How to diversify a Three Fund Portfolio in 8 steps

[hdas]

Some people have asked me privately about portfolio construction, in this post I'd like to share my current implementation. The 8 steps are merely to show the progression and effects of each change, also it can be customized to each person belief system.

Considerations:

1.) Balance Risk On/Off based on personal context. For this example I use my own, (80/20)
2.) Have some international risk (bond and stock) exposure
3.) Maximize Diversification given an objective criterion, enter Diversification Ratio.

The diversification ratio is the ratio of the weighted average of volatility divided by the portfolio volatility

4.) Achieve significant exposure to all equity style factors (Market, Quality, Size, Value, Momentum, BAB) [1] See exposure (loadings) with its T stat.
5.) No leverage
6.) Try to get closer to risk parity [2]

Step 1: Start with your default option of 3 Fund Portfolio [Diversification Ratio = 1.11]:

US Equities Total Market (VTI) - 50%
International Equities Total Market (VXUS) - 30%
US Fixed Income - Total Bond Market - (BND) - 20%

Step 2: Add US Long Term Treasuries [Diversification Ratio = 1.31]:

US Equities Total Market (VTI) - 50%
International Equities Total Market (VXUS) - 30%
US Long Term Treasury - (VGLT|EDV) - 20%

Step 3: Add US Equities Size/Value Tilt [Diversification Ratio = 1.35]:

US Equities Total Market (VTI) - 25%
US Equities - Small Cap Value (IJS|VIOV) - 25%
International Equities Total Market (VXUS) - 30%
US Long Term Treasury - (VGLT|EDV) - 20%

Step 4: Add US Equities Momentum Factor Exposure [Diversification Ratio = 1.38]:

US Equities - Momentum (MTUM) - 25%
US Equities - Small Cap Value (IJS|VIOV) - 25%
International Equities Total Market (VXUS) - 30%
US Long Term Treasury - (VGLT|EDV) - 20%

Step 5: Add Emerging Market Bonds [Diversification Ratio = 1.45]:

US Equities - Momentum (MTUM) - 25%
US Equities - Small Cap Value (IJS|VIOV) - 25%
International Equities (VXUS) - 15%
Emerging Market Bonds (EMB|VWOB|VEGBX|LEMB) - 15%
US Long Term Treasury - (VGLT|EDV) - 20%

Step 6: Incorporate Low Volatility Funds[3] (US and International) [Diversification Ratio = 1.49]:

US Equities - Momentum (MTUM) - 10.625%
US Equities - Low Volatility (USMV) - 10.625%
US Equities - Small Cap Value (IJS|VIOV) - 10.625%
US Equities - Small Cap Low Volatility (XSLV) - 10.625%
Global Low Volatility Equities (VMNVX) - 15%
International Equities Total Market (VXUS) - 8.5%
Emerging Market Bonds (EMB|VWOB|VEGBX|LEMB) - 15%
US Long Term Treasury - (VGLT|EDV) - 20%

Step 7: Optimize exposure in US equities[1] (Equity Style Factor Premia) [Diversification Ratio = 1.49]:

US Equities - Momentum (MTUM) - 8.36%
US Equities - Low Volatility (USMV) - 8.36%
US Equities - Small Cap Value (IJS|VIOV) - 10.14%
US Equities - Small Cap Low Volatility (XSLV) - 10.14%
Global Low Volatility Equities (VMNVX) - 15%
International Equities Total Market (VXUS) - 14%
Emerging Market Bonds (EMB|VWOB|VEGBX|LEMB) - 14%
US Fixed Income - (VGLT|EDV) - 20%

Step 8: Optimize exposure in International Equities [Diversification Ratio = 1.54]:

US Equities - Momentum (MTUM) - 8.36%
US Equities - Low Volatility (USMV) - 8.36%
US Equities - Small Cap Value (IJS|VIOV) - 10.14%
US Equities - Small Cap Low Volatility (XSLV) - 10.14%
Global Low Volatility Equities (VMNVX) - 15%
International Developed Small (ISCF) - 7%
Emerging Market Stocks (VWO|DGS) - 7%
Emerging Market Bonds (EMB|VWOB|VEGBX|LEMB) - 14%
US Long Term Treasury - (VGLT|EDV) - 20%

Notes:

[1]

[2]

[3] The low volatility exposure perhaps could be simplified to two funds.

Sector Exposure:



Cheers

[305pelusa]

I'm assuming you're generating numbers and optimizing by looking at underlying index numbers. Otherwise there's only 5 years of data because of XSLV I think.

Any ways, what was the St. Dev of that final portfolio in Step 8? I plugged it in to PV and got 7.48% but again, that's just 5 years. Is that about correct?

Thanks.

[TropikThunder]

[3] The exposure perhaps could be simplified to two funds.

Cheers

I notice (almost) all of those are ETF’s. How many people have a brokerage option in their workplace retirement plans?

[hdas]

I'm assuming you're generating numbers and optimizing by looking at underlying index numbers. Otherwise there's only 5 years of data because of XSLV I think.

Any ways, what was the St. Dev of that final portfolio in Step 8? I plugged it in to PV and got 7.48% but again, that's just 5 years. Is that about correct?

Thanks.

>> I didn't optimize in the traditional sense (finding the weights that maximize the diversification ratio), rather followed the path of the 8 steps, and incorporating the constrains of my own portfolio (em bonds, VMNVX, %Bonds). I was gadly surprised, at seeing that my holdings, were close to optimal en DRatio sense. The other optimization was looking at the minimum amount of SCV that would give me significant exposure to value.

>> XSLV has 6y and 8 months of data......I use daily data resolution.

>> Your .stdev() data sounds about right.

Cheers

[hdas]

[3] The exposure perhaps could be simplified to two funds.

Cheers

I notice (almost) all of those are ETF’s. How many people have a brokerage option in their workplace retirement plans?

I meant, the low volatility exposure. Sorry about the typo.

[TropikThunder]

[3] The exposure perhaps could be simplified to two funds.

Cheers

I notice (almost) all of those are ETF’s. How many people have a brokerage option in their workplace retirement plans?

I meant, the low volatility exposure. Sorry about the typo.

