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BETA and R-Squared

Reading price charts which may give you direction in the market using established TA methodology
EssDeeAitch
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BETA and R-Squared

#201534

Postby EssDeeAitch » February 15th, 2019, 11:35 am

I have read a bit about BETA and R-Squared and their correlation but I don't intuitively get it.

I have just run a Morningstar X-Ray on my portfolio and is shows the following:-

Beta | 0.68
R-Squared | 0.71
Information Ratio | 0.91
Tracking Error | 4.88

My PF is 66% equities and 23% bonds, the balance being cash and others.
Can anyone help me and put it in simple terms as to what these values mean? Thanks in advance.

modellingman
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Re: BETA and R-Squared

#202737

Postby modellingman » February 20th, 2019, 8:28 pm

EssDeeAitch wrote:I have read a bit about BETA and R-Squared and their correlation but I don't intuitively get it.

I have just run a Morningstar X-Ray on my portfolio and is shows the following:-

Beta | 0.68
R-Squared | 0.71
Information Ratio | 0.91
Tracking Error | 4.88

My PF is 66% equities and 23% bonds, the balance being cash and others.
Can anyone help me and put it in simple terms as to what these values mean? Thanks in advance.


You have been underwhelmed with responses and I know very little about X-Ray other than the rather scant information I've been able to locate, such as this video. However, I do know a bit about statistical modelling, so here goes...

Part of what X-Ray allows you to do is to compare your portfolio's performance to that of a benchmark. Reading a bit between the lines, "performance" appears to equate to return in each of a series of periods. Here, it is easier for me to introduce a bit of notation. Use P for the performance of your portfolio over a period and use B to denote the performance of the benchmark over the same period. So for a given period, we have a pair of values P and B. I have no idea what length a period might be but the video talks about comparison over 3 year period, so if each period is a quarter then there would be 12 periods in the 3 years giving rise to 12 pairs of values and that's just about enough for the analysis set out next.

The 12 (or however many) pairs of values can be plotted not as time series but as a scatterplot. Rather than using time on the horizontal axis, use the benchmark performance values and use the portfolio performance values on the vertical axis. You end up with 1 point in the scatterplot for each pair of values, so 12 pairs of performance values (or whatever) gives you 12 points (or whatever) on your scatterplot. R-squared, alpha (which you don't specifically mention) and beta are related to drawing a "line of best fit" through such a scatterplot.

This "line of best fit" is created using a bit of mathematical statistics known as a simple linear regression model. The equation of the line is

P' = alpha + beta*B

where P' is an estimate (from the line of best fit model) of your portfolio's performance (P) when the benchmark performance has a value of B. alpha and beta are the values reported by X-Ray

The vertical distance between the line and any point in the scatterplot is the difference between your portfolio's actual performance (P) and the model's estimated value (P') for the benchmark value (B) corresponding to that point. Because the model line is a line of best fit, some points will be above the line (indicating the model has underestimated the portfolio's performance) and some below it (indicating overestimation). If the points are all close to the line of best fit, these estimation errors are small and, in effect, your portfolio's performance is closely related to the benchmark performance. If the points are not close to the line, then your portfolio is not closely related to the benchmark.

R-Squared is a statistical measure which, in effect, quantifies the closeness of your portfolio's performance to that of the benchmark. It is a number which, as I'm sure you have found out, varies between 0 and 1. A value of zero indicates no discernable relationship between your portfolio and the benchmark (*). A value of 1 indicates a perfect relationship, and in this case, all the points in the scatterplot would lie on the line of best fit. Your value of 0.71 indicates that there is some relationship between your portfolio and the benchmark. A statistician might say that 71% of the observed variation in your portfolio's performance over the 3 year (or whatever) period can be explained by variation in the performance of the benchmark over the same period.

