Equal Weighting (1/N) Versus Market Cap Weighting

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Park
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Equal Weighting (1/N) Versus Market Cap Weighting

Post by Park » Sun Sep 10, 2017 12:46 am

For DIY investors, two common approaches to asset allocation are equal weighting (1/N) and market cap weighting in one's portfolio.



The link below is to a review of an important paper that found that equally weighting assets, as an asset allocation strategy, compares favorably to much more complicated models. The benchmark strategy is a "naive portfolio in which a fraction, 1/N, of wealth is allocated to each of N assets available for investment at each rebalancing date".

http://www.cfapubs.org/doi/full/10.2469/dig.v39.n4.13

"The authors conclude that none of the optimizing strategies always dominate the benchmark in terms of the Sharpe ratio. In terms of CEQ returns, none of the optimal strategies are consistently better than the benchmark. In terms of turnover, only the value-weighted market portfolio strategy is better than the benchmark. The authors also suggest that the 1/N diversification strategy should serve as a benchmark for evaluating the performance of other optimal allocation strategies"



There are advantages to equal weighting. When subasset classes have similar expected returns, volatility and correlations, then the most efficient portfolio will be generated by equal weighting. Subasset classes may very well have dissimilar expected returns, volatility and correlations. However, it is difficult to predict what those variables will be. By equal weighting, you don't try to predict those variables, and may very well do better than those who engage in prediction.

Equal weighting may result in a tilt to small and value, which could increase return. To make an analogy, when one equal weights the stocks in the S&P500, one tilts to small and value. And when one equal weights country allocation, one tilts to small and value. A market cap portfolio is dominated by large growth companies, relative to the number of companies in the index. If one allocates 25% to large, 25% to large value, 25% to small and 25% small value, you'll have less exposure to large growth and small growth than the market portfolio.

Even without a tilt to small and value, equal weighting might increase return. Rick Ferri found that splitting EAFE into 50% Europe and 50% Pacific increased annualized return by 0.6% on a precost pretax basis.

Equal weighting forces you to rebalance. Rebalancing will decrease your risk, and will probably modestly increase your return within an asset class.




There are advantages to market cap weighting. You don't have to rebalance. This will result in less personal time spent on one's portfolio. Transaction costs will be lower. Also, occasionally rebalancing will decrease return. An example is rebalancing between Japanese and US stocks after 1990, with the example coming from William Bernstein.

There will be less turnover, resulting in decreased cap gains tax to pay. William Bernstein makes the point that rebalancing in a taxable account will decrease returns, if you have to sell stocks to rebalance. He also states that turnover precludes having large value funds, small value funds and small growth funds in taxable accounts. The most common reason for a stock to leave a small cap fund is that its price has increased, such that it is now a mid or large cap stock. Similarly, the most common reason for a value stock to leave a value fund is that its price has increased, such that it is now a growth stock. In both cases, that means cap gains tax to pay. In a taxable account, he only recommends tax managed small cap funds; he doesn't recommend market cap small cap funds. He recommends splitting international exposure into European, Pacific and emerging only if you can do so in a tax advantaged account. However, he also states that Vanguard European, Vanguard Pacific, Vanguard Emerging Markets and iShares MSCI Value (EAFE Value) are OK in taxable accounts, which may be at odds with what is previously written. It should be noted that the ETF wrapper can decrease the tax bill, as long as tax lots laws are in effect.

I mentioned previously that equal weighting may result in a tilt to small and value. There are those who make the argument that markets in the future will reward a tilt to small and value less. If the increased cost handicap of such tilts can't be overcome, then you'd be better off with market cap weighting.



Many use both equal weighting and market cap weighting. For example, they might split the total market into quartiles, but within each quartile, they use market cap weighting.



A related question is how much splitting you should do; what should N equal?. More splitting will result in more rebalancing opportunities. More splitting might also result in a greater tilt to small and value. But more splitting means more personal time, more transaction costs and more cap gains. Once again, taxation matters. More splitting also means more tracking error. Rick Ferri, in his book on asset allocation, states that the diversification benefit tends to diminish after seven.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by privatefarmer » Sun Sep 10, 2017 1:33 am

the biggest problem IMO is that this is all common knowledge now. If the market as a whole "tilts" even the slightest to small/value/equal weighting, it will cause the "small/value" stocks to become over valued and the "large growth" stocks to become under valued, relative to their "intrinsic value".

