You've made some great improvements.Barry Barnitz wrote:Hi all:

I have fleshed out the wiki page : Low Volatility Index returns - Bogleheads.

regards,

## Equal weighting or market cap weighting?

### Re: Equal weighting or market cap weighting?

### Re: Equal weighting or market cap weighting?

Barry ..

Very nice.

Some quick observations

Under “Portfolio Constraints …. “ for non optimized I think that perhaps it should read, “typically none; may be constrained by the number of holdings” (100 for SPLV, 200 for IDLV, etc.) Perhaps an asterisk or something.

For US data MSCI data is available on fact sheets into the 1990s. (adds a few years to data) ... the fact sheets are linked.

You miss 2012 data for SP BMI low vol, the data is found in the comparative table at bottom.

EDIT: I re-read the wiki - you did this. I think it would be useful to post some information on sector concentration (Specifically consumer defensive / utilities for LV indexes). IDK. Perhaps this is related / nearby the Portfolio Constraint listed above.I would be curious what Akiva thinks on this last point.

Very nice.

Some quick observations

Under “Portfolio Constraints …. “ for non optimized I think that perhaps it should read, “typically none; may be constrained by the number of holdings” (100 for SPLV, 200 for IDLV, etc.) Perhaps an asterisk or something.

For US data MSCI data is available on fact sheets into the 1990s. (adds a few years to data) ... the fact sheets are linked.

You miss 2012 data for SP BMI low vol, the data is found in the comparative table at bottom.

EDIT: I re-read the wiki - you did this. I think it would be useful to post some information on sector concentration (Specifically consumer defensive / utilities for LV indexes). IDK. Perhaps this is related / nearby the Portfolio Constraint listed above.I would be curious what Akiva thinks on this last point.

Maximize Diversification - Minimize Costs - Avoid Lotteries

### Re: Equal weighting or market cap weighting?

Barry -

Some fine point errors.

S&P MV index in 2002 is 2.52 mot 25.2 (I was questioning the dispersion).

The dispersion figures for international is wrong pre-2001

Global makets annual return data in the comparisons is incomplete

Some fine point errors.

S&P MV index in 2002 is 2.52 mot 25.2 (I was questioning the dispersion).

The dispersion figures for international is wrong pre-2001

Global makets annual return data in the comparisons is incomplete

Maximize Diversification - Minimize Costs - Avoid Lotteries

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### Re: Equal weighting or market cap weighting?

Thanks steve r!

I have corrected table typos and included a footnote chart showing the current sector breakdown of the S&P 500 LV index.

Note that I was able to manually extract the 2012 S&P BMI dev. LV index total return from the live 5yr chart on the S&P page. However the comparative index S&P tracks is quirky (dev. ex-US ex S. Korea) so I will have to wait for the next update of the factsheet for this data.

Low Volatility Index returns - Bogleheads

regards,

I have corrected table typos and included a footnote chart showing the current sector breakdown of the S&P 500 LV index.

Note that I was able to manually extract the 2012 S&P BMI dev. LV index total return from the live 5yr chart on the S&P page. However the comparative index S&P tracks is quirky (dev. ex-US ex S. Korea) so I will have to wait for the next update of the factsheet for this data.

Low Volatility Index returns - Bogleheads

regards,

### Re: Equal weighting or market cap weighting?

Nice job Barry. The "average" is also off on the int. developed column when only one thing is being averaged. Do consider adding 1999 and 2000 for the US MCI MV .. the data is on fact sheets.

On an investing theory side, I had hoped spelling the data out like this would help me develop a personal preference for LV or MV approach. It has not.

An apples to apples comparsison over 22 years with S&P

LV 8.92 (sd. dev 13.43)

MV 9.93 (sd. dev 13.77) ... this advantage disappears with MSCI MV data plugged in in the last 11 years. These two MV stategies seem to optimize with largely the same contraints. The MCSI USA index and SP500 are very similar. Intuitively I like the MV approach w/ constraints. That said, it does not seem to result in minimum volatility, perhaps because of the contstraints.