I’m not focusing on the volatility portion (just didn’t want to copy the whole OP in my comment). I’m commenting on the ETF part, since not everyone has a brokerage option and the system isn’t very helpful if one can only use it outside their workplace plan.

[hdas]

[3] The exposure perhaps could be simplified to two funds.

Cheers

I notice (almost) all of those are ETF’s. How many people have a brokerage option in their workplace retirement plans?

I meant, the low volatility exposure. Sorry about the typo.

I’m not focusing on the volatility portion (just didn’t want to copy the whole OP in my comment). I’m commenting on the ETF part, since not everyone has a brokerage option and the system isn’t very helpful if one can only use it outside their workplace plan.

That's a good point and seems like there's little to do with that kind of restriction. However, one take away from the exercise is to look at the many dimensions that one can manipulate to improve diversification, hopefully in at least one or 2 a person can find funds available in their retirement plan. Cheers

[YRT70]

[1]

Interesting portfolio. Can I make my own factor regression similar to that? Should I use PV? If so, what settings do you recommend?

[305pelusa]

I'm assuming you're generating numbers and optimizing by looking at underlying index numbers. Otherwise there's only 5 years of data because of XSLV I think.

Any ways, what was the St. Dev of that final portfolio in Step 8? I plugged it in to PV and got 7.48% but again, that's just 5 years. Is that about correct?

Thanks.

>> I didn't optimize in the traditional sense (finding the weights that maximize the diversification ratio), rather followed the path of the 8 steps, and incorporating the constrains of my own portfolio (em bonds, VMNVX, %Bonds). I was gadly surprised, at seeing that my holdings, were close to optimal en DRatio sense. The other optimization was looking at the minimum amount of SCV that would give me significant exposure to value.

>> XSLV has 6y and 8 months of data......I use daily data resolution.

>> Your .stdev() data sounds about right.

Cheers

Show me how you calculated DR. I'm almost positive it's wrong.

[hdas]

I'm assuming you're generating numbers and optimizing by looking at underlying index numbers. Otherwise there's only 5 years of data because of XSLV I think.

Any ways, what was the St. Dev of that final portfolio in Step 8? I plugged it in to PV and got 7.48% but again, that's just 5 years. Is that about correct?

Thanks.

>> I didn't optimize in the traditional sense (finding the weights that maximize the diversification ratio), rather followed the path of the 8 steps, and incorporating the constrains of my own portfolio (em bonds, VMNVX, %Bonds). I was gadly surprised, at seeing that my holdings, were close to optimal en DRatio sense. The other optimization was looking at the minimum amount of SCV that would give me significant exposure to value.

>> XSLV has 6y and 8 months of data......I use daily data resolution.

>> Your .stdev() data sounds about right.

Cheers

Show me how you calculated DR. I'm almost positive it's wrong.

I will once I get to the desk. Why are you positive about this?

[hdas]

[1]

Interesting portfolio. Can I make my own factor regression similar to that? Should I use PV? If so, what settings do you recommend?

This is for the us equity portion only. Just normalize the weights so it adds 100%. AQR Factors, Robust Regression, Include Quality and BAB. Cheers

[305pelusa]

I'm assuming you're generating numbers and optimizing by looking at underlying index numbers. Otherwise there's only 5 years of data because of XSLV I think.

Any ways, what was the St. Dev of that final portfolio in Step 8? I plugged it in to PV and got 7.48% but again, that's just 5 years. Is that about correct?

Thanks.

>> I didn't optimize in the traditional sense (finding the weights that maximize the diversification ratio), rather followed the path of the 8 steps, and incorporating the constrains of my own portfolio (em bonds, VMNVX, %Bonds). I was gadly surprised, at seeing that my holdings, were close to optimal en DRatio sense. The other optimization was looking at the minimum amount of SCV that would give me significant exposure to value.

>> XSLV has 6y and 8 months of data......I use daily data resolution.

>> Your .stdev() data sounds about right.

Cheers

Show me how you calculated DR. I'm almost positive it's wrong.

I will once I get to the desk. Why are you positive about this?

Two reasons.
1) If the standard dev. of the portfolio is around 7.5%, a DR of 3 means the weighted St. Dev of the holdings is 22.5%. I don't think a single one of the St. Dev. of the holdings is even that high (maybe VWO if that).
2) Another interpretation of DR is that the square is equivalent to the number of uncorrelated assets. If you hold two uncorrelated assets with identical St. Dev, the DR will be 1.41. Because 1.41^2 = 2.
A DR of 3 implies 9 different uncorrelated holdings. The portfolio has 9 holdings. Only way you get a DR=3 is if they are all completely uncorrelated. And I'm pretty sure they aren't.

I've been wrong before of course.

[bgf]

i dont think i'll ever understand how someone can do this but won't accept investing in individual companies. seems just as complicated, difficult, and unpredictable as trying to do fundamental, or, god forbid, even technical analysis.

[Elysium]

Step 1: Start with your default option of 3 Fund Portfolio

US Equities Total Market (VTI) - 50%
International Equities Total Market (VXUS) - 30%
US Fixed Income - Total Bond Market - (BND) - 20%

Step 2: Stop reading rest, you are all set

I made a small correction, highlighted above. Most people can stop there and they will end up doing better than anyone following the rest. This portfolio is 100% diversified and efficient.

[asif408]

As best I can tell, you start off with 30% of your portfolio in international stocks market cap weighted. Seems fine. At the end you end up with less in international stocks (20%, with half of that in low vol international), a heavy tilt to small stocks in the US, replace total bond with LTTs, and throw in some EM bonds in place of the international stocks you lost and some of the US stocks. By sector composition, you dramatically overweight REITs, and significantly underweight energy, healthcare, and tech.

Is the goal to show one way you would have done better in the last decade or so by tilting away from the standard 3 fund portfolio? Because the "optimized" portfolio appears just to be a performance chaser, that overweights US small stocks, underweights international stocks, and tilts the bonds to LTTs and EM bonds. If there is any mean reversion in the future, it's likely large cap, value, and international tilts, with short term Treasuries, will be the better from a diversification standpoint. So if I was designing a portfolio for future potential outperformance and "diversification", I would do precisely the opposite of what you did. Or just stick with step 1 and not try to chase performance.

