Understanding asset allocation, portfolio performance, and correlation coefficients

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tominsc
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Understanding asset allocation, portfolio performance, and correlation coefficients

Post by tominsc »

I’m just curious. I understand correlation coefficients in theory. They measure how much of the behavior of “something” can be attributed to “something else”. I also know that a well known idea about asset allocation is that it (the “something else” here) is the thing that affects portfolio performance (the “something”) to a higher extent than just about everything else that could be affecting portfolio performance.

That’s a convoluted way of saying, for example, “X % of the performance of any portfolio can be attributed to its asset allocation”. And THAT is a common layman’s way of saying “the correlation coefficient between portfolio performance and asset allocation is X %”.

So here’s my question. When academics study portfolio performance and asset allocation what data or numerical factors do they analyze mathematically to arrive at their conclusions? How do they do the math? What data do they use?

Decades ago, I read a book called Asset Allocation (by Gibson? not positive of the title or the author but this is close). And I understood it. But that was decades ago, so I fergit stuff!

Any ideas?
secondopinion
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by secondopinion »

tominsc wrote: Mon Apr 05, 2021 4:14 pm That’s a convoluted way of saying, for example, “X % of the performance of any portfolio can be attributed to its asset allocation”. And THAT is a common layman’s way of saying “the correlation coefficient between portfolio performance and asset allocation is X %”.

So here’s my question. When academics study portfolio performance and asset allocation what data or numerical factors do they analyze mathematically to arrive at their conclusions? How do they do the math? What data do they use?
That is not entirely correct; a correlation of 0.5 means that (0.5)^2 = 25% of the variance can be explained by the factors provided by a linear model.

I do not know where they precisely they do the numbers or get the data.

I would start with a basic statistics book first.
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alex_686
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by alex_686 »

I am not exactly sure what you are saying.

The most popular correlation coefficient is the stock/bond coefficient. In lay terms it is measuring how closely the 2 assets move together. In mathematically term it does not. It actually is measuring errors in difference between the linear relationship of 2 variables, in which we make lots of assumptions.

The lower the correlation the better the diversifier the assets are.

On the plus side correlation coefficients are easy to work with. You don't need much data to work with to generate actionable data.

Now, lets talk problems. For context, there is no generalized way of handling portfolio data. The problem is that things keep changing.

Correlations assume a stable linear relationship with constant volatility.

Any time there is a market crisis you are going to get volatility and correlations go to 1. Maybe it is because the 2 assets are moving in the same direction. Or maybe it is because you have garbage data so are getting garbage results.

Correlations are driven by the underlying economic structures. Economic structures change over time. So correlations change over time. A better answer is to use a jump diffusion. The downside to this is that you need graduate level math, fancy software, and lots of data.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by retired@50 »

For what it's worth, your memory seems to be pretty good. The book is called "Asset Allocation: Balancing Financial Risk" by Roger C. Gibson.

See link: https://www.goodreads.com/book/show/170 ... allocation

Regards,
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by nisiprius »

If you can get monthly or annual returns for two different asset classes, you can just load them into a spreadsheet and calculate the correlation coefficient yourself.

To be specific, when I do this myself, the numbers I get are the same as the numbers published by people who know what they're doing.

This is, of course, the sample correlation coefficient. If you make the usual pack of strong Statistics 101 assumptions--if you assume that the things you are looking at are, in fact, random variables whose characteristics do not change over time--then the sample correlation coefficient is an estimate of what you'd really like to know, the correlation coefficient between the two variables that are being sampled. The sad fact, which a lot of investment writers seem not to know or prefer to ignore, is that there is a heck of a lot of sampling error in the correlation coefficient; that is, the coefficient you calculate from the sample you draw can differ a lot from the "real" correlation coefficient.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by BJJ_GUY »

tominsc wrote: Mon Apr 05, 2021 4:14 pm I’m just curious. I understand correlation coefficients in theory. They measure how much of the behavior of “something” can be attributed to “something else”. I also know that a well known idea about asset allocation is that it (the “something else” here) is the thing that affects portfolio performance (the “something”) to a higher extent than just about everything else that could be affecting portfolio performance.

That’s a convoluted way of saying, for example, “X % of the performance of any portfolio can be attributed to its asset allocation”. And THAT is a common layman’s way of saying “the correlation coefficient between portfolio performance and asset allocation is X %”.

