Spotlight on the Sharpe Ratio: Part II

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This is the technical portion of this article trilogy on the Sharpe Ratio so be sure you’re armed with high-water pants and a pocket-pen protector. A slide rule is a bonus.

A little background
Modern Portfolio Theory (MPT) is a Nobel Prize winning financial concept that utilizes mathematical optimization algorithms to determine the best allocation of investments in a portfolio. For a selected desired overall rate of return, MPT will tell you how to allocate your capital among a diversified set of investment asset classes such that the variation in the periodic portfolio total returns as measured by the standard deviation (aka risk) is minimized.

Follow any discussion on MPT and you’ll run into the following graph on the Efficient Frontier:

This technical plot shows the current relationship between desired return and standard deviation. What it shows is that to get a higher return, you’d need to assume more risk (given by a higher standard deviation). The curve known as the “efficient frontier” represents those optimally efficient portfolios that provide the least amount of risk at a given level of return. The shaded area underneath the curve represents the possible space of less efficient allocations of portfolio assets. To obtain a return in the area above the efficient frontier is theoretically impossible with the given asset classes.*

Note that the efficient frontier is a curved convex line. This is a result of the lack of correlation between the various asset classes 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. 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 single asset class portfolios, at the high end with 100% the riskiest asset class and at the low end by mostly the least risky asset class. The convexity of the efficient frontier curve therefore graphically demonstrates the beneficial effects of diversification.

How the Sharpe Ratio fits into MPT
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. On the graph, the annual rate of return on a 90-day T-Bill is about 1.2% (those were the days!). This is considered as safe an investment for the short term as is possible to find 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%. 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 line of best capital allocation 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.

The market portfolio has the unique property that, taken in combination with a riskless cash component, it will produce an optimal return not on the efficient frontier and outside the suboptimal allocation space. It will lie on the line of best capital allocation. The combination will offer a larger return for a given amount of risk than any of the portfolios on the efficient frontier. In this way you can easily construct a composite portfolio that is still optimum but also has a reduced standard deviation. Unfortunately, this comes 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 an arbitrarily selected standard deviation of 2.0%. From the line of best capital allocation on the graph, such a portfolio returns about 3.2% and is constructed by reducing the component of optimally allocated market portfolio risky assets to 64% (3.2 / 5.0) x 100% and by adding a 36% (the rest) riskless cash component. Portfolios on the line of best capital allocation above the market portfolio can only be achieved by adding leverage to the market portfolio financed by borrowing at the riskless rate of return.

Tomorrow’s focus
In the final segment, we’ll look at Sharpe Ratios for optimally allocated portfolios in the current market environment.

*A traditional MPT portfolio allocates funds among the following asset classes: Large Company Stocks, Small Company Stocks, Long-Term Corporate Bonds, Long-Term Gov’t Bonds, Medium-Term Gov’t Bonds, T-bills, REITs, International Stocks, and International Bonds.

Note that the efficient frontier curve will be different assuming a different set of asset classes.

>>> Go to Part III

Comments

[je]

Hi Kris,
While MPT seemed very relevant to the "growing market" of the last few years and is undoubtedly an exciting numerical exercise, do you really think it is relevant today.

We are in a bear market for sure with banking problems through 2012, also a certainty, don't you think it is more prudent at this time to discuss the matters of events that will turn the market up or down.

Right now we appear to be near the start of a rapid leg down. We could go on up in the DOW prior to 10116 without breaking this pattern of downward expectations. I really appreciate your intelligence and maybe I am wrong but your portfolio analysis just seems like instruction for another time.

I would like to ask you, because I trust your analysis as an unbiased scientist, the probability of this being a true rally and not the Fed leading a pseudo-recovery in this financial warfare with the rest of the world.

Right now they are fighting rising interest rates which calls for a correction, not only to reduce rates but also to lower the market to pick up new money to continue the rally.

Robert Prechter says the rally is over and we are beginning our trip down to at least as low as March and eventually lower but although I too think we are going there, I am not so sure it will occur right now. Maybe (and this is where your excellence would be greatly appreciated) he is right under normal market circumstances but these are days of desperation and anything but normal and I truly believe that you could help if you would care to stick your neck out and do so by looking at the internals in such a way that the rest of us can not.

A lot of Prechters analysis hinges on investor sentiment (emotion), which if the PPT and the liquidity provided to the banks can control, as I suspicion they might. By these wild market swings - like an evangelical does with raising and lowering emotions before passing the offering plates-They can systematically control investor sentiment. And when they arrive around the 8800 level(where ever you consider the neckline of the inverse head and shoulders to be), the investor sentiment having been finessed to a low to a level that could enable a rally that could have a blow off top pretty quickly.(That head and shoulders pattern that I did comes in around 11,263 in the DOW)

Now you throw all of that in with the fact of the expiration of the Bush tax cuts which predict profit taking prior to the new year because of the fear of taxes being made retroactive to the beginning of 2010 and the possibility of selling off the market or could they - the PPG, Banks, Goldmans High frequency trading, etc - continue the market rebound.

Some of the evidence could be with how much money was out of the market and how much of that has returned and does it match market levels.

I just simply question the validity of the level of this rally being truly investor oriented and I think you have powerful insight to the likelihood of these things that I suspicion.
There are so many wave patterns that are incomplete at this point and they vote for a continued rally after a substantial correction.

I read that pattern failures are up to 25% recently compared to 10% in the 90's and we do not know the influence of high frequency trading on the statistics of pattern successes and failures, but we are dealing with probabilities and that is your game.

What is the probability of my suspicions being correct about the Fed leading the rally and this currently just being a correction and not the top as Prechter says and if there is investor selling prior to years end will the Fed still be able to lead the rally higher into January?

Will you help a fellow out?
je

2009 Nov 08 10:59 AM