Correlation and volatility reduction

[IndependentlyPoor]

Warning: Long and Geeky

I have been interested in "modern portfolio theory" for some time. Basically, it is the something-for-nothing aspect of reducing a portfolio's volatility be combining funds with negative correlations that intrigues me.

While there is nothing wrong with combining a money market fund or a short term bond fund with a high volatility stock fund, you miss out on the something-for-nothing aspect. When you combine two such funds, the resulting portfolio will always have greater volatility than the money market fund, along with the greater return (usually). You are just making a risk/return trade off, depending on the relative allocation of the "risky" and "safe" funds. This is a result on the two funds having essentially zero correlation.

However, if you combine two funds that are negatively correlated, you can (theoretically) produce a portfolio that has less volatility than either fund.

To test this out, I wrote a script that found the optimum allocation of all possible two-fund portfolios of Vanguard stock and bond funds (not including money market funds, because I couldn't get data for them), 700 some-odd portfolios. I defined optimum as the allocation that produced the minimum volatility, as measured by the standard deviation of the percentage change in NAV.

Here are the results from the portfolio that enjoyed the greatest reduction in volatility, the FTSE All-World ex-US Small-Cap Index (VFSVX), and the Long-Term Treasury Fund (VUSTX).

The graph shows the the Equivalent APR vs. the standard deviation. The minimum volatility of about 0.6% daily change in NAV occurs at about a 30/70 allocation. Now VFSVX has been around for less than a year, and has a spectacular rise, so a portfolio of these two funds is not a candidate for a core holding, but it does show how the risk return of portfolios comprised of funds with high negative correlations look.

Notice that the minimum volatility point is less than that for either fund by itself. A portfolio of 100% VFSVX had a 60%+ return last year, but also more than 1% standard deviation in the daily NAV, Ouch. VUSTX also had a good year with about 6% return and 0.8% standard deviation, but the 30/70 mix more than doubled the yield of a 100% VUSTX portfolio, while reducing its volatility. That is the something-for-nothing aspect that interested me.

On the other hand, portfolios constructed of positively correlated tended to look like this graph of portfolios of the Emerging Markets Index Fund and the High-Yield Bond Fund:

The minimum volatility occurred at 0% VEIEX and increased almost linearly with increasing VEIEX allocation.

Unfortunately, more popular combinations, such as Total Stock (VTSMX) and Total Bond (VBMFX) don't show much benefit in volatility reduction, and the minimum occurs at about 10% VTSMX.


All of this is sort of interesting, but the effects are small. The best standard deviation reduction in percent daily NAV change was about 0.2%... not earth shattering. It seems like that while the claims made for "modern portfolio theory" are true, the effect is just too small to be very important. No free lunch after all.

 

[ERD50]

Yes, and the problem is that while you can mathematically take advantage of the non-correlation, you need to find funds that are both non-correlated and provide an upward bias. I haven't looked through your examples in detail, but I suspect that your conclusion is correct - 'the effect is just too small to be very important.'.

But it is interesting and it is real - thanks for posting. Maybe we will stumble upon the magic combination here on the forum! Well, at least the historically magic combination...

-ERD50

 

[IndependentlyPoor]

Maybe we will stumble upon the magic combination here on the forum! Well, at least the historically magic combination...

-ERD50

Yep. Hours of grinding numbers with sophisticated software on a powerful computer and it tells you that you should have bought AAPL in 1985...

 

[Gone4Good]

It seems like that while the claims made for "modern portfolio theory" are true, the effect is just too small to be very important. No free lunch after all.

Correlations change over time, so you're not getting the same result that Markowitz did in the 1950's. We've seen correlations increase dramatically across asset classes recently. Even commodities are now reasonably positively correlated with equities. It's likely due to the "mutual-fundization" of investments. The great virtue of mutual funds is that it opens asset classes up to the masses. But when the masses herd in and herd out of everything at the same time you kind of loose the benefit you were striving for.

The one big stand-out in the last correction was treasury bonds. I suspect the flight to safety trade will almost always be in style when other assets are cracking. So instead of using the bond market index as a diversifier, or worse yet, a corporate bond fund, you probably want to use intermediate treasuries.

Oh, and don't forget the massive risk reduction you get because of the diversified equity market portfolio. Run those same calculations with an individual stock, try Citigroup for example, and see what you get.

