Portfolio Theory Part I: The Financial Strategy of Yesteryear

Portfolio Theory is a construct of Harry M. Markowitz from the University of Chicago that economically models a risk-return adjusted optimal portfolio based upon two primary factors: risk and correlation. Sounds like a beautiful thing, right? WRONG! Textbook models are just that, textbook and are not enirely relevant outside of academe. They serve as restricted versions of actual market processes and I will argue against four of Markowitz's flawed assumptions. Also, despite quantum leaps in practical market theory over the last 60 years, some in the financial industry still apply this dusty-old model when managing their client’s money! Why, I can only suspect but read on to find out where the problem starts...



First, let’s start from the top, so everyone here knows what Portfolio Theory is and what it looks like…

Portfolio Expected Return

So all that these letters are saying is, “A portfolio’s expected return is a weighted-average of its two assets”. Not so bad, right? Now let’s go further into what it assumes, as models are simplifications of reality and while this is not exhaustive these are some major ones and so let’s see what we’ve observed over the last six decades and where it departs from reality.


1. Risk-Aversion and Sigma
The model assumes that the investor is risk-averse, which minimizes the equation and assumes only one dimesnion. However, investors also seek to maximize returns and so they may be risk-seekiing or perhaps they're risk-neutral, like in the case of an index fund manager. Also, the risk they assume is a known "negative outcome", such as an option’s premium while uncertainty is the "possibility of an event occurring", an option expiring out-of the-money. In the model, sigma is treated as a risk of return not as volatility.

2. One Time Period and Historical Pricing
A one period model means that the portfolio is static and invested only once in which all price histories are ex ante. Thus, the assets are optimized on sampled data only with no out-sample and can lead to a multitude of fallacies when attempting to extrapolate meaningful analysis from the series.

3. Gaussian Distribution of Returns and Probability Beliefs
Asset returns by their nature are not normal and typically fall within three to six sigma according to the authors of The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin, and Reward . Furthermore, Goldstein and Taleb in their paper, We Don’t Quite Know What We’re Talking About When We Talk About Volatility explain the misunderstanding of asset volatility and its effect on the Gaussian curve which ties back into point one.

4. Bi-Variate and Correlation
The equation at the top breaks down into this when calculating the portfolio variance:

Portfolio Variance

The model works because it uses only two assets, the same goes for the correlation coefficient but in practice a portfolio consists of more than two assets. For example, say I’m a mutual fund manager and wanted to construct a portfolio of 225 different assets, how would this be done?

• I’d need to figure out how to weight 225 assets
• I then have to calculate 225 measures of volatility
• I'd be tasked to solve a correlation matrix of 255 simultaneous equations

Sure, computers can do all of that work but what I have ended up with? In my opinion, this is a cumulative effect of the previous flawed underlying assumptions. It's a compounding error that's minute in each instance but pervasive throughout the model. Once its scaled it produces greater erroneous and misleading results which ties back into point two about data misinterpretation.


Put my analysis to the fire, let's hear it monkeys! Do you agree, disagree, and what methods do you think are better suited to manage today’s portfolio?


In PART II, I will dig deeper into why using correlations may not work for diversification and the alternatives to Portfolio Theory, so stay tuned…

 
  1. You're actually trying to argue that the risk-aversion is a flawed assumption? This is news to me....

  2. By definition, any choice made in the present can only utilize information from the past. Just because this is a one-period model doesn't prevent investors from re-weighting (or completely changing) their investment portfolio later on.

  3. I agree with this. Most returns are left-skewed.

  4. Not sure what your opinions on beta are, but that was the purpose of that whole discussion.

Also, for the record, most managers use some variation of Fama-French, which is a three factor model (market return, market cap, book to market ratio).

Any model is a simplification of reality, as you stated above. Then, by definition, there are going to be some flaws. I have to seriously disagree with some of your underlying assumptions above though. If you think that most investors are risk-loving (or even risk neutral for that matter), I'd like to see how well you do at managing other people's money.

There is no spoon
 
Mr. Anderson:
1. You're actually trying to argue that the risk-aversion is a flawed assumption? This is news to me....
  1. By definition, any choice made in the present can only utilize information from the past. Just because this is a one-period model doesn't prevent investors from re-weighting (or completely changing) their investment portfolio later on.

  2. I agree with this. Most returns are left-skewed.

  3. Not sure what your opinions on beta are, but that was the purpose of that whole discussion.

Also, for the record, most managers use some variation of Fama-French, which is a three factor model (market return, market cap, book to market ratio).

Any model is a simplification of reality, as you stated above. Then, by definition, there are going to be some flaws. I have to seriously disagree with some of your underlying assumptions above though. If you think that most investors are risk-loving (or even risk neutral for that matter), I'd like to see how well you do at managing other people's money.

  1. risk-aversion empirically exists.

