Yes, you can diversify a market portfolio

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vineviz
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Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 6:00 pm

Several times in recent weeks I've seen a reply on the Bogleheads forum that said "you can't diversify a market portfolio" or something along those lines.

This isn't true.

I think that most people believe, implicitly or explicitly, that diversified portfolio is balanced with respect to risk. In other words, diversification has a general meaning that most people intuitively grasp ("don't put all your eggs in one basket"),

Without relying too heavily on theory, let's take a look at three charts the represent a stereotypical "balanced" portfolio: 60% stocks (VTSMX) and 40% bonds (VBMFX). Results are a backtest from Portfolio Visualizer from 01/2011 to 05/2018 (link at the end of this post).
  • The piechart on the left shows the portfolio in proportion to its dollar VALUE.
  • The piechart in the middle shows the portfolio in proportion to the dollar RETURNS over the past five years or so.
  • The piechart on the right shows the portfolio in proportion to the RISK each asset contributed.
Image

You can easily and quickly see that, from a risk perspective, the 60/40 portfolio is far from balanced: virtually all the risk is coming from one asset. I can't imagine anyone looking at the risk profile of this two-fund portfolio and concluding that there is no room for improving the diversification of risk.

The Vanguard Total Stock Market Index fund contains almost entirely U.S. large cap stocks. It holds many of them, in many different sectors, in proportion to their market capitalization, but represents only a slice of the variety of investable companies. VTSMX has virtually no exposure to the risk and style factors (small companies, value companies, quality companies, low beta stocks, etc.) that economists have identified as offering excess returns or having relatively low correlations with large cap U.S. stocks.

Likewise, the Vanguard Total Bond Market Index fund represents an impressive portfolio, but it does not have the level of exposure to credit risk and term risk it needs in order to balance the stocks held in VTSMX.

Making some changes aimed at diversification (replacing Total Bond Market Index with a long-term bond ETF and adding two small cap ETFS, one invested in U.S. companies and one in non-U.S. companies) makes a dramatic difference to the concentration of risk in this portfolio.

Image

There is no single quantitative definition of diversification that is equivalent to the Sharpe ratio (which represents the highest risk-adjusted expected return of any combination of risky assets and is presumed to be mean-variance optimized), but there are several measures that offer some utility: diversification ratio, minimum variance, inverse volatility, and effective numbers of bets.

I think, for advanced individual investors, the diversification ratio offers a balance of utility and ease of understanding. It basically measures the average volatility of each asset in the portfolio to the overall volatility of portfolio, and the higher the number the better the assets are working together to offset risk. The lowest possible figure is 1.0, and for portfolios of publicly traded stocks and bonds it would be hard to get the ratio above 2.0 (though higher numbers are theoretically possible).

The two-fund version of the 60/40 portfolio above has a diversification ratio of 1.19 while the four-fund version has a diversification ratio of 1.60, a significant improvement.

The diversification ratio also has the benefit/disadvantage of being agnostic to expected returns: it is constructed based entirely on volatility and correlation measures. It does not incorporate either historical returns or expected returns, which makes it less likely than a mean-variance measure (like the Sharpe ratio) to overweight last year's winners.

Link to Portfolio Visualizer backtest: https://www.portfoliovisualizer.com/bac ... tion8_2=40
Last edited by vineviz on Tue Jun 05, 2018 6:44 pm, edited 1 time in total.
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Re: Yes, you can diversify a market portfolio

Post by bloom2708 » Tue Jun 05, 2018 6:08 pm

Interesting. I do not think you are using the word "Diversification" in the correct context typically used here.

Can you explain this quote from your post?

"The Vanguard Total Stock Market Index fund is contains only U.S. large cap stocks."

It looks like you add a lot more risk (long bonds, small cap) and then spread the risk out and call that risk diversification. Not sure that makes sense.
Last edited by bloom2708 on Tue Jun 05, 2018 6:28 pm, edited 1 time in total.
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Re: Yes, you can diversify a market portfolio

Post by michaeljc70 » Tue Jun 05, 2018 6:24 pm

"The Vanguard Total Stock Market Index fund is contains only U.S. large cap stocks. It holds many of them, in many different sectors, in proportion to their market capitalization, but represents only a slice of the variety of investable companies. VTSMX has virtually no exposure to the risk and style factors (small companies, value companies, quality companies, low beta stocks, etc.) that economists have identified as offering excess returns or having relatively low correlations with large cap U.S. stocks."

I think there is a lot incorrect in that paragraph. Of course the TSM has value, quality and low beta stocks. It has small cap and mid cap stocks too.

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Re: Yes, you can diversify a market portfolio

Post by nisiprius » Tue Jun 05, 2018 6:52 pm

The Diversification Ratio® (it's apparently a registered trademark of TOBAM S.A.S.) is an interesting idea, but you probably shouldn't assume that it is the generally accepted standard measure of diversification.

The Diversification Ratio® was defined by Yves Choueifaty in 2008 as "the ratio of the portfolio’s weighted average volatility to its overall volatility." (Presumably lower diversification ratios are more diverse).

There are some logical issues with this definition. According to this definition, all of these portfolios have Diversification Ratios® of 1.00 and are therefore all equally diversified:

100% stocks
50% stocks, 50% cash
5% stocks, 95% cash

Maybe this does make sense, I'm not sure.

Notes:

Choueifaty is one of the principals of TOBAM S.A.S., a French asset management company, which offers services based on their trademarked strategies. Their site states:
Yves Choueifaty after years of academic research introduced a measure of diversification: the Diversification Ratio®. The details of this were initially published in 2006 in the United States Patent and Trademark Office (Choueifaty, “Methods and Systems for Providing an Anti-Benchmark Portfolio, May 2006) and later in 2008 in the Journal of Portfolio Management [Choueifaty & Al, “Toward Maximum Diversification” Fall 2008]. TOBAM’s Anti-Benchmark® strategy is based on the Maximum Diversification® approach, designed to maximize the degree of diversification when selecting the weighting of assets in the portfolio allocation process.
The definition of the Diversification Ratio® is from Properties of the Most Diversified Portfolio, Journal of Investment Strategies, Vol.2(2), Spring 2013, pp.49-70, by Yves Choueifaty, Tristan Froidure, and Julien Reynier, all of the firm TOBAM.

TOBAM states that
"TOBAM,” “MaxDiv,” “Maximum Diversification,” “Diversification Ratio,” “Most Diversified Portfolio,” “Most Diversified Portfolios,” “MDP” and “Anti-Benchmark” are registered trademarks
"Diversification Ratio®," is a French trademark, French trademark #3779111. I assume the others are, too. I'm not sure whether I, as a US resident, need to use the ® symbol with them.
Last edited by nisiprius on Tue Jun 05, 2018 7:23 pm, edited 1 time in total.
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Re: Yes, you can diversify a market portfolio

Post by nisiprius » Tue Jun 05, 2018 7:19 pm

Vineviz, how do you define "share of risk?" In a 60/40 portfolio of Total Stock and Total Bond, you are showing the standard deviation as 6.55%. According to PV, adding first 100% Total Stock and then 100% Total Bond as portfolio 3, I am getting 11.10% for 100% Total Stock, 2.87% for 100 Total Bond.

(The weighted average standard deviation is thus 60% * 11.10% = 6.66%, + 40% * 2.87% = 1.148%, = 7.81% for the weighted average. The Diversification Ratio® is thus 6.55% / 7.81% = 0.84).

To my thinking, Total Stock is contributing 6.66% / 7.81% = 85% of the "risk" and Total Bond is contributing 1.148% / 7.81% = 15% of the "risk," not zero.

How did you do the calculation?
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Re: Yes, you can diversify a market portfolio

Post by JoMoney » Tue Jun 05, 2018 7:42 pm

"diversification" and "risk" haves different meaning and intimation to different people.
One could also look at your suggestion and say, "This isn't true."
As far as the market portfolio goes, it is a point of diversification that everyone can hold in a way that can't make a portfolio better off without making some other portfolio worse off on those same terms. To hold some other weighting requires that you find counterparties with a preference to hold some opposing portfolio for whatever reason, and obviously your preferred holding would not be optimal for them.
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Re: Yes, you can diversify a market portfolio

Post by Taylor Larimore » Tue Jun 05, 2018 8:10 pm

VTSMX has virtually no exposure to the risk and style factors (small companies, value companies, quality companies, low beta stocks, etc.)
Bogleheads:

This is a common misconception. VTSMX (Vanguard Total Stock Market) holds the market weight in nearly all U.S. "risk and style factors small companies, value companies, quality companies, low beta stocks, etc."

