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## Portfolio Theory and Market Efficiency

Modern Portfolio Theory (MPT) is a theory for how investors can construct portfolios of assets to achieve their goals in terms of desired returns and acceptable risk. Previous to the development of this theory, investors focused on choosing a collection of individual assets with the best stand-alone risk and return characteristics without regarding the relationship between these assets. Rather than focus on selecting individual securities, MPT considers portfolios of securities and examines the return and risk of these portfolios. The theory was introduced by Nobel-prize winner Harry Markowitz in 1952.

The MPT has been heavily criticized recently. We believe that this criticism is valid in many cases. Some of these criticisms stem from assumptions of the MPT which have been shown to not be generally valid. Some of these assumptions include:

- Returns are normally distributed.
- The market is efficient (we’ll discuss this issue later).

- Costs, taxes, and transaction fees are ignored in the model.

Cumulatively the MPT critics argues that these assumptions are so fundamental to the mathematics of the model that the results are not meaningful since these assumptions are not valid. Despite these valid points of criticism, the model has introduced some key concepts into investing which we believe are valuable and contribute to the creation of a successful long-term portfolio. These concepts are: the value of diversification and the existence of an efficient frontier of investment portfolios. We’ll spend some time on these two concepts and leave it to the reader to do more research on MPT if he desires. Additionally, we’ll discuss another important theory, called Market Efficiency, and discuss its implications for the development of an investment strategy.

### Diversification

Markowitz used a mathematical analysis of diversification to illustrate the value of making investment decisions based on portfolios rather than individual assets. The key insight was that the behavior of the assets in relation to one another will have an impact on the overall performance characteristics of the portfolio. Investors will typically judge the performance of a portfolio based on two measures: return and variance. The variance is a measure of the volatility of the returns of an asset. An asset with low variance will always deliver very similar returns while high variance will cause the returns to fluctuate greatly on a period-to-period basis. A general principle in investing is that you must sacrifice return to get low variance (and vice-versa). An example would be U.S. Treasuries which have lower mean returns than stocks but have much less variance.

For a portfolio, the mean return will simply be the weighted average of the returns of each of the component assets. Thus, if you have a portfolio of 50% bonds with a mean return of 4% and 50% stocks with a mean return of 8%, then the mean return of your portfolio will be 6%. However, the calculation of the variance of the portfolio includes terms which represent the correlation (or covariance) of the individual assets to one another.

Diversification can actually produce a portfolio with lower variance than the simple weighted average of the variances of each of the component assets. By diversification, we mean the construction of a portfolio from assets which are not perfectly correlated. With proper diversification, it is possible to reduce the variance (or risk) of a portfolio while not sacrificing the expected return.

To achieve this goal of reduced portfolio variance while not sacrificing mean return, you want select assets that have identical or similar mean returns but are uncorrelated or negatively correlated. With this approach, you can create an overall portfolio with similar return and lower variance than the best performing asset in your portfolio. This is the power of diversification.

In general, the concept of diversification is applied to two areas when investors construct portfolios. First, it is applied within asset classes. For instance, the investor may want to invest in large cap U.S. equities. Since the investor is not a professional, he may not be able to determine which individual stocks have the highest rates of return or lowest variances. They may all look the same from his standpoint. However, since the stocks represent companies in different business sectors, they will not all be highly correlated. Thus, he can choose a collection of these stocks and benefit from diversification. The collection will have the mean return of an average individual stock but will have a much lower variance thanks to the power of diversification. Thus, it is much less risky to hold the portfolio of S&P 500 stocks instead of just holding one stock.

Secondly, the concept of diversification is applied between asset classes. Now, let’s assume that our investor purchases all the S&P 500 stocks. We can think of this as one asset with a given expected mean return and variance. Now, he may choose to add another asset to the portfolio such as a U.S. Treasury bond. This new asset will also have an expected mean return and variance. As we mentioned before, U.S. Treasuries have historically had lower mean returns than the S&P 500 but have much less variance. Additionally, let’s assume that they are completely uncorrelated. Now, the investor will be giving up some mean return by adding the Treasuries to his portfolio but will definitely reduce his variance. In practice, it usually turns out that adding a small amount of an uncorrelated asset can lead to a decent sized reduction in variance while only causing a small decrease in expected return. This tradeoff may be acceptable to many investors.

To illustrate, let’s use an example taken from “The Intelligent Asset Allocator” by William Bernstein. There are two assets: stocks and bonds. In any period, the stocks are equally likely to return either +30% or -10% (a geometric mean return of 8%) while bonds are equally likely to return either +10% or 0% (a mean return of 5%). We can see by the volatility of the possible returns that the stocks have higher variance. The stocks and bonds are completely uncorrelated. Now, we want to compare portfolios consisting of various percentages of these stocks and bonds from 100% stocks to 100% bonds. Let’s plot these portfolios with mean return on the y-axis and standard deviation of returns on the x-axis.

