Market cap weighting vs rebalancing

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mlz
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Market cap weighting vs rebalancing

Post by mlz » Wed Jul 11, 2018 3:24 pm

Short version: market cap-weighting a fund seems to run counter to the general principle that periodic rebalancing improves returns for a given level of risk. While market-cap weighting is great from an efficiency perspective, it seems sub-optimal in terms of risk-reward trade-offs. Are there other fundamental reasons to market-cap weight vs some other strategy?


Long version: It is generally recommended here and elsewhere to pick an asset allocation based on desired risk and return, and then occasionally re-balance to keep that allocation steady. This recommendation makes sense: otherwise, components will drift, and you will no longer have the same risk exposure. Such rebalancing should produce higher returns for a given level of risk.

This question of rebalancing vs not seems at a high level to be very similar to the debate over market-cap weighting in a fund vs constant weighting. Namely, a market-cap weighted fund is akin to no rebalancing, and a constant-weight fund is akin to periodic rebalancing.

In particular, it seems that the same logic could be applied to stocks within a fund or within the stock portion of an investor's portfolio. In theory, it seems a superior approach to market cap weighting would be to compute weights for the individual stocks to give an optimal expected return for a given level of risk. Then, maintain these weights through some rebalancing scheme. Of course, this requires good estimates for expected return, risk, and correlation of individual stocks.

Obviously, there are some reasons why such fine-grained rebalancing won't work in real life:
- It is harder, and therefore more expensive, to rebalance across 100s or 1000s of stocks than a few asset classes.
- The trajectories of individual stocks and correlations between individual stocks are poorly understood. This will lead to poor choices of weights, and therefore worse results

So it make sense to market-cap weight at a micro level where information is poor and rebalancing is expensive, and only rebalance at the macro-level. However, I fail to see why stocks/bonds/etc is the right level of granularity for rebalancing. True, these asset classes are delineated because they are fundamentally different assets. However, from an investor's perspective, if we ignore subtleties such as taxes and we automatically re-invest all dividends, then they really function equivalently. They just have different risk-reward trade-offs. Then it seems that rebalancing at this level of granularity is arbitrary and unlikely to result in the optimal portfolio. Why not market-cap weight between stocks and bonds as well? Or, within stocks, why not rebalance at a more fine-grained level such as by sector, by large/mid/small cap, etc?

After all that setup, finally the question: is there any fundamental reason rebalance a the level of granularity of stocks vs bonds etc, but not at some other level?

(I suppose one possible answer is that computing weights based on expected returns and correlations is implicitly being performed by the efficient market. This would render the whole question moot: the optimal weighting would happen to coincide with the market weighting. However, this would only be true if my risk tolerance happened to match the market's. For any other level of risk, my ideal weighting would be different)

jalbert
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Re: Market cap weighting vs rebalancing

Post by jalbert » Wed Jul 11, 2018 3:37 pm

After all that setup, finally the question: is there any fundamental reason rebalance a the level of granularity of stocks vs bonds etc, but not at some other level?
Yes, there is. The most impactful driver of risk and return of a stock and investment grade bond portfolio is the ratio of stocks to bonds. Because of different life circumstances and different emotional makeups of people, a 1-size-fits-all ratio of stocks to bonds is not possible for all individual portfolios. Analogous principles apply to institutional portfolios.

Holding the world market cap of stocks and bonds would just allow for one ratio of stocks to bonds at any point in time. Since this is untenable per the preceding paragraph, investors need to rebalance between stocks and bonds.