Beta is just the slope of the line of best fit. Even if R-Squared was 1, this does not necessarily mean that your portfolio tracks the benchmark exactly. That would only be the case if, in addition, alpha was 0 and beta was 1. In your case beta is 0.68. This means that if the benchmark's performance increases by say 5% (from B to 1.05*B), the model would estimate that your portfolio's performance should change by 5%*0.68 = 3.4% (so from P' to 1.034*P'). The interpretation of beta here is that it measures the volatility of your portfolio relative to the benchmark. Seems reasonable to me.

Hopefully, a real expert will be along sometime to give you a more definitive answer including, perhaps, better definitions of the performance values than I have been able to provide.

(*) Not quite true. But this can be left to another post if necessary.

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Re: BETA and R-Squared

#202747

Postby dspp » February 20th, 2019, 9:23 pm

Sorry, I have been travelling the last fortnight.

In addition to what MM has set out take a read of these, which set out pretty clearly (but using a variety of different words so you can find one to suit your tastes) the relationship between alpha and beta:

http://www.arborinvestmentplanner.com/a ... ment-risk/
https://www.investopedia.com/ask/answer ... d-beta.asp
https://en.wikipedia.org/wiki/Alpha_(finance)
https://en.wikipedia.org/wiki/Beta_(finance)

With a beta of 0.68 you are less volatile than the relevant (benchmark) market index , which is meaningless unless you also tell us what index you had selected to plug in (but probably not surprising given your bond/equity split). Similarly we don't know your alpha which is the other pertinent piece of information.

(one could deduce alpha from the Information Ratio I think, but I'm lazy)

regards, dspp

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Re: BETA and R-Squared

#202791

Postby EssDeeAitch » February 21st, 2019, 3:52 am

modellingman wrote:
EssDeeAitch wrote:I have read a bit about BETA and R-Squared and their correlation but I don't intuitively get it.

I have just run a Morningstar X-Ray on my portfolio and is shows the following:-

Beta | 0.68
R-Squared | 0.71
Information Ratio | 0.91
Tracking Error | 4.88

My PF is 66% equities and 23% bonds, the balance being cash and others.
Can anyone help me and put it in simple terms as to what these values mean? Thanks in advance.


You have been underwhelmed with responses and I know very little about X-Ray other than the rather scant information I've been able to locate, such as this video. However, I do know a bit about statistical modelling, so here goes.......



Many thanks for this, it has really helped. The Alpha was 6.26 (Sharpe of 1.53) and the benchmark is UK Large Cap (which I take to be the FTSE 100 as I have not been able to find "UK Large Cap" in Morningstar, HL or Trustnet tools). The time period selected was three years but I also have for five.

So in short, my portfolio had a lower volatility than the benchmark and returned 6.26% more than the benchmark with a 71% correlation to the benchmark (the selection of the appropriate benchmark is clearly key in assessing performance). Does that sum it up adequately?

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Re: BETA and R-Squared

#202792

Postby EssDeeAitch » February 21st, 2019, 3:56 am

dspp wrote:Sorry, I have been travelling the last fortnight.

In addition to what MM has set out take a read of these, which set out pretty clearly (but using a variety of different words so you can find one to suit your tastes) the relationship between alpha and beta:

http://www.arborinvestmentplanner.com/a ... ment-risk/
https://www.investopedia.com/ask/answer ... d-beta.asp
https://en.wikipedia.org/wiki/Alpha_(finance)
https://en.wikipedia.org/wiki/Beta_(finance)

With a beta of 0.68 you are less volatile than the relevant (benchmark) market index , which is meaningless unless you also tell us what index you had selected to plug in (but probably not surprising given your bond/equity split). Similarly we don't know your alpha which is the other pertinent piece of information.

(one could deduce alpha from the Information Ratio I think, but I'm lazy)

regards, dspp


Thanks for the links dspp, appreciated. I have had a good read up now and feel much more comfortable and my reply to MM states the missing data.

For me, the benchmark is a real issue (I raised the matter of benchmarks in another post) and I can see that the R-Squared goes a long way to validating the choice of benchmark.