On the international side MSCI does have notably lower volatility when compared to S&P DI, but with the traditional tradeoff in terms of lower return. Either way, eleven years of data is not enough.

More perplexing, the Russell-Axioma approach is similar to S&P LV, and it had even lower volatility and return - for BOTH U.S. and International.

Any method does reduce volatility. Why the LV returns are so different for Russell and S&P is beyond me. Likewise for the return differences between MSCI MV and S&P MV approaches.

Still learning!

On an investing theory side, I had hoped spelling the data out like this would help me develop a personal preference for LV or MV approach. It has not.

An apples to apples comparsison over 22 years with S&P

LV 8.92 (sd. dev 13.43)

MV 9.93 (sd. dev 13.77) ... this advantage disappears with MSCI MV data plugged in in the last 11 years. These two MV stategies seem to optimize with largely the same contraints. The MCSI USA index and SP500 are very similar. Intuitively I like the MV approach w/ constraints. That said, it does not seem to result in minimum volatility, perhaps because of the contstraints.

On the international side MSCI does have notably lower volatility when compared to S&P DI, but with the traditional tradeoff in terms of lower return. Either way, eleven years of data is not enough.

More perplexing, the Russell-Axioma approach is similar to S&P LV, and it had even lower volatility and return - for BOTH U.S. and International.

Any method does reduce volatility. Why the LV returns are so different for Russell and S&P is beyond me. Likewise for the return differences between MSCI MV and S&P MV approaches.

Still learning!

Maximize Diversification - Minimize Costs - Avoid Lotteries

### Re: Equal weighting or market cap weighting?

One little nit:Akiva wrote:You've made some great improvements.Barry Barnitz wrote:Hi all:

I have fleshed out the wiki page : Low Volatility Index returns - Bogleheads.

regards,

At the top comparing the two strategies, you say that MSCI does "mean-variance optimization". Strictly speaking they just do variance minimization. (They find the left-most point on the efficient frontier, and operation that doesn't require forecasting returns. This is not the same thing as assuming that the stocks all have the same return. If the stocks were perfectly uncorrelated, it would the same as finding the geometrically optimal portfolio under the assumption that they all have the same sharpe ratio, but since they share risk-factor exposure, I'm not sure this holds with what they are actually doing. I think that conceptually they are equalizing the dollar value of your risk exposure to each of the factors in their risk model, most of which are known to be "unpriced" risk factors.)

And one suggestion:

At the bottom of each table listing the returns of each index, you have 5 and 10 year standard deviations. I think you should also include 5 and 10 year geometric average returns and then 5 and 10 year sharpe ratios using that data. (The point is to be able to compare them to the regular index rather than against each other as you allow for further down on the page. You can then see if the returns are more or less than the historic 2% negative return to residual volatility would account for and how much or little the volatility reduction helps your Sharpe ratio.)

I think some downside-risk metric like Sortino or SDR Sharpe would interesting as well because these indexes might have different skew than the underlying market-cap ones. My personal preference would be to use SDR Sharpe over Sortino because SDR Sharpe and normal Sharpe are on the same scale and can thus be compared. So it is easy to see the effect of skew and whether the skew is positive or negative.

I don't know what the overall skew situation is, but for the MSCI ACWI the skew is basically the same. The geometric sharpe ratio is ~.5 for the MV and ~.23 for the underlying (MV is 114% better). SDR Sharpe is ~.43 for the MV and .2 for the underlying (MV is 116% better). So both returns series have negative skew and roughly the same amount of it.

FWIW, the relative improvement here is absolutely phenomenal and much better than I would have intuitively expected. Even if you go in and subtract 2% from all of the MV indexes returns (and thus assume that the volatility factor will have 0 expected returns going forward). The lower volatility of the MV index still results in your geometric returns being slightly higher and your Sharpe ratio being about 60% better.

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### Re: Equal weighting or market cap weighting?

Thanks akiva:

I wordsmithed the summary description to

regards,

I wordsmithed the summary description to

The minimum volatility index seeks to lower volatility by using an optimization of variance minimization. The low volatility indexes employ a non-optimized approach based on historical volatities.