I keep coming back to the thought that the market knows best. It's good that active managers/hedge funds tilt this way and that as they make the market efficient so that us market-cap index fund holders end up w/ a better return.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by AlohaJoe » Sun Sep 10, 2017 4:19 am

Park wrote:
Sun Sep 10, 2017 12:46 am
Equal weighting may result in a tilt to small and value, which could increase return.
Equal weighting may result in a tilt relative to market capitalisation weighting but that's not what they measured against. Equal weighting almost certainly under-weighted small and value relative to model #1 (sample-based mean-variance). But they didn't report on the factor weights of the other methodologies so whether they result in a "tilt" or not is really just speculating.

In section 6.1 they briefly look at cross-sectional factors and argue that tilting towards factors isn't a clear win over simple equal-weighting.
So, for both datasets, the Sharpe ratio of the Brandt, Santa-Clara, and Valkanov strategy is higher but statistically indistinguishable from that of the 1/
To make an analogy, when one equal weights the stocks in the S&P500, one tilts to small and value.
They looked at 14 different asset allocation models, so repeated reference to the market cap seems like a strawman argument. As I said before, they looked at things other than market cap allocations in their comparison.
Equal weighting forces you to rebalance. Rebalancing will decrease your risk, and will probably modestly increase your return within an asset class.
There are arguments about both the risks & returns of rebalancing. In any case, rebalancing is not part of the results of this paper, which are entirely about estimation errors (and lower turnover). As they point out, we need centuries of data in order for the estimation errors not to completely swamp the results of our model:
Based on parameters calibrated to US stock-market data, we find that the critical length of the estimation window is 3000 months for a portfolio with only 25 assets, and more than 6000 months for a portfolio with 50 assets. The severity of estimation error is startling if we consider that, in practice, these portfolio models are typically estimated using only 60 or 120 months of data.
A related question is how much splitting you should do; what should N equal?
This is a great question and something I struggle with when thinking about equal-weighting. Should you equal weight between stocks and bonds? That means you have 50% stocks and 50% bonds. Or should you pick 3 assets classes -- US, International, and Bonds -- in which case you have 66% stocks and 33% bonds? Should you equal weight between equity & REITs? Between EM and US? Between high-yield bonds and US treasuries?

It feels hard to say "yes" to much of that.

Finally, there's a recent paper (November 2015 is the last draft I've seen) from Stivers & Sun on "Mitigating Estimation Risk in Asset Allocation: Diagonal Models Versus 1/N Diversification" which is a reply to the paper you linked. You'll need to read their full paper for how diagonal models work but their conclusion is that we don't need centuries of data to reduce the estimation errors...
Using five different data sets of disaggregate portfolio returns over the 1926-2012 period, we find that 1/N is generally not optimal when compared with these simple diagonal strategies.
They also have a literature survey that refers to other papers that reply to the original "1/N is efficient" claim.

My personal takeaway from all of this is that asset allocation ("should I have 0% REITs or 15% REITs?") is complete overrated and over-discussed. No one on Bogleheads (not even advisors like Ferrir and Swedroe) is doing anything remotely close to what academics currently believe is necessary to handle estimation errors. Virtually any reasonable asset allocation is likely to work out at least okay-ish and we won't know for 50 years which one is better.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by nisiprius » Sun Sep 10, 2017 6:16 am

There are such huge logical and implementation problems to equal weighting that I find it hard to take it seriously.

The first is that it is illogical and that it is very dependent on arbitrary categorizations. For example, suppose you had a stock allocation of $100,000 and you decided, in 2016, to equal weight all industry sectors (which is a frequently suggested strategy). Before August 31st, 2016, you would have put 10% or $10,000 each into information technology, health care, financials, consumer discretionary, consumer staples, industrials, energy, utilities, materials, and telecom. But on August 31st, Dow Jones and MSCI decided that REITs, which had been included in "financials," ought to be considered a separate sector of their own. So, boom! overnight, the equal weighting changed from 10% each of ten sectors to 9.09% each of eleven sectors, and now your allocation to "classic" financials almost doubled, from 10% to 18.18%. Actually it is worse than those numbers suggest, because REITs constitute only about 1/5th of "classic" financials, so before August, your 10% allocation to "financials" was 8% non-REITs, 2% REITS. Overnight, you needed to more than quadruple your REIT allocation.

