 50% he takes a relatively small hit (about 5% of total assets) and is all too happy to be able to finally increase his equity allocation.
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Post by DP »

Hi,
The problem with a valuation approach is that the market can stay overvalued, or undervalued for a very long time. I only skimmed the paper but it appears to be based entirely on US equities. An interesting measure of the approach would be to apply it to equities from other countries. I suspect Japan and Germany in particular would not fare so well ... but then neither did buy and hold.

Don
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Post by richard »

The proper way to value a stock is based on its future cash flows, not its past. We say that PE10 is a valuation ratio, but it's not a ratio you would use if you were trying to actually value a stock. It's only useful to the extent past earnings are a good proxy for future earnings. There is no way to be comfortable that past earnings will be a good predictor of future earnings.

Stock returns are dependent on future prices and future dividends. Given how low dividends are these days, returns are largely a function of future prices. Future prices are dependent on future p/e ratios. Even if we were able to predict future earnings, without the ability to predict future p/e ratios, we will not be able to predict prices and therefore returns.

PE10 may be the best valuation metric we have (or not), but there's no reason to have confidence that it will be a good enough tool.
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Post by 555 »

Okay, let's make things simple. I just want to buy low and sell high. Can someone tell me how to do that?
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Post by letsgobobby »

DP wrote:Hi,
The problem with a valuation approach is that the market can stay overvalued, or undervalued for a very long time. I only skimmed the paper but it appears to be based entirely on US equities. An interesting measure of the approach would be to apply it to equities from other countries. I suspect Japan and Germany in particular would not fare so well ... but then neither did buy and hold.

Don
without knowing any hard figures about Japan, I suspect one employing TA would have completely been out of equities by the late 80s. Only recently has Japan's PE or PE10 become remotely reasonable (this is just off the top of my head recall). So the question becomes: did a Japanese investor employing this model get out too early - ie, in the 1950s or 1960s? Because getting out any time in 1985 or later hardly looks "too early" based on a snapshot of the Nikkei over that time frame.

I don't dismiss what you are saying at all. it is possible that PE10 or any other valuation measure would work extremely poorly in the case of seismic economic change: ie, Germany post WWI, Japan post WWII, etc.
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Post by grayfox »

555 wrote:Okay, let's make things simple. I just want to buy low and sell high. Can someone tell me how to do that?
That sounds pretty easy to accomplish.

Just buy right now, immediately, at the market price.
The put in a GTC limit order to sell at any higher price.

When the limit order executes, voila! You bought low and sold high.

The higher the sell price, the longer, on average that you will have to wait before it executes. If it's one penny higher, it may execute within minutes. If it is double, you will have to wait longer--months, years or decades.

This sounds like a foolproof plan. Buy now and put in a limit order for 2x. What am I missing?
Last edited by grayfox on Tue May 31, 2011 1:55 pm, edited 1 time in total.
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Post by runthetrails »

555 wrote:
grayfox wrote:
555 wrote:Could someone please explain what the fuss is all about.
"It is quite ironic that you would question this market timing startegy because it is well known that you yourself are a timer.
Probably the most well-known timer there is, short of Big Ben. Every EE knows exactly what I'm talking about.
http://en.wikipedia.org/wiki/555_timer_IC "
That's actually very funny :lol: though I had to click in the link to figure out why :oops:
I actually always assumed (without really paying attention to whether it was consistent with your posts) that your screen name meant you were a "timer" for this reason. Radio Shack sold me a bunch of these ICs.
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Post by Dick Purcell »

Et tu, Rodc ?

I think there is something to look at regarding stock-market valuation. But to me the notion of setting a rule for future cliff-like leaps and plummets, based on criteria ignoring the investor’s cash flow plan and goals, is not the rational response. The paper you cite is an interesting complement to Wade’s, but they both are about that forget-the-plan-and-goals, leaps-and-plummets approach. To me all they both say is that there could be something here regarding valuations, for which we need a different, sensible approach.

I think the sensible approach is suggested by that graph on Bob90245’s website that I've referenced twice before. Higher current PE10 tends to portend lower stocks return-rate mean for the next ten years. Incorporate this in estimates for future return-rate probabilities, apply to the investor’s cash flow plan and goals via Monte, and you have the basis for decision now, focused on this investor’s dollar plan and goals.

For future decisions, don’t adopt preset rules now – look again when you know more, using the same dollar-goals-focused approach you are using now.