Unfortunately, adding all of this is way beyond my rather feeble spreadsheet skills. However, the spreadsheets are open for anyone to edit, so all that is necessary is for the skilled to select "view the spreadsheet in google" and begin editing. I have begun to provide CAGR data.And one suggestion:

At the bottom of each table listing the returns of each index, you have 5 and 10 year standard deviations. I think you should also include 5 and 10 year geometric average returns and then 5 and 10 year sharpe ratios using that data. (The point is to be able to compare them to the regular index rather than against each other as you allow for further down on the page. You can then see if the returns are more or less than the historic 2% negative return to residual volatility would account for and how much or little the volatility reduction helps your Sharpe ratio.)

I think some downside-risk metric like Sortino or SDR Sharpe would interesting as well because these indexes might have different skew than the underlying market-cap ones. My personal preference would be to use SDR Sharpe over Sortino because SDR Sharpe and normal Sharpe are on the same scale and can thus be compared. So it is easy to see the effect of skew and whether the skew is positive or negative.

regards,

### Re: Equal weighting or market cap weighting?

+1 (for both LV and MV approach).Akiva wrote:FWIW, the relative improvement here is absolutely phenomenal and much better than I would have intuitively expected. Even if you go in and subtract 2% from all of the MV indexes returns (and thus assume that the volatility factor will have 0 expected returns going forward). The lower volatility of the MV index still results in your geometric returns being slightly higher and your Sharpe ratio being about 60% better.

I would feel this way even if geometric return was slighltly lower (for a 30 +/1 reduction in volatility).

The fact that various researchers have found this back to 1926 (http://business.missouri.edu/yanx/resea ... 261962.pdf, in developed market, in emerging markets, in U.S. mid caps and in U.S. small caps (Wiki) is also

**absolutely phenomenal!**

The same cannot be said about equal weighting (topic on this thread), you may find higher returns, but also find higher risk than cap weight. Other index weightings (Risk weighting, fundamental, et.) may have higher returns, but certainly have higher volatility/risk than LV/MV weighting.

Maximize Diversification - Minimize Costs - Avoid Lotteries

### Re: Equal weighting or market cap weighting?

I'd do it for you, but I don't know how to create an area of the spreadsheet for "background calculations" that will feed into the numbers but not display on the webpage. Can you explain how to do this?Barry Barnitz wrote:Unfortunately, adding all of this is way beyond my rather feeble spreadsheet skills. However, the spreadsheets are open for anyone to edit, so all that is necessary is for the skilled to select "view the spreadsheet in google" and begin editing. I have begun to provide CAGR data.

regards,

### Re: Equal weighting or market cap weighting?

**1988 to 2012**MCSI Wold MV and other weight schemes data

A makeshift efficient frontier would clearly include MV ... (& World Quality Index and some others). I am convinced you can minimize volatility. I am less convinced that various weight schemes can predictably increase returns. Maybe, but maybe not.

Maximize Diversification - Minimize Costs - Avoid Lotteries

### Re: Equal weighting or market cap weighting?

As per Morningstar, VO tracks RSP very closely. When you do an equal weighting, it ends up like a mid cap fund...logical.

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### Re: Equal weighting or market cap weighting?

I have provided CAGR data for the tables. As far as I can surmise, the best way to handle computations that can't be derived by a single cell formula on the front tab is to create a new tab within the sheet (in google this is the + sign in the left bottom margin, which says "add sheet" when you hover your mouse over it) and then perform the necessary computations. The final computed cells can then be placed on the front tab by referencing the cell.Akiva wrote:I'd do it for you, but I don't know how to create an area of the spreadsheet for "background calculations" that will feed into the numbers but not display on the webpage. Can you explain how to do this?Barry Barnitz wrote:Unfortunately, adding all of this is way beyond my rather feeble spreadsheet skills. However, the spreadsheets are open for anyone to edit, so all that is necessary is for the skilled to select "view the spreadsheet in google" and begin editing. I have begun to provide CAGR data.

regards,

By the way, for ease of comparison, I added a chart of S&P 500 Index sector holdings to the footnote that provides a chart of the S&P 500 LV Index.