Now, people who are trying to attack indexing in general often complain about the (real but tiny) issue of front-running the S&P 500 by knowing in advance which stocks are going to be added to the S&P 500 and buying them up ahead of the index funds, but the addition of a fund to the S&P 500 typically represents about 0.02% of the index as a whole. Even if they were required to do it on the day it is added to the index (they are not), and even if that resulted in the price of the stock doubling, that would only cause a price elevation of 0.02% in the fund. By comparison, the transition in the GICS classification would require an equal-sector-weighed fund to go from 2% REITS to 9.09% REITS, i.e. add over 7% in REITS to the portfolio.

Similarly, a decision of equal weight the countries in an emerging markets fund would result in disruptive changes whenever the index provider changes its mind about whether a country should be classified as an emerging market or not.

Similarly, imagine the effect of mergers and spinoff. During the breakup of the Bell System, a single stock, AT&T, one stock, one ticker, was replaced by seven stocks, seven ticker symbols.

The second problem show up if you imagine trying to equal weight the total stock market. For decades, people bought S&P 500 cap-weighted index funds. The S&P 500 represents 80% of the market by cap-weight (and the remaining 20% has a 0.88 correlation with the S&P) and is thus a reasonable approximation to the total market, by cap-weight. However, the equal-weighted S&P 500 represents only about 15% of the total market by stock issue count, and cannot possibly be a good representation of the equal-weighted total market.

The third problem is best illustrated by looking at the actual composition of an ETF with the name "PowerShares Russell 2000 Equal Weight Portfolio," EQWS. The ETF has $21 million in total assets. An equal weight assignment to each of 2,000 stocks would imply an allocation of about $10,500 to each stock. However, it actually owns over $51,533 worth of the Boston Beer company, five times as much as expected, and $3,436 worth of Corvus Pharmaceuticals, a third the amount expected. The reason is that neither the ETF nor the index itself even tries to do what the name suggest--because even trying to equal-weighting any portfolio that includes small-caps is, well, crazy.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by rkhusky » Sun Sep 10, 2017 6:57 am

Another logic problem with equal weighting is conglomerates. Suppose a conglomerate owns 1000 brands. If the conglomerate split apart and formed 1000 separate companies, an equal weight fund would need to increase their share in those brands 1000-fold. Does that make sense? Why would those brands suddenly become worth 1000x more, not even considering any efficiencies that might come by being included in a conglomerate? The same logic issues come when conglomerates buy up other companies and equal weight funds must decrease their stake in those companies.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by AlohaJoe » Sun Sep 10, 2017 7:08 am

The research's claim is that all of these "logical" concerns don't appear to actually matter in the real world that the authors can tell. Simply reiterating them seems to be talking past the authors of the paper.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by Valuethinker » Sun Sep 10, 2017 7:39 am

AlohaJoe wrote:
Sun Sep 10, 2017 7:08 am
The research's claim is that all of these "logical" concerns don't appear to actually matter in the real world that the authors can tell. Simply reiterating them seems to be talking past the authors of the paper.
The real problem I see with equal weighting is the dealing costs.

Stocks have bid-ask spreads. To keep equal weighting, you have to keep dealing.

That could get very expensive.

Also, my understanding is a relatively small number of stocks drive the index rise? Most companies, the share price doesn't go anywhere (is range bounded, but you do get dividends). It's that 20-50 stocks in the S&P 500 that drive all the growth in the S&P 500?

You will be selling those, constantly. So you won't get the full benefit of the rises?

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by Lauretta » Sun Sep 10, 2017 7:47 am

Valuethinker wrote:
Sun Sep 10, 2017 7:39 am


Also, my understanding is a relatively small number of stocks drive the index rise? Most companies, the share price doesn't go anywhere (is range bounded, but you do get dividends). It's that 20-50 stocks in the S&P 500 that drive all the growth in the S&P 500?

You will be selling those, constantly. So you won't get the full benefit of the rises?
It's true that the performance of a stock index is affected significantly by a few stock
http://csinvesting.org/wp-content/uploa ... -Bills.pdf
I don't think your last sentence necessarily follows from these premises: it would seem to depend amongst other things on the time frame of your selling and rebalancing.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by BolderBoy » Sun Sep 10, 2017 9:24 am

Park wrote:
Sun Sep 10, 2017 12:46 am
The link below is to a review of an important paper that found that equally weighting assets, as an asset allocation strategy, compares favorably to much more complicated models.
This is the crux, isn't it? The BH philosophy is simplification.