Research sponsored by the Federal Reserve suggests that changes in the relative balance of supply and demand for bonds and growth investments is the market force that establishes real interest rates.
Risk is not a guarantor of return.

alex_686
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Re: Market cap weighting vs rebalancing

Post by alex_686 » Wed Jul 11, 2018 3:38 pm

There are 3 parts to your question.
mlz wrote:
Wed Jul 11, 2018 3:24 pm
(I suppose one possible answer is that computing weights based on expected returns and correlations is implicitly being performed by the efficient market. This would render the whole question moot: the optimal weighting would happen to coincide with the market weighting. However, this would only be true if my risk tolerance happened to match the market's. For any other level of risk, my ideal weighting would be different)
You have more or less hit it on the head here. The market, as measured by free float weight, is the most efficient. This does mark some broad assumptions on market efficient and risk tolerance but has proven empirically right. We can dig down into certain aspects such as value, size, REITs, etc. and find topics to debate about.

On customized risk, you are still best suited going top down. Bonds or Equity. This is a big lever. Or you could drop down a level. In Equity, do you focus on Size, Value, Momentum, or Low Beta factors? However each of these still remain a strong trace of market cap weights. The most correct answer would be to create a custom index that matches your risk profile. However this is out of reach of all except big institutional investors.

So that is for a efficient portfolio - highest return for a level of risk. What about rebalancing? That is a different story. You are going to want to read up on convex / concave rebalancing strategies. They are mostly about controlling risk. Some will return a rebalancing bonus in some market circumstances, while under-preforming in others.

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Phineas J. Whoopee
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Re: Market cap weighting vs rebalancing

Post by Phineas J. Whoopee » Wed Jul 11, 2018 3:41 pm

Hi mlz. I see you just joined yesterday. Welcome.

Optimization is quite a word. Seldom can one do that. You correctly note that in investing it's an impossibility in practice because we lack information about the future.

A cap-weighted stock portfolio needs no rebalancing within itself, because as the values of its components rise and fall the proportion of the portfolio rises and falls with them. There are no transaction costs and it's a feature unique to cap weighting.

We individuals, except for any businesses including real-estate rentals we run ourselves can only invest in that which is publicly traded.

The main lever we have to dial in an approximately appropriate level of risk for ourselves, based on careful consideration of our own personal circumstances, is the ratio between more risky assets, like stocks, and less risky ones, like fixed income. That's why it seldom makes sense to keep both stocks and fixed income at relative market weights to each other.

Sorry about the mixed metaphor.

If I've not explained well please say so and I'll try again.

PJW

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danielc
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Re: Market cap weighting vs rebalancing

Post by danielc » Wed Jul 11, 2018 5:18 pm

mlz wrote:
Wed Jul 11, 2018 3:24 pm
(I suppose one possible answer is that computing weights based on expected returns and correlations is implicitly being performed by the efficient market. This would render the whole question moot: the optimal weighting would happen to coincide with the market weighting. However, this would only be true if my risk tolerance happened to match the market's. For any other level of risk, my ideal weighting would be different)
Yes. In 1990 William Sharpe received the Nobel Prize in Economics in 1990 for his contributions to finance theory, and his biggest contribution is basically a theorem that says that the optimal portfolio for everyone is the market portfolio, plus the risk-free asset (Treasury Bills), in some ratio that matches their risk/return tolerance. The idea is illustrated in this plot:

Image

The dots, and everything inside the curved line, are the different equity portfolios you could have. The red dot is the "market portfolio". The risk-free asset (Treasury Bills) is at the bottom-left. The straight line shows the different combinations you could have of risk-free + market portfolio. No matter where your risk tolerance is, you will always get the best return if you move to the straight line. In other words, no matter what your risk tolerance is, the best portfolio for you will be some combination of T-Bills and the market portfolio.

mlz
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Re: Market cap weighting vs rebalancing

Post by mlz » Wed Jul 11, 2018 6:05 pm

Thanks everyone for the responses!