 

[kyounge1956]

IP, out of curiosity, of all the fund pairs you looked at, which one had the lowest standard deviation, and what was its return? Which pair had the highest return per unit of standard deviation? And how long a period did your data cover? i.e. is this the last year, last decade, longest available period for any given fund, or what? When I was following the Asset Allocation Tutorial thread, I looked into lots of different asset classes, but I was working from information published in books, and for many of the asset classes I wanted to look at there were only a few years of data, and sometimes I could not get data over the same period for all the asset classes I wanted to run through my spreadsheet. I take it this fund data is available online? I will have to check it out and see if I can improve on my current target allocation.

 

[IndependentlyPoor]

for many of the asset classes I wanted to look at there were only a few years of data, and sometimes I could not get data over the same period for all the asset classes I wanted to run through my spreadsheet. I take it this fund data is available online? I will have to check it out and see if I can improve on my current target allocation.

You have run into the same problem as I have. Comparisons between portfolios seems to be only valid if the time periods are the same. Unfortunately, many of the most interesting funds (promising for diversification) haven't been around for long. I wanted to run this scan over the broadest possible range of portfolios, so calculations for each pair are done over the maximum time frame for which data is available for both funds. The range of time frames is extreme. I should include the starting date in the results. Oops.

The data comes from Mathematica's server.

I should point out that I am doing this only as a substitute for crossword puzzles or knitting. I do not expect to discover anything significant about investing.

I'll try to post some more data that address your specific questions later. Right now my poor laptop is cranking away at another 770 portfolios.

 

[kyounge1956]

(snip)Oh, and don't forget the massive risk reduction you get because of the diversified equity market portfolio. Run those same calculations with an individual stock, try Citigroup for example, and see what you get.

Gone4good, is there a rule of thumb about how many individual stocks are required to reduce the volatility of the group to about that of the market as a whole? I vacillate between wanting to be an index investor (for all the usual reasons) and wanting to invest in individual stocks (to be more socially responsible).

Of course, there are such things as SRI funds, but I don't know if any of them filter for exactly the same set of things I'd want to, nor if I could do the filtering myself on a few stocks for less than the expenses of a fund. I've looked at a few of the Roman Catholic oriented funds, which are probably the closest thing available to what I would filter for, but as I recall the expense ratios were quite high, up to around 2.3% (this was some years ago and expenses may have changed since then). I also don't remember how actively the funds were managed. There might have been efforts toward market timing as well as the issues-based filtering, and I wouldn't want that in any case. Could an individual investor buy and hold a basket of individual, filtered stocks at lower cost than that?

 

[Gone4Good]

Gone4good, is there a rule of thumb about how many individual stocks are required to reduce the volatility of the group to about that of the market as a whole? I vacillate between wanting to be an index investor (for all the usual reasons) and wanting to invest in individual stocks (to be more socially responsible).

I seem to recall the magic number to be around 20. That assumes, of course, that those 20 stocks are pretty broadly diversified. It wouldn't do much good, for example, to own 20 makers of wind turbines.

 

[IndependentlyPoor]

Correlations change over time, so you're not getting the same result that Markowitz did in the 1950's. We've seen correlations increase dramatically across asset classes recently. Even commodities are now reasonably positively correlated with equities. It's likely due to the "mutual-fundization" of investments. The great virtue of mutual funds is that it opens asset classes up to the masses. But when the masses herd in and herd out of everything at the same time you kind of loose the benefit you were striving for.

The one big stand-out in the last correction was treasury bonds. I suspect the flight to safety trade will almost always be in style when other assets are cracking. So instead of using the bond market index as a diversifier, or worse yet, a corporate bond fund, you probably want to use intermediate treasuries.

Oh, and don't forget the massive risk reduction you get because of the diversified equity market portfolio. Run those same calculations with an individual stock, try Citigroup for example, and see what you get.

Yep, yep, and yep
Correlations do seem to have changed. In fact, they don't seem stable to me at all. Possibly so unstable as to call into question using them like this at all. Somewhere I have a bunch of plots of rolling 60 or 90 day correlations and they are all over the place.

The recent unpleasantness has tossed all this sort of analysis for a loop. Risk/return curves are upside down, with treasuries out performing stocks. (Well, at least that was true the last time I looked - a couple of thousand points ago.)