Best wishes.
Taylor
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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 8:14 pm

bloom2708 wrote:
Tue Jun 05, 2018 6:08 pm
"The Vanguard Total Stock Market Index fund is contains only U.S. large cap stocks."
It looks like I posted a version of the sentence that was garbled by an incomplete edit. I meant to say "contains almost exclusively" or some other qualifier. Obviously there are mid- and small-cap stocks in the index, just at such low weights as to be effectively absent.
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Re: Yes, you can diversify a market portfolio

Post by nisiprius » Tue Jun 05, 2018 8:16 pm

vineviz wrote:
Tue Jun 05, 2018 8:14 pm
bloom2708 wrote:
Tue Jun 05, 2018 6:08 pm
"The Vanguard Total Stock Market Index fund is contains only U.S. large cap stocks."
It looks like I posted a version of the sentence that was garbled by an incomplete edit. I meant to say "contains almost exclusively" or some other qualifier. Obviously there are mid- and small-cap stocks in the index, just at such low weights as to be effectively absent.
Just as they are in the market itself.
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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 8:19 pm

nisiprius wrote:
Tue Jun 05, 2018 6:52 pm
There are some logical issues with this definition. According to this definition, all of these portfolios have Diversification Ratios® of 1.00 and are therefore all equally diversified:

100% stocks
50% stocks, 50% cash
5% stocks, 95% cash

Maybe this does make sense, I'm not sure.
Investment portfolios, in MPT, by definition contain only risky assets. Cash, being riskless, is not used to compute the diversification ratio.

All three of these portfolios are 100% stock, and there is no way to compute the diversification ratio for them without knowing which stocks and their weights.
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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 8:19 pm

nisiprius wrote:
Tue Jun 05, 2018 8:16 pm
vineviz wrote:
Tue Jun 05, 2018 8:14 pm
bloom2708 wrote:
Tue Jun 05, 2018 6:08 pm
"The Vanguard Total Stock Market Index fund is contains only U.S. large cap stocks."
It looks like I posted a version of the sentence that was garbled by an incomplete edit. I meant to say "contains almost exclusively" or some other qualifier. Obviously there are mid- and small-cap stocks in the index, just at such low weights as to be effectively absent.
Just as they are in the market itself.
Sure.
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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 8:31 pm

Taylor Larimore wrote:
Tue Jun 05, 2018 8:10 pm
VTSMX has virtually no exposure to the risk and style factors (small companies, value companies, quality companies, low beta stocks, etc.)
This is a common misconception. VTSMX (Vanguard Total Stock Market) holds the market weight in nearly all U.S. "risk and style factors small companies, value companies, quality companies, low beta stocks, etc."
Actually, I think the misconception is yours.

It's true that there are small stocks, value stocks, low beta stocks, etc. in the index. That's not relevant because the index also contains an equal proportion of large stocks, growth stocks, high beta stocks, etc. Thus, the total market index (pretty much by definition) has virtually zero (net) exposure to those style and risk factors.

Code: Select all

Factor	Loading	Standard Error	t-stat	p-value
Market (Rm-Rf)	 1.00 	 0.00 	 318.38 	 -   
Size (SMB)	 -   	 0.00 	 (1.44)	 0.15 
Value (HML)	 0.02 	 0.00 	 4.22 	 -   
Alpha (α) bps	 (1.81)	 -   	 (2.09)	 0.04 
If you have one brokerage account that is long 100 shares of AMZN and another account that is short 100 shares of AMZN, you have no exposure to AMZN: the two positions cancel each other out.
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Re: Yes, you can diversify a market portfolio

Post by bloom2708 » Tue Jun 05, 2018 8:38 pm

vineviz wrote:
Tue Jun 05, 2018 8:31 pm
Taylor Larimore wrote:
Tue Jun 05, 2018 8:10 pm
VTSMX has virtually no exposure to the risk and style factors (small companies, value companies, quality companies, low beta stocks, etc.)
This is a common misconception. VTSMX (Vanguard Total Stock Market) holds the market weight in nearly all U.S. "risk and style factors small companies, value companies, quality companies, low beta stocks, etc."
Actually, I think the misconception is yours.

It's true that there are small stocks, value stocks, low beta stocks, etc. in the index. That's not relevant because the index also contains an equal proportion of large stocks, growth stocks, high beta stocks, etc. Thus, the total market index (pretty much by definition) has virtually zero (net) exposure to those style and risk factors.

Code: Select all

Factor	Loading	Standard Error	t-stat	p-value
Market (Rm-Rf)	 1.00 	 0.00 	 318.38 	 -   
Size (SMB)	 -   	 0.00 	 (1.44)	 0.15 
Value (HML)	 0.02 	 0.00 	 4.22 	 -   
Alpha (α) bps	 (1.81)	 -   	 (2.09)	 0.04 
If you have one brokerage account that is long 100 shares of AMZN and another account that is short 100 shares of AMZN, you have no exposure to AMZN: the two positions cancel each other out.
Now you are heading off into uncharted territory. You are essentially saying if I buy 1 share of the Total Stock Market, I own nothing. Because there are no tilts? You have to own extra of something to own it? The market is the market.
Last edited by bloom2708 on Tue Jun 05, 2018 8:48 pm, edited 1 time in total.
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Alexa9
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Re: Yes, you can diversify a market portfolio

Post by Alexa9 » Tue Jun 05, 2018 8:45 pm

I agree that Total Stock Market is not "diversified" by market cap. My profile picture is IMO >>>
or more towards value if desired. Some argue that "Large/mid/small caps" are arbitrary but the market cap weighting heavily favors large caps.
Yes the Total Stock Market fund is great for simplicity and cost but I don't think that it is ideal.

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Re: Yes, you can diversify a market portfolio

Post by Random Walker » Tue Jun 05, 2018 8:50 pm

I like this concept of diversification ratio = portfolio volatility / weighted average of portfolio components. As Larry Swedroe has written: “we need to look at diversification differently”. Diversification is not number of stocks. Rather it is more like independent sources of return. The expected return of a portfolio (simple average) is the weighted mean average expected return of the portfolio components. The actual compounded return of the portfolio (what we eat) will be less than the weighted average return of the components because of volatility. The SD of a portfolio will be less than the weighted average of the portfolio component SDs because of correlations less than 1. How a new potential investment affects a portfolio depends on its expected return, correlations, volatility, when correlations change, and of course costs. So when a potential portfolio addition has high expected returns, high volatility, low or negative correlation to other portfolio components, it has the potential to be an excellent addition.
The typical 60/40 portfolio has >90% of its risk wrapped up in equities. And if the equities are all TSM, then all that risk is wrapped up in a single factor, market beta. Diversification across independent sources of return improves portfolio efficiency lots. That improved efficiency is manifested by lower portfolio SD, compounded return closer to average return, smaller maximal drawdowns.

Dave

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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 8:51 pm

nisiprius wrote:
Tue Jun 05, 2018 7:19 pm
How did you do the calculation?
To be honest I don't calculate it manually, because it is time consuming and Portfolio Visualizer seems to do it competently.

To measure risk decomposition you need three numbers:
  • The weight of the asset in the portfolio
  • The beta of the asset to the portfolio
  • The standard deviation of the portfolio
https://cssanalytics.wordpress.com/2012 ... ributions/
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Misconception

Post by Taylor Larimore » Tue Jun 05, 2018 8:52 pm

vineviz wrote:Actually, I think the misconception is yours.
It's possible. :happy

Best wishes.
Taylor
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Re: Yes, you can diversify a market portfolio

Post by rkhusky » Tue Jun 05, 2018 8:57 pm

Total Stock Market has thousands of small-cap companies and only hundreds of large cap companies. By that measure, it is overweight in small caps.

Total Stock Market has equal percentages of the investable shares for all the companies in its portfolio. By that measure, it is perfectly diversified.

You can make up your own definition of diversity and call TSM whatever you want.

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Re: Yes, you can diversify a market portfolio

Post by Random Walker » Tue Jun 05, 2018 8:57 pm

Vineviz is correct about TSM having no net exposure to size or value factors. The exposure to small and value is exactly offset by exposure to big and growth by definition. The key is that size and value are defined as long/short portfolios, so the market factor is extracted. The key finding of FF is that after extracting market factor, size and value still represent unique and independent sources of risk. Independent of market and independent of each other. That means they have serious diversification potential for a portfolio.