So here’s my question. When academics study portfolio performance and asset allocation what data or numerical factors do they analyze mathematically to arrive at their conclusions? How do they do the math? What data do they use?

Decades ago, I read a book called Asset Allocation (by Gibson? not positive of the title or the author but this is close). And I understood it. But that was decades ago, so I fergit stuff!

Any ideas?
Without going into much math, it's probably worth noting something often misunderstood when it comes to using returns, correlations etc. in a portfolio construction exercise. A few notables:
- Expected return of the portfolio is the weighted average of the expected return of each sub-category (asset class)
- Expected portfolio volatility (standard deviation) is not an arithmetical calculation, so don't use a simple average.
- The correlation between asset classes doesn't provide adequate in information, alone, to determine the marginal diversification benefits of inclusion
- A low volatility asset class without a high correlation to global equities can serve as a portfolio volatility reducer, but likely has limited diversification benefits
- Diversification benefit to a portfolio accounts for pairwise correlations, which is why portfolio volatility cannot be calculated as the simple average of volatility of each asset class. (This is also why it is often incorrect to say asset class A is riskier than B based purely on the volatility profile; in fact, if non-correlated, the higher volatility can improve the portfolio level risk-adjusted return expectations.)

*** Critically, do not use historical inputs when building a forward looking model. History informs the estimates, but valuation and other variables (accounting for various environments and how they change inputs) should drive the forecasted characteristics.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by watchnerd »

retired@50 wrote: Tue Apr 06, 2021 5:46 pm For what it's worth, your memory seems to be pretty good. The book is called "Asset Allocation: Balancing Financial Risk" by Roger C. Gibson.

See link: https://www.goodreads.com/book/show/170 ... allocation

Regards,
I read that a long time ago. It was a good book.

One of the big take aways is that AA accounts for something like 96% of portfolio performance.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by BJJ_GUY »

watchnerd wrote: Wed Apr 07, 2021 11:29 am
retired@50 wrote: Tue Apr 06, 2021 5:46 pm For what it's worth, your memory seems to be pretty good. The book is called "Asset Allocation: Balancing Financial Risk" by Roger C. Gibson.

See link: https://www.goodreads.com/book/show/170 ... allocation

Regards,
I read that a long time ago. It was a good book.

One of the big take aways is that AA accounts for something like 96% of portfolio performance.
If you use passive index funds/ETFs then AA accounts should account for all performance.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by alex_686 »

BJJ_GUY wrote: Wed Apr 07, 2021 11:45 am If you use passive index funds/ETFs then AA accounts should account for all performance.
You still need to account for any rebalancing.

I also think the OP might be referencing how you make your asset selection. Stock/Bond. Domestic/International. Maybe factors.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by BJJ_GUY »

alex_686 wrote: Wed Apr 07, 2021 11:57 am
BJJ_GUY wrote: Wed Apr 07, 2021 11:45 am If you use passive index funds/ETFs then AA accounts should account for all performance.
You still need to account for any rebalancing.

I also think the OP might be referencing how you make your asset selection. Stock/Bond. Domestic/International. Maybe factors.
Agreed.

I was just pointing out that the way this is typically calculated is contrasting 1.) asset class targes & benchmark returns; versus 2.) Actual asset class exposure and returns. In this method of calculating returns being driven by asset allocation, as opposed to selection effect, then all return attribution should derive from asset allocation even if rebalancing causes the policy portfolio (benchmarks) to be different from the actual portfolio returns.

Of course, there are a ton of different ways to answer these questions. The other problem here is understanding what question is actually being asked can lead to different methods of solving for an answer. For the sake of simplicity, I used this two factor model, just to shed light on a basic method that can be utilized.

(For a totally different discussion to avoid getting off the rails, but the often quoted % of returns driven by asset allocation, aren't really the right interpretation, if I recall correctly. I believe that 95% number refers to the degree to which asset allocation explains the variability of a cross section of pension fund returns or something like that. Either way, especially when indexing, that nuance doesn't really matter -- and that was what I meant to point out with my post about indexing would mean 100% explained by asset allocation.)
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by alex_686 »

BJJ_GUY wrote: Wed Apr 07, 2021 12:16 pm (For a totally different discussion to avoid getting off the rails, but the often quoted % of returns driven by asset allocation, aren't really the right interpretation, if I recall correctly. I believe that 95% number refers to the degree to which asset allocation explains the variability of a cross section of pension fund returns or something like that. Either way, especially when indexing, that nuance doesn't really matter -- and that was what I meant to point out with my post about indexing would mean 100% explained by asset allocation.)
I have seen various studies over time where the generally number is 80%. This ranges from retail investors picking actively managed mutual funds or individual stocks to a fund's performance against its chosen index.