I should have noted that I'm much too lazy to try anything more adventurous than a few passive index funds :D. I have little interest in trying to highly optimize my own portfolio. However, I'm fascinated by the theory behind things (I joke with my wife that "I hate numbers, but love variables"). I'm really asking from the perspective of an amateur investing theoretician who's trying to understand why things are the way they are.

danielc wrote:
Wed Jul 11, 2018 5:18 pm
Yes. In 1990 William Sharpe received the Nobel Prize in Economics in 1990 for his contributions to finance theory, and his biggest contribution is basically a theorem that says that the optimal portfolio for everyone is the market portfolio, plus the risk-free asset (Treasury Bills), in some ratio that matches their risk/return tolerance.
In this case, the "market portfolio" would be the total of all investment vehicles (stocks/bonds/REITs etc) weighted by market cap, right? But if I wanted something riskier than this market portfolio this makes the assumption that I can borrow at the risk-free rate. But this hardly seems like a realistic assumption, at least not for an individual investor. Therefore, investors with higher risk appetite are going to need to open up the market portfolio and re-weight things to match their investment needs. Usually, it seems, people open up to the level of asset classes to re-weight. But this seems unlikely to yield the truly optimal portfolio.

Of course, as an individual investor, I can only invest in the vehicles available to me. However, a mutual fund company can do essentially whatever it wants. Vanguard, for example, has their "LifeStrategy" funds, which appear to target various risk/return profiles by providing different allocations between stocks and bonds. But again, it seems that balancing at the level of stocks and bonds is unlikely to be optimal. Instead, one can imagine a suit of funds which choose allocations at a more fine-grained level. In principle, it would seem such a strategy would give better returns than a simple weighting between stocks and bonds.

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Phineas J. Whoopee
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Re: Market cap weighting vs rebalancing

Post by Phineas J. Whoopee » Wed Jul 11, 2018 6:30 pm

If one, and let's confine ourselves to US stock exchanges for the moment although the reasoning transfers to many other markets, adopts an allocation other than the cap-weighted market portfolio one can expect gross returns different from it. They may be higher or lower. The ones who get less have their potentially higher returns taken by the ones who get more.

Colloquially, they ate our lunch.

But if one spends more money to invest in an other-than-market portfolio, one's net return is likely to be lower than that of index investors who spend the tiniest amount possible to participate in the market.

In aggregate the gross return of market participants will be the market return, but in aggregate the ones who spend more to participate will get a lower net return.

It's nothing beyond grade-school arithmetic. Addition and subtraction.

PJW

mlz
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Re: Market cap weighting vs rebalancing

Post by mlz » Wed Jul 11, 2018 7:06 pm

Phineas J. Whoopee wrote:
Wed Jul 11, 2018 6:30 pm
If one, and let's confine ourselves to US stock exchanges for the moment although the reasoning transfers to many other markets, adopts an allocation other than the cap-weighted market portfolio one can expect gross returns different from it. They may be higher or lower. The ones who get less have their potentially higher returns taken by the ones who get more.

Colloquially, they ate our lunch.

But if one spends more money to invest in an other-than-market portfolio, one's net return is likely to be lower than that of index investors who spend the tiniest amount possible to participate in the market.

In aggregate the gross return of market participants will be the market return, but in aggregate the ones who spend more to participate will get a lower net return.

It's nothing beyond grade-school arithmetic. Addition and subtraction.

PJW
Sure, but the market is made of of participants with varying risk tolerances. Some are fine with lower returns, but demand lower risk. Others are fine with large swings in balances, but insist on higher expected returns. Therefore, it doesn't seem quite fair to say that the riskier high-return investor ate the lunch of the safe low-return investor. They simply are getting different things out of the market. In other words, while absolute returns are a zero sum game relative to the market as a whole, expected risk-adjusted returns are not.

But let's say you are one of those investors with a different risk tolerance than the aggregate market. Typically, you would achieve your risk tolerance by re-weighting stocks vs bonds in your portfolio, tilting toward stocks or bonds as needed. Besides being the easy thing to do, I don't see fundamentally how this would magically give you the optimal portfolio for your level of risk tolerance.

alex_686
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Re: Market cap weighting vs rebalancing

Post by alex_686 » Wed Jul 11, 2018 9:15 pm

So, one of the tenets of modern financial theory is that all assets have the same risk adjusted returns. For example, if we assume that a 10 year government bond has the same risk adjusted return as the stock market then we can figure a few things out about investor's expected return and risk tolerance.