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Re: Yes, you can diversify a market portfolio

Post by MIretired » Tue Jun 05, 2018 8:59 pm

It's often mentioned that the total US stock index is the efficient mix of all US stocks. I wanted to check this using the Sharpe ratio, but didn't want to enter 1700+ stocks. So, I thought, I can get a good check using the efficient frontier of S&P500 (VFINX) vs extended market index (VEXMX). These constitute TSM by DJ S&P500 + DJ S&P completion.
https://us.spindices.com/indices/equity ... n-index-ci
The wiki suggests a good ratio of 80% 500/ 20% extended market.
https://www.bogleheads.org/wiki/Extende ... index_fund

And, sure enough, it lands at about 15% VEXMX/ 85% VFINX.
Edit: this test only considers size , really. Not value/growth, etc. So, for what it's worth.
https://www.portfoliovisualizer.com/opt ... tion2_1=20

PV also has a different mix for the 4 funder above of vineviz for it's max diversification test. Also has risk parity, min variance, etc.
It looks more like risk parity on PV.


https://www.portfoliovisualizer.com/opt ... tion4_1=25

I think vineviz and nisiprius' ref for the definition are the same; as in weighted average volatility in the numerator, and resultant (with non-correlation benefits) in the denominator, making the ratio 1.0 or higher.

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Re: Yes, you can diversify a market portfolio

Post by rkhusky » Tue Jun 05, 2018 9:02 pm

vineviz wrote:
Tue Jun 05, 2018 8:31 pm
Taylor Larimore wrote:
Tue Jun 05, 2018 8:10 pm
VTSMX has virtually no exposure to the risk and style factors (small companies, value companies, quality companies, low beta stocks, etc.)
This is a common misconception. VTSMX (Vanguard Total Stock Market) holds the market weight in nearly all U.S. "risk and style factors small companies, value companies, quality companies, low beta stocks, etc."
Actually, I think the misconception is yours.
Taylor accurately responded to your quote, where you defined the factors as small, value, quality, low beta. The FF factors are small-big, value-growth, etc.

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Re: Yes, you can diversify a market portfolio

Post by MIretired » Tue Jun 05, 2018 9:06 pm

What bothers me, still, about using just standard deviations, and anythig derived using SD, is they don't take into account the dispersion of SD. Do they? I mean skew and kurtosis and funny data series can have the same SDs but be very much less non-normal distributions. Hence I don't like to be made to wait unneccessarily long to get a possible benefit--even though the standard meaning of probabilty says its more likely to pay off; if it were a normal distribution (or more normal.)

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Re: Yes, you can diversify a market portfolio

Post by SimpleGift » Tue Jun 05, 2018 9:18 pm

Random Walker wrote:
Tue Jun 05, 2018 8:57 pm
The key finding of FF is that after extracting market factor, size and value still represent unique and independent sources of risk. Independent of market and independent of each other. That means they have serious diversification potential for a portfolio.
Certainly there is the potential for diversification by allocating to factors, but one should be careful not to assume that this somehow represents the "optimal portfolio." From an interview with Eugene Fama, in his own words (my bold):
Interviewer: Some people cite your research showing that value and small firms have higher average returns over time and they assume that you would recommend most investors have a big helping of small and value stocks in their portfolios. Is that a fair representation of your views?

Fama: Um, no. (Laughs) Basically this is a risk story the way we tell it, so there is no optimal portfolio. The way I like to talk about it when I give presentations for DFA or other people is, in every asset pricing model, the market portfolio is always an efficient portfolio. It’s always a relevant portfolio for an investor to hold. And investors can decide to tilt away from that based on their personal tastes.

But that’s what it amounts to. You can decide to tilt toward more value or smaller size based on your tastes for these dimensions of risk. But you needn’t do it. You could also decide to go the other way. You could look at the premiums and say, no, I think I like the growth stocks better. Then, as long as you get a diversified portfolio of them, I can’t argue with that either.

So there’s a whole multi-dimensional continuum here of efficient portfolios that anybody can decide to buy that I can’t quarrel with. And I have no recommendations about it because I think it’s totally a matter of taste. If you eat oranges and I eat apples I can’t really quarrel very much with that.
In short, one can diversify the market portfolio — but with humility, let's not assume a priori that it's always the superior choice.
Cordially, Todd

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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 9:25 pm

bloom2708 wrote:
Tue Jun 05, 2018 8:38 pm

Now you are heading off into uncharted territory. You are essentially saying if I buy 1 share of the Total Stock Market, I own nothing. Because there are no tilts? You have to own extra of something to own it? The market is the market.
I'm definitely not saying that you own nothing. TSM has a full, 100% exposure (aka 'beta') to the aggregate market risk premium and you certainly own that.

What you don't have is exposure to any other risk or style factors like the ones described by Fama/French, AQR, etc.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch

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Re: Yes, you can diversify a market portfolio

Post by Chan_va » Tue Jun 05, 2018 9:29 pm

Not sure I follow your argument. Lets look at 2 portfolio's

Portfolio 1: 100% VOO (S&P 500 ETF)
Portfolio 2: 50% VOO, 50% SSO (2x leveraged S&P 500 ETF)

Using your definition of diversification (weighted average of volatility/portfolio volatility), I get

Portfolio 1 diversity=1.0
Portfolio 2 diversity=1.04

Are you claiming that portfolio 2 is more diversified than portfolio 1? I am missing something here.

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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 9:32 pm

SimpleGift wrote:
Tue Jun 05, 2018 9:18 pm
In short, one can diversify the market portfolio — but with humility, let's not assume a priori that it's always the superior choice.
That's a good point, and I should have (and meant to) stress a related point: an maximally diversified portfolio is optimized only along whatever quantitative dimension(s) you are measuring, and a maximally diversified portfolio might NOT be optimal for any particular investor.

Indeed, because the diversification ratio does not make use of any information about historical or expected returns it is often the case that portfolios constructed to maximize it might fail miserably to produce an acceptable level of return for the investor if constraints are not applied.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch

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Re: Yes, you can diversify a market portfolio

Post by vineviz » Tue Jun 05, 2018 9:48 pm

Chan_va wrote:
Tue Jun 05, 2018 9:29 pm
Not sure I follow your argument. Lets look at 2 portfolio's

Portfolio 1: 100% VOO (S&P 500 ETF)
Portfolio 2: 50% VOO, 50% SSO (2x leveraged S&P 500 ETF)

Using your definition of diversification (weighted average of volatility/portfolio volatility), I get

Portfolio 1 diversity=1.0
Portfolio 2 diversity=1.04

Are you claiming that portfolio 2 is more diversified than portfolio 1? I am missing something here.
My calculations don't match yours: I calculate a DR of 1.0 for both portfolios. The problem might be in the 50/50 weighting: if you look at a long enough period of time with infrequent rebalancing, the initial 50/50 weights start to drift: the time-weighted weights are something different, so the calculation becomes inaccurate.

Take a look at a period from January to May of this year, with monthly rebalancing: https://www.portfoliovisualizer.com/bac ... tion2_3=50

VOO has a Stdev of 12.99%, SSO has a Stdev of 26.78%, and the portfolio has a Stdev of 19.89%.

Code: Select all

 (.5 * 12.99) + (.5 * 26.78) = 19.89
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Re: Yes, you can diversify a market portfolio

Post by JoMoney » Tue Jun 05, 2018 11:05 pm

vineviz wrote:
Tue Jun 05, 2018 9:25 pm
bloom2708 wrote:
Tue Jun 05, 2018 8:38 pm

Now you are heading off into uncharted territory. You are essentially saying if I buy 1 share of the Total Stock Market, I own nothing. Because there are no tilts? You have to own extra of something to own it? The market is the market.
I'm definitely not saying that you own nothing. TSM has a full, 100% exposure (aka 'beta') to the aggregate market risk premium and you certainly own that.

