Karmikel Investments - Modern Portfolio Theory

Modern portfolio theory is a disciplined, scientific approach to investing in a portfolio of assets. It utilizes statistical analysis, primarily expected return, variance, covariance, correlation coefficient, and standard deviation as a means to classify, estimate and control a portfolios risk and return.

Modern portfolio theory formally began with the academic work of Harry Markowitz. In 1952, Markowitz outlined a process of portfolio selection based on risk and return statistics. In 1990, he was awarded the Nobel Prize in economics for this pioneering work. Since 1952, a handful of other scholars (Cootner, Sharpe, Fama, French, et al.) have expanded vertically and laterally on Markowitz’s theory, leading to what has collectively become known as modern portfolio theory today.

Modern portfolio theory is rooted in the belief that security markets are efficient. An efficient market is one in which security prices are fairly valued and reflect all information at any given point in time. In other words, security prices will not stray by much or for long from the consensus of their return given their risk. As such, investors cannot consistently achieve abnormal returns. In fact, empirical research suggests that portfolio managers do not consistently achieved abnormal returns and that their returns fall into the category that would be obtained by random chance. This belief leads to the argument that it does not pay to try and beat the market. A more prudent approach is to keep trading and management costs low and invest in a properly diversified, efficient portfolio of assets. To quote Harry Markowitz from the beginning of his award winning paper, “The rule serves better, we will see, as an explanation of, and guide to, 'investment' as distinguished from 'speculative' behavior.” [5]

Market efficiency is divided up into three categories to facilitate empirical research. The three categories are weak, semi-strong, and strong form. The market is considered weak form efficient if security prices accurately reflect all historical information. A market with semi-strong efficiency has prices that fully incorporate all public information. A strong form efficient market is valued accurately for all public and private information. In a strong form efficient market, people or groups (e.g., market makers or corporate executives) with inside information could not earn a abnormal returns from their knowledge..

In 1970, Eugene F. Fama wrote a theoretical and empirical literature review on efficient markets [3]. In this review, he studied all three forms of efficient markets. He concluded that there was persuasive support for the markets being weak form efficient, especially for price or return movements greater than one day. There was some support for a day-to-day dependence in stock prices but it was not large enough to warrant calling the market inefficient or to generating profits that could offset the trading costs to take advantage of the dependence. For price or return movements greater than one day, he concluded that there was very little evidence to contradict that the market was not efficient in the weak form.

In the semi-strong form, Fama reviewed research that studied whether stock prices were fully valued before major news was released to the public. He studied papers by Fama, French, Fischer, and Roll [4], Ball and Brown [1], and Scholes [8] who investigated whether stock prices were fully valued before stock split announcements, annual earnings releases and new common stock issues, respectively. Fama noted that the aforementioned research had shown that in each of the cases, the stock was fully valued before the event.

A strong form efficient market, one that incorporates both private and public information, is the form that is least supported. Fama’s review found two deviations from the strong form efficient market. First, he found that research by Niederhoffer and Osborne [7] showed that specialists on the stock exchange floor could profit from information on their special order book, in particular their limit orders. Second, Fama reviewed Scholes [8] study, which revealed that corporate insiders have information that could lead to abnormal profits. However, since it is illegal for anyone to act on this information (insider trading), it can be concluded that overall markets are efficient.

Efficient markets are the foundation of modern portfolio theory because the belief in them leads to one other important thought: if markets are efficient and all securities are correctly valued at any point in time, then no one can obtain abnormal returns on a consistent basis. In short, researching individual stock fundamentals, charting past security price movements or timing one's market entry/exit will not lead to consistent above normal profits. In fact, this type of active investment increases trading and management costs which are usually not offset by increased profits.

Now let's discuss some general statistical measures [2] used in an uncertain environment first for a specific asset, and then for a portfolio of assets. Finally, we will discuss the general characteristics of a portfolio.

The primary statistical elements used are expected return, variance, covariance, correlation coefficient and standard deviation. The expected return of an asset is the weighted average of its possible outcomes, which is the probability of the event happening times the event return. The expected return for asset i is:

To measure the risk of an individual asset we inspect the dispersion of the asset's returns around the mean return of the asset. To do this, we use the variance and standard deviation. The variance (1.2) is the squared deviations of the returns from the mean return and the standard deviation (1.3) is the square root of the variance. The standard deviation is quite useful since it puts the dispersion (variance) in the same unit of measure as the original data.

In Table 2.1 we provide an example. Suppose we are comparing three different stocks and wanted to calculate their expected return. Let's create three economic outlooks (expansion, stagnation, and recession) and assign them a probability. Based on these assumptions, our expected returns would be 10%, 20% and 30% for asset 1, asset 2, and asset 3, respectively. Each asset's variance and standard deviation are also listed.

		Returns (%)
Event	Probability	Asset 1	Asset 2	Asset 3
Expansion	.33	14	28	42
Stagnation	.33	10	20	30
Recession	.33	6	12	18

Expected Return		10.00	20.00	30.00
Variance		10.67	42.67	96.00
Standard Deviation		3.27	6.53	9.80

The statistics presented above help explain individual assets. Now let's look at these statistics and add two other equations to allow us to investigate portfolios of assets. The equation to calculate a portfolio’s expected return is very similar to that of the equation used for a single asset. In the portfolio expected return calculation, we use the percent of total assets invested in each security as the weight.

