The coming year is certainly looking fairly heated in terms of market performance. As my previous post noted, the fundamentals appear to be on the up-side, so I thought it would be nice to round-off that little piece of balance-sheet action with some good ol’ quant flavored outlooks for 2011 and beyond.

Don’t worry, seat tables are already at the upright position, there’s no need for oxygen masks and forget about the bracing. This is simple stats based math, nothing too fancy, yet… Actually, the results discussed below were first reviewed with my 4!-Moments postings back in 2010, and I thought now would be a good time to rehash some of that data into a more consumer friendly format.

#### Curve Fitting:

Chart 1: Historical S&P500 Returns and Normal Curve with increased sample size

Okay, so there needs to be some explaining with this first graph. Chart 1 graphs one core data set and two statistical distributions based on parameters from that initial source data. Oh, and in good, completely unscientific style, I’ve actually opened up with my end-results chart (I need to make sure it stays visible on the front page).

#### The Data

The data covers the distribution of gross investment returns over the S&P500 benchmark for 1000 day periods. In other words, it represents the distribution count for 1000 business day trading windows between 1991 to 2010. Each trading window’s return was calculated as the gross percentage change over the benchmark between the opening date and the closing date, which was, you guessed it, 1000 days later than the opening date.

These returns were then tallied up and aggregated up into distribution count *buckets* and, hey, presto: you’ve got your distribution of returns for the period between 1991 to 2010.

Now, at this point, you might notice that the results weren’t that great. Actually, they appear a bit skewed towards the left with a fair amount of trading windows actually coming up below the 0% return line (in other words: you would have lost money over 4 years). But, and this is where things get a bit interesting, the data also seems to point at something else…

#### Past Beta but Future Alpha?

Now, based on the data, I’ve also *form fitted* a skewed distribution set over the observed sample size to give a bit of a template for describing the past twenty years of performance. This template was the Beta distribution curve (disambiguation: this is not the CAPM Beta metric but the statistical distribution set named Beta), which, as you can see, exhibits a very strong beta and low alpha ratio over the Min-Max range of the actually observed data. In other words, the data is heavily skewed towards the left with a thinner tail running over to the right.

Now, obviously, the Beta curve is not a 1:1 match to what happened with the S&P500 but I personally find the *naked-eye* match-o-meter to be pretty on the mark here. So, using this advanced qualitative *naked-eye* matchmaker further, let’s see if we can start seeing, or should I say foreseeing, a little bit more here.

Alright, brakes on, time to list out a few assumptions before stepping forward.

#### Assumption 1: Normality Of Returns Over the Benchmark Over the Long Run

Hmm, okay, so this one can pretty much make you stop reading this post now and think that I’ve just tried to oversimplify the whole thing a tid-tad too much. And, well, you would probably be right. However, there is some value in taking a pause and realising that we are looking at the returns of over 3700 distinct trading windows over the past twenty years. That these returns are, in theory, pretty much unpredictable and random. And that, therefore, they should, over a long enough time scale, exhibit a normally distributed shape in terms of their returns.

Anyways, that’s the opening assumption of this post: on a long enough timescale, returns over a benchmark will exhibit strong normally distributed returns.

#### Assumption 2: The Normality of Distribution can be Described by Two Parameters

Okay, now we are going to go a bit further and saying that only two parameter are required to describe this normal curve. Of course, I am speaking about the standard deviation of returns and the average of these returns. Both these metrics were calculated over the returns for the 1991 to 2010 period. The standard normal distribution function was used producing the characteristic bell shaped curve based on these two parameters.

#### Assumption 3: Standard Deviation and Average Returns Over the Past Twenty Years Will Remain Constant

Yes, yes, I know, here I am really pulling a thin line on unstable footing. Basically, I am saying that these past twenty years, and the relevant distributions they have produced, have provided a sufficiently relevant average and standard deviation measure that they can implicitly be used over an extended time frame.

This, therefore, brings me to the bell curve set in the background that actually presents a potentially normal distribution of returns for the period 1991 to 2030 (or 2010 plus another twenty years). So, basically, what I did here, is just double up the sample size for the distribution calculations, which produced results potentially covering the next twenty years.

#### Please, Little Prince, Draw Me a Bull-Market In a Normal Curve

Alright, so given the assumptions previously made, what can we observe? Assuming that the next twenty years should reach a normal distribution, then certainly, the set of returns should come forward and *fill-in* the curve. As you might be able to see from the graph, the majority of the curve left to fill is in the *positive* region.

True, there remains a little bit of poor returns back there on the left, but it is worth noting that the past twenty-years have been unusually volatile. Depending on whether this standard deviation holds up, we can see one of two things: either a flattening of the curve with more scope of some *fat-tails* or a potential *squeezing* with less variance coming through. But even if my previous assumption holds, the probability is still, normally speaking, higher on the upside than the downside.