And it may well be on track. I would like to hear back from the OP on what they are looking for. When I hear correlation coefficient my mind goes to asset allocation like stock / bond allocation. How do you select the best ratio between these 2? I is a important, complex, and ever shifting question.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by watchnerd »

BJJ_GUY wrote: Wed Apr 07, 2021 11:45 am
watchnerd wrote: Wed Apr 07, 2021 11:29 am
retired@50 wrote: Tue Apr 06, 2021 5:46 pm For what it's worth, your memory seems to be pretty good. The book is called "Asset Allocation: Balancing Financial Risk" by Roger C. Gibson.

See link: https://www.goodreads.com/book/show/170 ... allocation

Regards,
I read that a long time ago. It was a good book.

One of the big take aways is that AA accounts for something like 96% of portfolio performance.
If you use passive index funds/ETFs then AA accounts should account for all performance.
Even if all passive, there can be minor variances in correlation between a different index for the same macro segment.

e.g. Total Stock Market vs SP500, FTSE vs MSCI foreign indexes, etc.

And, of course, variances in expenses and index tracking.
Last edited by watchnerd on Wed Apr 07, 2021 2:00 pm, edited 1 time in total.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by Random Walker »

tominsc wrote: Mon Apr 05, 2021 4:14 pm Decades ago, I read a book called Asset Allocation (by Gibson? not positive of the title or the author but this is close). And I understood it. But that was decades ago, so I fergit stuff!

Any ideas?
Here is a short essay by Gibson which basically summarizes the book. It’s one of my very favorite pieces of investment reading. The specific asset classes mentioned are not what matters. It’s an appreciation for the general concept of combining less than perfectly correlated assets in a portfolio that matters. What a potential portfolio addition adds to a portfolio depends on expected return, volatility, correlations to other portfolio components.

https://ivinvestor.com/wp-content/uploa ... sting1.pdf

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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by nisiprius »

Couple of things.

1) Real-world correlations between asset classes, as measured with historical data, are unstable and fluctuate--just like return. To say that two asset classes "have" high correlation is the same kind of statement as saying that the ARK Innovation ETF "has" high return.

This is what the actual correlation between stocks and long-term government bonds has been, over 36-month periods:

Image

This is what the actual correlation between a commodities fund and stocks has been, over 36-month periods. Notice, in particular, the failure of the commodities fund to show low correlation with stocks during the global financial crisis:

source

Image

2) Because of this instability, using correlations in investing gets you into exactly the same issues as using trends or return. Are the correlations really changing, or is it just sampling error? If they are really changing, then you get into timing correlations, chasing low correlations, or trying to do fundamental or technical analysis of correlations.

3) Imperfect correlation, low or zero correlation, and negative correlation are not the same thing.

If it actually existed, an asset that combines positive return with persistent, reliable, robust negative correlation with stocks would be almost magic, because it could be used to erase or cancel out a lot of stock risk. The people following HEDGEFUNDIE believe that long-term Treasury bonds are exactly such an asset. I personally believe it is the financial equivalent of perpetual motion.

Low or zero correlation has some value, and probably justifies the use of intermediate-term bonds in a portfolio instead of saving accounts. Looking at past data, adding bonds would generally had added less risk to a portfolio that would have been expected based on its volatility (moderate to begin with).

It is often suggested that correlation need not be negative or zero to be valuable, and this is true. However, even when it would have improve a portfolio, correlations in the ballpark of 0.7 or so would not have improved it much, and it is always a balancing act with return itself. Over some stated period of time, a portfolio that includes an asset with lowish correlation would not necessarily have had higher risk-adjusted return. It is possible to find cases where "the magic happens," i.e. an asset with lower risk-adjusted return, nevertheless improves the risk-adjusted return of a portfolio when it is added to the portfolio--but it does not always happen.

4) The concept of correlation can be abused and touted as a "secret ingredient" to justify unnecessarily complex portfolios.