[long_gamma]

As other posted pointed out, your diversification ratio calculations seems to be off.

I just quickly plugged into my risk parity worksheet, It is giving me different results.

I used only last year (252 days) of daily prices, Correlation shown is EWMA weighted, but equal weighted shouldn't be too much different.

[long_gamma]

Currently I use modified RP version in one of my accounts. These have traditional assets that can be levered up easily. DR is much higher.



ES, RTY, ZT, ZB, CL, 6J and GC futures in place of the ETF's. Options are liquid for EEM, IYR. EFA can be spot or option, No other options other than spot for EMLC

[sabhen]

Thanks for sharing. But...

What are the underlying rationale framework and (back) testing data behind such a portfolio construction? Sharpe ratio?

Relying on beliefs is likely to be a poor strategy.

[columbia]

The logic here is that a “more diversified” portfolio is better than a three fund portfolio because _______?

If _______ = “it’s more diversified”, then I question the utility of enacting such a portfolio.

[Jebediah]

Why global low vol instead of international low vol?

Where's VPU? Where's GLD?

[UpperNwGuy]

Lots of folk on this board seem to treat diversification as the ultimate goal of investing.

[hdas]

I'm assuming you're generating numbers and optimizing by looking at underlying index numbers. Otherwise there's only 5 years of data because of XSLV I think.

Any ways, what was the St. Dev of that final portfolio in Step 8? I plugged it in to PV and got 7.48% but again, that's just 5 years. Is that about correct?

Thanks.

>> I didn't optimize in the traditional sense (finding the weights that maximize the diversification ratio), rather followed the path of the 8 steps, and incorporating the constrains of my own portfolio (em bonds, VMNVX, %Bonds). I was gadly surprised, at seeing that my holdings, were close to optimal en DRatio sense. The other optimization was looking at the minimum amount of SCV that would give me significant exposure to value.

>> XSLV has 6y and 8 months of data......I use daily data resolution.

>> Your .stdev() data sounds about right.

Cheers

Show me how you calculated DR. I'm almost positive it's wrong.

I will once I get to the desk. Why are you positive about this?

Two reasons.
1) If the standard dev. of the portfolio is around 7.5%, a DR of 3 means the weighted St. Dev of the holdings is 22.5%. I don't think a single one of the St. Dev. of the holdings is even that high (maybe VWO if that).
2) Another interpretation of DR is that the square is equivalent to the number of uncorrelated assets. If you hold two uncorrelated assets with identical St. Dev, the DR will be 1.41. Because 1.41^2 = 2.
A DR of 3 implies 9 different uncorrelated holdings. The portfolio has 9 holdings. Only way you get a DR=3 is if they are all completely uncorrelated. And I'm pretty sure they aren't.

I've been wrong before of course.

Thank you for the heads up (great peer review!). I had something funky in my code, some extra () and a missing sqrt(). I corrected the post. Directionally it doesn't change anything (meaning each change did add improve the ratio), but the value of the ratio was wrong. Cheers

[hdas]

As other posted pointed out, your diversification ratio calculations seems to be off.

I just quickly plugged into my risk parity worksheet, It is giving me different results.

I used only last year (252 days) of daily prices, Correlation shown is EWMA weighted, but equal weighted shouldn't be too much different.

Thanks, yes I corrected it. In your example, I guess the question is how much LTT you want to hold, I wanted to show the example with max 20%. Cheers

[andrew99999]

Challenge to OP - can you get 80% of the optimisation with no more than 5 funds?

[D-Dog]

Challenge to OP - can you get 80% of the optimisation with no more than 5 funds?

Just guessing, how about these five:
VFMF - vanguard multi factor
VMVFX - vanguard Global Min vol
VWO - vanguard emerg mkts
EDV - vanguard extended dur treasury
VWOB - vanguard emerg mkt bond

Seems to capture a lot of what the OP was going for in five funds.

[long_gamma]

In your example, I guess the question is how much LTT you want to hold, I wanted to show the example with max 20%. Cheers

Mine is a trading model, which starts with risk parity and levers up based on the trend and re-balances weekly. To answer the question, I am agnostic about the max weight and use whatever the model spits out. Japanese yen is much more diversifier than LTT

If I were to use this in investing world with a very long rebalance horizon, then I would base that on weights determined from some bootstrapping technique instead of arbitrary max and min weights.

[Uncorrelated]

Could you rationalize the MTUM recommendation?

I have read from other sources that momentum is difficult to capture, for example:

Despite the consensus on the existence of momentum, debates still exist over the cost-effectiveness of momentum strategies. There are many challenges in trying to estimate the cost of implementing any strategy historically, and this is particularly true with high turnover strategies. Since market microstructures have changed over time, trading cost estimates need to be date matched with the returns for each security at its transaction time. The estimation of trading costs, especially the implicit components such as bid-ask spread and market impact, can yield very different results depending on the model used and the assumptions made.

Lesmond, Schill, and Zhou (2004) used a price friction model to infer trading costs from daily return data. They found that the top and bottom momentum deciles tend to include stocks that are more expensive to trade, and as a result, their equally weighted long/short momentum portfolios failed to generate abnormal returns net of estimated trading costs during the sample period 1980–1998.

dimensional fund advisors, 2015 A Review of the Empirical Evidence on the Dimensions of Expected Stock Returns

According to fama & french 5 factor model + momentum, MTUM has a loading of -.14 size -.28 value and +.26 momentum. Coupled with the high turnover of MTUM (138% this year. Expected trading costs 0.25% to 1%), this results in a negative expected performance compared to a simple market weighted fund. Before factoring in tax efficiency.

Ben Felix from this youtube channel recommends 33% value, 33% small cap value and 33% broad market. See https://www.pwlcapital.com/resources/fa ... with-etfs/ for the paper. I don't think maximizing the diversification ratio is a good portfolio construction approach. The paper from Ben Felix describes an approach that I think works very well. The problem with diversification ratio is that you need to correctly estimate the covariance between the elements in your portfolio, but you need much more than 20 years of data to accurately estimate the covariance.