What you don't have is exposure to any other risk or style factors like the ones described by Fama/French, AQR, etc.
Just about every definition of diversification (as it relates to finance) that I've seen describes "diversification" as method to reduce risk. Within the paradigm that looks at "Risk Factors" as a thing, the market portfolio has neutralized them. Your suggestion is to increase the risk on those dimensions where the market portfolio doesn't bear those risks at all. You then attempt to use the Sharpe ratio to demonstrate reducing the risk, but that is in-congruent with the idea that those "Risk Factors" are actually a unique dimension of "risk", which would require a factor regression to see.
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Re: Yes, you can diversify a market portfolio

Post by Ben Mathew » Tue Jun 05, 2018 11:50 pm

vineviz wrote:
Tue Jun 05, 2018 6:00 pm

Without relying too heavily on theory, let's take a look at three charts the represent a stereotypical "balanced" portfolio: 60% stocks (VTSMX) and 40% bonds (VBMFX). Results are a backtest from Portfolio Visualizer from 01/2011 to 05/2018 (link at the end of this post).
  • The piechart on the left shows the portfolio in proportion to its dollar VALUE.
  • The piechart in the middle shows the portfolio in proportion to the dollar RETURNS over the past five years or so.
  • The piechart on the right shows the portfolio in proportion to the RISK each asset contributed.
Image

You can easily and quickly see that, from a risk perspective, the 60/40 portfolio is far from balanced: virtually all the risk is coming from one asset. I can't imagine anyone looking at the risk profile of this two-fund portfolio and concluding that there is no room for improving the diversification of risk.
If you combine a riskless asset (safe bonds) with a risky asset (stocks), 100% of the risk would come from stocks. This is normal. It does not prove that the portfolio is undiversified.
vineviz wrote:
Tue Jun 05, 2018 6:00 pm
The Vanguard Total Stock Market Index fund contains almost entirely U.S. large cap stocks. It holds many of them, in many different sectors, in proportion to their market capitalization, but represents only a slice of the variety of investable companies. VTSMX has virtually no exposure to the risk and style factors (small companies, value companies, quality companies, low beta stocks, etc.) that economists have identified as offering excess returns or having relatively low correlations with large cap U.S. stocks.

Likewise, the Vanguard Total Bond Market Index fund represents an impressive portfolio, but it does not have the level of exposure to credit risk and term risk it needs in order to balance the stocks held in VTSMX.

Making some changes aimed at diversification (replacing Total Bond Market Index with a long-term bond ETF and adding two small cap ETFS, one invested in U.S. companies and one in non-U.S. companies) makes a dramatic difference to the concentration of risk in this portfolio.

Image

There is no single quantitative definition of diversification that is equivalent to the Sharpe ratio (which represents the highest risk-adjusted expected return of any combination of risky assets and is presumed to be mean-variance optimized), but there are several measures that offer some utility: diversification ratio, minimum variance, inverse volatility, and effective numbers of bets.

I think, for advanced individual investors, the diversification ratio offers a balance of utility and ease of understanding. It basically measures the average volatility of each asset in the portfolio to the overall volatility of portfolio, and the higher the number the better the assets are working together to offset risk. The lowest possible figure is 1.0, and for portfolios of publicly traded stocks and bonds it would be hard to get the ratio above 2.0 (though higher numbers are theoretically possible).

The two-fund version of the 60/40 portfolio above has a diversification ratio of 1.19 while the four-fund version has a diversification ratio of 1.60, a significant improvement.
You could make the argument that the Vanguard Total Stock Market Index fund is not fully diversified. To show that, you would need to show that by adding other funds you can reduce risk without reducing return. But the diversification measure you are using is very problematic--for example, if you combine different separate assets into a single fund, your diversification ratio would drop to 1. That sounds like a bad thing according your measure when nothing has changed for the portfolio--it's the same assets with the same risk and return.

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Re: Yes, you can diversify a market portfolio

Post by vineviz » Wed Jun 06, 2018 12:23 am

JoMoney wrote:
Tue Jun 05, 2018 11:05 pm
Just about every definition of diversification (as it relates to finance) that I've seen describes "diversification" as method to reduce risk.
Indeed. The literature is full of definitions of "diversification", many of them imprecise and some of them contradictory. Even defining "risk" is risky.

I tend to favor a definition that describes diversification as a method to MANAGE uncertainty, because the goal in constructing any given portfolio might not (or might) be to minimize uncertainty.
JoMoney wrote:
Tue Jun 05, 2018 11:05 pm
You then attempt to use the Sharpe ratio to demonstrate reducing the risk, but that is in-congruent with the idea that those "Risk Factors" are actually a unique dimension of "risk", which would require a factor regression to see.
I'm not sure I agree that this is how I used the Sharpe ratio in my post, but perhaps I wasn't careful enough in my language. I merely meant to hold up the Sharpe ratio as an example. Given a certain set of assumptions (a la CAPM), you can optimize a portfolio by maximizing a single number (the Sharpe ratio). Yet even with just two dimensions, historically we've not had an equivalent way to measure diversification, since in a one-factor world the mean/variance optimized portfolio isn't necessarily the same as the minimum variance portfolio. And once you start thinking about additional factors, neither of those portfolios might be the maximally diversified portfolio.

As Fabozzi wrote in 2010:
A critical weakness of mean-variance analysis is the use of variance as a measure of risk. In some sense, risk is a subjective concept and different investors adopt different investment strategies in seeking to realize their investment objectives (Holton 1997; Siu et al. 2001; and Boyle et al. 2002), and hence the exogenous characteristics of investors mean that probably no unique risk measure exists that can accommodate every investor’s problem (Ortobelli et al. 2005).
The main point I was trying to make, through all this, is that a portfolio built with index funds (like VTSMX and VBMFX) can still inadequately diversified. Even the market portfolio can be further diversified.
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Re: Yes, you can diversify a market portfolio

Post by vineviz » Wed Jun 06, 2018 12:41 am

Ben Mathew wrote:
Tue Jun 05, 2018 11:50 pm
If you combine a riskless asset (safe bonds) with a risky asset (stocks), 100% of the risk would come from stocks. This is normal.
Sure, but the bonds in VBMFX are far from riskless (at least as they'd be defined in MPT). And I think that many investors intuitively think about diversification as BALANCING their risks (e.g. spreading out their bets), so I think there is utility in illustrating that a fund like VBMFX isn't spreading out the risks so very much after all.
Ben Mathew wrote:
Tue Jun 05, 2018 11:50 pm
You could make the argument that the Vanguard Total Stock Market Index fund is not fully diversified. To show that, you would need to show that by adding other funds you can reduce risk without reducing return.
Imposing the requirement that return can't be reduced would ensure that the portfolio is mean-variance optimal, but that's not the same thing as saying it is maximally diversified. There is nothing that requires the most diversified portfolio to be mean-variance optimal, and nothing that requires the mean-variance optimal portfolio to be maximally diversified (unless you use a circular definition for that word "diversified"). In fact, if you are working under strict CAPM assumptions, which make the market portfolio the tangency portfolio the mean-variance optimal portfolio is guaranteed to NOT be the minimum variance portfolio, for example.
Ben Mathew wrote:
Tue Jun 05, 2018 11:50 pm
But the diversification measure you are using is very problematic--for example, if you combine different separate assets into a single fund, your diversification ratio would drop to 1. That sounds like a bad thing according your measure when nothing has changed for the portfolio--it's the same assets with the same risk and return.
The diversification ratio is inherently a relative measure of diversification, not an absolute one, so it does have limits for sure and this is one of them: it's not a very useful way of evaluating a portfolio that contains only one asset, and its also not terribly obvious how you'd use it to compare two portfolios that have no overlapping assets.
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Re: Yes, you can diversify a market portfolio

Post by jalbert » Wed Jun 06, 2018 2:56 am

I assume it was an oversight using VTI, the ETF share class of total US stock in one portfolio giving it an ER advantage over VTMSX (investor shares of same fund with higher ER) used in the other portfolio.

Backtests can suffer from substantial sample bias and certainly a non-random 7-year sample of consecutive returns often is highly biased. I’m not taking sides on the two portfolios, but it turns out the higher return and Sharpe ratio in the portfolio you propose as more diversified arises from the use of long bonds, as it was a period where changes in interest rates favored longer durations, a significant sample bias in favor of the portfolio with the long bond fund.

If we use the same bond fund for both portfolios, say BLV for both, then the balanced portfolio with just the total US stock market index fund dusted the alternative portfolio on pretty much all risk and return sample statistics:

https://www.portfoliovisualizer.com/bac ... tion4_2=40

This is not a surprise because the biased historical sample period selected for the backtest is one where US stocks outperformed non-US stocks, and the value risk premium was not rewarded.
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Re: Yes, you can diversify a market portfolio

Post by Beehave » Wed Jun 06, 2018 6:52 am

To me, this discussion is in the weeds, the cause being a peculiar use of the term "diversification."