  2. you can extend to multiple periods relatively easily, but its not necessary if your investments are liquid and have multiple cycles (super-cycles) of historical data (i.e. not RMBS circa 2007)

  3. Yes. But as Nassim Taleb likes to point out, there is no reason we need to use a normal distribution today.

  4. People (with computers) do this all the time. Most equity/bond fund managers get an email that contains this info at the end of each trading day. Options traders and other running more sophisticated strategies have even better tools to match the complexity of their portfolios.

On Fama-French/Four Factor Models: You can calculate all the coefficients and throw them in a table very easily.

 
  1. I'm not sure your explanation makes any sense. "Risk" in Modern Portfolio Theory stems from the implication that there is a relationship between the volatility of returns and the expected value of returns. We now know this to be false. Especially with research in Minimum Variance Portfolios (See Eric Falkenstein's new book "The Missing Risk Premium")

  2. Again, I'm not sure what you're arguing. It is well known that historical volatility is a good indicator of future volatility, but not the same for returns. What you're suggesting sounds like you want an out of sample error feedback loop, but that really has nothing to do with Markowitz or portfolio construction.

  3. No contemporary argues that asset returns follow a normal distribution.

  4. There are no "compounding" errors in the situation you suggest. You can generate portfolio weights for 255 stocks for a targeted volatility level by using a Genetic Algorithm in a matter of seconds—with great accuracy. And all this will end up proving is what we already know from point one.

For your part 2: You're going to have to make a highly convincing argument to the Minimum Variance/Minimum Correlation crowd. A recent white paper from David Varadi (CSS Analytics) has a strong case for minimum correlation. My brief personal research in minimum variance shows potential for consistent index beating returns with a weekly re-balance.

 
Best Response

From the title of the post, I assumed that this would be an interesting walk through the modeling tools used portfolio managers through the past century or so. Sadly, I was mistaken and instead this is another missive demonstrating how little even knowledgable people know about modeling. Depressing. Since it appears other commentors have noted the errors in #1-#4, I'll point out the errors that came straight out of the gates:

Markowitz's portfolio theory is beautiful. The construction of a portfolio by comparing asset risk and their correlation of returns was absolute brilliance. Prior to MPT this was a bizzare notion and it wasn't unusual to see portfolios built from single assets.

All models, each and every one of them, ever built for any discipline ever, now and for the forseeable future, utilize assumptions that create models that "serve as restricted versions of actual [whatever you're modeling] processes". This is akin to complaining about how all planes are restricted versions of spaceships because lazy engineers keep ignoring friction and drag.

The last point of note, is that the assumptions utilized by MPT are there to keep the mathematics manageable. I mean, let's be serious, if you wanted to nit pick the assumptions that are the least related to reality, the four you selected aren't even close to the worst. "Any investor can lend and borrow an unlimited amount at the risk free rate of interest" is the overall #1 far and away. But, "Investors have an accurate conception of possible returns" is definitely making strides in ridiculousness.

"My caddie's chauffeur informs me that a bank is a place where people put money that isn't properly invested."
 

"Nearly every assumption going into this model is not true. However that is not the point. The point is does the model, when empirically tested, teach us something about the world. The answer is a definitive yes." - Eugene Fama

Agree with the above posters. The biggest assumptions underlying MPT (complete agreement among investors with regard to expected returns, standard deviations, and covariances; unlimited borrowing and lending at the risk free rate, et.al.) are obviously not true. CAPM has proven to not completely hold, which has helped fuel the fire to multi-factor models like Fama-French which hold more, but still not completely. Fama-French has an R-squared in the mid-nineties which is pretty darn good imo.

 

Question: Everyone agrees that returns are not normally distributed, but then what do you use as your proxy for risk? Since standard deviation will either underestimate or overestimate variability for any non-symmetrically distributed variable.

 
2.71828:
Question: Everyone agrees that returns are not normally distributed, but then what do you use as your proxy for risk? Since standard deviation will either underestimate or overestimate variability for any non-symmetrically distributed variable.

Use the standard deviation. The normal distribution is an awful, terrible, and totally inappropriate distribution for returns, but it's better than the rest of them (for most purposes.) Just remember, the minute someone builds a perfect, unicorn & rainbow shitting model for stock returns, we're all out of a job :).

"My caddie's chauffeur informs me that a bank is a place where people put money that isn't properly invested."
 

You could use an Anderson-Darling Test and see if the data is normally distributed, which works really well for this as well as for the presence of other distributions. However, using an un-treated sigma in the event of leptokurtic distributions will produce erroneous and misleading results in your risk statistics. Using the appropriate distribution should give more accurate representation. If the data meet all the criteria for normality then using sigma should be basically sufficient (see tomorrow’s article for more).

Who Am I? | See what GMngmt is all about at About.Me
 
GMngmt:
You could use an Anderson-Darling Test and see if the data is normally distributed, which works really well for this as well as for the presence of other distributions. However, using an un-treated sigma in the event of leptokurtic distributions will produce erroneous and misleading results in your risk statistics. Using the appropriate distribution should give more accurate representation. If the data meet all the criteria for normality then using sigma should be basically sufficient (see tomorrow’s article for more).
Who Am I? | See what GMngmt is all about at About.Me
 

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