Your portfolio contains bonds, but you are not using bond factors. Even if your factor exposure image was run only on equity, some forms of equity (REIT in particular) have non-trivial amounts of TERM exposure.

Portfolio 1 and 2 do not have the same expected risk, which makes comparison awkward. Every subsequent portfolio takes on more risk. Adding exposure to a factor increases the reward as well as the risk of the portfolio, to compensate you should increase the amount of treasuries in your portfolio. (or simply accept more risk. But why do you start off with a portfolio of 20% bonds if you can handle that much risk?).

I would probably start with portfolio 1, replace VTI with VFMF and call it a day. That gets you to similar expected performance (based on factor regression) only 3 funds.


"risk pairity" is quickly becoming my most hated term in finance. If you want true "risk parity" then you should diversify in such a way that each factor contributes the same amount of risk. But since each factor has different expected returns, there is no reason to build such a portfolio. If you calculate risk parity based on live portfolio performance, then you need hundreds of years of performance before your results become statistically significant.

[Forester]

Could you rationalize the MTUM recommendation?

...
Ben Felix from this youtube channel recommends 33% value, 33% small cap value and 33% broad market. See https://www.pwlcapital.com/resources/fa ... with-etfs/ for the paper. I don't think maximizing the diversification ratio is a good portfolio construction approach. The paper from Ben Felix describes an approach that I think works very well. The problem with diversification ratio is that you need to correctly estimate the covariance between the elements in your portfolio, but you need much more than 20 years of data to accurately estimate the covariance.

Ben Felix is a big CAPM cheerleader. I don't rate his analysis of factor investing.

MTUM is a good fund because it places a premium on liquidity and isn't as exposed to stale momentum as something like QMOM. It's more like a megacap S&P fund with a momentum tilt.

Articles on the momentum-low vol barbell: https://www.fortunefinancialadvisors.co ... m-barbell/ https://www.fortunefinancialadvisors.co ... i-indices/

[Uncorrelated]

Could you rationalize the MTUM recommendation?

...
Ben Felix from this youtube channel recommends 33% value, 33% small cap value and 33% broad market. See https://www.pwlcapital.com/resources/fa ... with-etfs/ for the paper. I don't think maximizing the diversification ratio is a good portfolio construction approach. The paper from Ben Felix describes an approach that I think works very well. The problem with diversification ratio is that you need to correctly estimate the covariance between the elements in your portfolio, but you need much more than 20 years of data to accurately estimate the covariance.

Ben Felix is a big CAPM cheerleader. I don't rate his analysis of factor investing.

MTUM is a good fund because it places a premium on liquidity and isn't as exposed to stale momentum as something like QMOM. It's more like a megacap S&P fund with a momentum tilt.

Articles on the momentum-low vol barbell: https://www.fortunefinancialadvisors.co ... m-barbell/ https://www.fortunefinancialadvisors.co ... i-indices/

Why would you want to have a megacap fund with a momentum tilt when it comes with large negative size and value exposure? There are well-designed mutlifactor and small cap value funds (such as those from DFA) that have larger momentum exposure than MTUM plus positive size/value exposure. Because you target multiple factors with one fund, the transaction costs are also lower.

I can't take that blog seriously. What a surprise that a fund with negative value and negative size exposure did well the last 2 decades.

[stan1]

Demographics is a big factor that we don't often capture in these diversification discussions (3 fund vs diversification). I think people come in with their personal experiences and apply them to everyone else.

We have a number of posters who are 70+ years of age. Because its Bogleheads and we have a pretty specific demographic here many are millionaires having been frugal often with good incomes their entire working life. Some have pensions. Some transferred risk to institutions by buying insurance products like annuities and LTCI decades ago which turned out in retrospect to be very good market timing. Maybe their portfolio withdrawal rate is close to zero (they just reinvest dividends and even RMDs or gift to future heirs). Social Security, Medicare, historically low tax rates, and even quantitative easing have served as a backstop along with decades of relative peace and prosperity that have seen the US economy grow in extraordinary ways. They have been very fortunate, and making the assumption "don't worry it will all turn out OK" did in fact turn out to be a good one. My FIL who passed away this year at 94 (Battle of the Bulge veteran) did very, very well for himself as a high school graduate who went on to a mid-level management position at a Fortune 500 company.

Then we have posters who are under 50. Their end of life is hopefully 30-70 years away. They can focus on earning income and savings rate. Intellectual capital is their biggest asset which gives future earning potential. They have time to recover if something really bad happens even if recovery takes decades. Financial independence might be a goal but it often means continuing work in another way (see last weeks' Rick Ferri Podcase on the FIRE movement).

But what about people between 50-70 or a true early retiree by choice or necessity? Many fewer have pensions or insurance products like their parents and even if they do the institution backing the product has set it up to take on lower risk. Maybe they will plan to work longer but sometimes that doesn't work out due to health or declining value of intellectual capital. Will SS and Medicare be there in the same way as it was for their parents? Institutions are not standing by to accept risk (we are on our own in ways our parents were not). Many will still live long lives due to good health care. Will stocks and bonds perform the same way in the next 50 years as they did in the last 50 years? Assuming that everything will turn out OK over a 30-50 year timeline seems risky, yet if you are living on your investments now you have to withdraw and might not have time to recover in a big downturn (especially if recovery takes decades not a few years). This group sees volatility as a big risk, and the ways to transfer that risk to government or corporate institution with deep pockets are closing.

Will a three fund or two fund portfolio be good enough? Will a portfolio such as hdas posted yield a materially different result? I don't know, but the benefits of diversification away from the Top 25 companies by market cap passes an easy logical test for me.

[305pelusa]

Why would you want to have a megacap fund with a momentum tilt when it comes with large negative size and value exposure? There are well-designed mutlifactor and small cap value funds (such as those from DFA) that have larger momentum exposure than MTUM plus positive size/value exposure.

Do you have any examples other than DFA? Like an ETF? I haven't found any with a MUM load as high as MTUM thus far. Then again I haven't looked that hard haha. It'd be really helpful if you had one in mind.

Thank you very much!

[Forester]

Could you rationalize the MTUM recommendation?