My recent runs tell me that if I had just put everything into health care and long term treasuries I would be rich now.

Running portfolios of individual stocks would be interesting, except that I am drawing a blank on which to run as examples. Suggest some ticker symbols, and I'll run'em. Give me two lists, 5 or 10 symbols each, and my script will run find the optimum allocation for each portfolio composed on one stock from one list and the other from the second list.

 

[Gone4Good]

Yep, yep, and yep
Correlations do seem to have changed. In fact, they don't seem stable to me at all. Possibly so unstable as to call into question using them like this at all.

I personally think they're unstable enough so that using them to "optimize" a portfolio is pointless. Some folks tie themselves in knots trying to construct the perfect portfolio and completely overlook the fact that the benchmarks they're using to build that portfolio are variable.

 

[kyounge1956]

You have run into the same problem as I have. Comparisons between portfolios seems to be only valid if the time periods are the same. (snip)

I was only guessing that the data should all come from the same time period. It seemed like mixing apples and oranges otherwise. Seems my intuition was correct, since as G4G has pointed out, the correlations can change over time. The references I was using said the same thing.

I should point out that I am doing this only as a substitute for crossword puzzles or knitting. I do not expect to discover anything significant about investing.

I'll try to post some more data that address your specific questions later. Right now my poor laptop is cranking away at another 770 portfolios.

You can do the crossword while your laptop is slaving away. And if you ever have any questions about knitting, don't hesitate to ask. I'm an old hand.

P.S. Can your laptop compare groups of three funds, or only pairs?

 

[IndependentlyPoor]

P.S. Can your laptop compare groups of three funds, or only pairs?

Yep, it can do threebies. Actually, any number is theoretically possible, it just becomes a hassle to program. I'll try to post some results with consistent time periods.

Right now my overworked MacBook is optimizing portfolio 588 out of 770.

There is a big difference between optimizing for minimum risk (lowest standard deviation) and return/risk. Right now I am just looking at minimum risk. These days (after the crash) looking at risk/return tends to just tell you to buy treasuries. Maybe that is changing some now that the market is recovering.

 

[kyounge1956]

Yep, it can do threebies. Actually, any number is theoretically possible, it just becomes a hassle to program. I'll try to post some results with consistent time periods.

I found the formula in a book and made spreadsheets to do the calculations. The most asset classes I can handle at one time is 6, which is the most I have simultaneous data for anyhow.

Right now my overworked MacBook is optimizing portfolio 588 out of 770.

I can almost here the numbers crunching!

There is a big difference between optimizing for minimum risk (lowest standard deviation) and return/risk. Right now I am just looking at minimum risk. These days (after the crash) looking at risk/return tends to just tell you to buy treasuries. Maybe that is changing some now that the market is recovering.

Are treasury bonds also the lowest standard deviation overall? Maybe the highest return/risk isn't actually what I meant. When I was looking for a target allocation I picked a maximum standard deviation and then looked for the combination of assets that would give me the most return without exceeding the volatility ceiling. Then I plugged that return into the online Monte Carlo modeler I use to make sure it was adequate for portfolio survival. It's probably time to check again and see if I'm on course or not.

 

[kyounge1956]

I personally think they're unstable enough so that using them to "optimize" a portfolio is pointless. Some folks tie themselves in knots trying to construct the perfect portfolio and completely overlook the fact that the benchmarks they're using to build that portfolio are variable.

I have to plead a little guilty there, but I hope they're good enough to get me in the ballpark, or for broad generalizations like "any significant allocation to such and such an asset class makes the portfolio more volatile than I care for", or "....makes it return less than I need it to", or "...doesn't make much of a difference either way".

 

[IndependentlyPoor]

Okey dokey, I generated 770 portfolios, each composed of one Vanguard stock fund and one other Vanguard fund (no money market funds though). For each portfolio, I found the asset allocation that produces the minimum volatility.

Here is the amount of volatility reduction achieved in minimum volatility portfolios plotted against correlation. It is clear that something is affecting volatility other than correlation.


The best way to view the data is to download the Mathematica viewer. It is free, works on both Mac and PC, and is easy to install. Here is the link.