Just to illustrate this, you can see below the same graph as above only this time with a normal curve using the actual sample count (ie set to fit the 1991 to 2010 period). This chart more explicitly depicts how skewed the last twenty-years were compared to *normal* expectations:

Chart 2: 1991 to 2010 Distrbution of Returns With Normal Comparative

#### Criticising Assumption 1

A quick jab to myself here: but are stock market returns actually normally distributed over the long-run? If this were the case, wouldn’t we be able to take advantage over this?

There are two questions listed out here. On the first point, I would say yes, on the very long-run, markets are surprisingly *normal*, but the question then becomes: how long is a long-run? As for the next point, can we take advantage over this, well… It becomes tricky here. The data reviewed so far cover 1000 business trading day holding periods, which is roughly equal to just under 4 year investment windows.

A lot can happen over 4 years: politics, war, marriage, birth, taxes, funerals, natural disaster, scientific discoveries, economic and cultural leaps forwards, shock crashes, bankruptcies, lottery winners, etc. In other words, to take a long time frame and say: the odds are in my favor is a bit too easy a spot to take, as Benjamin Graham once said: “Markets can remain irrational longer than an investor can remain solvent.”

However, this being said, if we take the period 1954 to 2010, the furthest I’ve tried to research, then we get something looking like the following graph: an extremely *normal* set of returns.

#### Caveat Emptor

Alright, now, obviously any financial analysis based largely on inferences from past data will need to draw the following warning line: *Past Returns Are No Guarantee Of Future Performance*. This warning sort of goes without saying but it is still worth saying, none the less.

#### In Conclusion

Right, so there’s some statistical back-log data that would *seem* to suggest that we *might*, just *might*, be entering into something looking like one of those multi-year cyclical bull runs. Okay, so now, am I alone to think this at the moment?

Apparently not, actually a number of economists heading to Davos at the moment seem to be putting forward just such a thesis: that the structural enhancement of the emerging markets (or was that emerging maturing markets) is going to offer the world a unique economic push only observed three times over the past century. You can read more about that story over on Bloomberg.

So here’s to being bullish for the coming years 🙂

#### Data Information

You can review the source data to the graphs right here: source data (warning 4.2 MB file). Also, please note that the S&P500 returns used in the analysis above did not include dividend yields and reinvestment but was adjusted for stock-splits and other related capital adjustments.

## Bullish Normality

Jan 26th, 2011 by Tariq Scherer

The coming year is certainly looking fairly heated in terms of market performance. As my previous post noted, the fundamentals appear to be on the up-side, so I thought it would be nice to round-off that little piece of balance-sheet action with some good ol’ quant flavored outlooks for 2011 and beyond.

Don’t worry, seat tables are already at the upright position, there’s no need for oxygen masks and forget about the bracing. This is simple stats based math, nothing too fancy, yet… Actually, the results discussed below were first reviewed with my 4!-Moments postings back in 2010, and I thought now would be a good time to rehash some of that data into a more consumer friendly format.

## Curve Fitting:

Chart 1: Historical S&P500 Returns and Normal Curve with increased sample size

Okay, so there needs to be some explaining with this first graph. Chart 1 graphs one core data set and two statistical distributions based on parameters from that initial source data. Oh, and in good, completely unscientific style, I’ve actually opened up with my end-results chart (I need to make sure it stays visible on the front page).

## The Data

The data covers the distribution of gross investment returns over the S&P500 benchmark for 1000 day periods. In other words, it represents the distribution count for 1000 business day trading windows between 1991 to 2010. Each trading window’s return was calculated as the gross percentage change over the benchmark between the opening date and the closing date, which was, you guessed it, 1000 days later than the opening date.

These returns were then tallied up and aggregated up into distribution count

bucketsand, hey, presto: you’ve got your distribution of returns for the period between 1991 to 2010.Now, at this point, you might notice that the results weren’t that great. Actually, they appear a bit skewed towards the left with a fair amount of trading windows actually coming up below the 0% return line (in other words: you would have lost money over 4 years). But, and this is where things get a bit interesting, the data also seems to point at something else…

## Past Beta but Future Alpha?

Now, based on the data, I’ve also

form fitteda skewed distribution set over the observed sample size to give a bit of a template for describing the past twenty years of performance. This template was the Beta distribution curve (disambiguation: this is not the CAPM Beta metric but the statistical distribution set named Beta), which, as you can see, exhibits a very strong beta and low alpha ratio over the Min-Max range of the actually observed data. In other words, the data is heavily skewed towards the left with a thinner tail running over to the right.Now, obviously, the Beta curve is not a 1:1 match to what happened with the S&P500 but I personally find the

naked-eyematch-o-meter to be pretty on the mark here. So, using this advanced qualitativenaked-eyematchmaker further, let’s see if we can start seeing, or should I say foreseeing, a little bit more here.Alright, brakes on, time to list out a few assumptions before stepping forward.