...
Ben Felix from this youtube channel recommends 33% value, 33% small cap value and 33% broad market. See https://www.pwlcapital.com/resources/fa ... with-etfs/ for the paper. I don't think maximizing the diversification ratio is a good portfolio construction approach. The paper from Ben Felix describes an approach that I think works very well. The problem with diversification ratio is that you need to correctly estimate the covariance between the elements in your portfolio, but you need much more than 20 years of data to accurately estimate the covariance.

Ben Felix is a big CAPM cheerleader. I don't rate his analysis of factor investing.

MTUM is a good fund because it places a premium on liquidity and isn't as exposed to stale momentum as something like QMOM. It's more like a megacap S&P fund with a momentum tilt.

Articles on the momentum-low vol barbell: https://www.fortunefinancialadvisors.co ... m-barbell/ https://www.fortunefinancialadvisors.co ... i-indices/

Why would you want to have a megacap fund with a momentum tilt when it comes with large negative size and value exposure? There are well-designed mutlifactor and small cap value funds (such as those from DFA) that have larger momentum exposure than MTUM plus positive size/value exposure. Because you target multiple factors with one fund, the transaction costs are also lower.

I can't take that blog seriously. What a surprise that a fund with negative value and negative size exposure did well the last 2 decades.

What matters is results and MTUM delivered. It's a pepped-up S&P 500 fund which is fine. No megacap momentum fund will get the academic momentum return of 15%p.a. or whatever it's just not realistic.

[305pelusa]

>> I didn't optimize in the traditional sense (finding the weights that maximize the diversification ratio), rather followed the path of the 8 steps, and incorporating the constrains of my own portfolio (em bonds, VMNVX, %Bonds). I was gadly surprised, at seeing that my holdings, were close to optimal en DRatio sense. The other optimization was looking at the minimum amount of SCV that would give me significant exposure to value.

>> XSLV has 6y and 8 months of data......I use daily data resolution.

>> Your .stdev() data sounds about right.

Cheers

Show me how you calculated DR. I'm almost positive it's wrong.

I will once I get to the desk. Why are you positive about this?

Two reasons.
1) If the standard dev. of the portfolio is around 7.5%, a DR of 3 means the weighted St. Dev of the holdings is 22.5%. I don't think a single one of the St. Dev. of the holdings is even that high (maybe VWO if that).
2) Another interpretation of DR is that the square is equivalent to the number of uncorrelated assets. If you hold two uncorrelated assets with identical St. Dev, the DR will be 1.41. Because 1.41^2 = 2.
A DR of 3 implies 9 different uncorrelated holdings. The portfolio has 9 holdings. Only way you get a DR=3 is if they are all completely uncorrelated. And I'm pretty sure they aren't.

I've been wrong before of course.

Thank you for the heads up (great peer review!). I had something funky in my code, some extra () and a missing sqrt(). I corrected the post. Directionally it doesn't change anything (meaning each change did add improve the ratio), but the value of the ratio was wrong. Cheers

Yes your numbers seem much more reasonable.

The last thing I'll say is that Diversification Ratio is a relative measurement, not an absolute one. So calculating the DR of the portfolio in Step 8, comparing it to the 3-Fund, and concluding the final portfolio is more diversified isn't actually correct. You can't compare DRs between portfolios of different assets entirely. Certainly not when you're increasing the number of assets since we know it's not associative.

D-dog showed the discrepancy back in this thread:
viewtopic.php?t=275899&start=50

Ex: You could substitute VTI for a representative weighting in Vanguard sector funds and the 3-Fund portfolio (now the 12+ Fund portfolio) will have a higher DR. Even though it's basically the same underlying holdings, it'll look "more diversified". And your final portfolio won't "look as good in comparison"; even though, again, it's the exact same holdings.

Just beware of the shortcomings of the metric and don't use it in comparisons where it's not accurate (like this thread seems to suggest).

Cheers mate

[Uncorrelated]

What matters is results and MTUM delivered.

All-in amazon!

I'm sure you don't really believe that. What matters is expense ratio, factor exposure, transaction costs and tracking error. What you call results is what I would describe as recency bias.

[Uncorrelated]

Why would you want to have a megacap fund with a momentum tilt when it comes with large negative size and value exposure? There are well-designed mutlifactor and small cap value funds (such as those from DFA) that have larger momentum exposure than MTUM plus positive size/value exposure.

Do you have any examples other than DFA? Like an ETF? I haven't found any with a MUM load as high as MTUM thus far. Then again I haven't looked that hard haha. It'd be really helpful if you had one in mind.

Thank you very much!

I think I used the wrong factor set to do my analysis, the fund i was looking at (VFMF, Vanguard U.S. Multifactor ETF) has higher momentum exposure than MTUM with AA's factor set, but not with FF or ARQ.

The DFA approach is to screen for negative momentum before looking at size/value. This avoids the large negative momentum exposure that some value stocks have (value and momentum are negatively correlated). I think this is a much more sensible approach than targeting momentum directly. After all, it's the overall portfolio factor exposure that counts.

[hdas]

Show me how you calculated DR. I'm almost positive it's wrong.

I will once I get to the desk. Why are you positive about this?

Two reasons.
1) If the standard dev. of the portfolio is around 7.5%, a DR of 3 means the weighted St. Dev of the holdings is 22.5%. I don't think a single one of the St. Dev. of the holdings is even that high (maybe VWO if that).
2) Another interpretation of DR is that the square is equivalent to the number of uncorrelated assets. If you hold two uncorrelated assets with identical St. Dev, the DR will be 1.41. Because 1.41^2 = 2.
A DR of 3 implies 9 different uncorrelated holdings. The portfolio has 9 holdings. Only way you get a DR=3 is if they are all completely uncorrelated. And I'm pretty sure they aren't.

I've been wrong before of course.

Thank you for the heads up (great peer review!). I had something funky in my code, some extra () and a missing sqrt(). I corrected the post. Directionally it doesn't change anything (meaning each change did add improve the ratio), but the value of the ratio was wrong. Cheers

Yes your numbers seem much more reasonable.