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Re: BETA and R-Squared

#202893

Postby modellingman » February 21st, 2019, 12:11 pm

EssDeeAitch wrote:So in short, my portfolio had a lower volatility than the benchmark and returned 6.26% more than the benchmark with a 71% correlation to the benchmark (the selection of the appropriate benchmark is clearly key in assessing performance). Does that sum it up adequately?


It's getting there.

From the helpful links that dspp posted, the performance values are clearly percentage returns but over what length period is still not clear. I'll return to the importance of this in a moment.

Let's deal with your "6.26% more than the benchmark" first. A property of the line of best fit is that it always passes through the centroid of the data. So recalling that the data comprises a number of pairs of values and we can abbreviate a general pair to (B, P), the performance values of the benchmark and portfolio, respectively. The centroid is simply the point (B^, P^), where B^ is the average of all the B values in the pairs of values and P^ is the average of the P's. Because, the centroid is always on the line of best fit, it follows that

P^ = alpha + beta*B^

The P^ value here is the average return from your portfolio per period over the 3 (or whatever) years and, the important thing to note is that it is derived from actual data rather than estimates. So the equation is telling you how your average return per period (P^) relates to B^, the average return per period from the benchmark. The point being that the difference between your average return and the benchmark's (ie P^ - B^) is not simply alpha as your statement implies, rather it is

P^ - B^ = alpha + beta*B^ - B^ = alpha + (beta-1)*B^

So although alpha is part of the difference, there is also that additional bit (beta-1)*B^. Since your beta is less than 1, then beta-1 is negative, -0.32. If the average benchmark performance (B^) has been positive over the 3 year period then (beta-1)*B^ will be negative so the difference between your portfolio's return and that of the benchmark will be less than alpha. On the other hand, if B^ is also negative then the difference will be greater than alpha. Alpha on its own isn't the whole story, despite what some authors may claim. Strictly, alpha is the expected return from your portfolio when the benchmark return is zero.

Ok, onto that 71% correlation. Correlation, in the statistics world, has a precise meaning which is related to (one might almost say correlated with) the everyday usage. In statistics, correlation (or Pearson's Correlation Co-efficient [PCC] to give it the full monty) is a measure of the linear relationship between paired variables. It varies between -1 and +1, and is often abbreviated as the Greek letter rho or the Roman letter, r. Perhaps you can see where this is headed. It turns out that in a simple linear regression model, r-squared is the square of the PCC between the two variables. So, to be strictly accurate, the correlation [PCC] is the square root of the r-squared value, ie 84% rather than 71%.

l will now return to that point about the lengths of the periods over which the returns B and P are measured. Beta is a dimensionless number. However, alpha is not. It has dimensions of % return per period. So understanding the period length is vital to correctly understanding alpha. It is also possible that an alpha value which has been derived from say quarterly data is then restated as an annualised value for reporting purposes. I simply don't know enough about MPT and its accepted norms and conventions to say if this is the case, but it is another gotcha to be aware of. If values are restated for reporting purposes then the analysis above about P^ - B^ would need to take this into account by converting alpha back to a per period basis.

Finally, dspp's references have also lead me to the definitions of tracking error and the information ratio. Going back to the pairs of values (B, P), the tracking error is simply a measure of how much the difference, P-B has varied over the observed pairs of values. The measure of variation used is the standard deviation. If these differences were Normally distributed about the mean difference (estimated as P^-B^) then in a large sample of observations around two-thirds of the differences should lie within one standard deviation of the mean and 95% within two standard deviations. The information ratio is simply the mean difference divided by the tracking error. Statistician's use the term "normalisation" to describe this process of dividing something by its standard deviation. The resulting value has a standard deviation of 1, which is quite handy. Quite what MPT uses it for I haven't investigated but a statistician would use it for testing a hypothesis such as whether the mean difference is significantly different from zero. A value of 0.91 is too low to reject the no difference hypothesis, but the hypothesis test sets a high bar and if the number of data points is low, the chances of false negative are considerable (see viewtopic.php?f=9&t=14959#p187124 for more about significance, hypothesis testing, etc). The differences P-B, have an estimated mean value of 0.91*4.88 = 4.44 and standard deviation of 4.88, so that should provide some additional insight into the relationship between your portfolio's returns and that of the benchmark.