I need a drink.
Wolfram Mathematica Player: Download
Each point represents a portfolio with the asset allocation chosen to produce the minimum volatility. The charts are interactive. When you hover the cursor over a point, a popup displays the asset allocation and return for that portfolio.
Here is the link for the interactive chart:
http://www.wolfram.com/solutions/in...=1734456552&filename=allStockBond+publish.nbp

If you want to see the shape of the risk/return curves underlying each minimum volatility portfolio, download and view this interactive chart.
http://www.wolfram.com/solutions/in...836087723&filename=allStockBond+2+publish.nbp
The popups look like this.

They show the asset allocation and the return of the minimum volatility portfolio, in addition to the risk/return curve of several asset allocations. The minimum volatility point is the leftmost point on the popup graphs.

 

[Midpack]

During the meltdown of the 2008-09 I would agree that correlations broke down and everything declined. However, I still think it's wise to be aware of the correlation of all assets and to try to utilize some negative correlation. While it may not pan out all the time, what is the downside of attempting this strategy to reduce volatility? It has worked (to some extent) for me most of the time.

The Benefits of Low Correlation - JOI Articles - IndexUniverse.com - ETFs, Index Funds, Indexes

 

[Texas Proud]

I seem to recall the magic number to be around 20. That assumes, of course, that those 20 stocks are pretty broadly diversified. It wouldn't do much good, for example, to own 20 makers of wind turbines.

When I took portfolio analysis in college.... it was 30... who knows now...


I always find it interesting that some people say the S&P 500 is not diversified... it is... it might not be the portfolio you want, but it IS diversified...

 

[IndependentlyPoor]

Here are the top few portfolios sorted by return/volatility.
Again, please note that these portfolios were calculated with different time periods. I don't know if there is any useful information here at all other than that bonds have been good recently.

Here is a pdf with the 100 best return/volatility ratios.
View attachment correlation and risk reduction.pdf

Finally, I should apologize for the misleading titles on the popup graphs in the previous post. The graphs all all labeled at the top with something like 63/37 VMNFX/VIPSX. The 63/37 is the asset allocation of the minimum volatility VMNFX/VIPSX portfolio, but the graph is of the risk/return of various allocations of VMNFX/VIPSX from 100/0 to 0/100.

 

[audreyh1]

When I took portfolio analysis in college.... it was 30... who knows now...


I always find it interesting that some people say the S&P 500 is not diversified... it is... it might not be the portfolio you want, but it IS diversified...

I think when they say that, they mean S&P500 is all US-based large-cap stocks. If you want small-cap, mid-cap or foreign stocks, you're not going to find them in the S&P500.

Audrey

 

[ziggy29]

Yep. Hours of grinding numbers with sophisticated software on a powerful computer and it tells you that you should have bought AAPL in 1985...

Even if you missed that go-round, you still had another shot around 1997 or so, when the company was in a Gil Amelio-led death spiral and the stock was basically wallpaper. Of course, you'd need to know Steve Jobs would come back and resurrect the product line, or else it would have felt like investing in Enron.

 

[Nords]

I think when they say that, they mean S&P500 is all US-based large-cap stocks. If you want small-cap, mid-cap or foreign stocks, you're not going to find them in the S&P500.

I think it was Frank Armstrong who said "If everything in your portfolio is going up at the same time... then you're not diversified!"

 

[ERD50]

Even if you missed that go-round, you still had another shot around 1997 or so, when the company was in a Gil Amelio-led death spiral and the stock was basically wallpaper. Of course, you'd need to know Steve Jobs would come back and resurrect the product line, or else it would have felt like investing in Enron.

This was my big contrarian move, and it paid off big time for me.

Seemed like *everyone* I knew, and every news article was claiming that Apple was dead. You never saw "Apple" without the preface, 'beleaguered', 'troubled', etc.

But I was aware that they had a loyal following, and with the right moves they could still capitalize on that. It was still a risk, but at 'wallpaper' prices, I was up for it.

Funny thing was, a techno-friend of mine who had released applications for Apple computers, and was a big fan, said that he and a friend of his who worked for Apple at the time *sold* their stock when the iMac was announced. They felt that the iMac didn't really compete with the cheap PCs of the time. I told him he was right, but.... there were loyal Apple fans that had held off buying a computer for a long time, because Apple's desktop offerings were just not attractive to the home user. But they weren't quite ready to jump ship yet. So while the iMac was limited, and still relatively high $, it had an eager market ready, willing and able to drive up profits for Apple.