Equal weighting is more complex than a basic 3-Fund portfolio so equal weighting is already starting out down the trail of complexity and as the authors pointed out, may not provide significant advantages anyway.

It is an interesting academic article but what of its value in the real world?
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by nisiprius » Sun Sep 10, 2017 10:04 am

AlohaJoe wrote:
Sun Sep 10, 2017 7:08 am
The research's claim is that all of these "logical" concerns don't appear to actually matter in the real world that the authors can tell. Simply reiterating them seems to be talking past the authors of the paper.
Well, I keep clicking on "full text" and getting an abstract instead, so I'd appreciate it if someone would point me to the real full text because as it is I can't tell exactly what kind of portfolio the authors are talking about or what the more-complex-but-no-better portfolios are.

Meanwhile, also in the "real world," we find that the Guggenheim S&P 500 Equal Weight ETF, RSP (blue) has, over its lifetime, had a virtually identical Sharpe ratio as the Vanguard 500 Index Fund, VFIAX (red) i.e. any supposed outperformance is completely explained by higher risk:

Source

Image


Source

Image

And the PowerShares Russell 2000 [so-called] Equal Weight ETF, EQWS, blue, has had virtual identical standard deviation compared to the Vanguard Russell 2000 ETF, VTWO, red--yet has had slightly lower return and thus a slightly lower Sharpe ratio.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by nisiprius » Sun Sep 10, 2017 10:32 am

The actual paper appears to be here: Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?

1) The authors state explicitly:
We wish to emphasize, however, that the purpose of this study is not to advocate the use of the 1/N heuristic as an asset-allocation strategy
2) Unlike the implication of the thread title, the paper does not compare equal weight to market cap weighting. Market cap weighting is not one of the strategies used in the comparison. The strategies actually compared are:
List of various asset-allocation models considered

Naive

0. 1/N with rebalancing (benchmark strategy) Classical approach that ignores estimation error
1. Sample-based mean-variance

Bayesian approach to estimation error

2. Bayesian diffuse-prior
3. Bayes-Stein
4. Bayesian Data-and-Model

Moment restrictions

5. Minimum-variance
6. Value-weighted market portfolio
7. MacKinlay and Pastor’s (2000) missing-factor model

Portfolio constraints

8. Sample-based mean-variance with shortsale constraints
9. Bayes-Stein with shortsale constraints
10. Minimum-variance with shortsale constraints
11. Minimum-variance with generalized constraints

Optimal combinations of portfolios

12. Kan and Zhou’s (2007) “three-fund” model
13. Mixture of minimum-variance and 1/ N
14. Garlappi, Uppal, and Wang’s (2007) multi-prior model
3. The purpose of the paper seems to be exploring the use of MPT-informed asset allocation. Thus, approach #1 is to take the actual past data, calculate the actual means, standard deviations and correlations between the assets, construct the efficient frontier, and use the tangent portfolio. It's often been observed that this produces crazy results. In practice, all of the parameters used for MPT fluctuate wildly and, within the MPT model framework, are evidently subject to huge sampling error. The straightforward application of MPT, using the actual past data, typically recommends huge allocations, often leveraged allocations, to whatever did best during the sample period, and tiny allocations, often short positions, to whatever did poorly.

The paper is comparing the results of various attempts to "tame" the bad behavior and apply MPT in a practical way to the real world.

The paper says there's no evidence that any of them is any better than equal weighting. That is, the basic thrust of the paper is to challenge (and/or debunk) the application of MPT to real-world portfolio construction.

It does not attempt to address whether equal weighting is better than cap-weighting, and it explicitly says that the authors do not advocate the use of equal weighting.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by nisiprius » Sun Sep 10, 2017 10:37 am

Oh, for what it's worth: the portfolios examined by the authors are alternative weightings of portfolios in which the assets are:
Table 2
1 Ten sector portfolios of the S&P 500 and the US equity market portfolio
2 Ten industry portfolios and the US equity market portfolio
3 Eight country indexes and the World Index...
and five more. Thus, for example, portfolio 1 in the real world would be a portfolio consisting of ten individual-sector ETFs plus a total market ETF, and the question is how best to assign relative allocations to the ten ETFs. In the specific case of portfolio 1, I guess the "cap-weighted" allocation would be 0% to each of the sectors and 100% to the total U.S. equity market, and that is not one of the allocations considered by the authors.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by patrick013 » Sun Sep 10, 2017 2:24 pm

I think people who equal weight consider all of their stock
fund or stock choices excellent and would like to give them
all equal chance to perform.