If alpha hasn't been restated then you can plug the mean difference value of 4.44 into the equation above and with a bit of algebraic jiggery-pokery work out the P^ and B^ values. I make them 5.69 for B^ and 10.13 for P^, which leaves me wondering whether the P, B and difference values might also have been annualised prior to model fitting, etc.

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Re: BETA and R-Squared

#202935

Postby dspp » February 21st, 2019, 2:03 pm

Thank you mm.

Over on Portfolio Management & Review there was a post today by forrado candidly pointing out that his/her (?) portfolio's return is "reverting to the mean", or in his case to slightly less than the mean, measured over an 8-year period. See viewtopic.php?f=56&p=202927#p202927 .

This is relevant because in the end alpha and beta are about performance & risk, and where you are with respect to the efficient frontier. The alpha is essentially your stock (or bond) picking skill, and the beta is essentially the general market return. It would seem that over the 8-year period forrado has returned a negative alpha. He has returned about 6% less than the market over a 8-year period, so say a alpha of -1% per year in approximate numbers.

I must make sure I have something similar set up in my end year spreadsheets that I typically run in April, as I use the end March (-ish) as my year end. Otherwise, as it is so easy to do, I am fooling myself but no-one else.

regards, dspp

(PS. I have just updated my earlier post from yesterday with the links in it to give the correct wiki links, as for some reason the trailing ')' was not being auto-detected)

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Re: BETA and R-Squared

#202969

Postby EssDeeAitch » February 21st, 2019, 4:38 pm

Again, thank you both for a comprehensive and detailed review of the topic. It is not as straight forward for me as I would like but no worry as I do grasp the concepts well enough.

All part of the learning process and it helps having people like you both prepared to spend the time helping.

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Re: BETA and R-Squared

#203117

Postby modellingman » February 22nd, 2019, 10:14 am

dspp wrote:Thank you mm.

Over on Portfolio Management & Review there was a post today by forrado candidly pointing out that his/her (?) portfolio's return is "reverting to the mean", or in his case to slightly less than the mean, measured over an 8-year period. See viewtopic.php?f=56&p=202927#p202927 .

This is relevant because in the end alpha and beta are about performance & risk, and where you are with respect to the efficient frontier. The alpha is essentially your stock (or bond) picking skill, and the beta is essentially the general market return. It would seem that over the 8-year period forrado has returned a negative alpha. He has returned about 6% less than the market over a 8-year period, so say a alpha of -1% per year in approximate numbers.


I agree broadly with what you are saying but think there is an important proviso that beta needs to be fairly close to 1. As EssDeeAitch's number's indicate, his alpha is 6.26, whereas his performance relative to his benchmark is 4.44 (what I previously termed P^ - B^). As I hope is clear from the argument presented in my second post, this difference arises because his beta differs from 1.

I might be wrong, but I have strong sense that in both your comments quoted above and in lootman's post on the forrado thread you have highlighted, the terms "alpha" and "beta" are being using as a shorthand for "stock picking ability" and "general market return". I can understand entirely how this arises, particularly when as with forrado's candid post, there is very little information beyond just a single pair of return values over an 8 year period. However, the alpha's and beta's derived from Morningstar's X-Ray appear to me a bit more nuanced than their shorthand counterparts.

From a statistical perspective, whilst the use of a simple linear regression model is an obvious way exploring the relationship between portfolio and benchmark returns, there are some traps to be wary of.