Assume a buy-and-hold overall methodology. The core idea is that over time the market rises; but with jagged, sometimes severe bumps. The objective is to survive the bumps and profit from the long-term overall rise in market value. The "Classic Boglehead Strategy" ("CBS") is to invest in bonds, US stocks, and international stocks (ex-US) in some reasonable, fixed proportion, and then rebalance periodically with new or existing funds to maintain the balance. The investments in the CBS are typically total market (market-weighted) stock and bond mutual funds or ETFs.

What I glean from the OP's writing is that a barbell strategy for this implementation is somehow supposed to be superior to a market-weighted strategy. Specifically, it is suggested that adding tilts such as long-term bonds and small caps adds diversification and lowers risk. I would argue that these tilts are no different and no better (over time) than simply altering the stock-to-bond ratio of the overall CBS portfolio. These tilts add complexity, not diversification.

For simplicity's sake, suppose I'm holding a CBS-type portfolio comprised of 50% total global stock market and 50% total bond. Let's call this 50-50 portfolio "CBS1." Now, someone comes along and tells me I could increase diversification by tilting from my under-diversified, market-weighted CBS1 portfolio by adding a small cap fund, say for 10% of my stock holdings (which is 5% of my entire portfolio). My new portfolio is 5% small-cap, 45% total stock, and 50% bond. Let's call this new portfolio of holdings "Diversified Portfolio 1" ("DP1").

Without doing any fancy back-testing analysis and math, I'll make a bold assertion here:

THEOREM 1:
There is some CBS portfolio, call it "CBS2" with n% stock and 100-n% bonds that will perform exactly the same within any time-frame as DP1. My guess is that over a long time frame (for any reasonably long buy-and-hold lifetime), a 55% stock-45% bond portfolio would approximate DP1. In other words, for all intents and purposes, over time, CBS2 = DP1. One could back-test to see what the exact ratio is, but I'm not sure what the point would be because the task is purely dependent on the arbitrary decision that 10%-worth of small-cap was the right amount to add to CBS1 n the first place to "fix its diversification problem."

What the theorem states is that DP1 is not really diversifying, it is simply (in effect) tilting stocks in favor of bonds.

Now, to approximate the OP's complete scenario, suppose we attempt further diversification to DP1 by adding 10%-worth of long-term bonds to our total bond fund. We have now created DP2 which is 5% small cap, 45% total stock, 5% long bond, and 45% total bond fund.

THEOREM 2:
There is some CBS portfolio, call it "CBS3" which over time performs exactly the same as DP2. Here's the kicker, it sure looks to me as if CBS3 = CBS1.
In other words, adding the long-term bond tilt to the small-cap stock tilt has put you right back where you started from.

The same logic applies to any tilt. Add value stocks to the initial 50-50 portfolio so that it is 5% value, 45% total stock, and 50% bond? Probably almost exactly the same as a pure 47% stock, 53% bond portfolio. Etcetera.

To me, these tilts are just fancy ways of redistributing the stock-to-bond allocation. They are not diversifiers. Diversifiers are cash, commodities, real estate, TIPS, pensions, annuities, maintaining skills and employment over time, caring family, and so on. Diversification is important. Believing you are accomplishing it by tilting from a market-weighted base position is self-deception.

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Re: Yes, you can diversify a market portfolio

Post by vineviz » Wed Jun 06, 2018 7:07 am

jalbert wrote:
Wed Jun 06, 2018 2:56 am
I assume it was an oversight using VTI, the ETF share class of total US stock in one portfolio giving it an ER advantage over VTMSX (investor shares of same fund with higher ER) used in the other portfolio.
It wasn't an oversight, per se. Two of the ETFs I used in the diversified portfolio are only available in mutual fund form as Institutional shares at Vanguard, but the ETF versions are readily available. You're right that I could have used VTI/BND instead of VTSMX/VBMFX for the base fund, but the mutual fund version seemed to be more familiar to people here. It was definitely a judgement call that I debated. I ran VTI/BND also in my backtest, and it was within 5bps per year of VTSMX/VBMFX with no real improvement on any other measure. It's basically a wash.
jalbert wrote:
Wed Jun 06, 2018 2:56 am
Backtests can suffer from substantial sample bias and certainly a non-random 7-year sample of consecutive returns often is highly biased. I’m not taking sides on the two portfolios, but it turns out the higher return and Sharpe ratio in the portfolio you propose as more diversified arises from the use of long bonds, as it was a period where changes in interest rates favored longer durations, a significant sample bias in favor of the portfolio with the long bond fund.
On the one hand, although my diversified portfolio did actually outperform the VTSMX/VBMFX portfolio in the backtest you'll note that I am specifically NOT making a claim that a more diversified portfolio will outperform an under diversified one in absolute returns. Often it will not, even though in this period it did not. It is interesting that academic studies (using much more robust methodologies than I have here) have observed that rolling diversification-optimized portfolios have, perhaps counter-intuitively, regularly had higher ex-post Sharpe ratios than portfolios that were specifically mean-variance optimized. The theory of whether that is expected or might hold generally is beyond the scope of my post.

On the other hand, it's no accident that I view VBMFX/BND as a weak functional choice in a portfolio of risky assets: it doesn't take much risk. VBMFX turns out to have relatively little credit OR term factor exposure compared to other choices. It has less term risk than Vanguard Intmdt-Term Trs ETF (VGIT) and less credit risk than Vanguard Short-Term Corporate Bond ETF (VCSH). VCLT, BLV, VCIT, and BIV each has more combined credit & term factor exposure than VBMFX/BND has. This is one of the main reasons that the bond fund in the VTSMX/VBMFX portfolio contributes very little to the risk allocation in a portfolio with VTSMX/VTI: it's just not strong enough to provide the risk diversification that I think investors assume it is providing. You're definitely right than in, in another period, taking on such risk might hurt performance.

I didn't specify this, but I think the traditional notion of the capital allocation line (where you optimize a portfolio of risky assets, then adjust returns by holding or lending a riskless asset) makes a lot of sense.

I also think a lot of investors view the bond portion of their portfolio as simultaneously providing safety (stereotypically the role of cash or a riskless asset) while also balancing the risk of the equity portion (stereotypically the role of an uncorrelated risky asset). Perhaps a fund like VBMFX/BND succeeds at doing both for many investors, but in some ways it does neither especially well.

IMHO, VBMFX/BND is too risky to be considered "safe" but not risky enough to provide "risk balance". I'd rather see investors (including myself) do a better job of optimizing risk in their portfolio of risky assets while holding more truly riskless assets (if their circumstances drive it) outside of that portfolio.
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Re: Yes, you can diversify a market portfolio

Post by vineviz » Wed Jun 06, 2018 7:23 am

Beehave wrote:
Wed Jun 06, 2018 6:52 am
I glean from the OP's writing is that a barbell strategy for this implementation is somehow supposed to be superior to a market-weighted strategy. Specifically, it is suggested that adding tilts such as long-term bonds and small caps adds diversification and lowers risk. I would argue that these tilts are no different and no better (over time) than simply altering the stock-to-bond ratio of the overall CBS portfolio. These tilts add complexity, not diversification.
I'm definitely not advocating that any one approach is superior to any other. My primary goal, actually, was to address the false belief that the market portfolio CAN not be further diversified.

Whether it SHOULD be further diversified is, of course, a matter of investment policy that each individual investor must address for themselves. It's my opinion the they can do so only if they understand the risks they are taking, and my experience is that many investors don't fully understand them.

In my own investment planning, I definitely try to balance the behavioral goal of simplicity (for all the reasons that Bogleheads believe you should) with the financial goal of risk optimization and return optimization. I definitely agree that a typical three- or four-fund CBS portfolio approach is the appropriate level of complexity for many investors, even though I am comfortable taking on a little more complexity to make sure my portfolio reflects my beliefs and values about the financial markets.

[begin Soapbox] There is a TON of practical and theoretical research on risk allocation and portfolio optimization that most CBS practitioners will never want to look at, and I'm fine with that. At the same time, it does rankle a bit to have some of the most sophisticated discoveries in modern portfolio theory dismissed pejoratively as "adding tilts". [end Soapbox]
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Re: Yes, you can diversify a market portfolio

Post by nisiprius » Wed Jun 06, 2018 9:00 am

If I understand the concepts, I am thinking that this might be angels-on-the-head-of-a-pin stuff.