The last thing I'll say is that Diversification Ratio is a relative measurement, not an absolute one. So calculating the DR of the portfolio in Step 8, comparing it to the 3-Fund, and concluding the final portfolio is more diversified isn't actually correct. You can't compare DRs between portfolios of different assets entirely. Certainly not when you're increasing the number of assets since we know it's not associative.

D-dog showed the discrepancy back in this thread:
viewtopic.php?t=275899&start=50

Ex: You could substitute VTI for a representative weighting in Vanguard sector funds and the 3-Fund portfolio (now the 12+ Fund portfolio) will have a higher DR. Even though it's basically the same underlying holdings, it'll look "more diversified". And your final portfolio won't "look as good in comparison"; even though, again, it's the exact same holdings.

Just beware of the shortcomings of the metric and don't use it in comparisons where it's not accurate (like this thread seems to suggest).

Cheers mate

Very interesting, the good thing about the exercise above is that the diversification wasn't a mindless optimization routine, it also was congruent with:

>> Adding assets that have different properties (ie. EM Bonds)
>> If one believes in equity style factors, it was diversified across those for US equities (Quality, Size, Value, MOM, BAB)
>> The holdings weights were very close to risk parity without an huge treasury position and, like most of those optimizations end up.

Will keep looking for ways to objectively measure diversification. Cheers

[305pelusa]

I will once I get to the desk. Why are you positive about this?

Two reasons.
1) If the standard dev. of the portfolio is around 7.5%, a DR of 3 means the weighted St. Dev of the holdings is 22.5%. I don't think a single one of the St. Dev. of the holdings is even that high (maybe VWO if that).
2) Another interpretation of DR is that the square is equivalent to the number of uncorrelated assets. If you hold two uncorrelated assets with identical St. Dev, the DR will be 1.41. Because 1.41^2 = 2.
A DR of 3 implies 9 different uncorrelated holdings. The portfolio has 9 holdings. Only way you get a DR=3 is if they are all completely uncorrelated. And I'm pretty sure they aren't.

I've been wrong before of course.

Thank you for the heads up (great peer review!). I had something funky in my code, some extra () and a missing sqrt(). I corrected the post. Directionally it doesn't change anything (meaning each change did add improve the ratio), but the value of the ratio was wrong. Cheers

Yes your numbers seem much more reasonable.

The last thing I'll say is that Diversification Ratio is a relative measurement, not an absolute one. So calculating the DR of the portfolio in Step 8, comparing it to the 3-Fund, and concluding the final portfolio is more diversified isn't actually correct. You can't compare DRs between portfolios of different assets entirely. Certainly not when you're increasing the number of assets since we know it's not associative.

D-dog showed the discrepancy back in this thread:
viewtopic.php?t=275899&start=50

Ex: You could substitute VTI for a representative weighting in Vanguard sector funds and the 3-Fund portfolio (now the 12+ Fund portfolio) will have a higher DR. Even though it's basically the same underlying holdings, it'll look "more diversified". And your final portfolio won't "look as good in comparison"; even though, again, it's the exact same holdings.

Just beware of the shortcomings of the metric and don't use it in comparisons where it's not accurate (like this thread seems to suggest).

Cheers mate

Very interesting, the good thing about the exercise above is that the diversification wasn't a mindless optimization routine, it also was congruent with:

>> Adding assets that have different properties (ie. EM Bonds)
>> If one believes in equity style factors, it was diversified across those for US equities (Quality, Size, Value, MOM, BAB)
>> The holdings weights were very close to risk parity without an huge treasury position and, like most of those optimizations end up.

Will keep looking for ways to objectively measure diversification. Cheers

I have no problem with the methodology and I agree with the addition of LT bonds, and EM bonds (not sure about the Low Vol funds from a diversification perspective but you know me, I'm not sold on those).

It's just that as you add funds, or split holdings, the DR will go up. So it's impossible to judge whether you're actually improving things; at least not with DR.

Something else to keep in mind is that diversification across different factors might well be useful. But you will NOT see that via the St Dev. That's because factor risk isn't represented in the St Dev; otherwise it wouldn't be a different factor all together. I only recently learned this.

Writes Fama:
"This brings up a tricky aspect of the research. Let’s review. Fama and French identified three independent sources of risk in stock market returns. For these risks to be truly independent, we expect them not to manifest themselves in the same way. If the return differences could all be explained by a shared source of risk like standard deviation, we’d be back to a single-factor model."

https://www.ifa.com/academic-papers/mul ... a_2006.pdf

So maybe there's Diversification from different factors. It certainly sounds logical (although that doesn't make it true). But even if there were benefits, I'm not sure you could see it with standard measurements and tools like DR. I haven't come up with a good alternative yet, still reading away haha.

[305pelusa]

Why would you want to have a megacap fund with a momentum tilt when it comes with large negative size and value exposure? There are well-designed mutlifactor and small cap value funds (such as those from DFA) that have larger momentum exposure than MTUM plus positive size/value exposure.

Do you have any examples other than DFA? Like an ETF? I haven't found any with a MUM load as high as MTUM thus far. Then again I haven't looked that hard haha. It'd be really helpful if you had one in mind.

Thank you very much!

I think I used the wrong factor set to do my analysis, the fund i was looking at (VFMF, Vanguard U.S. Multifactor ETF) has higher momentum exposure than MTUM with AA's factor set, but not with FF or ARQ.

The DFA approach is to screen for negative momentum before looking at size/value. This avoids the large negative momentum exposure that some value stocks have (value and momentum are negatively correlated). I think this is a much more sensible approach than targeting momentum directly. After all, it's the overall portfolio factor exposure that counts.

Mkay. Yeah the ideal is for the value tilt to not introduce much negative momentum. Some funds are pretty good at this even without screens (like the S&P600 Value funds), but they invariably bring a little. I've been looking for a strong MOM fund to counteract it but MTUM seems best thus far.

The "problem" of Multi-factor fund is they bring much less MOM. That is ok because since they don't bring negative value/size, you could just add more of it and achieve a similar exposure. But I do have a preference for value exposure from SC because the premium is a little stronger there historically.

