Wednesday, May 7, 2008

Modern Portfolio Theory Part V: The Sharpe Ratio

In the April 21-24 blogs, one of my guest contributors, Professor Pat, tackled the esoteric subject of Modern Portfolio Theory (MPT). Using data from 1927 to the present, he derived an asset allocation table that an investor can use to construct his or her own optimally allocated investment portfolio according to an investor's requirements for portfolio return versus risk. Today, he's extending his presentation to include the Sharpe Ratio and how that can be used to further increase portfolio returns while simultaneously minimizing risk. Take it away, Professor!

This installment in the Modern Portfolio Theory (MPT) series will discuss the concept of the efficient frontier, the Sharpe ratio as applied to MPT and how to add a riskless cash component to your portfolio. The discussion is applicable to interest rates and optimum portfolio allocations and returns current to today.

The Efficient Frontier
In Part III of this series (see April 23 blog) a table was presented entitled ”Optimum Asset Allocation Among Investment Classes” whose first two columns were Required Return and Standard Deviation. Let's now plot those two columns (and many more data points in between) on the graph shown below. The resulting magenta line is known as the “efficient frontier” as is represents those optimally efficient portfolios that provide the least amount of risk for each level of return in today's market. The shaded area underneath the curve represents the possible space of less efficient allocations of portfolio assets.

Note that the efficient frontier is a curved convex line. This is a result of the lack of correlation (see Part IV, April 24 blog) between the various assets in the portfolio and how that results in an optimum portfolio characterized by an overall standard deviation less than it would be for one of completely correlated assets. Remember, for uncorrelated assets when some are up in value others are down in value and the overall portfolio therefore exhibits reduced variation. For a portfolio of totally correlated assets a combined linearly weighted standard deviation would result and the efficient frontier curve would be straight rather than curved. It is the existence of optimum allocations that pulls the line to the left for a lower standard deviation at each point that creates that convex shaped curve. The end points of the curve do not benefit from this left pulling because they are dominated by portfolios containing a high percentage of a single asset class--at the high end of the curve, the optimum portfolio is composed of 100% in Small Stocks and at the low end by 93% in Treasuries. The convexity of the efficient frontier curve therefore graphically demonstrates the beneficial effects of diversification.

Adding a Cash Component and The Sharpe Ratio
The Sharpe Ratio, S, is named after William Sharpe, who won the 1990 Nobel Prize for his work on the Capital Asset Pricing Model which shows how the market prices individual securities in relation to their asset class. Here the discussion is limited to the Sharpe Ratio which, for a particular investment, is a direct quantitative measure of reward to risk. It is a measure of how much excess return a portfolio provides above a riskless investment considering the additional risk it entails. It is defined as:

S = (Portfolio Average Rate of Return – Current Rate of Return of a Riskless Investment) / Standard Deviation

Let's look at the line of best capital allocation on the graph above. This line represents the best way to incorporate a riskless cash component into an already optimum portfolio thereby reducing the risk even further albeit at a reduced overall rate of return. As of this writing, the annual rate of return on a 30-day United States Treasury Bill is about 1.2%. This is considered the safest investment for the short term and is considered to be as riskless as cash. The line on the graph therefore starts at 1.2% plotted at zero standard deviation (it's riskless!) and is drawn so that it contacts the efficient frontier curve at a point such that the slope of the line is a maximum. This is the point on the efficient frontier with the greatest Sharpe Ratio and is known as the market portfolio. The slope of the line is the Sharpe Ratio for the market portfolio and the current riskless rate of return. As seen on the plot above, the market portfolio provides about a 5.0% return which has a standard deviation of 3.4%. You can confirm this in the table from Part III. The Sharpe Ratio for this investment is therefore (5.0 – 1.2) / 3.4 = 1.12 which, as we have noted, is also the slope of the line of best capital allocation.

Note that the black line is straight because the riskless investment has no standard deviation and there is no better “optimum” allocation that will produce a resultant combined reduced standard deviation that would produce a convex curve like that for the optimum allocations of the other investment classes.

Investments anywhere on the line of best capital allocation give you the overall return and standard deviation of an optimally allocated investment combined with an additional riskless cash component (the optimum allocation for the 5.0% return already has about a 70% 30-day T-bill component which in the long term has a nonzero standard deviation but that is immaterial for this analysis of the current state of interest rates and the market). In this way you can easily construct a composite portfolio that is also optimum but that has a reduced standard deviation resulting from adding a riskless cash component. This is of course at the penalty of a lower overall portfolio return. The percentage of cash to include in the composite portfolio is easy to determine since the standard deviation is reduced linearly according to the slope of the line (the Sharpe Ratio). For example, a portfolio of assets at risk returning 5.0% with a standard deviation of 3.4% can be reduced to one with a standard deviation of 2%. From the graph, such a portfolio returns about 3.2% and is constructed by reducing the component of optimally allocated risky assets to 64% (3.2 / 5.0) x 100% and by adding a 36% (the rest) cash component.

Increasing Potential Returns Through Margin Borrowing
Instead of adding a cash component to your portfolio you can choose to borrow against the market portfolio and realize increased potential returns. Naturally, doing this also increases risk.Whereas a portfolio with an added cash component and lower risk resides on the line of best capital allocation at or below the market portfolio, a portfolio that has been borrowed against will reside on the line of best capital allocation above the market portfolio. A higher risk, but still optimum, portfolio can be formed by borrowing against the market portfolio on margin (assume the rate of borrowing is also at the riskless rate of interest). You then reinvest the borrowed cash right back in the market portfolio in the same proportion as assets that portfolio already holds. So now you have a portfolio allocated exactly the same as the market portfolio but it now has a higher value. But you have margin debt to pay at a lower rate than the portfolio is expected to produce on average. The standard deviation of returns this portfolio will exhibit is greater than the market portfolio as seen as you climb up the line of best capital allocation. Hopefully the result will be positive. This expanded portfolio residing on the line of best capital allocation which has a slope equal to the Sharpe Ratio exists in a region above the market portfolio contact point on the efficient frontier. It is there on that line that you can decide the standard deviation and thus the amount of additional risk you'll have to take in order to achieve a higher return.

Dr. Kris Note: I know that these discussions on MPT might be flying over the heads of many of you, but it is something that every serious investor who does his own portfolio asset allocation should try to understand because these tools can be used to effectively increase returns while minimizing risk. I think that's a good enough reason to merit these discussions.

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