We need to bring in covariance to assist in the calculation of the portfolio variance. The covariance measures how closely the returns are associated with each pair of securities. It is calculated as:

One very useful tool in statistics is the ability to standardize the covariance of two securities. Dividing the covariance by the product of each security's standard deviation will set the covariance in the range from –1 to +1. This is called the correlation coefficient.

The covariance is used in the variance calculation. To calculate the variance of a portfolio we use the following equation:

Notice in this equation that the right side is multiplied by the covariance. If the covariance between all pairs of securities in the portfolio is zero, then the variance of the total portfolio is simply the sum of the variances of the individual securities times the square of the amount invested in each (i.e., the left side of the equation only). If we then go one step further and assume that the investor places an equal amount of money in each of N securities and we substitute 1/N into the equation for the proportion invested in each, then the equation becomes:

With this situation and formula 1.7, if we combine enough independent assets the portfolio variance will approach zero. This is mathematically possible to achieve but in the real world most securities have a slightly positive covariance and it would be impossible to have enough independent assets to make this occur. However, I point it out to show the power of properly selecting low correlation assets for a portfolio. I will discuss graphically the effects of variance and covariance on a portfolio in the next section below on Markowitz’s portfolio selection theory.

To calculate the standard deviation we take the square root of the variance. The standard deviation is very useful because it puts the variance results into the unit of measure from the original population. In this case, it converts the variance into a percent of return unit allowing the user to readily identify the dispersion range. For example, Asset 1 in Table 2.1 has a variance of 10.67 and a standard deviation of 3.27. The variance is the magnitude of the dispersion but the standard deviation of 3.27% puts the variance into the same unit as the original data.

Besides believing in efficient markets, the second most important foundational element of modern portfolio theory is the portfolio selection criteria outlined in 1952 by Harry Markowitz [6]. In this paper, Markowitz argued that it was wrong to assume that rational investors just wanted to maximize expected return with complete disregard for risk. Rather investors wanted a stock that maximized expected return for a given risk level. This is illustrated by Figure 1 which shows four different securities plotted on a risk return graph. Markowitz claimed investors would choose security B rather than security A because security B has a higher expected return for the same amount of risk taken. Likewise, an investor would use the same argument to choose security C over security D. In this example, securities B and C are more efficient than securities A and D.

Markowitz took this philosophy one step further and applied it to create efficient portfolios. He used mathematical statistics to create an efficient frontier on which optimal (efficient) portfolios lie. Consider a portfolio made up of only two securities. Based on the expected return, standard deviation and covariance between the two assets (Asset A and Asset B), Figure 2 can be constructed. The black line represents the different combinations of the two assets. All portfolios on line segment TB are efficient. That is, no other portfolios offer higher returns for the given level of risk. The portfolios on segment TA are inefficient: you could invest in the portfolios on segment TB and achieve higher expected returns for the same risk. This same process can be carried out for a portfolio of unlimited securities, although the calculations are more extensive.

As noted by Markowitz, proper diversification cannot be achieved without selecting statistically uncorrelated assets. Simply allocating assets to many different securities is not enough to build a properly diversified portfolio. For example, one could assume that a portfolio comprised of 30 different stocks was diversified. Suppose, however, that all 30 stocks were in the U.S. banking industry. The portfolio selection process must involve studying covariances between each of the security sets that make up the portfolio. One cannot simply pick assets at random, combine them together in the appropriate percentage to maximize the expected return and consider this an efficient portfolio. The two steps to creating an efficient portfolio are selecting a group of assets that have low covariances among themselves and then determining the percentage of total assets to allocate in each asset.

The variance is such an important concept that Markowitz gave an example in his research paper. He showed that combining two portfolios with an equal variance created a new portfolio with less variance than either of the original portfolios. The risk has to be equal or less than either of the original portfolios. The only way to have an equal risk is if the assets are perfectly positively correlated (+1). The greater the correlation coefficient approaches a perfectly negatively correlation (-1) the greater the reduction in the portfolio’s risk. Markowitz illustrated this principle using mathematic formulas and plotting them on an efficient frontier chart. Here we will use a table to present the some of our own data.

In Table 2.2 notice that both assets by themselves have the same expected return, variance and standard deviation over the four year investing period. Now look at what happens when an investor splits his or her money and buys equal portions of both assets. Notice that the portfolio has the same expected return. However, this same return is achieved at a lower risk (standard deviation) than either asset alone.

Let's present a graphical presentation of the effects of different correlations' factors on a two-asset portfolio. Figure 3 shows a graph of two securities, securities A and B. The green line represents the two assets at a correlation coefficient of 1. A correlation coefficient of 0 is diagramed as the blue line and a –1 correlation by the yellow line. Notice that if two assets are perfectly negatively correlated (-1) there is one combination of the two assets that has zero standard deviation and will offer a known return of approximately 14% in any given situation. Also notice that all combinations offer higher returns and less risk than investing in asset A alone. This graphically represents the power of proper assets selections and combinations to create an efficient portfolio. The characteristics are very similar for portfolios comprised of more assets, only the mathematics become more rigorous.

