August 15, 2013

Harry Markowitz's Broken Toy: A Journey Through Modern Portfolio Theory (Part I)

One of the most important contributes that academia brought to Wall Street is surely Harry Markowitz's modern portfolio theory.

The theory is pretty simple: create a basket of (presumably) uncorrelated securities and you can maximize return and minimize risk. The two risks the model tries to reduce are the unsystematic one along with the company's. The magic of the model lies in the diversification into stocks (or other asset classes) with different volatility and that are little or negatively correlated. Markowitz's model has a series of assumptions:
  • Markets must be somewhat efficient or able to absorb new information quickly and completely;
  • Investors have free access to new and correct information;
  • Investors try to maximize return and minimize risk;
  • Investors make their investment decisions according to the excepted return of the security and its volatility (as measured by the standard deviation);
  • Investors are supposed to act rationally in order to maximize their utility given their level of income;
At time t for an investment X, we define the moments of a normal distribution as follows:

Moments of a normally distributed random variable.*

Let's assume, for the sack of simplicity, that an investor wants to invest in stocks (through the S&P 500) and government bonds (the model is also applicable to 2+ asset classes). We can create several portfolios assigning different weights to stocks and bonds (i.e. long 90% equities and 10% bonds or alternatively short -50% equity and long 150% bonds) and then use the mean-variance analysis to compute expected return and volatility: 

The model, however, has several limitations:
  • Volatility, as well as correlation, is supposed to stay constant;
  • Returns are assumed not to violate the assumptions of normality;
  • Securities can be bought and sold in the market without encountering any liquidity squeeze;
  • Investor are supposed to act rationally;
  • Investors are supposed to have the same investment horizon;
  • There are no transaction costs;

Several (if not each) of the model's assumptions have been violated during recent years. Correlations among asset classes broke in 2008 and tended to one as it usually happens in highly volatile markets (Robert Frey has a post on this). The normality behavior of stocks and bonds has been questioned as well as the rationality of investors. Whoever used Markowitz's model (banks, retail and institutional investors) incurred large losses as markets turned bearish. 

Correlation changes over time and with dramatic effects on the portfolio frontier. In the graph below, I plotted three portfolio frontiers according to three different correlations: the historical correlation between stocks and bonds from 1926 to 2012  (approximately 1.4%) and two hypothetical new correlations: +0.6 and -0.8. 

Any portfolio along the three frontiers is optimal considering the trade-off between risk and return, given the weights assigned to stocks and bonds. The green arrows indicate how risk and return move along the frontiers as correlation changes (I am assuming the investor is merely interested in having a portfolio with minimum risk and maximum return).

Stocks might violate the normality assumptions. I calculated daily returns for IBM from 1962 to 2012 and the frequency distribution is presented below. You may be thinking "Hey! It really looks like a bell curve. IBM returns can be normally distributed" and you would be wrong! Excess kurtosis (a measure of the "peakedness" of the probability distribution of a real-valued random variable) is over 10. Not a good sign. 

IBM Daily Returns 1962 - 2012

Excess Kurtosis: 10.42
In terms of implementation this is a big problem as numerous asset sub-classes like options and interest rate derivatives are nonlinear. The Gaussian nature of the fluctuations of the underlying assets and the non-linear dependence of the price of the derivatives are major obstacles for the implementation in Markowitz's model.

The moments of the normal distribution change: Correlation changes over time, as well as volatility, thus the moments of the normal distribution change too. As moments change over time, the shape of the distribution changes giving different estimates about the future depending on the historical sample used, with considerable impact on the location and shape of the portfolio frontier.

I created two portfolio of stocks for two different time intervals: 1926-1966 and 1967-2012. The frequency distributions of returns are presented below:

LFT Portfolio (1926-1966) - Mean Return: 12.4%, SD 23.5%, Skewness -0.2, Kurtosis -0.38
RT Portfolio (1967-2012) - Mean Return: 10.8%, SD 17.6%, Skewness -0.7 Kurtosis 0.12
Tails are also affected: a 3-standard deviation move from the mean has probability of 5% in the 1926-1966 portfolio and 0.5% in the 1967-2012 one.

Markowitz's model gives a false sense of risk control to investors. The optimization process and the normal distributions indirectly "promote" the underestimation of risk with non-optimal portfolio allocation. However, (partial) solutions for these problems have been found. More on that in the following weeks. Stay tuned!

*In reality the variance is one of the moments of a normal distribution, not the standard deviation. However, I defined the latter for reasons the reader will understand when reading the post.


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