I did a first-cut preliminary test, and it suggests to me this valuation-adjusted approach is worth doing – especially in the retired phase, when in the worst cases portfolio value is highest in the earlier years.

I started with Wade’s basic example from that other thread on updating the Trinity study – stocks/bonds 50-50, data 1926-2009, retirement plan of 30 years, each year withdraw 4% of initial value inflation-adjusted. I used $1 million initial balance, $40k/year inflation-adjusted withdrawal. This is my base plan. Monte shows it to have 93% success rate, 7% risk of running out of money.

For my test, I note from Bob90245’s graph that the slope of next ten years’ return rate relative to current PE10 is minus ½. For every +1 in PE10, next ten years’ return rate tends to be ½% lower. His measure of return rate is stocks relative to bonds, but just for a first preliminary test I say let’s assume the same slope for stocks vs. zero.

Well – let’s look at a situation in which PE10 is now 8 above average. I’ll look at the effect of reducing my assumption for stocks arithmetic mean by 4% for the next ten years.

Here it is in a Pathfinder graph. Blue is the base case, red is the same plan and goals with the PE10-based adjustment:

[img]<table><tr><td><a%20href="https://picasaweb.google.com/lh/photo/q ... tr></table>[/img]

Key to the symbols is in the graph's upper left. For either plan, for any year, probability is 50% of being above the horozontal line and 95% of being above the bottom of the thin vertical line.

The PE10-based adjustment has a significant effect. Success rate is dropped from 93% to 84%. Or stated the other way, risk of failure is multiplied by about 2 ½, from 7% to 16%. From further testing, I found that to make the plan with the PE10 adjustment appear as safe as it appeared without that adjustment, I’d have to reduce my annual draw from 4% to less than 3.5%.

To apply this approach, I’d need the slope from a graph a little different from Bob90245’s, showing stocks return rate versus 0% instead of versus bonds. But this is the way I’d incorporate the meaning of PE10. No wild leaps and plummets of change in asset allocation, no setting rules in concrete now to blindly apply in future years, and most important I can apply it and see it for my particular dollar plan, goals, and priorities.

On that last point, I gotta say -- I'm appalled that there's all this discussion without considering the investment purpose -- make the particular investor's particular dollar plan offer best prospects for her particular dollar goals.

Dumb wide receivers in pro football learn to keep their eyes on the ball. Why can't we?

Dick Purcell
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Post by 555 »

runthetrails wrote:
555 wrote:
grayfox wrote:
555 wrote:Could someone please explain what the fuss is all about.
"It is quite ironic that you would question this market timing startegy because it is well known that you yourself are a timer.
Probably the most well-known timer there is, short of Big Ben. Every EE knows exactly what I'm talking about.
http://en.wikipedia.org/wiki/555_timer_IC "
That's actually very funny :lol: though I had to click in the link to figure out why :oops:
"I actually always assumed (without really paying attention to whether it was consistent with your posts) that your screen name meant you were a "timer" for this reason. Radio Shack sold me a bunch of these ICs."
Now that I think about it, I did dabble a bit in electronics as a kids so I'm sure I soldered a few of these things to circuit boards at some point. But I'd totally forgotten that, and my "name" 555 is for a totally different reason.
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Post by Rodc »

Hi Dick,

I'll have to look closer when less tired.

Certainly looking at goals is best. Adjusting along the way.

Only so many hours in a day so not every little thing I've thrown together over the years is complete. Someone was pushing q-timing as the greatest thing since sliced bread some time back so I took a quick look, and found, well maybe or maybe not.

Like Wade's look at using P/E10, my look at using q is just a quickie good enough to post on a forum sort of deal, but not something you'd publish as real work.

I think maybe better than Monte is more along the lines of Wade's work that showed when valuations are high, not only do returns tend to be low going forward, but they were high up to that point so you likely have more than expected. So you can naturally reduce stocks to reduce unnecessary risk which in effect does the same thing as P/E10 market timing.

This really only matters once you have a pretty good stash, and simply reevaluating need for risk, how close you are to goals, etc should take care of things, I would think. I'm not sure actively trying to time markets based on only one measure (ignoring bond yields) of valuations (and a backward looking one at that), no look at goals (as you wisely note), wild moves into and out asset classes (with commensurate risk of being wildly wrong) is in anybody's best interest.