I think more research is req'd. Do the statisticians actually
believe some of the things they say ? So far EW has shown
extra returns for large cap and dividend funds but not so
much for mid and small cap if any. So more stocks in large
cap have contributed to total return than in MC and SC per
observation then I would suspect, rather than contribution
to total return dependent on market cap. Don't know about Intl.

Time will tell.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by TD2626 » Sun Sep 10, 2017 6:54 pm

In my opinion, people advocating "equal weight" are mistaken. I think the terminology used is misleading.

Basically, cap weighting is the equal weighting that many desire. You're basically equal weighting by percentage stake in companies when cap weighting.

Why? A cap weight fund buys (roughly) the same stake in every company. A large Total Stock fund might buy a 1% stake in every company in the US. An equal percentage stake - that's equal weighting in my mind. (I get that that's not the common usage of the term).

Also, why would one want to put the same number of dollars in General Electric as in (insert any random microcap startup)? The (1/N) equal weighting strategy would do this. If you truly want to put the same number of dollars in someone's longshot startup as in America's largest companies, reread the post that ended with this line:
nisiprius wrote:
Sun Sep 10, 2017 6:16 am
trying to equal-weighting any portfolio that includes small-caps is, well, crazy.
A reductio ad absurdum may hammer this home:
Imagine if they did this in venture capital.
Imagine if they did this on Shark Tank.
A venture capitalist would say "I equal weight. I have $100,000,000. I will give $1,000,000 to the first 100 companies to make their pitches to me, without regard to the quality of their company's ideas". You can see how this might cause issues for the VC investor.

It's probably not quite as crazy with publicly traded stocks trading on major exchanges like the NYSE or the NASDAQ - as these companies have gotten past the VC stage and are at least (one would hope) functional companies with reasonable prospects. The NASDAQ won't list a penny stock.

Anyway, as far as I know, it's generally accepted that if a small-cap tilt is desired, one would do it via tilting (Total Stock + Small Cap fund) or via a slice-and-dice (and overweighting small cap). An equal weight portfolio isn't special, it just has a small-cap tilt that can be replicated in many other ways. It may be worth looking over Vanguard's views on factor-based investing, here: https://personal.vanguard.com/pdf/ISGFB ... Online.pdf

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by Lauretta » Sun Sep 10, 2017 6:57 pm

Lauretta wrote:
Sun Sep 10, 2017 7:47 am
Valuethinker wrote:
Sun Sep 10, 2017 7:39 am


Also, my understanding is a relatively small number of stocks drive the index rise? Most companies, the share price doesn't go anywhere (is range bounded, but you do get dividends). It's that 20-50 stocks in the S&P 500 that drive all the growth in the S&P 500?

You will be selling those, constantly. So you won't get the full benefit of the rises?
It's true that the performance of a stock index is affected significantly by a few stock
http://csinvesting.org/wp-content/uploa ... -Bills.pdf
I don't think your last sentence necessarily follows from these premises: it would seem to depend amongst other things on the time frame of your selling and rebalancing.
I have thought about it and your argument really doesn't seem to make sense to me. Imagine that the stock that contributes most to the index is the one with the lowest market cap in the index. Then by holding equal weight you will actually have more of that stock and thus a greater benefit from its rise.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by AlohaJoe » Mon Sep 11, 2017 4:48 am

nisiprius wrote:
Sun Sep 10, 2017 10:32 am
2) Unlike the implication of the thread title, the paper does not compare equal weight to market cap weighting. Market cap weighting is not one of the strategies used in the comparison.
I think strategy #6 (value-weighted market portfolio) is what we here call "market cap weighting". But other than that I totally agree with you; the thread subject and the OP's characterisation of the research are, charitably, extremely misleading. Re-reading my original comment I didn't make my feelings on that point clearly enough.
It does not attempt to address whether equal weighting is better than cap-weighting, and it explicitly says that the authors do not advocate the use of equal weighting.
Agreed (though I think some of that is typical academic hedging); the authors are just saying in essence "if your asset allocation strategy can't consistently beat this super-dumb 1/N equal weights strategy that everything thinks is for Dumby McDumboheads...well, maybe you should rethink whether your strategy is as smart as you think it is".