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

Of course the ideal is a fund as MOM loaded as MTUM without the negative size/value. But I'm still looking haha

[hdas]

Mkay. Yeah the ideal is for the value tilt to not introduce much negative momentum. Some funds are pretty good at this even without screens (like the S&P600 Value funds), but they invariably bring a little. I've been looking for a strong MOM fund to counteract it but MTUM seems best thus far.

The "problem" of Multi-factor fund is they bring much less MOM. That is ok because since they don't bring negative value/size, you could just add more of it and achieve a similar exposure. But I do have a preference for value exposure from SC because the premium is a little stronger there historically.

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

Of course the ideal is a fund as MOM loaded as MTUM without the negative size/value. But I'm still looking haha

I think I mentioned this to you before, but this is the reason I hold XSLV, it is itself a true multifactor fund:

[hdas]

I have no problem with the methodology and I agree with the addition of LT bonds, and EM bonds (not sure about the Low Vol funds from a diversification perspective but you know me, I'm not sold on those).

I think its worthwhile for you to look into it. Here's a good place with references (this is all before Low Vol exploded)

It's just that as you add funds, or split holdings, the DR will go up. So it's impossible to judge whether you're actually improving things; at least not with DR.

Totally agree, I don't think I added a single fund that was just for the sake of it.

MTUM - MOM loading, I guess you could replace with VUG or QQQ (the latter messes up your value and BAB)
ISCF - I think there's evidence of the larger corr of Large International with US and the particular local risk exposures to small companies.
VIOV - XSLV - Value and Size loadings
Low Vol Funds - Perhaps we agree to disagree on these for now.

Something else to keep in mind is that diversification across different factors might well be useful. But you will NOT see that via the St Dev. That's because factor risk isn't represented in the St Dev; otherwise it wouldn't be a different factor all together. I only recently learned this.

Writes Fama:
" we expect them not to manifest themselves in the same way. If the return differences could all be explained by a shared source of risk like standard deviation, we’d be back to a single-factor model."

https://www.ifa.com/academic-papers/mul ... a_2006.pdf

So maybe there's Diversification from different factors. It certainly sounds logical (although that doesn't make it true). But even if there were benefits, I'm not sure you could see it with standard measurements and tools like
DR.

Interesting, I certainly can see this is true with Low Vol, where the risk is def not manifested in the SD, and the way low vol underperforms is different yet rather predictable.

I haven't come up with a good alternative yet, still reading away haha.

Have you look into complexity theory stuff......entropy and what not?

Cheers

[hdas]

As best I can tell, you start off with 30% of your portfolio in international stocks market cap weighted. Seems fine. At the end you end up with less in international stocks (20%, with half of that in low vol international), a heavy tilt to small stocks in the US, replace total bond with LTTs, and throw in some EM bonds in place of the international stocks you lost and some of the US stocks. By sector composition, you dramatically overweight REITs, and significantly underweight energy, healthcare, and tech.

Is the goal to show one way you would have done better in the last decade or so by tilting away from the standard 3 fund portfolio? Because the "optimized" portfolio appears just to be a performance chaser, that overweights US small stocks, underweights international stocks, and tilts the bonds to LTTs and EM bonds. If there is any mean reversion in the future, it's likely large cap, value, and international tilts, with short term Treasuries, will be the better from a diversification standpoint. So if I was designing a portfolio for future potential outperformance and "diversification", I would do precisely the opposite of what you did. Or just stick with step 1 and not try to chase performance.

I think you bring up a good point about exposure to international stocks, specifically un-hedged exposure. I disagree with you regarding the "performance chasing" element, non of this exercise included information regarding returns, just volatility and correlation. That being said the analysis was contrived by my own holdings and preferences. Cheers

[Uncorrelated]

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

That is not true. Two funds with the same value exposure have the same expected excess factor performance. But it is easier to build a small cap fund with strong value exposure than it is to build a large cap fund with strong value exposure because, as you say, the value effect is stronger in small caps.

Note that two funds with same same factor exposure might have widely different tracking errors, but there is no reliable way to predict future tracking error. The reason why you don't go all-in on small cap value momentum low-volatility funds is that the tracking error is too large because there are only a few stocks with positive exposure to all factors at the same time. (idiosyncratic risk)

For example, XSLV has only 118 holdings which translates into high idiosyncratic risk. Even if value consistently outperforms blend, a fund with a value exposure of 0.4 can easily under perform the wider market because of tracking error.

[305pelusa]

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

That is not true. Two funds with the same value exposure have the same expected excess factor performance.

Construct a long-short portfolio based on FF BtM methodology for SC stocks. Look at its returns.

Construct a long-short portfolio based on BtM in LC stocks. Look at its returns.

Historically, the former portfolio had a higher return. How do you explain it? I think by definition they both have a value exposure of 1.0 no?

[Uncorrelated]

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

That is not true. Two funds with the same value exposure have the same expected excess factor performance.

Construct a long-short portfolio based on FF BtM methodology for SC stocks. Look at its returns.

Construct a long-short portfolio based on BtM in LC stocks. Look at its returns.

Historically, the former portfolio had a higher return. How do you explain it? I think by definition they both have a value exposure of 1.0 no?

HML is defined as:

HML = 1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth).

I suspect, but have not verified, that SC value has a value of exposure around 1.5 and LC value has a value exposure 0.5. This also explains how it is possible to get a value exposure above 0.5 with a long-only portfolio.

[305pelusa]

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

That is not true. Two funds with the same value exposure have the same expected excess factor performance.

Construct a long-short portfolio based on FF BtM methodology for SC stocks. Look at its returns.

Construct a long-short portfolio based on BtM in LC stocks. Look at its returns.

Historically, the former portfolio had a higher return. How do you explain it? I think by definition they both have a value exposure of 1.0 no?

HML is defined as:

HML = 1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth).

I suspect, but have not verified, that SC value has a value of exposure around 1.5 and LC value has a value exposure 0.5. This also explains how it is possible to get a value exposure above 0.5 with a long-only portfolio.

I see. It's not that the value premium is bigger in small cap. It's that small caps have a higher exposure to the value factor itself. Which are equivalent statements if you think about it but I hadn't made the connection.

Thank you!