regards,

### Re: Equal weighting or market cap weighting?

Re: this chart.steve r wrote:1988 to 2012MCSI Wold MV and other weight schemes data

AFAIK, the value weighted and "quality" indexes are totally proprietary and thus really aren't distinguishable from some scheme that tried to find undervalued / overvalued stocks and adjusts the weights accordingly. I can't find anything scholarly on the "quality" index, but fundamental indexing has been discussed in this thread and it seems to me that, like equally weighted indexing, the differences are largely caused by small / value tilting. Whatever isn't explained by those factors could probably be accounted for by reference to the industries it over/under weights relative to the cap weighted index and how they've done recently. (The same issues apply to the risk weighted index.) The where fundamental indexing could arguably be better is that it doesn't necessarily keep your small/value tilts constant, and if it can vary the exposure in a way that lines up with positive and negative returns to those factors, then you'd come out ahead.

Anyway, in case anyone is still interested in fundamental indexing, here is a longish paper on how it works:

http://papers.ssrn.com/sol3/papers.cfm? ... _id=604842

The academically interesting thing is that MV doesn't have a size or value tilt and still manages to to dominate (be both up and to the left of) the market cap weighted one. So presumably the efficient frontier would include MV, value, and small-cap portfolios at a minimum. (Though an interesting open question is whether you can do the MV thing in a way that incorporates a value and small-cap tilt.)A makeshift efficient frontier would clearly include MV ...

I'll try to add those calculations this afternoon.Barry Barnitz wrote:As far as I can surmise, the best way to handle computations that can't be derived by a single cell formula on the front tab is to create a new tab within the sheet (in google this is the + sign in the left bottom margin, which says "add sheet" when you hover your mouse over it) and then perform the necessary computations. The final computed cells can then be placed on the front tab by referencing the cell.

### Re: Equal weighting or market cap weighting?

It would certainly include MV on an Ef. Frontier ... the small and value has been debated on this site to death ... Personally, I am more confident on the ability to produce a low volatility basket of stocks than the ability to outperform with various tilts. I looked at the data, looked at VG small value versus small growth, looked further back in time, read the threads, etc ... just not convinced. I am also not convinced MV or LV will lead to higher returns.Akiva wrote:[The academically interesting thing is that MV doesn't have a size or value tilt and still manages to to dominate (be both up and to the left of) the market cap weighted one. So presumably the efficient frontier would include MV, value, and small-cap portfolios at a minimum. (Though an interesting open question is whether you can do the MV thing in a way that incorporates a value and small-cap tilt.)

I am only convinced MV or LV will lead to lower volatility. I certainly hope for higher returns continuing in the future but do not expect them. Bought some ACWV ... plan to buy more in March.

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### Re: Equal weighting or market cap weighting?

I believe that from Modern Portfolio Theory (MPT) there is a portfolio known as the Tangency Portfolio which lies on the efficient frontier and which has the highest Sharpe Ratio. Every investor is supposed to hold a combination of the risk-free asset and Tangency Portfolio. On the Capital Asset Line (CAL) that goes between the risk-free asset and the Tangency Portfolio.nisiprius wrote: 3) In the words of Jeremy Siegel--let me note upfront that he says the stated assumptions arenottrue and that "fundamentally-weighted" indexes are better--but, nevertheless:Equal weighting is one of an infinite number of non-market-cap weightings. It's not clear what special properties it's supposed to have. I cannot for the life of me see some obvious reason why I'd want to own the same dollar value of Zimmer stock and Exxon Mobil stock. It just seems goofy to me. Can anyone give any reason other than "it outperformed the cap-weighted S&P?"Capitalization-weighted indexes... under certain assumptions give investors the "best" tradeoff between risk and return. That means that for any given risk level, these capitalization-weighted portfolios give the highest returns, and for any given return, these portfolios give the lowest risk. This property is called mean-variance efficiency.

from Modern portfolio theory on Wikipedia

BTW, the Sharpe Ratio is the slope of the CAL so it should be obvious from the graph that Tangent Portfolio has the highest slope, or highest Sharpe Ratio.