Several points are illustrated by this graph. First, we see that starting with a portfolio of 100% stock and adding bond will reduce the variance of the portfolio. The loss of return is fairly small when bond is first added to the portfolio. Secondly, on the other side, adding stock to a portfolio of bonds will actually increase the return and reduce the variance until a decent amount of stock has been added. Another increasing point is to notice that the portfolio which consists of 50% stock and 50% bonds is not on the midpoint of a line connecting the 100% stock and 100% bond portfolios. It is actually above this line and to the left. This is the benefit of diversification. The same principle applies when we make more sophisticated assumptions concerning the distribution of returns for the assets within the portfolio. The important thing is that the assets not be fully correlated.

We think that diversification is a key principle that should be used by all long-term investors. It allows the investor to reduce the variance of his portfolio’s returns while hopefully not sacrificing too much in expected return.

### Efficient Frontier

Our simple graph leads us to another concept in MPT: the existence of an Efficient Frontier. If we considered a set of asset classes and allowed investors to construct any possible portfolio consisting of these assets, then we could plot the return and standard deviation for each portfolio. There will be a set of portfolios which achieve the highest possible return for each level of standard deviation (or alternatively, the lowest standard deviation for each level of return). This set of portfolios forms the Efficient Frontier. This is where the investor wants his portfolio to be.

The chart below demonstrates the Efficient Frontier. The line represents the Efficient Frontier. All possible portfolios will lie on that line or below and to the right.

One major problem with the Efficient Frontier is that you cannot know which portfolios will lie on the efficient frontier before you invest. The set of portfolios on the Efficient Frontier must be calculated with either historical results or estimates of the returns, variances and covariances of all asset classes. Unfortunately, returns, variances and covariances of asset classes will change over time. The idea is still a useful concept and you can use your own estimates or historical data to find portfolio which you believe will be near the Efficient Frontier. That is the best that we can hope for when

### Market Efficiency

There is an ongoing debate between academics and industry insiders about the ability of individuals to pick stocks whose returns will consistently beat the average of whatever segment of the market in which they invest. These debates center around the idea of Market Efficiency. The theory of Market Efficiency states that all information about a company or asset will be built into its price and the future movement of the stock is basically a random walk that can’t be predicted. If a positive movement could be predicted then the market would drive up the price and any possible gain would be crushed. Academics have produced many studies where they show that many high-performing mutual fund managers have benefited from luck much more than skill. The industry insiders (especially mutual fund managers) argue that this theory is incorrect. They argue that their analysis and investing techniques can put them at an advantage over the average investor. Thus, they think that consistent winning stock pickers exist.

There are two main problems that face the individual investor. First, does you believe that the market is really completely efficient? If no, then there should be an opportunity to beat the market. However, how do you identify the stock pickers who will be successful in the future?

Based on our experience and research, we believe that the market is not completely efficient. Although most of the hard data in the debate favors the Efficient Market theory, we believe that there is a *small* set of investment professionals who have the ability to beat the market over the long-term. However, from the investor’s standpoint, it is very difficult to identify these managers. Academics have performed many studies where they seek to find characteristics of successful stock pickers (so they can predict who will be successful in the future). However, they have failed to devise a method for identifying successful managers in advance. We don’t think that the individual investor will be successful at identifying these managers.

### Mutual Funds and the Efficient Market

To compound the problem that it is very difficult to identify a successful manager, if you invest in actively traded mutual funds, you will be fighting an uphill battle to just match the performance of the market. The problem is that mutual funds incur costs that cause their net returns to lag the market. The current mutual fund industry is so massive that it can’t be expected to beat the market – it is the market! Thus, it makes intuitive sense that the mutual funds would lag the market by the amount of their expenses. In fact, typically 80+% of all mutual funds will lag the market (or relevant sector) returns on an annual post-cost basis. When you increase the length of the period over which you compare mutual funds to the market returns, the percentage of mutual funds which under-perform increases to over 90+% for a 5 year period.

Actively managed mutual funds create expenses for investors in various ways. Let’s describe some of them.

**Management fees and marketing expenses:**These expenses which are generally captured in the expense ratio. These expenses will directly reduce the your return.**Commissions:**Fees charged to the fund when they make trades of the underlying assets in their portfolio.**Bid-ask spreads:**Whenever trades are executed, the fund must pay more for the asset than they could sell it for. This spread is essentially lost by the investor.

**Market impacts:**Buying an asset will cause its price to increase. Thus, the total cost of buying an asset will turn out to be larger than the current price times the number of shares desired.

Studies have been conducted that show that mutual funds tend to lag the market by a little bit more than their total expenses. In fact, the mutual funds with the highest expense ratios tend to have the worst pre-expense returns. That’s amazing! Even with more money to spend on research, they perform the worst.

The story actually gets worse. Actively managed mutual funds have other tax problems that we explain elsewhere on this website.

These ideas about Market Efficiency and the problems of active mutual funds point the investor towards developing a strategy which involves investments in low-cost index funds or exchange-traded funds (ETFs).

### Resources

**Moneychimp:**The moneychimp.com gives a fairly detailed overview of many of the components of MPT.

**Wikipedia:**The wikipedia has an entry on Modern Portfolio Theory.

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