I don't think think this particular bit of work helps anyone.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.
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Post by letsgobobby »

this approach has helped me a great deal for 15 years so I will continue to use it. I explicitly reject the "cliff-like" adjustments and recommend a more subtle shifting approach. When I posted extensively in 2007-08 on the fatwallet forum I repeatedly said that a valuation based metric could be wrong, could be completely misleading, etc. But the argument isn't that low PE10 guarantee better returns, only that they increase the likelihood of better returns.

There are only 2 possibilities. A below average PE10 either stays below average forever, which it has never done; or it reverts to the mean (goes up). At PE10 10, there are several instances where PE10 has fallen further. That simply means that very few people should have 100% equities at PE10 10. There are no instances of PE10 5 going to PE10 2.5. Therefore, a young person might want 100% equities at PE10 5. It doesn't guarantee success; it simply makes it more likely.
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Post by Dick Purcell »

Rodc –

1. That effect of high PE10 indicating likely lower return-rate arithmetic mean for the next ten years is IMPORTANT ! Especially for retirees considering safe withdrawal rate.

Remember that prior discussion where you pointed out that there’s so much uncertainty about stocks’ return-rate mean that it could well be 2% below our estimate? WellSir, for a retiree intending to do safe withdrawal rate in a plan similar to what I graphed above, PE10 being 8 above average at the start may be just as bad.

In my previous-message example, stocks return-rate mean 2% below estimate produces a 15% failure rate. PE10 above average by 8 produces a 16% failure rate.

(That’s assuming the slope (minus ½) in Bob90245’s graph for stocks minus bonds is about the same for stocks minus zero.)

2. But what’s important about PE for real investors with dollar plans and goals, especially in the retirement phase, is not jumping up and down in radical switches of allocations – it’s the effect on the investor’s cash flow. (Wade brought out this cash flow effect back in that earlier long discussion about Trinity update.)

For the example described and graphed in a message just a couple of notches above, I sought optimal allocation for the base case without the PE10 adjustment, and also for the case with the PE10 adjustment. For each, the optimal portfolios are in the range of 30% to 40% stocks. The case with the PE10 adjustment likes the stock % slightly lower than the case without the PE10 adjustment, but the differences are so tiny it’s not worth the bother of adjusting.

On the other hand, as reported in my previous message, incorporating the PE10 adjustment can have big effect on the investor’s “safe” withdrawal budget. In my somewhat arbitrary example, PE10 up at 8 above average reduces “safe” withdrawal rate by almost 15%, from 4% down to just under 3.5%.

3. I’m not sure how you meant that “better than Monte” line, but the effect you described would automatically be included if you do an annual Monte. After a big PE runup, for the next year’s Monte start with the higher current portfolio value resulting from the runup.

Dick Purcell
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Post by wade »

Dick, I'll be sure to read your messages more fully next week.

For now, I'm just wondering if I went about presenting this all in the wrong way.

More simply, your strategic asset allocation depends on:

1. individual circumstances and constraints (which includes goals)
2. long-term capital market expectations

if you believe that valuations affect long-term capital market expectations, then you should take valuations into account when determining your strategic asset allocation.

People can have tactical asset allocation changes when the change is made for short-term capital market expectations.

But strategic asset allocation is not something that is fixed once and never subject to any changes. Whenever either of the two items on that list change, your strategic asset allocation should be updated.

Perhaps that's really all there is to say. The question then is just: how do you change your strategic asset allocation?

In this regard, perhaps Dick's Monte Carlo approach provides a very viable system for doing that, as it provides the link between individual's circumstances/constraints/goals to the proper asset allocation given long-term capital market views.
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Re: Valuations and timing

Post by Noobvestor »

Mel Lindauer wrote:
Lauren Vignec wrote:Hello Everyone,

At the beginning of this year I told everyone I work with, "Last year turned out to be great. But valuations on stocks are high now, and interest rates on bonds are low. Don't expect good returns anywhere in the near future."

It's not just that "valuations matter". If we talk about investing to anyone, we have a responsibility to explain to people when expected returns are low, as they still are now for both stocks and bonds.

As far as timing goes? Well, the all-or-nothing approach will never work. It's too stressful to change allocations that dramatically--no one is going to do it in real life. And in order to follow the strategy outlined in Pfau's paper, an investor would have had to stay out of stocks for the entire decade of the 1990's. And the whole time, they would have had to have the willingness and emotional fortitude to push all their money into the stock market as soon as Great Depression II hit. No one is like that.