Then the next step is to poke holes in the above theory, looking for anomalies that offer superior risk adjusted returns. The Free Lunch.
mlz wrote:
Wed Jul 11, 2018 6:05 pm
In this case, the "market portfolio" would be the total of all investment vehicles (stocks/bonds/REITs etc) weighted by market cap, right? But if I wanted something riskier than this market portfolio this makes the assumption that I can borrow at the risk-free rate. But this hardly seems like a realistic assumption, at least not for an individual investor.
But it is not that unrealistic assumption. Should you be 100% equities? Should you carry a large mortgage on your home and have the extra cash flow go to investment - or pay-down that mortgage? And once your account gets to 100k you can start getting really cheap leverage thanks to options and futures.
mlz wrote:
Wed Jul 11, 2018 7:06 pm
But let's say you are one of those investors with a different risk tolerance than the aggregate market. Typically, you would achieve your risk tolerance by re-weighting stocks vs bonds in your portfolio, tilting toward stocks or bonds as needed. Besides being the easy thing to do, I don't see fundamentally how this would magically give you the optimal portfolio for your level of risk tolerance.

So let us look at this. Of the four main factors - size, low beta, momentum, and value - momentum and value offer higher returns and higher risk. There is a debate if these last 2 offer a premium - a free lunch. Or if one could get the same risk / return by leveraging the market portfolio. Once again, lots of interesting debates around this subject.

rkhusky
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Re: Market cap weighting vs rebalancing

Post by rkhusky » Wed Jul 11, 2018 10:02 pm

While one can determine the optimal portfolio for the past, an optimal portfolio for the future is unknowable.

There is no guarantee that rebalancing will produce higher returns. Rebalancing could produce worse returns than not rebalancing.

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danielc
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Re: Market cap weighting vs rebalancing

Post by danielc » Thu Jul 12, 2018 12:14 am

mlz wrote:
Wed Jul 11, 2018 6:05 pm
In this case, the "market portfolio" would be the total of all investment vehicles (stocks/bonds/REITs etc) weighted by market cap, right?
The market portfolio refers to equities, which includes REITs. Bonds are very different instruments with very different risk characteristics than equities. I am positive that the CAPM would not work for the total stocks+bonds market. For example, it is not possible to hold a bond forever, and it is not possible for the price of a bond to go up above a certain price. Also, the US bond market is dominated by US gov't bonds, many of which would actually be classified as the risk-free asset. When people make their portfolios, they often treat their bond fund as a rough proxy for the risk-free asset (I actually think this is a big mistake, but nevermind that).

Even if we stick to equities, the CAPM explains most but not all of the return of different portfolios. So later, Eugene Fama and Kenneth French expanded the CAPM to add two more variables (known as "factors") based on certain properties of stocks. The CAPM already explained most of the portfolio performance, and the new "factors" from Fama and French made the model fit better. Now everyone is going crazy with "factors" so it seems like everyone and their dog is "discovering" new factors. Last I heard there were a few thousand new "factors", each one trying to predict another ounce of portfolio performance. There is a very real risk that most of these "factors" are just the result of data-mining, and there is a risk that some factors might be real but after they've been discovered they'll be eliminated by market arbiteurs.

So... what should you believe? Well, everyone agrees that the simple CAPM idea of mixing the risk-free asset with the market portfolio (of stocks) is fundamentally right and is the main thing that determines your risk and return. Many people (but not everyone) also accept the two extra factors from Fama and French, and use them to make small adjustments (or "tilts") in their portfolio. Few people go beyond that.