Now my question is: What is the Tangency Portfolio in real world? Is it cap-weighted Total Stock Market Index or cap-weighted Total World Stock Market Index? Or is it something else?

Gott mit uns.

### Re: Equal weighting or market cap weighting?

A Capitalgrayfox wrote:I believe that from Modern Portfolio Theory (MPT) there is a portfolio known as the Tangency Portfolio which lies on the efficient frontier and which has the highest Sharpe Ratio. Every investor is supposed to hold a combination of the risk-free asset and Tangency Portfolio. On the Capital Asset Line (CAL) that goes between the risk-free asset and the Tangency Portfolio.

**(not**

*Allocation**Asset*) Line (CAL) runs from the risk-free asset to any risky asset. The CAL with the highest Sharpe ratio runs from the risk-free asset to the Tangency Portfolio; this particular CAL is called the Capital

**Line (CML).**

*Market*Something else.grayfox wrote:Now my question is: What is the Tangency Portfolio in real world? Is it cap-weighted Total Stock Market Index or cap-weighted Total World Stock Market Index? Or is it something else?

It won't be just the stock market; it'll be stocks and bonds and real estate and baseball cards and so on.

And it almost certainly won't be market-cap weighted (or price-weighted or equal-weighted for that matter). It'll be a minimum-variance portfolio whose weights will depend on the expected returns, volatilities, and correlations of returns of every investable asset; there's no reason to believe that that portfolio will have weights even remotely close to market-cap weights.

Simplify the complicated side; don't complify the simplicated side.

### Re: Equal weighting or market cap weighting?

Too bad my mother threw away my baseball card and comic book collections. Who would have thought I needed them to build the optimal portfolio for retirement?magician wrote:Something else.grayfox wrote:Now my question is: What is the Tangency Portfolio in real world? Is it cap-weighted Total Stock Market Index or cap-weighted Total World Stock Market Index? Or is it something else?

It won't be just the stock market; it'll be stocks and bonds and real estate and baseball cards and so on.

Seriously though, in the real world, say an IRA at Vanguard, where the choices are mutual funds and ETFs. What is the closest approximation to the Tangency Portfolio?

Gott mit uns.

### Re: Equal weighting or market cap weighting?

I was reading the discussion about minimum variance portfolio. It sounds like an interesting idea. Who wouldn't want to minimize the volatility of their portfolio returns?

The expected returns are muA and muB.

The standard deviations are sigmaA and sigmaB, and the covariance is sigmaAB.

Find the weights xA and xB that minimizes portfolio variance, sigmaP^2.

sigmaP^2 = xA*sigmaA^2 + xB*sigmaB^2 + 2*xA*xB*sigmaAB

s.t. xA + xB = 1

After a bunch of math, the result was

xA = (sigmaB^2 - sigmaAB) / (sigmaA^2 + sigmaB^2 - 2*sigmaAB)

xB = 1 - xA

Notice that the answer doesn't depend on the expected returns, only the variances and covariance.

This is excellent, because you don't have to estimate the expected return.

Also observe that the equation for weight in A has sigmaB ^2 in the the numerator.

Suppose asset A is stocks that has sigmaA = 20% and asset B is CDs that have sigmaB = 0 and CD returns are uncorrelated with stock returns, so sigmaAB=0.

xA = (0 - 0)/(.20^2 + 0 - 0) = 0 and xB = 1 - 0 = 1

The minimum variance portfolio wlll be 100% in CDs.

Wow! All that math to show what your grandmother already knows!

It seems like it only makes sense to find the minimum variance portfolio of all the risky assets. Otherwise all the weight goes to the very low volatile assets like cash, CDs, ST bonds, etc.

It sound like a good idea, but like RodC said, it probably doesn't make sense to include bonds in the computation. Othewise the solution is everything in the low volatile assets and nothing in the high volatile assets.

In fact I tried finding minimum variance portfolio for VTSMX (stocks) and VBMFX (bonds) over some period and the answer was something like 97% in bonds VBMFX and 3% in stocks VTSMX.