However, the advice to change allocations in smaller increments based on valuations is more plausible. The problem here is that people will lower their stock allocations when markets get more richly valued, but they won't increase their allocations when markets drop. Then again, this is the same problem people have with buy-hold-rebalance. In general, investors have a tougher time buying stocks after market drops than they do selling them when markets go up.

And in any case, the current expected returns of both stocks and bonds are low by historical standards. Normally that may not have been an issue, but it is certainly an issue now! So I'm left thinking along exactly the same lines as Dick Purcell and a few of the other posters--yes valuations matter, and no, market timing isn't going to help people.

L
Well said, Lauren.
I don't get it - my return on bonds and stocks has been pretty darned good this year,which seems near-term enough for me :?:
"In the absence of clarity, diversification is the only logical strategy" -= Larry Swedroe
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Re: Valuations and timing

Post by Lauren Vignec »

Noobvestor wrote:
I don't get it - my return on bonds and stocks has been pretty darned good this year,which seems near-term enough for me :?:
Hello NI,

This year has turned out better than I expected (so far.) That often happens. Now imagine that you were trying to make timing decisions based on valuations, even small timing decisions, like shifting some of the portfolio here and some there. How much patience would you have?

L
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Re: Valuations and timing

Post by letsgobobby »

Lauren Vignec wrote:Hello Everyone,

At the beginning of this year I told everyone I work with, "Last year turned out to be great. But valuations on stocks are high now, and interest rates on bonds are low. Don't expect good returns anywhere in the near future."

It's not just that "valuations matter". If we talk about investing to anyone, we have a responsibility to explain to people when expected returns are low, as they still are now for both stocks and bonds.

As far as timing goes? Well, the all-or-nothing approach will never work. It's too stressful to change allocations that dramatically--no one is going to do it in real life. And in order to follow the strategy outlined in Pfau's paper, an investor would have had to stay out of stocks for the entire decade of the 1990's. And the whole time, they would have had to have the willingness and emotional fortitude to push all their money into the stock market as soon as Great Depression II hit. No one is like that.

However, the advice to change allocations in smaller increments based on valuations is more plausible. The problem here is that people will lower their stock allocations when markets get more richly valued, but they won't increase their allocations when markets drop. Then again, this is the same problem people have with buy-hold-rebalance. In general, investors have a tougher time buying stocks after market drops than they do selling them when markets go up.

And in any case, the current expected returns of both stocks and bonds are low by historical standards. Normally that may not have been an issue, but it is certainly an issue now! So I'm left thinking along exactly the same lines as Dick Purcell and a few of the other posters--yes valuations matter, and no, market timing isn't going to help people.

L
- not the entire 90s - would have been progressively less invested in stocks through the late 90s, but never zero. that would be gambling.

- would not have pushed all their money into stocks 3/9/9. rather, would have been increasing stock allocations through late 2008 - early 2009.

- it's true this approach requires extraordinary discipline and patience. so does running the Ironman. that doesn't make it impossible. but before you head out on the Kona asphalt for that 112 mile bike ride, you better damn know what you're made of.
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Post by letsgobobby »

just found this quote by larry swedroe in another thread and it captures what I've been trying to say, precisely:

"The vast majority of time I assume prices are rational and efficient. That is within very wide bands of valuations. I cannot tell whether p/es of say 20 or 6 are right or not. But I do know that at 40 in a bull/bubble market is not right. Now I might be wrong timing an exit but I am willing to live with that risk."

Don't necessarily agree with his numbers - I'd probably use 12 and 25 rather than 6 and 20 as my examples - but certain extreme valuations are clearly 'wrong' which impacts expected returns and even if timing is a bit off I am willing to take the risk.
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Post by lazyday »

Interesting quote. Is PE the only measure he mentioned?

Of course, leaving when PE is outside the "very wide bands of valuations" such as at PE of 40, is very different than what the paper suggests. Also very different than being out of US stocks today, or in the mid 2000's.

Being out of stocks then or today, IMO, is disagreeing with his statement, not agreeing with it.

Maybe he was talking about 2000.
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Post by Dick Purcell »

My analyses of implications of P/E or PE10 make these two things jump out:

1. The principal thing you should adjust in response to PE10 is not your asset allocation but your future cash flow plan.

When I kicked PE10 up for a test retirement-years plan, the analysis showed only miniscule changes in comparisons of asset allocations, but dramatic reduction in safe withdrawal rate.