I chose two assets, not for realism or practical benefit, but to give a nice fat bulgy efficient frontier that illustrates the concepts well. I compared three ways to choose a stock/bond allocation: traditional MPT tangent portfolio, maximum Diversification Ratio®, and "risk parity."

[Note: the horizontal blue line is misplaced. The diagram will be corrected).
Image

This is a familiar efficient frontier diagram for stocks and long-term bonds, with the efficient frontier in black and the MPT optimum "tangent portfolio" in yellow. More details below. The tangent portfolio is about 43/57. (By the way, for stocks and bonds, over most time periods, it almost always has a lower stock allocation than 60/40 and the history and rationale behind 60/40 are mysterious).

I added some blue lines to show, conceptually, how you would maximize the diversification ratio.

The dotted blue line connects the green 100%-bond and red 100%-stock portfolios. The blue arrow points to the dot representing a 70/30 stocks/bonds portfolio. It's a horizontal line, and the other end touches the dotted blue line at a point representing the weighted average standard deviation and return of a 70/30 stocks/bonds portfolio.

Maximizing the diversification ratio involves sliding that horizontal blue arrow up and down to find the place where the ratio of the standard deviations, curved line divided by dotted blue line, is a minimum. I haven't done a precise calculation of the exact but it is between 40/60 and 42/58. Call it 41/59.

Now, for a third method, let's consider "risk parity." (This is a slippery concept. It's not clear whether it means "equalize the weighted standard deviations of any desired set of assets" or whether it means "equalize the weighted standard deviations of a set of four specific assets, including stocks, bonds, commodities and foreign currencies" or whether it means "whatever Ray Dalio's intuition guides him to do in one of his hedge funds.") But in any case, the portfolio that equalizes the weighted standard deviation of stocks and bonds, shown in purple, is about 41/59. Vineviz has instructed me that as a matter of fact risk parity is the same as maximum diversification ratio in the case of a portfolio of two assets.

So the question is this. Are the differences between
  • 41/59 (maximum diversification ratio),
  • 41/59 (risk parity), and
  • 43/57 (MPT tangent portfolio
a big deal?

The real problem is that even (say) Y - 1926 years of past data only give us the roughest indication of what will happen over the next thirty years. In general, how confident can we ever be that a forward-looking prediction of the best allocation, using method X is really going to be much better than one using method Y?

How much conviction can one muster that one is any better than the others... and how much better is it? Enough to give reasonable assurance that the advantage will actually show up in the real world, using mutual funds and ETFs available to the ordinary retail investor, over the next thirty years?

All of them fit with the general idea of having decent allocations of stocks and bonds, inside the range 75/25 to 25/75. Notice that the efficient frontier and similar curves are gentle around the optimum; the optimum is broad and shallow, and small departures from the optimum produce only small degradations on return.

Vanguard provides only four different choices of asset allocations in their LifeStrategy funds: 80/20, 60/40, 40/60, 20/80. That is, they don't think investors can calibrate the allocation then want to within better than ±10%.

If you believe everything is perfectly precise and predictable than of course you can compound the small difference between Your Favorite Trademarked® method and Some Boring Traditional method out for thirty years and make it look big, but I don't believe that.

For example, with different endpoints, the MPT tangent portfolio shifts around by much more than the differences between those three allocations. What you actually experience over some future unknown time period is like to be far more influenced by the luck of that time period than the method of choosing an allocation. For 1985-2000 (sixteen years), the tangent portfolio was 79/21; for 1993-2008, 23/77. I'm pretty sure that the variations between the "maximum diversification ratio" allocations, and the "risk parity" allocations, would be large, too.

Notes on the diagram:

X axis is standard deviation (a measure of volatility or risk), Y axis is return.
The green dot is bonds as exemplified by the Vanguard Long-Term Treasury Fund.
The red dot is stocks, as exemplified by the Vanguard 500 index fund.
The time period is 1985-2017.
The black curve is the efficient frontier and the dots mark portfolios that are, top to bottom, 90/10, 80/20, 70/30, and so forth.
The blue dot is the riskless asset.
The red line is the set of risk-return combinations available using only stocks.
The yellow dot is the traditional MPT tangent portfolio, optimum under one set of assumptions, with the angle between the red and yellow lines representing the degree of improvement.
Last edited by nisiprius on Wed Jun 06, 2018 1:49 pm, edited 3 times in total.
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Re: Yes, you can diversify a market portfolio

Post by vineviz » Wed Jun 06, 2018 10:53 am

nisiprius wrote:
Wed Jun 06, 2018 9:00 am
If I understand the concepts, I am thinking that this might be angels-on-the-head-of-a-pin stuff.
I’m out sailing today, but I do want to briefly observe that there must be an error in your calculations somewhere.

By construction, with a two-asset portfolio the formulas for maximum diversification ratio and risk parity must produce the same portfolio in which the asset weights are simply the inverse of their relative volatility.

If you’re getting different portfolios from those approaches then there has to be an error.

With three or more assets the portfolios diverge.

In some conditions, especially ones approaching the textbook example you present, some of those portfolios will approximate the mean-variance optimal portfolio. Quite often they do not, however.
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Re: Yes, you can diversify a market portfolio

Post by Epsilon Delta » Wed Jun 06, 2018 11:13 am

Beehave wrote:
Wed Jun 06, 2018 6:52 am

For simplicity's sake, suppose I'm holding a CBS-type portfolio comprised of 50% total global stock market and 50% total bond. Let's call this 50-50 portfolio "CBS1." Now, someone comes along and tells me I could increase diversification by tilting from my under-diversified, market-weighted CBS1 portfolio by adding a small cap fund, say for 10% of my stock holdings (which is 5% of my entire portfolio). My new portfolio is 5% small-cap, 45% total stock, and 50% bond. Let's call this new portfolio of holdings "Diversified Portfolio 1" ("DP1").

Without doing any fancy back-testing analysis and math, I'll make a bold assertion here:

THEOREM 1:
There is some CBS portfolio, call it "CBS2" with n% stock and 100-n% bonds that will perform exactly the same within any time-frame as DP1.
I believe that the factors tilters, and their empirical evidence, explicitly claim theorem 1 is false.

They say that an appropriately tilted portfolio has a higher Sharpe ratio than the market portfolio. That means that given any CBS-type portfolio there is a tilted portfolio that is better for both expected return and variance. Better on both dimensions is not performing exactly the same. Since changing the CBS portfolio trades off s.d. and expected return (improving one at the expense of the other) there can be no other CBS-type portfolio that is exactly the same.

I square this by claiming there are multiple dimensions of risk. One is s.d. (or at least s.d. is a good proxy for it) others are unknown. When we squeeze s.d. by tilting some unknown form of risk increases. Having different amounts of different risks is not exactly the same. It is a matter of taste which risk is preferred.

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Re: Yes, you can diversify a market portfolio

Post by David Jay » Wed Jun 06, 2018 11:22 am

Epsilon Delta wrote:
Wed Jun 06, 2018 11:13 am
I square this by claiming there are multiple dimensions of risk. One is s.d. (or at least s.d. is a good proxy for it) others are unknown. When we squeeze s.d. by tilting some unknown form of risk increases. Having different amounts of different risks is not exactly the same. It is a matter of taste which risk is preferred.
As a process engineer, I have always been concerned that using SD as a proxy for "risk" (as is commonly done in financial analysis) severely limits our understanding of risk. On the order of "assume a spherical cow".
Prediction is very difficult, especially about the future - Niels Bohr | To get the "risk premium", you really do have to take the risk - nisiprius

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Re: Yes, you can diversify a market portfolio

Post by nisiprius » Wed Jun 06, 2018 12:05 pm

David Jay wrote:
Wed Jun 06, 2018 11:22 am
Epsilon Delta wrote:
Wed Jun 06, 2018 11:13 am
I square this by claiming there are multiple dimensions of risk. One is s.d. (or at least s.d. is a good proxy for it) others are unknown. When we squeeze s.d. by tilting some unknown form of risk increases. Having different amounts of different risks is not exactly the same. It is a matter of taste which risk is preferred.
As a process engineer, I have always been concerned that using SD as a proxy for "risk" (as is commonly done in financial analysis) severely limits our understanding of risk. On the order of "assume a spherical cow".
Demonstrate to me that there are real-world situations in which standard deviation produces seriously different results from other reasonable differences of other forms of "risk." For example, many people (including myself!) feel that maximum drawdown is a more useful definition of "risk" than standard deviation--but when I selected thirteen mutual funds, representing asset classes with as wide a range of risk as I could think of, "risk" as measured by standard deviation was a very good guide to "risk" as measured by maximum drawdown. In short, risk is risk and most forms of risk go along with other forms of risk... I think.