mlz wrote:
Wed Jul 11, 2018 6:05 pm
But if I wanted something riskier than this market portfolio this makes the assumption that I can borrow at the risk-free rate. But this hardly seems like a realistic assumption, at least not for an individual investor.
That is correct on both counts. You can use a margin account to borrow a little, and you can use index options to effectively leverage a lot. But both options cost more money than the risk-free rate, and have various other caveats and complexities. So in practice leveraging is not practical. But also in practice, a 100% equity portfolio is already VERY RISKY and it is not advisable for most people. A typical reasonable portfolio is only around 60% equities.
mlz wrote:
Wed Jul 11, 2018 6:05 pm
Therefore, investors with higher risk appetite are going to need to open up the market portfolio and re-weight things to match their investment needs. Usually, it seems, people open up to the level of asset classes to re-weight. But this seems unlikely to yield the truly optimal portfolio.
That is indeed more or less guaranteed to be very sub-optimal. Once you are past 100% equities, going toward riskier stocks usually means that you are accepting a significantly higher risk for a very tiny amount of return. You are not being well compensated for the risk you are taking. So reasonable people never go that route.
mlz wrote:
Wed Jul 11, 2018 6:05 pm
Of course, as an individual investor, I can only invest in the vehicles available to me. However, a mutual fund company can do essentially whatever it wants.
A mutual fund company cannot do whatever it wants. This is actually the key difference between a hedge fund and a mutual fund. Hedge funds have very few restrictions on them and they are allowed to do dangerous and stupid things like buying options and borrowing to invest. Mutual funds cannot do either of those things; all they can do is basically buy stocks and bonds. Having said that, the track record for hedge funds is really miserable. They don't work. People who invest in them basically ALWAYS get a worse performance that if they had just bought the market portfolio. So they get exactly zero net return for the significantly higher risk that they get.

Famously, in 2007 Warren Buffet made a bet with an asset manager from Protégé Partners. The bet was that the asset manager could pick any list of hedge funds that he wanted, and Buffet bet $1 million dollars that after 10 years those hedge funds would not perform as well as the market portfolio (using the S&P 500 as a proxy). That bet was won by Buffet last year and the $1M were donated to charity. The difference between the profesionally hand-picked hedge funds and the market portfolio was astounding. The hedge funds made a return of 2.2% while the simple plain boring S&P 500 had a return of 7.1%. Keep this story in mind next time someone tells you that they can beat the market.
mlz wrote:
Wed Jul 11, 2018 6:05 pm
Vanguard, for example, has their "LifeStrategy" funds, which appear to target various risk/return profiles by providing different allocations between stocks and bonds. But again, it seems that balancing at the level of stocks and bonds is unlikely to be optimal. Instead, one can imagine a suit of funds which choose allocations at a more fine-grained level. In principle, it would seem such a strategy would give better returns than a simple weighting between stocks and bonds.
The LifeStrategy funds from Vanguard are probably all very close to optimal. They should all lie very close to the straight line that I showed in the plot.

mlz
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Re: Market cap weighting vs rebalancing

Post by mlz » Mon Jul 16, 2018 9:03 am

Thanks for the info!

Here is an additional point of confusion: Consider the following two possible markets:
- Market 1: The market consists of just two companies, A and B. Both companies are quite large, and have identical projected returns x% per year, since they are both considered identically risky. We will assume that the success probabilities of both companies are largely independent. I'm going to assume for now that this means their returns are independent.
- Market 2: Again, just two companies, A and B. Again, both have independent returns x%. However, this time A is a large company whose products have broad appeal, while A has a more niche product and therefore is quite small.

Suppose I hold the market portfolio. Then in Market 1 I would hold similar amounts of both companies. In Market 2, I would hold far more of company A than B. Notice that in either case, no matter my asset allocation, my expected (arithmetic) returns are identical. In either case, the optimal portfolio will consist of equal parts A and B since this minimizes variance (and hence maximizes geometric returns). In the case of Market 1, this would just so happen to be the Market portfolio. However, in Market 2, the market portfolio would be highly suboptimal.

So this suggests that the Market weight is not guaranteed to be the optimal portfolio. But of course, in Market 2, if the Market were efficient any investor would ditch the market portfolio and tilt toward company B. This would then cause Company B's price to increase, reducing future returns. This would then reduce the weight of company B in the optimal portfolio until an equilibrium is reached and the Market portfolio matches the optimal portfolio.