5) If an asset cannot boast either high return or high risk-adjusted return, it will sometimes be touted as having low correlation. Turning this around, if low correlation is being touted, one should look carefully to see if it is only being mentioned because there is nothing better to say about it.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by Random Walker »

To amplify a couple of BJJ’s points.
The expected return of a portfolio is the weighted simple average of the portfolio components.
When the correlation between portfolio components is less than 1, the expected volatility is less than the simple weighted mean standard deviation of the individual assets. The greater the weight, the greater the volatility, and the lower the correlations, the greater the diversification benefit derived from a potential portfolio addition.

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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by Random Walker »

nisiprius wrote: Wed Apr 07, 2021 2:04 pm Couple of things.

This is what the actual correlation between stocks and long-term government bonds has been, over 36-month periods:

Image
This is a great chart. It certainly demonstrates the fact that correlations change over time. Nonetheless I do think it’s fair to think about average correlations over time. With regard to bonds specifically, I think this chart shows something very important. Correlations between stocks and bonds overall tend I think to be uncorrelated or low positive correlation. But when stocks head south in a big way, that’s when the correlation to bonds tends to turn sharply negative: just when you would want it to. Notice the downward blue spikes around 1929-30, 2000-01, 2008, and 2020.

Dave
Last edited by Random Walker on Wed Apr 07, 2021 2:24 pm, edited 1 time in total.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by BJJ_GUY »

watchnerd wrote: Wed Apr 07, 2021 1:58 pm
BJJ_GUY wrote: Wed Apr 07, 2021 11:45 am
watchnerd wrote: Wed Apr 07, 2021 11:29 am
retired@50 wrote: Tue Apr 06, 2021 5:46 pm For what it's worth, your memory seems to be pretty good. The book is called "Asset Allocation: Balancing Financial Risk" by Roger C. Gibson.

See link: https://www.goodreads.com/book/show/170 ... allocation

Regards,
I read that a long time ago. It was a good book.

One of the big take aways is that AA accounts for something like 96% of portfolio performance.
If you use passive index funds/ETFs then AA accounts should account for all performance.
Even if all passive, there can be minor variances in correlation between a different index for the same macro segment.

e.g. Total Stock Market vs SP500, FTSE vs MSCI foreign indexes, etc.

And, of course, variances in expenses and index tracking.
You're conflating two distinct concepts. The returns of a portfolio of broad index funds will, by definition, be entirely explained by asset allocation decisions. If my global equity portfolio is invested in VT, then my benchmark should be the index that ETF tracks (otherwise, why on earth would you make things confusing with a different benchmark?). If you want to be super picky about it, I guess you could apply the tracking error to selection effect, but that really doesn't make logical sense.

Actual returns will not perfectly match the policy portfolio, but the difference is still explained by allocation decisions when every investment is in passive index funds.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by watchnerd »

BJJ_GUY wrote: Wed Apr 07, 2021 2:21 pm

You're conflating two distinct concepts. The returns of a portfolio of broad index funds will, by definition, be entirely explained by asset allocation decisions. If my global equity portfolio is invested in VT, then my benchmark should be the index that ETF tracks (otherwise, why on earth would you make things confusing with a different benchmark?). If you want to be super picky about it, I guess you could apply the tracking error to selection effect, but that really doesn't make logical sense.

Actual returns will not perfectly match the policy portfolio, but the difference is still explained by allocation decisions when every investment is in passive index funds.
Well, you can read the book and then report back on what you think he means.

It's been over 10 years since I read it.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by BJJ_GUY »

watchnerd wrote: Wed Apr 07, 2021 2:29 pm
BJJ_GUY wrote: Wed Apr 07, 2021 2:21 pm

You're conflating two distinct concepts. The returns of a portfolio of broad index funds will, by definition, be entirely explained by asset allocation decisions. If my global equity portfolio is invested in VT, then my benchmark should be the index that ETF tracks (otherwise, why on earth would you make things confusing with a different benchmark?). If you want to be super picky about it, I guess you could apply the tracking error to selection effect, but that really doesn't make logical sense.

Actual returns will not perfectly match the policy portfolio, but the difference is still explained by allocation decisions when every investment is in passive index funds.
Well, you can read the book and then report back on what you think he means.