Dec 2004 - Dec 2017
Standard deviation, max drawdown

St. dev. Max drawdown
CASHX 0.50% 0.00%
VBMFX 3.22% -3.99%
VWEHX 8.19% -28.90%
FNMIX 9.59% -26.04%
VTSMX 14.18% -50.89%
VJPNX 16.80% -54.96%
VGTSX 17.65% -58.50%
LMVTX 18.04% -68.91%
GLD 18.21% -42.91%
DFSTX 18.79% -55.02%
PCRIX 19.87% -63.49%
VEIEX 22.29% -62.70%
ULPIX 27.79% -81.36%

Image
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garlandwhizzer
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Re: Yes, you can diversify a market portfolio

Post by garlandwhizzer » Wed Jun 06, 2018 12:13 pm

There is a big difference between risk and volatility. Academic theory can reduce volatility by optimizing diversification across various equity factors, exposure to INTL equity and bonds, using differing bond types and asset types in a mixture to optimize Sharpe ratio and reduce portfolio volatility based on academic backtesting. Risk is something else. It becomes most important when there is market panic in a severe crash like 1929-32 and recently 2007-9. When risk/panic rises to these massive levels, essentially all assets except high quality bonds trend toward a correlation of 100%. Beta, small, value, momentum, quality, even low volatility, as well as all INTL equity tend to tank simultaneously in that circumstance. Hiding out in equity diversifying factors or INTL simply doesn't work when you need it most. Value gets killed during crashes usually more so than beta. Momentum gets utterly destroyed in the recovery if long/short MOM is used. What has worked in these circumstances is one thing: US Treasuries and highest quality credit instruments. The most important aspect of severe risk control is quality bond exposure. Period. Other additions are optional.

There are some asset classes (commodities, MMF) which help a portfolio during an long term inflationary disaster but they come at a price, decreased expected long term portfolio returns. A diversified portfolio such as the one described may make the portfolio ride smoother with reduced volatility, but when it really matters in the case of a deep recession or depression, it won't bail you out. Likewise protection from inflation by diversifying into commodities for example does give some inflationary protection but it reduces long term portfolio returns. The things that protect you from recession/depression (quality bonds) get killed during inflation and the things that protect you from inflation (commodities, MMF) lose big time when the opposite (recession/depression) occurs. Each additional piece added to portfolio construction to reduce volatility comes at an price. Whether it's worth it or not is up to individual investor rather than some academic study that claims to have it all figured out.

Garland Whizzer

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nisiprius
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Re: Yes, you can diversify a market portfolio

Post by nisiprius » Wed Jun 06, 2018 12:14 pm

vineviz wrote:
Wed Jun 06, 2018 10:53 am
nisiprius wrote:
Wed Jun 06, 2018 9:00 am
If I understand the concepts, I am thinking that this might be angels-on-the-head-of-a-pin stuff.
I’m out sailing today, but I do want to briefly observe that there must be an error in your calculations somewhere.
I didn't do calculations, I judged by eye. I was wrong. You are correct. The maximum Diversification Ratio® for the data I plotted occurs somewhere between 42/58 and 40/60.

I've edited my post above accordingly.

However, it strengthens my point: in this (reasonable) case, the differences between three different methods of choosing an allocation lead to differences in allocation that are unlikely to be important. In order to make a case for diversification ratio being of practical importance, it seems to me that you need to show not only that it leads to a different allocation from MPT, but that it is quite different... and that the results of that difference in allocation are almost certain to be quite different... and that the difference is almost certain to be an improvement.

This is the corrected diagram:
Image

And the calculation of diversification ratio for closely-spaced allocations:

σ1 = 16.560, σ2 = 11.715
100/0, DR = 1.000, weighted = 16.560, actual = 16.560
98/2, DR = 0.985, weighted = 16.463, actual = 16.215
...
50/50, DR = 0.695, weighted = 14.137, actual = 9.829
48/52, DR = 0.691, weighted = 14.040, actual = 9.698
46/54, DR = 0.687, weighted = 13.943, actual = 9.583
44/56, DR = 0.685, weighted = 13.847, actual = 9.485
42/58, DR = 0.684, weighted = 13.750, actual = 9.405
40/60, DR = 0.684, weighted = 13.653, actual = 9.342
38/62, DR = 0.686, weighted = 13.556, actual = 9.298
36/64, DR = 0.689, weighted = 13.459, actual = 9.273
...
2/98, DR = 0.971, weighted = 11.812, actual = 11.464
0/100, DR = 1.000, weighted = 11.715, actual = 11.715
Last edited by nisiprius on Wed Jun 06, 2018 1:58 pm, edited 6 times in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

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David Jay
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Re: Yes, you can diversify a market portfolio

Post by David Jay » Wed Jun 06, 2018 12:16 pm

nisiprius wrote:
Wed Jun 06, 2018 12:05 pm
David Jay wrote:
Wed Jun 06, 2018 11:22 am
Epsilon Delta wrote:
Wed Jun 06, 2018 11:13 am
I square this by claiming there are multiple dimensions of risk. One is s.d. (or at least s.d. is a good proxy for it) others are unknown. When we squeeze s.d. by tilting some unknown form of risk increases. Having different amounts of different risks is not exactly the same. It is a matter of taste which risk is preferred.
As a process engineer, I have always been concerned that using SD as a proxy for "risk" (as is commonly done in financial analysis) severely limits our understanding of risk. On the order of "assume a spherical cow".
Demonstrate to me that there are real-world situations in which standard deviation produces seriously different results from other reasonable differences of other forms of "risk." For example, many people (including myself!) feel that maximum drawdown is a more useful definition of "risk" than standard deviation--but when I selected thirteen mutual funds, representing asset classes with as wide a range of risk as I could think of, "risk" as measured by standard deviation was a very good guide to "risk" as measured by maximum drawdown. In short, risk is risk and most forms of risk go along with other forms of risk... I think.

Dec 2004 - Dec 2017
Standard deviation, max drawdown

St. dev. Max drawdown
CASHX 0.50% 0.00%
VBMFX 3.22% -3.99%
VWEHX 8.19% -28.90%
FNMIX 9.59% -26.04%
VTSMX 14.18% -50.89%
VJPNX 16.80% -54.96%
VGTSX 17.65% -58.50%
LMVTX 18.04% -68.91%
GLD 18.21% -42.91%
DFSTX 18.79% -55.02%
PCRIX 19.87% -63.49%
VEIEX 22.29% -62.70%
ULPIX 27.79% -81.36%

Image
In my post, I was leaning more towards conceptual categories of risk (more like Bernstein's "Deep Risk"), not mathematical representations of risk.
Prediction is very difficult, especially about the future - Niels Bohr | To get the "risk premium", you really do have to take the risk - nisiprius

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David Jay
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Re: Yes, you can diversify a market portfolio

Post by David Jay » Wed Jun 06, 2018 12:20 pm

garlandwhizzer wrote:
Wed Jun 06, 2018 12:13 pm
There are some asset classes (commodities, MMF) which help a portfolio during an long term inflationary disaster...
Bernstein suggests that equities may be the best tool against hyper-inflation (ne: "Deep Risk").
Prediction is very difficult, especially about the future - Niels Bohr | To get the "risk premium", you really do have to take the risk - nisiprius

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Re: Yes, you can diversify a market portfolio

Post by Beehave » Wed Jun 06, 2018 12:53 pm

vineviz wrote:
Wed Jun 06, 2018 7:23 am
Beehave wrote:
Wed Jun 06, 2018 6:52 am
I glean from the OP's writing is that a barbell strategy for this implementation is somehow supposed to be superior to a market-weighted strategy. Specifically, it is suggested that adding tilts such as long-term bonds and small caps adds diversification and lowers risk. I would argue that these tilts are no different and no better (over time) than simply altering the stock-to-bond ratio of the overall CBS portfolio. These tilts add complexity, not diversification.
I'm definitely not advocating that any one approach is superior to any other. My primary goal, actually, was to address the false belief that the market portfolio CAN not be further diversified.

