This Page Contains an Article on The Omega Return

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Short Term Risk-adjusted Performance:

a Style Based Analysis

 

"out of oceans of ashes, emerged islands of order" -Arcadia

A major problem in performance measurement is how to estimate the risk the manager took over relatively short term intervals of five years or less. The calculations of risk and return used in performance measurement are very sensitive to the time period over which they are calculated. D.S. Hsu [1986] has shown that summary statistics from discrete time intervals are very unstable. It is common knowledge that the average return for the past five years is a poor predictor of the next five year return. Hsu showed that five year standard deviations varied from twenty-five per cent to less than ten per cent.

Evidence that current performance measures do not adequately address this time sensitivity problem is provided in a recent Wall Street Journal Article [4/5/96] claiming: the NASD has been trying to "prod" Morningstar into providing one-year risk-adjusted performance measures. Morningstar’s opposition is due in part to the difficulty in calculating risk for such a short interval. Obviously, a one year interval would exacerbate the time sensitivity problem. This paper draws on the style analysis research of Bill Sharpe [1992] to provide a risk-adjusted return that is independent of the time interval chosen. The great benefit of style analysis is that twenty years or more of style data is readily available from a number of sources. The risk associated with a given style can be applied to all managers employing that style.

Methodology:

In this paper we assume investors have no demonstrable ability to determine if the market is fairly valued, overvalued, or undervalued. Monthly returns were used on style indexes, e.g., large growth, large value, small growth, and small value. Style benchmarks were then generated for each manager using a factor analytic model. The methodology for determining the factors and their weights has been discussed at length in the literature and is not really essential to this paper. We assume the reader has a working knowledge of style analysis and recommend Coggin & Fabozzi [1995] for an excellent overview.

Exhibit 2

Exhibit 2 presents some interesting statistics of each style for comparative purposes. On average, one would expect to earn 11.3% by investing in the S&P 500. But, 10% of the time, returns should fall below -8.7%. The large growth style has about the same expected return as the S&P 500, but its inherent uncertainty has 16% more downside risk, with a 10% chance that returns will be below -10.4%. Large value has a higher expected return and less risk than the S&P 500. Small growth also has a higher expected return than the S&P 500 but almost twice as much downside risk. Most important, these risk measures are not time sensitive. They are estimates of the inherent risk associated with that particular style.

Exhibit 2 does imply that small value stocks should perform better than the other styles, but it does not mean that a large growth manager could not earn higher returns while taking no more risk than a large value manager. The large growth manager’s selection and timing skills might result in more draws from the right half of the distribution while the large value manager’s selections resulted in more draws from the left.

Therefore, we need to know if managers take more or less risk than is inherent in their styles. This is similar to determining whether a manager systematically took more or less risk than the market. We propose a ratio of the manager’s risk to the style benchmark risk to capture this relationship. If risk is measured as standard deviation, for a well diversified portfolio it is very similar to beta, except the denominator is the standard deviation for the manager’s style benchmark instead of the market. To avoid confusion with the CAPM beta, we will precede the symbol for beta with the word "style" to identify it as a "style beta."

1.                          Style b = risk_mgr / risk_benchmark

Without this adjustment factor, the assumption would be that the style benchmark risk is identical to the manager’s risk. In those cases where the R² is one, the style b will also be one. When the R² is less than one, the manager’s risk will be different than the style risk. The style b attempts to account for this difference.

Because the style b is used to adjust the style risk, it should be in the same units as the style risk, e.g., standard deviation or downside deviation. Standard deviation was used to measure risk as far back as 1789 by a British actuary named Tetans. More recently, measures of downside risk have become popular. Manager’s want to present their past performance in a risk-return framework that does not treat the high returns they earn as risky. Also, many investors do not view returns above the return they must earn at minimum in order to accomplish their goal as risky. Both manager’s and said investors tend to agree that risk has to do with the returns that fall below this minimal acceptable return. This accounts for the reason that "downside risk" is gaining acceptance with both managers and investors. The methodology described here is equally applicable to both measures of risk,.