It's been over 10 years since I read it.
I'm not sure what you mean. I wasn't commenting on a book
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by watchnerd »

BJJ_GUY wrote: Wed Apr 07, 2021 3:15 pm
watchnerd wrote: Wed Apr 07, 2021 2:29 pm
BJJ_GUY wrote: Wed Apr 07, 2021 2:21 pm

You're conflating two distinct concepts. The returns of a portfolio of broad index funds will, by definition, be entirely explained by asset allocation decisions. If my global equity portfolio is invested in VT, then my benchmark should be the index that ETF tracks (otherwise, why on earth would you make things confusing with a different benchmark?). If you want to be super picky about it, I guess you could apply the tracking error to selection effect, but that really doesn't make logical sense.

Actual returns will not perfectly match the policy portfolio, but the difference is still explained by allocation decisions when every investment is in passive index funds.
Well, you can read the book and then report back on what you think he means.

It's been over 10 years since I read it.
I'm not sure what you mean. I wasn't commenting on a book
But I was upthread.

So if you want to deconstruct the number, it's not mine to begin with.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by BJJ_GUY »

watchnerd wrote: Wed Apr 07, 2021 3:19 pm
BJJ_GUY wrote: Wed Apr 07, 2021 3:15 pm I'm not sure what you mean. I wasn't commenting on a book
But I was upthread.

So if you want to deconstruct the number, it's not mine to begin with.
Sorry, misunderstanding. I forgot you initially quoted the study from the book. :oops:

I'm not trying to argue with the point being made in the book (or the reference to a study in the book). All I am saying is that if we're talking about a portfolio of passive index funds then that study does not apply. This does not at all contradict your quote from the book.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by tominsc »

OP here. Been busy. I need more time to sort through the excellent dialogue.

Looks like I asked a provocative question. I’m glad I started a spirited dialogue. All my life I’ve felt that given the choice I’d rather be provocative than right!

So, let me try to restate my initial concern, and then let the dialogue continue.

Whenever I read any statement such as “portfolio performance depends on how the portfolio is allocated among asset classes”, I think that a simple two-factor calculation of correlation coefficient has been performed between two variables, one representing the portfolio performance, or return, and the other representing in some numerical way the asset allocation of the whole portfolio.

Maybe my confusion arises because the computation of correlation coefficient isn’t a simple two-factor exercise in this case.

I know how to do the math to figure the correlation between two numerical things, like average returns of stocks vs average returns of bonds.

I don’t know how to do a mathematical computation of correlation coefficient between two things that aren’t both the same kind of data, like returns of stocks or bonds over the same time period—such as, say, average returns of a PORTFOLIO comprised of MANY asset classes simultaneously vs some numerical parameter that represents the ASSET ALLOCATION of the portfolio, the percentages of each asset class comprising the portfolio. I.e., how do you represent numerically the allocation of any combination of several asset classes comprising the portfolio, so you can plug the numerical representation of the allocation into the equations for computing correlation coefficient and calculate how the portfolio performance depends on how the whole portfolio is allocated?

Did that help at all? 😳😆🤨😱😰😆
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by alex_686 »

tominsc wrote: Fri Apr 09, 2021 4:05 pm Maybe my confusion arises because the computation of correlation coefficient isn’t a simple two-factor exercise in this case.

I know how to do the math to figure the correlation between two numerical things, like average returns of stocks vs average returns of bonds.

I don’t know how to do a mathematical computation of correlation coefficient between two things that aren’t both the same kind of data, like returns of stocks or bonds over the same time period—such as, say, average returns of a PORTFOLIO comprised of MANY asset classes simultaneously vs some numerical parameter that represents the ASSET ALLOCATION of the portfolio, the percentages of each asset class comprising the portfolio. I.e., how do you represent numerically the allocation of any combination of several asset classes comprising the portfolio, so you can plug the numerical representation of the allocation into the equations for computing correlation coefficient and calculate how the portfolio performance depends on how the whole portfolio is allocated?

Did that help at all? 😳😆🤨😱😰😆
I would classify correlation coefficients as the shallow end of the pool. That it is simple is both a blessing and a curse. Thinking about the correlation coefficient between 2 assets is a valuable exercise. However, it rests on a fair number of simplifying assumptions. A good example is that most of the inputs are not stable from secular period to secular period.