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by Park » Mon Sep 11, 2017 8:24 am

First of all, I haven't read the paper. As someone who could charitably be described as an amateur investor, I wouldn't understand it. I rely on others to interpret such publications for me.

Why did I cite the paper? I noticed that a common strategy is equal weighting at the subasset class level. I didn't know the rationale for it, so I spent a little time looking into it. Among those who consider equal weighting at the subasset class level to be an acceptable option, they frequently cite that publication.

The question of whether I was being misleading by citing that paper has been raised. I apologize profusely, if I was being misleading at all.

Could this thread focus on what I hoped it would, the relative merits of equal weighting vs. market cap weighting at the subasset class level? Or perhaps a discussion of diagonal strategies, which I've never heard of before?

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by AlohaJoe » Mon Sep 11, 2017 9:27 am

Park wrote:
Mon Sep 11, 2017 8:24 am
Could this thread focus on what I hoped it would, the relative merits of equal weighting vs. market cap weighting at the subasset class level? Or perhaps a discussion of diagonal strategies, which I've never heard of before?
I found the merits of 1/N weighting persuasive when thinking about my own portfolio design. I don't follow it in everything but I use it as a default starting point and then think about why I want to change away from it.

When people design an asset allocation they have to answer questions like "how much EM should I hold?" I reckon most people can't really give you a good explanation for how they arrived at the number they did. If pressed it would probably be something like, "I read it in Swensen's book". Or maybe "I read it in Random Walk Down Wall Street but that number felt too high/low so I adjusted it by 5%".

As an example, if you look at the 3-fund portfolio it is 40% US, 20% international, 40% bonds. But...why 20%? Why not 25%? Or 30%? Or 15%? There are very rarely good, satisfying answers to those kinds of questions. I mean, heck, why not 22.7%? It's not like we aren't all using spreadsheets and calculators that can handle arbitrary precision these days.

And there are a fair number of well-known portfolios that do subscribe to a 1/N weighting scheme in whole or part. The Permanent Portfolio has 4 assets and 25% of each one. William Bernstein's "No-Brainer" portfolio has the same 4x25 equal weighting (though with a different set of assets). Paul Merriman's "Ultimate Buy & Hold" appears to take a semi-equal weighting approach with most assets held at 6% (10 slices @ 6% = the 60% equity portion of the portfolio, one asset gets "two slices" and is at 12%). Meb Faber's "Ivy Portfolio" is 5x20. The "Golden Butterfly" is also 5x20. Bill Schulteis's "Coffeehouse Portfolio" uses 10% slices of 6 assets to form the equity portion (6x10 = 60%). Craig Israelson's "7Twelve Portfolio" has slices of 8.3% each on the equity side.

In McClung's Living Off Your Money he invents a "Harvesting Ratio" metric and compares a number of well-known lazy portfolios and ends up coming out in favor of equally-weighted stock allocations (though he leaves it up to you how many assets you want to equally weight across, 2x50, 4x25, 5x20, 8x12, 10x10).

You'll notice that equal-weighting is most often applied to the equity side. But sometimes you'll see it on the bond side as well. I think it is Swensen who suggested 50% Treasuries and 50% TIPS. Maybe in another decade, as international bonds go mainstream, we'll start seeing 3x33 bond portfolios of 33% treasuries, 33% TIPS, 33% international?

All of those asset allocations take a traditional view of asset classes -- they are build from things like "small cap value" and "REITs". If you are more factor-inclined then it isn't clear how an equal-weighting approach would apply. Do you try to build a portfolio that is 0.3 beta, 0.3 HML, 0.3 SMB? Or maybe 0.2 term, 0.2 credit, 0.2 beta, 0.2 HML, 0.2 SMB? None of those are really right, I don't think.