This one time, in portfolio camp, we had to derive the minimize variance portfolio of two assets. A and B.Rodc wrote: If the results are completely known already the method results in no significant new information; we all knew min variance stocks/bonds portfolio was historically about 30/70.

And yes I have strayed from one topic to a related topic; conversations are like that.

The more I think about this the more nonsensical it seems to be.

If one is building a diversified portfolio of stocks and bonds and one ignores expected returns and only looks at variance that completely ignores the rather important fact that stocks have a much higher expected return than bonds. That makes no sense at all. Might be valuable for a hedge fund that is willing to take on leverage, though may well blow up in their face, but is of no value to anyone here.

If one applies this, like the iShares funds to only a collection of say a couple hundred stocks it makes sense, especially if one sticks to large caps, to assume they have the same expected return so you can ignore expected returns. But then you must estimate expected variance (risk) of each individual company going forward (and correlation), like somehow this is a constant, like markets don't change and like this is not swamped by noise. Boy does that sound like a shady business. I might buy you have a hope, sort of or in a relative sense, to estimate the future variance of a large well defined sector (say stocks vs bonds, or large stocks vs small), but of individual companies? No way does that make sense.

The expected returns are muA and muB.

The standard deviations are sigmaA and sigmaB, and the covariance is sigmaAB.

Find the weights xA and xB that minimizes portfolio variance, sigmaP^2.

sigmaP^2 = xA*sigmaA^2 + xB*sigmaB^2 + 2*xA*xB*sigmaAB

s.t. xA + xB = 1

After a bunch of math, the result was

xA = (sigmaB^2 - sigmaAB) / (sigmaA^2 + sigmaB^2 - 2*sigmaAB)

xB = 1 - xA

Notice that the answer doesn't depend on the expected returns, only the variances and covariance.

This is excellent, because you don't have to estimate the expected return.

Also observe that the equation for weight in A has sigmaB ^2 in the the numerator.

Suppose asset A is stocks that has sigmaA = 20% and asset B is CDs that have sigmaB = 0 and CD returns are uncorrelated with stock returns, so sigmaAB=0.

xA = (0 - 0)/(.20^2 + 0 - 0) = 0 and xB = 1 - 0 = 1

The minimum variance portfolio wlll be 100% in CDs.

Wow! All that math to show what your grandmother already knows!

It seems like it only makes sense to find the minimum variance portfolio of all the risky assets. Otherwise all the weight goes to the very low volatile assets like cash, CDs, ST bonds, etc.

It sound like a good idea, but like RodC said, it probably doesn't make sense to include bonds in the computation. Othewise the solution is everything in the low volatile assets and nothing in the high volatile assets.

In fact I tried finding minimum variance portfolio for VTSMX (stocks) and VBMFX (bonds) over some period and the answer was something like 97% in bonds VBMFX and 3% in stocks VTSMX.

Gott mit uns.

### Re: Equal weighting or market cap weighting?

Well, if you are putting in assets like CDs that don't have nominal volatility but have real volatility, then you have to do the calculation in real terms to get a meaningful result.grayfox wrote:Suppose asset A is stocks that has sigmaA = 20% and asset B is CDs that have sigmaB = 0 and CD returns are uncorrelated with stock returns, so sigmaAB=0.

xA = (0 - 0)/(.20^2 + 0 - 0) = 0 and xB = 1 - 0 = 1

The minimum variance portfolio wlll be 100% in CDs.

Wow! All that math to show what your grandmother already knows!

Also, the technical reason for why MV works for stocks is because "beta" only predicts aggregate differences in returns between stocks and bonds but not cross-sectional differences in stock returns. (So all stocks are in some rudimentary sense "the same" except for their volatility and correlations.)

If you just run a numeric optimization (i.e. the formula above using historic values for variances and covariances) on total stock market and 5 year treasuries in nominal terms using recent data, I get ~85% bonds and 15% stocks. (FWIW, if you plug in actual historic returns and do mean variance optimization the same way, it only gets slightly better for stocks because they've done so poorly relative to bonds in recent history.)In fact I tried finding minimum variance portfolio for VTSMX (stocks) and VBMFX (bonds) over some period and the answer was something like 97% in bonds VBMFX and 3% in stocks VTSMX.