[305pelusa]

Mkay. Yeah the ideal is for the value tilt to not introduce much negative momentum. Some funds are pretty good at this even without screens (like the S&P600 Value funds), but they invariably bring a little. I've been looking for a strong MOM fund to counteract it but MTUM seems best thus far.

The "problem" of Multi-factor fund is they bring much less MOM. That is ok because since they don't bring negative value/size, you could just add more of it and achieve a similar exposure. But I do have a preference for value exposure from SC because the premium is a little stronger there historically.

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

Of course the ideal is a fund as MOM loaded as MTUM without the negative size/value. But I'm still looking haha

I think I mentioned this to you before, but this is the reason I hold XSLV, it is itself a true multifactor fund:

I briefly looked at XLSV back when you first brought it up to me. I used the data for its underlying index since 2009. Ten years is way too short of a time period to judge factor exposure but it's better than the 6 years that you'd get from PV.

I used FF 5 factor regression to find:
Beta = 0.7
SmB = 0.66
HmL = 0.08
RmW = 0.25
CmA = 0.13

With an alpha of 0.22, albeit statistically insignificant (p-value of 0.13). The R^2 was 0.92, which tells me the above does a much better explanatory job than PV seemed to do for your screenshot, although I still think it's rather low. More data would be better of course. With the understanding that this is way too little data to form good conclusions:

Conclusion
I've said it before and I'll say it again: I think XLSV looks to be a pretty stacked fund. I personally do not tilt to RmW or CmW. I tilt to size and value, in which case this fund is not particularly worthwhile since it has a very light value loading. Also, I tilt to factors to increase the risk/return of my stock side so funds that have less beta would be counterproductive. Yes, I know that less beta means just as good returns historically (low vol anomaly) but this is another subject I'm not sold on yet (you know this).
Overall, it's not a fund that fits well with my portfolio and targets. That said, for someone looking to tilt towards these various factors, and bullish on the BAB anomaly, it looks like an excellent choice.

Just some thoughts.

Have you look into complexity theory stuff......entropy and what not?

Cheers

I have not yet aside from some pretty preliminary stuff (the actual calculation of entropy for instance). It's on my to-do list, hopefully I'll get to it eventually haha.

Cheers mate.

[asif408]

I disagree with you regarding the "performance chasing" element, non of this exercise included information regarding returns, just volatility and correlation. That being said the analysis was contrived by my own holdings and preferences. Cheers

Although you didn't include return information, your own holdings and preferences are overweighted towards the recent winners. Hopefully for you they will continue to be.

[klaus14]

Mkay. Yeah the ideal is for the value tilt to not introduce much negative momentum. Some funds are pretty good at this even without screens (like the S&P600 Value funds), but they invariably bring a little. I've been looking for a strong MOM fund to counteract it but MTUM seems best thus far.

The "problem" of Multi-factor fund is they bring much less MOM. That is ok because since they don't bring negative value/size, you could just add more of it and achieve a similar exposure. But I do have a preference for value exposure from SC because the premium is a little stronger there historically.

In other words, given two portfolios with identical factor loadings, I have a preference for the one that has mostly SCV (like SLYV). Say 80% SLYV/20% MTUM vs 50% SLYV/50% MF. They might have similar loadings, but I would prefer the former.

Of course the ideal is a fund as MOM loaded as MTUM without the negative size/value. But I'm still looking haha

I think I mentioned this to you before, but this is the reason I hold XSLV, it is itself a true multifactor fund:

I briefly looked at XLSV back when you first brought it up to me. I used the data for its underlying index since 2009. Ten years is way too short of a time period to judge factor exposure but it's better than the 6 years that you'd get from PV.

I used FF 5 factor regression to find:
Beta = 0.7
SmB = 0.66
HmL = 0.08
RmW = 0.25
CmA = 0.13

With an alpha of 0.22, albeit statistically insignificant (p-value of 0.13). The R^2 was 0.92, which tells me the above does a much better explanatory job than PV seemed to do for your screenshot, although I still think it's rather low. More data would be better of course. With the understanding that this is way too little data to form good conclusions:

Conclusion
I've said it before and I'll say it again: I think XLSV looks to be a pretty stacked fund. I personally do not tilt to RmW or CmW. I tilt to size and value, in which case this fund is not particularly worthwhile since it has a very light value loading. Also, I tilt to factors to increase the risk/return of my stock side so funds that have less beta would be counterproductive. Yes, I know that less beta means just as good returns historically (low vol anomaly) but this is another subject I'm not sold on yet (you know this).
Overall, it's not a fund that fits well with my portfolio and targets. That said, for someone looking to tilt towards these various factors, and bullish on the BAB anomaly, it looks like an excellent choice.

Just some thoughts.

Have you look into complexity theory stuff......entropy and what not?

Cheers

I have not yet aside from some pretty preliminary stuff (the actual calculation of entropy for instance). It's on my to-do list, hopefully I'll get to it eventually haha.

Cheers mate.

Did you check XSLV sector composition? It is:
Financials 43%
Real Estate 25%


Basically a sector fund. I would avoid.

[Uncorrelated]

I did some quick calculations and came to the conclusion that diversification ratio is thoroughly broken. For example take the following portfolio: VTI (50%) + walmart (50%)

Walmart has a correlation of .26 with VTI. This portfolio has a higher sharpe ratio than just VTI and a diversification ratio of 1.25.

Of course, the idea that a portfolio of 50% Walmart is better diversified than VTI is completely nonsensical. I hope that this example is sufficient to explain the dangers of backtesting.



To add to asif408: this is 100% performance chasing. You just defined performance as low correlation instead of high return.

[Hydromod]

I did some quick calculations and came to the conclusion that diversification ratio is thoroughly broken. For example take the following portfolio: VTI (50%) + walmart (50%)

Walmart has a correlation of .26 with VTI. This portfolio has a higher sharpe ratio than just VTI and a diversification ratio of 1.25.

Of course, the idea that a portfolio of 50% Walmart is better diversified than VTI is completely nonsensical. I hope that this example is sufficient to explain the dangers of backtesting.



To add to asif408: this is 100% performance chasing. You just defined performance as low correlation instead of high return.

Just out of curiosity, what part of the exercise contained backtesting? I missed it somehow.