## Archive for November, 2009

### Part II: The Sharpe Ratio and Modern Portfolio Theory

Friday, November 6th, 2009

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.

### Honing in on the Sharpe Ratio – Part I

Wednesday, November 4th, 2009

I’ve been reading that investment professionals still use the Sharpe Ratio to evaluate fund performance which led me to ask the question of some of my peers as to what they consider to be a “good” Sharpe Ratio. The answers I received were unexpectedly all over the map; some considered anything over 1 to be good. Others thought that only double digit Sharpe Ratios were worth bothering with while still others said that the Sharpe Ratio had no impact on their investment decisions.

This left me even more confused and I decided to explore the Sharpe Ratio in detail so that I could make my own informed decision concerning its validity. To help me in my investigation, I called on Professor Pat as he’s the resident StockMarketCookBook expert on all things related to portfolio theory.

In only two days, he presented me with an exceptional treatise on the subject of the Sharpe Ratio. The only problem I had with it was the length which I felt was too long for one blog so I broke it up into three distinct parts which will be run over the next few days. What follows is essentially his writing with some of my own thoughts tossed in.

Definition of The Sharpe Ratio
The Sharpe Ratio is named after William Sharpe, a 1990 Economics Nobel Laureate who won it for his work on the Capital Asset Pricing Model (known as CAPM) which shows how the market prices individual securities in relation to their asset class. Here the discussion is focused on the Sharpe Ratio which for a particular investment is a direct quantitative measure of reward to risk.

Sharpe devised the ratio which he called the ‘reward to variability ratio’ in 1966. It later became known as the Sharpe ratio as other economists and financial professionals attributed the ratio to him. It is a measure of how much excess return an investment provides over and above a riskless investment (e.g., T-bills) considering the additional risk (σ, the standard deviation of the returns on the investment under evaluation) that is entailed. It is mathematically defined as the following:

Sharpe Ratio = [Investment Rate of Return – Risk-free rate of return] / σ

The Sharpe ratio is there used to assess how well an investment compensates an investor for assuming additional risk. Higher values of the Sharpe ratio are considered to be better than lower ones. The investment community generally considers values over 1.0 to be good, over 2.0 to be very good, and over 3.0 to be excellent, but these can vary depending on the current financial climate. [Note: As of this writing, the risk-free rate is almost zero which means that the Sharpe Ratio is essentially just the investment return rate divided by the investment risk.]

Riskless Investment Returns
The measure used in calculating the rate of return of a riskless investment is typically short-term (90-day or less) U.S. Treasury Bills. This investment is considered to be as safe an investment as you can possibly find, and will exhibit no variation in base value during the holding period. Only the rate of interest will change as the bonds mature and are rolled over.

Other measures of what is considered “risk-less” can produce significantly different Sharpe Ratios. Long-term government bonds are not an appropriate measure for a risk-less security as market interest rate changes can significantly alter their values and in some cases can drive down the total return to very small levels or even produce losses.

Hedge Funds and the Sharpe Ratio
Implicit in the Sharpe Ratio is the assumption that returns on the investment follow a normal distribution (i.e., the bell-shaped curve). While this is a very good assumption for equity based stock index funds that are large and liquid it is not so good for strategically managed hedge funds that employ dynamic trading techniques, illiquid investments, or highly leveraged instruments such as options. For example, a hedge fund strategy that sells deep out-of-the-money options will show a higher than average Sharpe ratio–that is, until the market unexpectedly moves counter to the prevailing trend and the fund is hit with large losses.

Because of the Sharpe Ratio’s mathematical nature, high values must either employ a high return and/or a low risk. As history has shown, high return/low risk situations can’t be sustained for long periods of time.

Here’s an actual example of a supposedly stable fund with a high Sharpe Ratio whose demise nearly destabilized the global financial markets.

In the 1990’s, Long Term Capital Management, the bond arbitrage hedge fund touted as mathematically safe due to the supposed low probability of incurring significant overall losses on its massive portfolio, showed a very high Sharpe Ratio of 4.35. The fund managers employed highly leveraged strategies to make money on small arbitrage spreads. A financial crisis in Russia caused by defaults on government bonds and a resultant flight to quality resulted in massive losses for the fund which controlled nearly 5% of the world’s fixed income market.

Unable to make the loan payments on the huge debt incurred to finance its leveraged positions, they faced collapse. Had they gone into default it would have caused a worldwide financial meltdown. They were bailed out by the Federal Reserve and other creditors and taken over in 1998. So, what else is new?

Hedge fund illiquidity also works to distort the Sharpe Ratio. Investments in real estate, private equity, or mortgage backed securities for which there is no ready market can appear to be less volatile which helps their Sharpe Ratios. Fund managers tend to price these securities in a way that is, of course, favorable to the fund’s statistics and can produce an artificially high value for the Sharpe Ratio.

[Note: Illiquidity relates to how assets are valued. When there is no ready market how do you determine what something is worth? The recent mortgage crisis is an example where fund managers were valuing their own assets arbitrarily. The rate of return on an investment is related to how its value fluctuates. If it does not fluctuate then the standard deviation is low and the Sharpe Ratio is high. If that lack of fluctuation is artificial then the Sharpe Ratio is also artificial.]

Summary
Many hedge funds that currently report a high Sharpe Ratio may be employing strategies that are manageable at their current size but as they grow those same techniques may prove impractical and the Sharpe Ratio could drop dramatically. As a result, the Sharpe Ratio may not always be a good measure for evaluating a hedge fund’s risk/reward relationship.

To get a better picture of a fund’s risk/reward profile, investment professionals look at other aspects of performance including maximum portfolio draw-down and statistical measurements such as kurtosis and skewness. The Sortino Ratio, a modification of the Sharpe Ratio, focuses on downside volatility. Some think that it’s a more accurate representation of hedge fund risk but it, too, is subject to many of the same criticisms as the Sharpe Ratio. As one professor of risk management so aptly said, “Risk is one word, but it is not one number.” [Ref. 1]

Tomorrow we’ll look at the Sharpe Ratio in the context of Modern Portfolio Theory, so dust off your propeller hats!

References
1. “The Sharpe Ratio can oversimplify risk,” Investopedia.com.
2. “Risk gets riskier,” Hal Lux, Institutional Investor, October 2002.

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