You can use correlation matrixes and portfolio optimization techniques for multiple asset classes. You can find Excel examples out there. Just google "excel portfolio optimization correlation matrix". This is still pretty simple statically and investment theory stuff. While a valuable intellectual exercise it is almost a worthless technique. Small changes in the inputs (return, volatility, correlation) can lead to radically different portfolios. "Corner solutions" are common, where one asset dominates another. So it takes wisdom and experience to run these.

Does that help?

Now, this is all at the shallow end of the pool. 200 level statistics, 300 level Finance, or 1st year MBA stuff. Unfortunately the next step is a pretty big one. You learn about all of the assumptions behind the models and why they are so dodgy. Graduate level math is where the real answers are - and even those are dodgy without wisdom and experience.
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tominsc
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by tominsc »

Alex-686,

Thank you! This is the sort of guidance I was looking for! I understand matrixes (matrices) and types of computations using them, etc, so now I have a better idea how it’s done when you have a lot more variables than just two asset classes at once for example.

As for the importance of the underlying assumptions, I understand exactly what you mean by that!

Thx again, one and all.
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Re: Understanding asset allocation, portfolio performance, and correlation coefficients

Post by vineviz »

tominsc wrote: Fri Apr 09, 2021 4:05 pm OP here. Been busy. I need more time to sort through the excellent dialogue.

Looks like I asked a provocative question. I’m glad I started a spirited dialogue. All my life I’ve felt that given the choice I’d rather be provocative than right!

So, let me try to restate my initial concern, and then let the dialogue continue.

Whenever I read any statement such as “portfolio performance depends on how the portfolio is allocated among asset classes”, I think that a simple two-factor calculation of correlation coefficient has been performed between two variables, one representing the portfolio performance, or return, and the other representing in some numerical way the asset allocation of the whole portfolio.

Maybe my confusion arises because the computation of correlation coefficient isn’t a simple two-factor exercise in this case.

I know how to do the math to figure the correlation between two numerical things, like average returns of stocks vs average returns of bonds.

I don’t know how to do a mathematical computation of correlation coefficient between two things that aren’t both the same kind of data, like returns of stocks or bonds over the same time period—such as, say, average returns of a PORTFOLIO comprised of MANY asset classes simultaneously vs some numerical parameter that represents the ASSET ALLOCATION of the portfolio, the percentages of each asset class comprising the portfolio. I.e., how do you represent numerically the allocation of any combination of several asset classes comprising the portfolio, so you can plug the numerical representation of the allocation into the equations for computing correlation coefficient and calculate how the portfolio performance depends on how the whole portfolio is allocated?

Did that help at all? 😳😆🤨😱😰😆
Clear as mud.

Seriously, I'm not entirely sure I understand your concern but maybe the following will help.

I suspect the performance attribution analysis you're thinking of is actually comparing two SETS of computations: one is evaluating the variation in performance WITHIN each asset class and the other is is evolution in the variation in performance BETWEEN each asset class. By comparing these two types of computations, the performance of the portfolio can be attributed (in whole or part) to the choice of asset CLASSES versus the choice of ASSETS.

Imagine a simple world in which there are only two asset classes (stocks and bonds) and only two mutual funds in each class. Let's say the funds are as follows:

iShares Core S&P Total US Stock Mkt ETF (ITOT)
iShares Core S&P 500 ETF (IVV)
iShares US Treasury Bond ETF (GOVT)
iShares 7-10 Year Treasury Bond ETF (IEF)

It should be readily apparent that the allocation between the funds within each asset class (e.g. ITOT vs IVV or GOVT vs IEF) is going matter MUCH less than the allocation between asset classes (e.g. stocks vs bonds).

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Any 50/50 stock bond split is going to have very similar performance. Those four combinations will all have virtually identical returns, volatility, drawdowns, etc.

50% ITOT / 50% GOVT
50% IVV / 50% GOVT
50% ITOT / 50% IEF
50% IVV / 50% IEF

On the other hand, the two other combinations (100% stocks or 100% bonds) will have GREATLY differing returns and volatility.

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A quick-and-dirty analysis of these figures would compare the standard deviation of returns for the first four portfolios (0.15%) to the standard deviation of returns for the final two portfolios (9.44%). In this contrived case, you could say that asset allocation accounted for over 98% of the performance outcomes (1 minus (0.15 divided by 9.44)).
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch
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