The important response was not to jump toward or away from stocks, but to cut back on world cruises.

2. For responding to P/E, a cliff policy does not make sense. Any response to P/E high or low should be incremental.

If PE10 above average by 12 calls for X change in plan, then PE10 above average by 6 calls for just about ½ X change in plan. It does not make sense to figure 39 is fine but 40 and I do something radical, or 24 is fine but 25 and I do something radical.

If with a given retirement plan, PE10 of 24 makes the safe withdrawal rate considerably lower, but you don’t reduce it because your cliff-jump tripwire is set for 25, your risk of spending your final years in that cardboard box behind the pool hall is higher than planned.

You can do both of these things -- view and respond to PE10 effects incrementally, and see what they portend for your outlook for future asset allocations, returns and cash flows -- by integrating the historical relationship between PE10 and next Y years' stock return-rate mean into your analyses.

Dick Purcell
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Post by Rodc »

Dick Purcell wrote:My analyses of implications of P/E or PE10 make these two things jump out:

1. The principal thing you should adjust in response to PE10 is not your asset allocation but your future cash flow plan.

When I kicked PE10 up for a test retirement-years plan, the analysis showed only miniscule changes in comparisons of asset allocations, but dramatic reduction in safe withdrawal rate.

The important response was not to jump toward or away from stocks, but to cut back on world cruises.

2. For responding to P/E, a cliff policy does not make sense. Any response to P/E high or low should be incremental.

If PE10 above average by 12 calls for X change in plan, then PE10 above average by 6 calls for just about ½ X change in plan. It does not make sense to figure 39 is fine but 40 and I do something radical, or 24 is fine but 25 and I do something radical.

If with a given retirement plan, PE10 of 24 makes the safe withdrawal rate considerably lower, but you don’t reduce it because your cliff-jump tripwire is set for 25, your risk of spending your final years in that cardboard box behind the pool hall is higher than planned.

You can do both of these things -- view and respond to PE10 effects incrementally, and see what they portend for your outlook for future asset allocations, returns and cash flows -- by integrating the historical relationship between PE10 and next Y years' stock return-rate mean into your analyses.

Dick Purcell
Makes good practical sense.

The only thing I would add is that in addition to cutting spending in retirement with high PE10 (etc.) is if in accumulation you can work on either making extra sure you retire without debt to make cutting back on spending easier, it to increase savings in anticipation of low returns.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.
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Post by wellmoneyed »

I tried to read the entire thread - I'll admit to a little skimming. :D

One thing I did not see brought up is the dependence of PE10 as a "constant". Historically some number being high and some number being low.

Looking forward I can see some game changing technology (Artificial Intelligence, Robotics). It seems like you could make a reasonable case for PE being shifted up permanently or at least for a long time if new technology fundamentally alters our average growth rate. If this is true, then it seems like this strategy would end with growth going off the charts and you sitting on the sidelines waiting (forever) for PE's to come back down.
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Post by yobria »

Dick Purcell wrote:My analyses of implications of P/E or PE10 make these two things jump out:

1. The principal thing you should adjust in response to PE10 is not your asset allocation but your future cash flow plan.

When I kicked PE10 up for a test retirement-years plan, the analysis showed only miniscule changes in comparisons of asset allocations, but dramatic reduction in safe withdrawal rate.

The important response was not to jump toward or away from stocks, but to cut back on world cruises.
Depends on why the PE changed. If the P doubled, I'd rebalance to bonds and increase the number of cruises taken (I'm richer).

If the E dropped by half, I'd cut back on cruises - I invested in businesses that are now less profitable than before.

In general, I'd be careful about adjusting AA based on any patterns, be they cup and saucer, PE/10, January Effect, "lottery effect" (avoid small growth), etc. They tend to stop working about the time everyone starts discussing them.

Nick
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Post by bob90245 »

yobria wrote:In general, I'd be careful about adjusting AA based on any patterns, be they cup and saucer, PE/10, January Effect, "lottery effect" (avoid small growth), etc. They tend to stop working about the time everyone starts discussing them.
I beg to disagree regarding P/E10 patterns stop working. It's like saying we will never experience a tech bubble like the late 1990's again. Maybe not while the current crop of investors have fresh memories. But maybe the generation of investors that come after us.
Ignore the market noise. Keep to your rebalancing schedule whether that is semi-annual, annual or trigger bands.
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