OTOH, if you did it in real terms, or had a larger data set (that say went back to the 60s and 70s so that you didn't have one long bull-market in bonds), then I think you'd get a more aggressive stock allocation from the same procedure. (And this is setting aside statistical problems with using simple historic estimates for your projection of future variances, etc.)

Still, for a pretty wide range of reasonable estimates about what returns, volatilities, and correlations will be, it will show that even in mean-variance terms, you want a relatively small allocation to stocks. (I gave an example towards the end of this thread.)

Finally, I should mention that I took the limited data available for the iShares funds and looked to see if the "VIX trick" worked for them. It seems that the volatility reduction from the trick is entirely separate from the volatility reduction from the MV procedure and that the two things can be combined. This is based on an extremely limited data set however.

### Re: Equal weighting or market cap weighting?

Yes, maybe with CDs it would be better to do calculations using real returns.Well, if you are putting in assets like CDs that don't have nominal volatility but have real volatility, then you have to do the calculation in real terms to get a meaningful result.

But, here's a case I ran to find minimum variance portfolio using monthly Yahoo data.

3-fund Portfolio: VTSMX, VGTSX, VBMFX

Dec-1996 to Dec-2012

Here are inputs:

Code: Select all

```
StartDate Dec-1996
EndDate Dec-2012
VTSMX VGTSX VBMFX
AvgRet 0.06142 0.04746 0.05824
SD 0.16958 0.19128 0.03514
Correlation Matrix
VTSMX VGTSX VBMFX
VTSMX 1.00000
VGTSX 0.87068 1.000000
VBMFX -0.05143 -0.002638 1.000000
```

U.S. and International stocks have high correlation, and both are more or less uncorrelated with total bond market.

So how will it mix them to get the lowest volatility?

Results:

Code: Select all

```
Portfolio weights:
VTSMX VGTSX VBMFX
0.0504 0.0000 0.9496
E[R] 5.839%
SD 3.29%
```

Almost all bonds, with a pinch of U.S. stocks.

Last edited by grayfox on Mon Feb 18, 2013 9:34 pm, edited 1 time in total.

Gott mit uns.

### Re: Equal weighting or market cap weighting?

I don't see your point. I already told you that if you do it with nominal returns over that time period bonds are going to win out whether you do variance minimization or a full mean-variance optimization. Even if you do it with real returns, I think bonds are going to come out heavily favored because the average stock market premium was so bad for the time period in question. The optimization doesn't lie; if you had used that allocation over this time period, you'd have gotten much lower variance and reasonably good returns. But what are you thinking that this exercise will prove?grayfox wrote:Yes, maybe with CDs it wold be better to do calculations using real returns.Well, if you are putting in assets like CDs that don't have nominal volatility but have real volatility, then you have to do the calculation in real terms to get a meaningful result.

But, here's a case I ran to find minimum variance portfolio using monthly Yahoo data.

### Re: Equal weighting or market cap weighting?

I think it shows that if you use both stocks and bonds and form a min variance portfolio, almost everything goes into the bonds.

Because the bonds have about 1/5 the volatility of stocks, the solution is to weight bonds heavily to get lowest volatility.

I'm thinking it would make more sense to form a min variance portfolio only with stocks.

Then you mix the min-variance-stock portfolio with bonds to get the complete portfolio.

Because the bonds have about 1/5 the volatility of stocks, the solution is to weight bonds heavily to get lowest volatility.

I'm thinking it would make more sense to form a min variance portfolio only with stocks.

Then you mix the min-variance-stock portfolio with bonds to get the complete portfolio.

Gott mit uns.

### Re: Equal weighting or market cap weighting?

You can't take data in which bonds have had nothing but a long, massive bull market and make sensible conclusions about portfolio construction. I also pointed out above in the thread that using simple historic values for volatility, etc. would give you weird results if you just plugged it into an optimizer.grayfox wrote:I think it shows that if you use both stocks and bonds and form a min variance portfolio, almost everything goes into the bonds.

Because the bonds have about 1/5 the volatility of stocks, the solution is to weight bonds heavily to get lowest volatility.