If applied to the popular Sharpe ratio, the style b should be a ratio of standard deviation of the manager divided by the standard deviation of the manager’s style.  The style beta would then be used to adjust the standard deviation of the manager's style benchmark.

2.                          (Return - Rf) / ( Style b (Style s ))

Whether variance, standard deviation, downside deviation, or downside variance is used, style betas should be calculated over an interval of time that captures the relationship of the manager’s risk visa vis the style. We recommend the interval cover an entire cycle (up and down) for the style.

The denominator of equation 2 will be called the style adjusted (SA) risk. For the remainder of this article, we will focus on the style adjusted downside risk, or SAD risk. Because we intend to employ SAD risk in a formula requiring downside variance, the style b in SAD risk is a ratio of the downside variance of the manager to the downside variance of the style.

In the Fishburn utility function, a multiple of the SAD is subtracted from the manager’s realized return to produce a risk adjusted return. The end result we are looking for is a style risk-adjusted return that utilizes the bootstrap procedure and downside risk. It would be cumbersome to try to identify this with an acronym, therefore we propose calling it the omega return, omega being at the end of the Greek alphabet. This will also help to differentiate it from the risk-adjusted return (RAR) we used in previous articles. The equation for the omega return with downside risk is:

(3)             W Return = R - A [SAD]

where

W = The Greek letter omega

R = The geometric average return realized by the manager.

A = An indicator of the investors degree of risk-aversion

SAD = the style beta times the style downside variance

Equation 3 addresses the time sensitivity problem, in that the return could be calculated for a one-year interval; the style beta could be calculated over some representative interval, say, five years; and the style risk could be estimated by bootstrapping twenty years or more of monthly returns. Thus, the style risk would be independent of the time interval used to calculate the omega return.

Excess Return:

There are two excess returns worth knowing about: the risk-adjusted excess return that was earned, and the risk-adjusted excess return that could have been earned. By comparing the mean from the bootstrap of the manager data to the mean of the bootstrap of the style benchmark data one can estimate the excess return the manager could have been expected to earn relative to the style benchmark.. This will be referred to as the bootstrap alpha (boot a).

(4)                  boot a = _bm_m - _bm_s

where

boot a = the excess return that could have been earned.

_bm_m = the arithmetic mean from the bootstrap of manager data.

_bm_s = the arithmetic mean from the bootstrap of style data.

The excess return that was actually earned on a risk adjusted basis is the differencebetween the omega return the portfolio manager earned and the omega return for the style benchmark. We will refer to this as the omega excess (W excess).

  5.          W Return of manager - W Return of style benchmark

The interval chosen to calculate excess returns should be the same and should be chosen judiciously. Also, the interval should be the same for all managers under consideration. If the manager’s performance is dependent on the manager’s style, one or both of these excess return measures might provide clues as to future performance, relative to the style.

Empirical Results Revised 11/97:

If a manager is in the top quartile based on Omega Excess returns, what is the chance that he will be in the top quartile next year? Beginning in 1981 and using the previous five years to calculate the statistics, 70% of the managers who were in the first quartile, remained in the first quartile the following year (see Exhibit 3). The results were the same for a three year interval. Many services show the past three year and five year results; so, presumably, investors invest on this basis. Only 2% of managers in the first quartile dropped to the fourth quartile. On the other hand, 69% of the managers who were in the fourth quartile remained in the fourth quartile in the following year.

Exhibit 3

In the November 10^th issue of P&I, we showed how the Omega excess return appeared to have some predictive power. If a firm was in the top quartile in a given year, there was a 70% chance that it would be in the top quartile the following year. This is quite surprising, but could one have made more money with this information than simply following a naive strategy? To answer this question we compare the results for four different strategies.

The first strategy assumes the investor chooses the fund with the highest Omega excess return at the end of each calendar year and holds that until the end of the following year. This will be referred to as the Top Omega strategy. The second strategy assumes the investor simply invests an equal amount in all of the funds and holds that portfolio. The third strategy assumes the investor chooses the top quartile of funds based on Omega excess returns. The final strategy simply selects the fund with the highest raw return at the end of each year.