In the paragraphs above, we have shown what an efficient portfolio constructed using risky securities means and gave an example of how combining the right assets can reduce the risk and increase the expected return of a portfolio. Now lets bring riskless lending and borrowing into the discussion. An example of riskless lending is buying U.S. Government securities. The expected rate of return on the riskless asset, by definition, would be the rate of interest received. Since the standard deviation on the riskless asset is zero it will intercept the Y-axis on the risk return chart at the riskless rate of interest. Combining the riskless asset with a portfolio of risky assets produces a straight line (Equation 1.9)[9]. If an investor contributes a percent of their investments in the riskless (A) asset and the rest in the risky portfolio (B) the expected return of the combinations will be a straight line:

E_Rc is the expected return for the portfolio of risky and riskless asset combination

s_Rc is the standard deviation for the portfolio of risky and riskless asset combination

but since the risk of the risk free asset is zero (s_Rf = 0), the equation reduces to:

Thus, all combinations of the risky portfolio and the riskless asset falls on a straight line. In figure 4, line RV represent combinations of the riskless asset and risky portfolio A. Line RU is combinations of the riskless asset and risky portfolio B. However, line RS, which dissects risky portfolio P, is most optimal that lies. Combinations along line RS offer the most expected return for a given standard deviation and offer the most incremental increase (greatest slope) in expected return for a given increase in standard deviation.

To borrow assets is the opposite of lending assets, so let the amount of assets allocated to the riskless portfolio (a) in equations 1.10 and 1.11 be negative. If we take the money borrowed and reinvest it in either of the risky portfolios (A, B, or P), the line would extend to the right. In Figure 5, borrowing is represented by segments AV, BU and PS. Lending is represented by segments RA, RB and RP. As was the case with lending, line RS dominates all other lines and would be the preferred combination of riskless asset and risky asset whether lending or borrowing.

In the riskless lending and borrowing scenario just presented, if riskless borrowing is not realistic, investors will still choose a portfolio along line RP. If riskless borrowing is possible in some obscure case, investors would choose a portfolio from line segment PS making line segment RS valid. Summarizing, if a portfolio of risky assets is combined with a riskless asset, there will be only one risky portfolio that is efficient from all the different combinations of risky assets chosen.

So why did it take so long for Markowitz’s groundbreaking theory on efficient portfolios and Sharpe’s addition of a riskless asset to Markowitz’s efficient portfolio to be accepted? First, in the terms of theories, fifty years is not really that long of time. New theories are not generally immediately accepted as acceptable; they must be scrutinized from all angles buy many different minds before they can gain credibility. Second, there are a lot of calculations required to supply input into these models and there were no affordable high speed computers back in the 1950's. Let's consider the analysis required to review 250 stocks as possible candidates for selection into the efficient portfolio we want to create. The inputs required for Markowitz’s model are expected returns, standard deviations, and correlation coefficients. This would require 250 expected returns, 250 standard deviations and 31,125 (N(N-1)/2 correlation coefficients, one for each pair of securities) correlation coefficients. Not only would 31,625 calculations have to be performed at the time of portfolio implementation but they would also need to be performed at selected time intervals to perform analysis and keep the portfolio efficient.

1. Ball, Ray, and Phillip Brown. “An Empirical Evaluation of Accounting Income Numbers.” Journal of Accounting Research, (Autumn, 1968), 159-78.

2. Elton, Edwin J., & Gruber, Martin J. Modern Portfolio Theory and Investment Analysis. New York: John Wiley & Sons, 1981.

3. Fama, Eugene. “Efficient Capital Markets: A review of Theory and Empirical Work.” The Journal of Finance, Vol. 25, No. 2, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York, N.Y. December, 28-30, 1969. (May, 1970), pp. 383-417.

4. Fama, Eugene, Lawrence Fischer, Michael C. Jensen and Richard Roll. “The Adjustment of Stock Prices to New Information." International Economic Review, Vol. 10, No. 1. (Feb., 1969), pp. 1-21.

5. Markowitz, Harry. “Portfolio Selection.” The Journal of Finance VOL 7, No. 1, (Mar., 1952), pp. 77-91.

6. ---. Portfolio Selection: Efficient Diversification of Investment (New York: John Wiley & Sons, 1959).

7. Niederhoffer, Victor and M. F. M. Osborne. “Market Making and Reversal on the Stock Exchange.” Journal of the American Statistical Association, (December, 1966), pp. 897-916.

8. Scholes, Myron. “A Test of the Competitive Hypothesis: The Market for New Issues and Secondary Offerings.” Unpublished PH.D thesis, Graduate School of Business, University of Chicago, 1969.

9. Sharpe, William F. “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk.” Journal of Finance, Vol. 19, no. 3 (September 1964), pp. 425-442.