One issue is the number of data points used. The alpha and beta values derived from such a model are themselves only estimates of the "true" but unknown values of alpha and beta, in much the same way that a sample average is an estimate of the true but unknown mean value of the population from which the sample is drawn. A small sample produces an estimate of a population mean with a relatively wide confidence interval and the rule of thumb, as I'm sure you know, is that halving the confidence interval requires a quadrupling of sample size. Broadly it is similar with the estimates of alpha and beta: if the number of data points is low, then the confidence intervals of their estimates will be quite wide. If the confidence interval of alpha includes zero and/or the confidence interval of beta includes 1, then any interpretation of the estimates of alpha and beta becomes a bit meaningless, and perhaps the most sensible approach in this case would be to simply compare just two returns from the entire period and adopt the shorthand approach noted above.

With investment data, unlike many other data sets, the number of data points can be increased by measuring returns over a shorter period. Unfortunately, this isn't a great solution to the problem of having too few data points. The technical issues centre on something called auto-correlation, successive observations increasingly lack independence (in a statistical sense) from each other as the measurement interval is reduced. Whilst this problem can be overcome, it requires more than just a simple linear regression approach and, because not that much new information is being added by reducing the measurement period, not that much will be gained in terms of reducing confidence intervals of estimates.

Finally, even if a long comparison period is used with longish measurement intervals (so addressing the problems of number of data points and auto-correlation) there is a question mark about whether a single linear model relating portfolio returns to benchmark returns is applicable over the entire period. It did strike me that a possible explanation for forrado's situation where he had a period of outperforming his benchmark followed by one where clearly he was not, might be because something had caused a discontinuity of some sort in the (statistical) relationship between his portfolio and his chosen benchmark, ie a single linear model over the entire period is not an appropriate model to use in this case.

Overall, what I think I'm saying is that without access to the underlying data (and the ability to play around with it) I'd be wary about reading too much into X-Ray's alpha and beta estimates.

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Re: BETA and R-Squared

#203226

Postby dspp » February 22nd, 2019, 3:40 pm

mm,

Thank you.

The reversion-to-the-mean portfolio I used to give an estimated 8-year alpha example of "approx minus 1% per annum" was actually forrado's over on viewtopic.php?f=56&p=202927#p202927. I didn't use EssDeeAitch's number's at all as I was too uncertain of his units to run the numbers the way you did. Sorry if I confused anyone.

Your posts are terrific. They very much remind me of when I was a junior engineer wandering into a geological scientist's office asking for some pressure data. The answer would always be fantastic, but come with enormous numbers of caveats and error bars and cautions, and after about an hour I'd extract myself from their office muttering something along the lines of "er thanks, so you reckon it might be between 549.645 and 1427.6778 bar, so in that case I'll work on 1500 bar and we're safe. " :)

You are very right that I am using alpha and beta in the more generic sense, but I don't think that is at variance with their usual definitions - though it might be (as you point out) somewhat at variance with how Morningstar has spat out the data for EssDeeAitch. Or it might not be.

I'd be very interested to see EssDeeAitch's numbers when he/she has figured them all out and got them into the right units and etc. It is a learning exercise for us all. Every day is a school day.

regards, dspp

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Re: BETA and R-Squared

#203625

Postby TheMotorcycleBoy » February 25th, 2019, 8:58 am

Hi,

This is just a quick question and hopefully not too off-topic. I've read several of the earlier posts, especially MMs where the concept of a best fit linear equation

P' = alpha + beta*B'

can be established by comparison of the real world P and B pairs. I assuming that the R squared entity is perhaps a standard derivation measure, but I must confess I didn't follow this on in depth, or start reading the post introducing the averages and the centroids.