I explained above why this happens to work from a theoretical standpoint.I'm thinking it would make more sense to form a min variance portfolio only with stocks.

Ideally you would estimate expected returns for both stocks and bonds and expected variances and covariances and then do a mean-variance optimization. If you "estimate" by plugging in recent historical values, you come out very heavy with bonds. OTOH, even if you make some reasonable assumptions, you still get far more weight on bonds than many people allocate for...Then you mix the min-variance-stock portfolio with bonds to get the complete portfolio.

### Re: Equal weighting or market cap weighting?

FYI:

Indexuniverse March 2013- David Blitz - How smart is 'smart beta'?

- Low-Volatility Indexes

- Maximum Sharpe Ratio Indexes

- Momentum Indexes

- Equally weighted indexes

With regard to the latter:

Equally weighted indexes do not invest in the smallest and most illiquid stocks that generate the size effect:

Indexuniverse March 2013- David Blitz - How smart is 'smart beta'?

- Fundamentally Weighted IndexesSummary

In smart-beta indexes—such as fundamentally weighted and minimum-volatility indexes—stock weights are based not on their market capitalizations, but on some alternative formula. We have argued that the added value of smart-beta indexes comes from systematic tilts toward classic factor premiums that are induced by these alternative weighting schemes. We also showed that smart-beta indexes are not specifically designed for harvesting factor premiums in the most efficient manner, but primarily for simplicity and appeal. For a number of popular smart-beta indexes, we have discussed the main pitfalls, and how investors may capture factor premiums more efficiently by addressing these concerns. Finally, it is important to remember that although passive management can be used to replicate smart indexes, investors should realize that, without exception, smart indexes themselves always represent active strategies

- Low-Volatility Indexes

- Maximum Sharpe Ratio Indexes

- Momentum Indexes

- Equally weighted indexes

With regard to the latter:

Equally weighted indexes do not invest in the smallest and most illiquid stocks that generate the size effect:

Thus, equally weighted indexes are better described as strategies that try to exploit a possible difference in return between large stocks and even larger stocks. Equally weighted indexes are thus able to profit only partly, at best, from the small-cap effect considered in the literature...

frequent rebalancing is required to maintain equal weights. As this rebalancing involves selling recent winners and buying recent losers, this tends to go against the momentum effect..

In our view, therefore, a traditional capitalization-weighted (buy-and-hold) index of true small stocks is a more appropriate and also a more efficient way to capture the small-cap premium...

prior username: hafis50

### Re: Equal weighting or market cap weighting?

First off, contra the implications of your post, he article does not argue that minimum volatility indexes are weighting the standard 4 factors in a different way. (Though as the Boggleheads wiki points out, if you add a 5th "intrinsic volatility" factor, then you can account for much of the low volatility fund's returns.)hafius500 wrote:FYI:

Indexuniverse March 2013- David Blitz - How smart is 'smart beta'?

- Fundamentally Weighted IndexesSummary

In smart-beta indexes—such as fundamentally weighted and minimum-volatility indexes—stock weights are based not on their market capitalizations, but on some alternative formula. We have argued that the added value of smart-beta indexes comes from systematic tilts toward classic factor premiums that are induced by these alternative weighting schemes. We also showed that smart-beta indexes are not specifically designed for harvesting factor premiums in the most efficient manner, but primarily for simplicity and appeal. For a number of popular smart-beta indexes, we have discussed the main pitfalls, and how investors may capture factor premiums more efficiently by addressing these concerns. Finally, it is important to remember that although passive management can be used to replicate smart indexes, investors should realize that, without exception, smart indexes themselves always represent active strategies

- Low-Volatility Indexes

- Maximum Sharpe Ratio Indexes

- Momentum Indexes

- Equally weighted indexes

Also, I disagree with the claim that article makes that the minimum volatility indexes are "black box" indexes. You can do the same basic thing (and get only slightly different weights) with the statistical methods I talked about in this thread. And the BARRA model is constructed using well known statistical principles and doesn't itself count as a "black box" since you can replicate it's results for yourself.