Whether it SHOULD be further diversified is, of course, a matter of investment policy that each individual investor must address for themselves. It's my opinion the they can do so only if they understand the risks they are taking, and my experience is that many investors don't fully understand them.

In my own investment planning, I definitely try to balance the behavioral goal of simplicity (for all the reasons that Bogleheads believe you should) with the financial goal of risk optimization and return optimization. I definitely agree that a typical three- or four-fund CBS portfolio approach is the appropriate level of complexity for many investors, even though I am comfortable taking on a little more complexity to make sure my portfolio reflects my beliefs and values about the financial markets.

[begin Soapbox] There is a TON of practical and theoretical research on risk allocation and portfolio optimization that most CBS practitioners will never want to look at, and I'm fine with that. At the same time, it does rankle a bit to have some of the most sophisticated discoveries in modern portfolio theory dismissed pejoratively as "adding tilts". [end Soapbox]
I was completely unfamiliar with portfolio theory before reading your reply. So thank you for your thought-provoking original post and now your equally informative reply. My response, I guess, was not directed at a portfolio theory-based statement. In portfolio theory, as I now understand it (and did not before) "risk" is a measure of volatility during a span of time. For me, my understanding of risk is "given some particular starting time (usually, right now), what is the chance that I will be unhappy with my assets at some (possibly unspecified) future time? " Now, it is possible that the volatility of the assets I hold will affect my happiness. I might become depressed if they are low. I might sell them out of fear because of volatility. In ththose cases I guess I am concerned with volatility, so it is a risk factor. But then again, I could be perfectly content with the volatility. If a stock or fund is gyrating upwards, the risk measured by its degree of deviation from a norm that it is exceeding is something I am quite happy to experience. So in general, my concept of risk is quite different than what portfolio theory portrays as risk.

One net effect of this is that my "tilt" comment is referring to my concept of risk -- possibly not portfolio theory's measure. So, no insult or belittling comment is or was intended. I did not even really know what portfolio theory was when I write it! :oops: :sharebeer

What I want to think about now are the examples I constructed for the sake of argument suggesting the similarity between different pairs of portfolios; for example a portfolio allocation comprised of 15% small stock, 35% total stock, 15% long bond, 35% total bond fund versus, say, the classic 60 stock/40 bond market-weighted portfolio allocation. I suspect that over the long term their outcomes will be similar(?), but now understanding a bit about portfolio another question would be that for whatever ratio does make the outcomes similar over a longish duration, which would be the more volatile and by how much during that timeframe?

And again, I still need to wrestle with the concept of risk as volatility. I'm not sure it really resonates with me. If the long-term trend is up, volatility with rebalancing in the interim is my friend - - at least I think so!

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Re: Yes, you can diversify a market portfolio

Post by jalbert » Wed Jun 06, 2018 2:00 pm

A 7 year biased sample doesn’t tell us which portfolio is better by any measure, and it is far from a trivial matter to devise an experimental design to tell which is better by whatever measure you prefer.

I don’t particularly like long duration bonds unless they are being used to match to the duration of liabilities. But in that case, there is no term risk because the liabilities and bond portfolio move in lockstep as rates rise and fall.

The problem with long duration bonds for investment portfolios is that the institutional investors who drive the market for them are insurance companies and pension funds and other players who need to cover long duration liabilities. Because they are matching duration to liabilities, they are not taking term risk, so they do not require to be compensated for term exposure. That is why the yield curve slope flattens so much as you move out on the term structure (even when the yield curve has average or above average slope by historical standards).

If institutional buyers of 20 year bonds expected to be fully compensated for inflation and term risk at that duration, yields would be much higher at those durations.

If you use diversification ratio rather than some measure of risk-adjusted return as a measuring tool for a diversification benefit it would seem to be imperative to ensure risk was adequately rewarded for the individual asset classes. If you use risk-adjusted return, a poorly rewarded asset class would be a drag on portfolio risk-adjusted return and the diversification benefit would have to be compelling enough to overcome the low risk-adjusted return of the individual asset. But diversification ratio by itself won’t protect you from choosing an individual asset class in which risk is not adeqately rewarded, such as long bonds.
Last edited by jalbert on Wed Jun 06, 2018 2:24 pm, edited 1 time in total.
Risk is not a guarantor of return.

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Re: Yes, you can diversify a market portfolio

Post by Epsilon Delta » Wed Jun 06, 2018 2:20 pm

nisiprius wrote:
Wed Jun 06, 2018 12:05 pm
David Jay wrote:
Wed Jun 06, 2018 11:22 am
Epsilon Delta wrote:
Wed Jun 06, 2018 11:13 am
I square this by claiming there are multiple dimensions of risk. One is s.d. (or at least s.d. is a good proxy for it) others are unknown. When we squeeze s.d. by tilting some unknown form of risk increases. Having different amounts of different risks is not exactly the same. It is a matter of taste which risk is preferred.
As a process engineer, I have always been concerned that using SD as a proxy for "risk" (as is commonly done in financial analysis) severely limits our understanding of risk. On the order of "assume a spherical cow".
Demonstrate to me that there are real-world situations in which standard deviation produces seriously different results from other reasonable differences of other forms of "risk."
First I'm not sure if David was agreeing with me, but I agree with him. But note that spherical cows can be useful when considering things that cows and spheres share, like volume and surface area. They are less useful if you care about things like legs.

Second people throw around s.d. as if it completely said what they are talking about. It doesn't. You have to say s.d. of what. Daily returns? Yearly returns? Decade returns? It is open if these are the same or different things. Is there a central tendency/reversion to the mean/stationary process? We don't have enough data so it's mostly assumption. MPT itself only deals with a single period, lots of people carelessly use yearly results to optimize over decades or assume that optimizing for a particular fixed time horizon optimizes for all time horizons.

People can care about having enough money next year and having enough money in thirty years. These are different formulations of risk, and we don't know if they are the same thing.

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Re: Yes, you can diversify a market portfolio

Post by SimpleGift » Wed Jun 06, 2018 2:54 pm

Beehave wrote:
Wed Jun 06, 2018 12:53 pm
What I want to think about now are the examples I constructed for the sake of argument suggesting the similarity between different pairs of portfolios; for example a portfolio allocation comprised of 15% small stock, 35% total stock, 15% long bond, 35% total bond fund versus, say, the classic 60 stock/40 bond market-weighted portfolio allocation.
Perhaps a helpful way to frame and think about your question is in terms of classic portfolio selection theory (as best I understand it!). In the two charts below, we plot all of the combinations of risky assets, with their expected returns on the vertical axis and their volatilities (standard deviations) on the horizontal axis.

The blue line is the efficient frontier, which represent those combinations of risky assets with the highest expected return for each given level of volatility (standard deviation). Clearly, there is a continuum of many efficient portfolios — but for each individual investor, there's one optimal portfolio, determined by one's individual risk/return preference.

The market portfolio is simply that optimal portfolio on the continuum that represents the choices of all investors in aggregate.
Simplified in the Forum vernacular, there are many efficient roads to Dublin, but one optimal route for each traveler. Hope this helps!
Cordially, Todd

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patrick013
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Re: Yes, you can diversify a market portfolio

Post by patrick013 » Wed Jun 06, 2018 3:13 pm

I always thought when bonds were involved the efficient frontier
was the guiding number. Or at least some concern with lowering
beta thru bond portfolio inclusion.

Several studies have reached the conclusion that in an all stock
portfolio around 30 stocks there is a definite reduction in volatility,
and around 1000 stocks there is no further statistical reduction in
volatility, when the portfolio is equal weighted. I think it is hard to
construct a portfolio with negative correlations and good returns as
the lowered correlations always seem to decrease return as well.

So to get risk/reward it seems plausible to surrender to market beta
and weight mid and small cap at least to give them a chance at
contributing to total return while lowering portfolio beta with bonds.
But it looks like everybody has a different way of doing the same thing.
Last edited by patrick013 on Wed Jun 06, 2018 3:17 pm, edited 1 time in total.
age in bonds, buy-and-hold, 10 year business cycle

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Re: Yes, you can diversify a market portfolio

Post by DaufuskieNate » Wed Jun 06, 2018 3:14 pm

Totally agree with SimpleGift's observation about individual risk preferences. Would only add that these preferences can be determined by understanding one's need, willingness and ability to take risk.

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