But this seems to raise a couple weird results:
- First, it says that company B's returns are dependent on company A's returns even though their success probabilities are independent (thus violating the assumption stated above). This doesn't seem to make sense in a rational world.
- Second, now company B's returns are lower. However, the probabilities it succeeds is the same as for company A, which should mean that B has the same risk. But then this seems violates the principle that all companies have the same risk adjusted returns.

What am I missing?

pascalwager
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Re: Market cap weighting vs rebalancing

Post by pascalwager » Tue Jul 17, 2018 11:34 am

You wouldn't need to use the fixed VG LS funds to approximate Sharpe and they wouldn't really be adequate as market proportions are variable. You could use VG TWS and perhaps the upcoming VG global bond fund if it follows a world market AA.

Sharpe doesn't expect the market portfolio to meet every investor's needs. It's most suitable for retirees and a young investor could use his Adaptive Asset Allocation (AAA) Formula 15 to determine the stock/bond ratio for greater risk.

The average investor is a large entity with a longer time horizon than a retiree, so I use the VG Short-Term Bond Index fund as a replacement for his bonds. If VG had a short-term int'l bond fund, I would probably also use that. (I asked Sharpe about this issue and he said that the risk-free Treasury assets might reduce these concerns.)

Read Sharpe's RISMAT, Section 7 for his market portfolio construction.
Preferred AA: Total US and foreign stock markets and short-term Treasury fixed income

pascalwager
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Re: Market cap weighting vs rebalancing

Post by pascalwager » Wed Jul 18, 2018 2:03 am

mlz wrote:
Wed Jul 11, 2018 6:05 pm
Thanks everyone for the responses!

I should have noted that I'm much too lazy to try anything more adventurous than a few passive index funds :D. I have little interest in trying to highly optimize my own portfolio. However, I'm fascinated by the theory behind things (I joke with my wife that "I hate numbers, but love variables"). I'm really asking from the perspective of an amateur investing theoretician who's trying to understand why things are the way they are.

danielc wrote:
Wed Jul 11, 2018 5:18 pm
Yes. In 1990 William Sharpe received the Nobel Prize in Economics in 1990 for his contributions to finance theory, and his biggest contribution is basically a theorem that says that the optimal portfolio for everyone is the market portfolio, plus the risk-free asset (Treasury Bills), in some ratio that matches their risk/return tolerance.
In this case, the "market portfolio" would be the total of all investment vehicles (stocks/bonds/REITs etc) weighted by market cap, right? But if I wanted something riskier than this market portfolio this makes the assumption that I can borrow at the risk-free rate. But this hardly seems like a realistic assumption, at least not for an individual investor. Therefore, investors with higher risk appetite are going to need to open up the market portfolio and re-weight things to match their investment needs. Usually, it seems, people open up to the level of asset classes to re-weight. But this seems unlikely to yield the truly optimal portfolio.

Of course, as an individual investor, I can only invest in the vehicles available to me. However, a mutual fund company can do essentially whatever it wants. Vanguard, for example, has their "LifeStrategy" funds, which appear to target various risk/return profiles by providing different allocations between stocks and bonds. But again, it seems that balancing at the level of stocks and bonds is unlikely to be optimal. Instead, one can imagine a suit of funds which choose allocations at a more fine-grained level. In principle, it would seem such a strategy would give better returns than a simple weighting between stocks and bonds.
What you call an "optimal" portfolio, Sharpe calls "betting". He calls the market portfolio "investing" along with other equity/bond ratios if they have a clear rationale.

His practical version of the market portfolio uses the following four Vanguard funds: total stock market, total international stock, total bond market, and total international bond. Or, but at greater ER, you could use Vanguard Total World Stock fund in place of the two US and non-US stock funds.
Preferred AA: Total US and foreign stock markets and short-term Treasury fixed income

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