I've not used the mentioned X-ray analysis, but I have been unitising our foli since 31st December 2018, at which point I set a unit price of 1.000. I'm recalculating the unit price on the last weekend of the each month. This weekend I calculated our (accumulation) unit's price to be 1.085, but wondered what I should benchmark this performance against. I did observe that the FTSE350 index has risen by about 7.2% in the same interval, but then the thought occurred to me, that this index is only one of market capitalisation, and does not include growth due to dividends, whereas the unitising process I apply to our foli does.

So, I wondered, is there a source of FTSE350 (I think that's what our foli is closest too) marketcap+dividend combined index figures out there that I could use to measure our foli against?

many thanks, and apologies if this post is misplaced,
Matt

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Re: BETA and R-Squared

#203633

Postby tjh290633 » February 25th, 2019, 9:22 am

You need the Total Return version of the FTSE350. If you search for "World Markets at a glance" you will find a PDF from the FT and the UK Actuaries Indices are in one of the tables. The TR version is the right hand column.

There may be other sources, and the FT daily issues are behind a paywall.

TJH

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Re: BETA and R-Squared

#203639

Postby TheMotorcycleBoy » February 25th, 2019, 9:47 am

tjh290633 wrote:You need the Total Return version of the FTSE350. If you search for "World Markets at a glance" you will find a PDF from the FT and the UK Actuaries Indices are in one of the tables. The TR version is the right hand column.

There may be other sources, and the FT daily issues are behind a paywall.

TJH

Thanks TJH, I couldn't find what I was after with your search criteria string. Instead I ended up googling with

ftse 350 "total return index"


First hit was was, and yes the TR column was on the right :)
https://www.ftse.com/products/indices/uk

Given me 7107.21 for last weekend.

Anyone got backdated values for Jan 25th 2019, Dec 31st 2018 ;)

many thanks, Matt

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Re: BETA and R-Squared

#203649

Postby dspp » February 25th, 2019, 9:59 am

try downloading the pdf

https://www.ftse.com/Analytics/Factshee ... nual=False

I think you can then back it out from the knowledge of the current ones, or get pretty close by using the graph

dspp

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Re: BETA and R-Squared

#203706

Postby tjh290633 » February 25th, 2019, 2:02 pm

Looking at your link, the TR column is not on the right. There is a tab on the right and, if you click on that, the TR data comes up in every column.

TJH

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Re: BETA and R-Squared

#203708

Postby EssDeeAitch » February 25th, 2019, 2:10 pm

dspp wrote:
I'd be very interested to see EssDeeAitch's numbers when he/she has figured them all out and got them into the right units and etc. It is a learning exercise for us all. Every day is a school day.

regards, dspp


Hi dspp.

I cant figure out any numbers TBH, all I can do is to report them as is. I feel a bit dim but the conversation has left me a few concepts short if you catch my drift.

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Re: BETA and R-Squared

#203718

Postby TheMotorcycleBoy » February 25th, 2019, 2:31 pm

EssDeeAitch wrote:
dspp wrote:
I'd be very interested to see EssDeeAitch's numbers when he/she has figured them all out and got them into the right units and etc. It is a learning exercise for us all. Every day is a school day.

regards, dspp


Hi dspp.

I cant figure out any numbers TBH, all I can do is to report them as is. I feel a bit dim but the conversation has left me a few concepts short if you catch my drift.

Have you started unitising yet? TBH I think that's an interesting and useful exercise in itself.

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Re: BETA and R-Squared

#203742

Postby EssDeeAitch » February 25th, 2019, 4:09 pm

TheMotorcycleBoy wrote:
EssDeeAitch wrote:
dspp wrote:
I'd be very interested to see EssDeeAitch's numbers when he/she has figured them all out and got them into the right units and etc. It is a learning exercise for us all. Every day is a school day.

regards, dspp


Hi dspp.

I cant figure out any numbers TBH, all I can do is to report them as is. I feel a bit dim but the conversation has left me a few concepts short if you catch my drift.

Have you started unitising yet? TBH I think that's an interesting and useful exercise in itself.


Yes I have. It's a good thing to do and not difficult


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