And the rest, as they say, is history

edit_ I guess my big buys were in 1998, near the lows IIRC...

05/15/98 AAPL B 14.875
12/01/98 AAPL B 15.875
12/09/98 AAPL B 16.000



-ERD50

 

[samclem]

Gone4good, is there a rule of thumb about how many individual stocks are required to reduce the volatility of the group to about that of the market as a whole?

It is more stocks than most people realize. Here's what William Bernstein said:

One of the most dangerous investment chestnuts is the idea that you can successfully diversify your portfolio with a relatively small number of stocks, the magic number usually being about 15. For example, Ben Graham, in The Intelligent Investor, suggests that adequate diversification can be obtained with 10 to 30 names. In a classic piece in Journal of Finance in 1968, Evans and Archer found that portfolios with as few as 10 securities had risk, measured as standard deviation, virtually identical to that of the market. Over the decades, the "15-stock diversification solution" has become enshrined in various texts and monographs . .
. . .
[in recent years] a grossly disproportionate fraction of the total return came from a very few "superstocks" like Dell Computer, which increased in value over 550 times. If you didn’t have one of the half-dozen or so of these in your portfolio, then you badly lagged the market. (The odds of owing one of the 10 superstocks are approximately one in six.) Of course, by owning only 15 stocks you also increase your chances of becoming fabulously rich. But unfortunately, in investing, it is all too often true that the same things that maximize your chances of getting rich also maximize your chances of getting poor.
If the O’Neal data are generalizable to stocks, and I believe that they are, then even 100 stocks are not nearly enough to eliminate this very important source of financial risk.

I vacillate between wanting to be an index investor (for all the usual reasons) and wanting to invest in individual stocks (to be more socially responsible).

You could just invest in index funds and send the money you saved in expenses to your favorite causes. It's a lot more direct and probably will do a lot more good, and you'll even get a tax write-off. Those "bad companies" are still going to do "bad stuff" whether you happen to own 3.254 shares of their stock or not.

 

[kyounge1956]

(snip) I vacillate between wanting to be an index investor (for all the usual reasons) and wanting to invest in individual stocks (to be more socially responsible). (snip)

(snip)You could just invest in index funds and send the money you saved in expenses to your favorite causes. It's a lot more direct and probably will do a lot more good, and you'll even get a tax write-off. Those "bad companies" are still going to do "bad stuff" whether you happen to own 3.254 shares of their stock or not.

I agree, bad companies would still be doing their bad stuff. The difference is, I'd no longer be directly benefiting from their badness. That's the thing I'd like to eliminate. I can't do away with it altogether, because most likely the pension fund I will be getting most of my income from owns stock in some of those very same companies. But still, I would prefer to reduce the portion of my income that could be described as "ill-gotten gains" to the absolute minimum.

I have a co-worker who's a Muslim, and that religion forbids loaning money at interest. The pension fund has lots of bonds in its holdings. That's a real dilemma for him.

 

[ziggy29]

I agree, bad companies would still be doing their bad stuff. The difference is, I'd no longer be directly benefiting from their badness. That's the thing I'd like to eliminate. I can't do away with it altogether, because most likely the pension fund I will be getting most of my income from owns stock in some of those very same companies. But still, I would prefer to reduce the portion of my income that could be described as "ill-gotten gains" to the absolute minimum

This is, in a nutshell, why "socially responsible investing" is problematic. Everyone's definition of what is "socially responsible" is somewhat different and no mutual fund can possibly meet anyone's criteria exactly.

So that's why I tend to focus not on the investment side, but on the return on investment side of the ledger. It's much easier to choose where to donate some of my profits in a way that *does* mirror my values than it is to choose where to invest it with my 100% blessing.

Let's say I support cause X. Building a diversified portfolio with mutual funds or ETFs is nearly impossible without holding some companies which are anti-X or whose policies undermine X. Hopefully if I'm in support of X, the direct donations I make in support of X will more than offset the profits I've made from the anti-X portion of my portfolio. The net result, hopefully, is that my money is working on the side of X.

Not perfect, but IMO most practical.