One of the benefits in my mind of an equal-weighting approach is that it forces you to really think about whether an asset belongs in your portfolio. On Bogleheads we often see people, often new to investing, who propose a portfolio with 2% this or 4% that. Or even something like 30% bonds, 60% US, 10% international. Or people asking about putting 5% or 8% of their portfolio in something like AQR's multi-premium or a market neutral fund. I think there are real questions about the effectiveness of such a dilute position on a portfolio.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by Park » Tue Sep 12, 2017 7:42 am

AlohaJoe wrote:
Sun Sep 10, 2017 4:19 am
Equal weighting forces you to rebalance. Rebalancing will decrease your risk, and will probably modestly increase your return within an asset class.
There are arguments about both the risks & returns of rebalancing.
I would agree with that. Campbell Harvey is an author on a paper that found that rebalancing can increase your risk:

https://papers.ssrn.com/sol3/Papers.cfm ... id=2488552

http://www.marketwatch.com/story/the-hi ... 2014-12-09

"Whenever the market is in a longer-term up or down trend, rebalancing actually increases risk, particularly downside risk. That’s because, when you rebalance, you take money away from the better-performing asset class and reinvest it in the poorer-performing one. If those two asset classes’ relative strength persists after the rebalancing, as they often do, you’ll end up worse off than if you had not rebalanced...A similar process is present during protracted bear markets in stocks. During those declines, any rebalancing takes money away from bonds and invests more in stocks, only to have equities decline even more...Professor Harvey, in an interview, stressed that the answer depends on your tolerance for risk and the time horizon of your investments. If you are willing to undertake more risk and have a very long-term horizon of at least 10 to 20 years, then rebalancing is an appropriate strategy. That’s because, over long periods of time, a periodically rebalanced portfolio is likely to earn enough to compensate you for its greater risk. However, Harvey said, rebalancing becomes less appropriate to the extent you are a conservative investor and have an investment horizon shorter than 10 years."

That's why my plan is to rebalance every 3 years.

Rebalancing between stocks and bonds will most likely decrease your long term returns.

Rebalancing between asset classes within stocks may modestly increase your returns. William Bernstein has written about this.

However, I have seen the argument made that if a subasset class outperforms, then such rebalancing may decrease return. That's been mentioned previously in this thread, when it comes to rebalancing between Japanese and US stocks from 1990 to 1999. Once again, I got that from William Bernstein.

Paul Merriman has also made an argument against rebalancing between asset classes within stocks.

http://paulmerriman.com/why-rebalancing-huge-mistake/

He looked at the historical performance of a portfolio divided into quartiles between US large, US large value, US small and US small value. He looked at the results from 1963-2013 of rebalancing between the quartiles versus not. Not rebalancing resulted in an increased return of 1.1%. He didn't find evidence that not rebalancing increased risk.

"To test my theory further, I looked at what happened to these four asset classes in the four years 1929 through 1933. That should be a severe test. I was a bit surprised to find that the compound return (actually compound loss) was almost identical whether or not the asset classes were rebalanced. Rebalanced each year, the portfolio lost 76.9%; without rebalancing it lost 76.6%.

Of course, that assumes that you had 25% in each subasset class in 1929. If you hadn't been rebalancing for some time prior to 1929, you may have had considerably more than 25% in SCV in 1929. And I believe that SCV did the worse of the 4 subasset classes from 1929-1933.

I mention rebalancing, because it's part of a 1/N portfolio, but most likely plays a lesser role in a market cap portfolio.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by nisiprius » Tue Sep 12, 2017 8:33 pm

AlohaJoe wrote:
Mon Sep 11, 2017 9:27 am
...When people design an asset allocation they have to answer questions like "how much EM should I hold?"...
But equal weighting does not tell you that, either. The result of equal weighting depends entirely on the decisions you make up-front about how to categorize things.

If you categorize investments as "U.S. stocks and international stocks then equal weighting leads you to an allocation of 1/2 U.S. stocks, 1/2 international stocks.

If you categorize them as "U.S. stocks, developed markets, and emerging markets" then equal weighting leads you to an allocation of 1/3 each... which is 1/3 U.S., 2/3 international.

If you categorize them as "U.S. stocks, developed markets, emerging markets, and frontier markets," that leads you to an allocation of 1/4 each... which is 1/4 U.S., 3/4 international. It also leads to 1/4 frontier markets which is... crazy.

If you categorize them as "U.S. stocks, German stocks, French stocks, Swiss stocks, Swedish stocks, Portuguese stocks, Italian stocks, Singapore stocks, Russian stocks, Australian stocks, Brazilian stocks, Mexican stocks, Chilean stocks, .... " then you end up with, I dunno, 1/50 U.S. stocks, 49/50 international stocks.