## Assumption 1: Normality Of Returns Over the Benchmark Over the Long Run

Hmm, okay, so this one can pretty much make you stop reading this post now and think that I’ve just tried to oversimplify the whole thing a tid-tad too much. And, well, you would probably be right. However, there is some value in taking a pause and realising that we are looking at the returns of over 3700 distinct trading windows over the past twenty years. That these returns are, in theory, pretty much unpredictable and random. And that, therefore, they should, over a long enough time scale, exhibit a normally distributed shape in terms of their returns.

Anyways, that’s the opening assumption of this post: on a long enough timescale, returns over a benchmark will exhibit strong normally distributed returns.

## Assumption 2: The Normality of Distribution can be Described by Two Parameters

Okay, now we are going to go a bit further and saying that only two parameter are required to describe this normal curve. Of course, I am speaking about the standard deviation of returns and the average of these returns. Both these metrics were calculated over the returns for the 1991 to 2010 period. The standard normal distribution function was used producing the characteristic bell shaped curve based on these two parameters.

## Assumption 3: Standard Deviation and Average Returns Over the Past Twenty Years Will Remain Constant

Yes, yes, I know, here I am really pulling a thin line on unstable footing. Basically, I am saying that these past twenty years, and the relevant distributions they have produced, have provided a sufficiently relevant average and standard deviation measure that they can implicitly be used over an extended time frame.

This, therefore, brings me to the bell curve set in the background that actually presents a potentially normal distribution of returns for the period 1991 to 2030 (or 2010 plus another twenty years). So, basically, what I did here, is just double up the sample size for the distribution calculations, which produced results potentially covering the next twenty years.

## Please, Little Prince, Draw Me a Bull-Market In a Normal Curve

Alright, so given the assumptions previously made, what can we observe? Assuming that the next twenty years should reach a normal distribution, then certainly, the set of returns should come forward and

fill-inthe curve. As you might be able to see from the graph, the majority of the curve left to fill is in thepositiveregion.True, there remains a little bit of poor returns back there on the left, but it is worth noting that the past twenty-years have been unusually volatile. Depending on whether this standard deviation holds up, we can see one of two things: either a flattening of the curve with more scope of some

fat-tailsor a potentialsqueezingwith less variance coming through. But even if my previous assumption holds, the probability is still, normally speaking, higher on the upside than the downside.Just to illustrate this, you can see below the same graph as above only this time with a normal curve using the actual sample count (ie set to fit the 1991 to 2010 period). This chart more explicitly depicts how skewed the last twenty-years were compared to

normalexpectations:Chart 2: 1991 to 2010 Distrbution of Returns With Normal Comparative

## Criticising Assumption 1

A quick jab to myself here: but are stock market returns actually normally distributed over the long-run? If this were the case, wouldn’t we be able to take advantage over this?

There are two questions listed out here. On the first point, I would say yes, on the very long-run, markets are surprisingly

normal, but the question then becomes: how long is a long-run? As for the next point, can we take advantage over this, well… It becomes tricky here. The data reviewed so far cover 1000 business trading day holding periods, which is roughly equal to just under 4 year investment windows.A lot can happen over 4 years: politics, war, marriage, birth, taxes, funerals, natural disaster, scientific discoveries, economic and cultural leaps forwards, shock crashes, bankruptcies, lottery winners, etc. In other words, to take a long time frame and say: the odds are in my favor is a bit too easy a spot to take, as Benjamin Graham once said: “Markets can remain irrational longer than an investor can remain solvent.”

However, this being said, if we take the period 1954 to 2010, the furthest I’ve tried to research, then we get something looking like the following graph: an extremely

normalset of returns.## Caveat Emptor

Alright, now, obviously any financial analysis based largely on inferences from past data will need to draw the following warning line:

Past Returns Are No Guarantee Of Future Performance. This warning sort of goes without saying but it is still worth saying, none the less.## In Conclusion

Right, so there’s some statistical back-log data that would

seemto suggest that wemight, justmight, be entering into something looking like one of those multi-year cyclical bull runs. Okay, so now, am I alone to think this at the moment?Apparently not, actually a number of economists heading to Davos at the moment seem to be putting forward just such a thesis: that the structural enhancement of the emerging markets (or was that emerging maturing markets) is going to offer the world a unique economic push only observed three times over the past century. You can read more about that story over on Bloomberg.

So here’s to being bullish for the coming years 🙂

## Data Information

You can review the source data to the graphs right here: source data (warning 4.2 MB file). Also, please note that the S&P500 returns used in the analysis above did not include dividend yields and reinvestment but was adjusted for stock-splits and other related capital adjustments.

Background LinksPosted in Analysis, Market Comments | Tagged in , 1954 to 2010 stock market prices, Benchmark returns, Beta Distribution, bull market, Capital Markets, Curve fitting, Distribution graphs, Equities, future performance, Historical Returns, Normal Distribution, Parametric modelling, S&P500, statistical inference, Variance