The results are shown in the graph on this page, which assume no taxes and no transaction costs. Not surprisingly, the worst results were obtained by investing in the fund with the highest raw return in the previous year, i.e., the return unadjusted for risk, or anything else. This is consistent with other studies warning against buying last year's winner. Still, a cumulative return of 796% equates to a fifteen year compound growth rate of 15.7%. By simply holding all the funds that were in business in 1980, and were still in business at the end of 1996, one could have increased the cumulative return to 943%.

Now, let's take a close look at how the Top Omega strategy appreciated 1,444% over this period for a compound growth rate of 20%. At the end of 1980, the investor would have invested in Fidelity Magellan fund until the end of 1986. Then, IDS New Dimension fund replaced Magellan, followed by Growth Fund of America, Janus, four years in Fidelity Contra fund, and two years in Keystone Small Company Growth fund. Neuberger & Berman Partners replaced Keystone for 1997. Investing in last year's top Omega fund did not always produce the best results. In 1996 it was the worst. Nevertheless, the best results over fifteen years were produced by the top Omega excess strategy, and the second best results by the top quartile of Omega excess funds

The evidence for the past fifteen years suggests the Omega excess return captured something that allowed one to capitalize on an inefficiency in the market. However, like every other means for exploiting inefficiency, it can only work so long as it is not widely used. Although, I only calculate these results for P&I, the details of how to calculate it are on www.pionline.com. The statistics were generated by Bernardo Kuan at DAL Investment Co., San Francisco.

                                                       Exhibit 4

Summary & Conclusions:

How managers performed in the past should include a measure of risk as well as return, no matter how short the interval. Unfortunately, the problem of estimating risk is severely biased by the time interval chosen for the data. The shorter the time interval, the greater the problem. The application of style analysis makes it possible to measure risk independent of the time interval chosen to calculate the return. Consequently, risk-adjusted returns for periods as short as one-year are feasible. Meaningful results were obtained even though the interval chosen for the style b was not over a complete market cycle. We believe this approach satisfies the demands of the NASD for short term performance measures without compromising the ethics of our profession.

We have presented a methodology for calculating risk-adjusted returns that we believe can be easily understood by most investors. While the methodology for calculating the omega return is fairly rigorous, anyone could understand the end result: TRP earned an average of 26% for the last five years; after adjusting for risk, they earned 24%. More sophisticated investors may find value in the excess return measures presented here.

References

Balzer, Les. "Measuring Investment Risk: A Review," Journal of Investing, Fall 1994.

Clements, Jonathon. "We Got the List Down to 81 Funds; The Final Decisions Are Up to You,"

in the Wall Street Journal, Section C1, 4/16/96.

Coggin, T. Daniel, and Frank J. Fabozzi. "The Handbook of Equity Style Management, " Frank J.

Fabozzi Associates, 1995

Damato, Karen. "Morningstar Edges Toward One-Year Ratings," Wall Street Journal, Section

C1, 4/5/96.

Effron, Bradley, and Robert Tibshirani. "An Introduction to the Bootstrap". London: Chapman

and Hall, 1993

Fama, Eugene F., and Kenneth R. French. "Size and Book to Market Factors in Earnings and

Returns," Journal of Finance, March 1995

Fishburn, Peter. "Mean-Risk Analysis with Risk Associated with Below Market Returns"

American Economic Review, March 1977

Hsu, D.A. "The Behavior of Stock Returns: Is it Stationary or Evolutionary? " Journal of

Financial and Quantitative Analysis, March 1984

Sharpe, William F. Asset Allocation: "Management Style and Performance Measurement."

Journal of Portfolio Management, Winter 1992.

Sortino, Frank A., and Lee N. Price, "Performance Measurement in a Downside risk

Framework." Journal of Investing, Fall 1994.

Sortino, Frank A., and Hal J. Forsey, "On the Use and Misuse of Downside Risk." Journal of

Portfolio Management, Winter 1996.

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