Modern Portfolio Theory – Part One

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Our risk/return series continues with a review of Modern Portfolio Theory (MPT). We’ve already looked at alpha, beta, efficient markets, and returns.  Our ultimate goal is to evaluate the role of alpha in hedge fund profitability, how to replicate hedge fund results without needing alpha, and finally how you can start your own cutting-edge hedge fund using beta-only replication techniques.

MPT suggests that a portfolio can be optimized in terms of risk and return by carefully mixing individual investments that have widely differing betas. Recall that beta is return due to correlation with an overall market (known as systematic risk). To take a trivial example, if you hold equal long and short positions in the S&P 500 index, your portfolio would have an overall beta of (.5 * 1 + .5 * -1) = 0.  There would be no risk, but in this case, there would be no return either, except for the slow drain of commissions, fees, etc.

A more desirable portfolio would have a low or zero beta (uncorrelated to the overall market) but with a positive net return in excess of the risk-free rate.  MPT posits that this is possible through careful selection of the individual securities within the portfolio. MPT was invented in 1952 by Harry Markowitz, and has received much study and criticism over the years.  It has been extended by Post-Modern Portfolio Theory, challenged by Arbitrage Pricing Theory, and denigrated by Behavioral Finance. We’ll look at of all of these in late installments. Nonetheless, MPT is the starting point for understanding many essential investing concepts.

Diversification is the key to MPT: adding an asset with a beta less than one to your portfolio will lower the portfolio’s volatility. The portfolio volatility is equal to the square of the total weighted sum of the individual asset volatilities (as measured the asset’s standard deviation of returns over time). This measure of volatility is called variance. Assets are weighted based on their market value contribution (i.e. shares x price) to the overall portfolio.

MPT makes a number of simplifying assumptions. Recall from a previous blog that the Efficient Market Hypothesis (EFH), based on the Random Walk Theory, states that the market is a reasonable speculation, with the same odds of winning and losing.  Marginal supply and marginal demand determine a securities price. MPT makes the assumption that the EFH is true.

Another assumption is that investors tend to be risk-adverse: they want to be paid for taking on additional risk. After all, some are content to take no risk and collect the risk-free rate. Why take on additional risk if there is no reward for doing so?  Of course, this also means that an investor will need to accept additional risk if he or she wants a higher expected return. The corollary to this assumption is that rational investors, if presented with two portfolios with different levels of risk but the same expected return, will choose the less risky portfolio – it has a more attractive risk-return profile.

If a theory is only interested in the portfolio’s mean return and its standard deviation of return, then the portfolio can be plotted as two-dimensional quadratic utility function, with expected return on the y axis and the volatility (i.e. the standard deviation of returns) on the x axis.  Each assets contribution to the overall risk of the portfolio is a function the asset’s beta (market correlation).  The lower the asset’s beta, the less volatile the portfolio will be after the asset is added.

When a risk-free asset is mixed with a risky asset, the effect is a linear change (a straight line in risk-return space).  That means a straight line connects the risk free rate from the y axis to a risky asset’s (x, y) location, and any point on the line indicates the relative amount of risk-free asset added.  At the y axis, it’s all risk-free; at the risky asset’s (x, y) coordinates, there is no risk-free portion.  Any coordinate on the connecting line can be reached by the appropriate addition of risk-free asset to the portfolio. This connecting line is called the Capital Allocation Line (CAL), as shown in the following diagram:

Capital Allocation Line

The CAL shows the ratio of reward to variability, and its slope is the incremental change in that ratio. Note that by borrowing money, an investor can buy addition risk-free asset and extend the CAL, albeit at a lower slope.

We’ll continue our discussion of Market Portfolio Theory next time when we take up the Efficient Frontier.

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Copyright 2010 Eric Bank, Freelance Writer

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