If you categorize them as U.S. large-cap growth, U. S. large-cap blend, U.S. large-cap value, U.S. mid-cap growth, U.S. mid-cap blend, ... U.S. small-cap value, and international" then you end up with 90% U.S., 10% international.

Explain to me what the logical solution to this problem could be.

The interesting thing about all this is that if you use exactly the same categorizations but cap-weight them, all of these turn out to be identical... and all of them are equivalent to simply buying one fund, the Vanguard Total World Stock Index Fund.
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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by Park » Thu Sep 14, 2017 7:36 am

This thread is really about how much you want to emphasize market cap indexing versus equal weighting.

Nisiprius is pointing out that market cap indexing is when one decides N=1.

Those who follow the Fama French model divide the market by small/large and value/growth. That generates 4 categories, and N=4. But within those 4 categories, it's common to use market cap indexing. There's an increasing role for equal weighting, but market cap indexing is still playing a very important role.

AQR has a paper on international investing, and they make a case for equal asset allocation between countries. They state that this is for institutional investors; I think retail investors would find that to be a very difficult model to emulate. So as Nisiprius points out, N might equal 50 in that case. Once again, market cap indexing is advocated for each N. There's an increasing role for equal weighting, but market cap indexing is playing an important role.

Those would buy equal weighted S&P500 funds are deciding that N=500. Now it is a question entirely of equal weighting; market cap indexing doesn't play a role.

The greater the N, the greater the tilt to small and value there will tend to be.

The greater than N, the more agnostic you are about predicting returns, volatility and correlation. With equal weighting, you are assuming that returns/volatility/correlations are equal, which results in equal weighting being optimal.

If one assumes that the wisdom of crowds is optimal when it comes to prediction, then market cap indexing is the strategy of choice.

But the greater the N, the greater the costs, personal time and taxes there will be.

Compared to other asset allocation models, it is interesting that 1/N does reasonably well, regardless of what you decide N should be. The same could be said of market cap indexing. Once again, intelllectual simplicity in asset allocation tends to be a virtue.

How much should N be? It depends on how many distinct segments you think there are within the market cap universe, and what you think the optimal prediction method is. The best answer may be a mix of equal weighting and market cap weighting. In asset allocation, diversification is usually a good idea. Diversification based on how you predict makes sense.

However, the increasing operational complexity of increasing N becomes an issue. So for the vast majority of retail investors, somewhere between 1 to perhaps 9 is the answer for N.
Last edited by Park on Thu Sep 14, 2017 8:41 am, edited 1 time in total.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by rkhusky » Thu Sep 14, 2017 8:33 am

Park wrote:
Tue Sep 12, 2017 7:42 am
I mention rebalancing, because it's part of a 1/N portfolio, but most likely plays a lesser role in a market cap portfolio.
There is no need to rebalance in a fully market cap portfolio, unless new investments appear or old investments disappear, then a single rebalance is necessary to account for the change.

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Re: Equal Weighting (1/N) Versus Market Cap Weighting

Post by Park » Sat Sep 16, 2017 12:01 pm

I read the paper; thank you Nisiprius for providing the link.

The following are some excerpts, that I understood, from the paper:

"A second reason why the 1/N rule performs well in the datasets we consider
is that we are using it to allocate wealth across portfolios of stocks rather
than individual stocks. Because diversified portfolios have lower idiosyncratic
volatility than individual assets, the loss from naive as opposed to optimal
diversification is much smaller when allocating wealth across portfolios. Our
simulations show that optimal diversification policies will dominate the 1/N
rule only for very high levels of idiosyncratic volatility"

"the value-weighted market portfolio has a lower Sharpe ratio than
the 1/N benchmark in all the datasets, which is partly because of the small-
firm effect" (value weighted means market cap weighted)

"in terms of turnover, only the “vw” strategy, in which the investor holds the
market portfolio and does not trade at all, is better than the 1/N strategy"

My conclusions are that if N gets too high, then idiosyncratic volatility becomes an issue. Sharpe ratio is better for 1/N vs. market cap, but turnover is lower for market cap.

Another variable that might determine the relative merits of market cap vs. 1/N is how much tilt to small and value you want. 1/N will give you a tilt to small and value; that's probably where the higher Sharpe ratio of the 1/N comes from. But if you already have a tilt to small and value in the funds you use, the tilt of a 1/N portfolio may be less important. Conversely, if the funds you use are market cap weighted, the tilt of a 1/N portfolio may be more important.

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