The popularity of the standard deviation in investing dates back to 1973 and the publication of Burton Malkiel's legendary book, A Random Walk Down Wall Street, which suggested that the prices of stocks are random and independent. According to this theory, the likelihood of a higher price is no more probable than a lower price. These random walk assumptions yielded a formula for computing the volatility of an investment, which is described as the standard deviation, or SD, multiplied times the square root of the number of time periods in question. For example, an "annualized SD" would require the daily SD to be multiplied by the square root of the number of trading days in the year--approximately 255--yielding a square root of approximately 16.
Standard deviations, then, can give us some idea of a range of returns over a given period. Frequently, this range of returns for a period is described using a "normal distribution" as shown in the chart at right.
One of the problems of this methodology is that it tells us very little about the day-to-day or week-to-week price behavior of an investment. And over the course of my nearly 30 years as an investment advisor, I have become less and less satisfied with this method of calculating risk. In fact, the analysis of investments to fit the personalities and risk tolerance characteristics of clients has presented a continuing challenge. As a result of investment surprises--those larger than expected price movements--I began to look for a method to screen investments for more useful return and volatility characteristics.
During an intensive period of studying numerous mutual funds and portfolios, I was struck by a simple graph registering the performance of different mutual funds over several years. The graph titled "Daily Price Changes 3/00-6/06" on page 58 illustrates the price fluctuations of several funds over more than six years. It is apparent from the graph which funds provided the greatest and least growth, and which funds provided the greatest and least volatility. But how could one pick the fund that gives the greatest growth for the least volatility? And, if one could make such a choice, how might one generalize the method to allow selection from among hundreds of alternatives?
The answer: By measuring the total of price changes without respect to positive or negative movement, one can obtain a picture of total volatility. This procedure eliminates several of the problems with measuring volatility using the annualized SD. While the standard deviation assumes independence and randomness, there is data that suggests that neither assumption holds. Additionally, the annualized SD does not provide adequate information about short-term volatility. And finally, annualized SDs typically are measured from January through December, which does not necessarily reflect the typical investor experience. In this method, the sum of the absolute values of price changes can be measured over any period without the need to annualize and makes no assumptions about randomness and independence. Far simpler than computing SD's, beta, or the Sharpe ratio, it gives any investor access to a powerful tool that reveals the total volatility.
The table below titled "Absolute Volatility" illustrates the calculation of this value for a particular investment. In order to eliminate pages of data, I have simply shown the price fluctuations over the last few days of a six-year period. This value can then be compared to any index or alternative investment for the same period. Note that, while the actual changes from day-to-day may be positive or negative, the total absolute volatility grows daily. In the example, the total volatility as of August 31, 2006 is 1,470.78 percent for the period beginning March 1, 2000.
(% change Aug 30 = (33.03-33.09)/33.09 =-0.18%, absolute = + 0.18%)
The total return provides a simple method of comparing investment alternatives without concern for compounding methodology or annualization. Total return is the sum of all the dividends, capital gains and increase in value compared to the initial investment. For example, if on Jan. 6, 2003, one invested $10,000 and on Oct. 16, 2006 that investment was worth $14,000, the total return would be 40 percent. There is no annualizing, nor does the calculation take time into account. When this total return is divided by the total volatility described previously, a useful ratio emerges--the Total Return to Volatility Index (TRV):
Total Return to Volatility Index (TRV)
TRV (for any period) = Total Percentage Return
Total Absolute % Change
The equation is simple: divide the total return by the total volatility. That is, divide the total percentage change of the investment by the total percentage price change over the same period, whether days, weeks, months or years.
For example, during the particularly volatile period from March 1, 2000 through Aug. 31, 2006, the Vanguard S&P 500 Index Fund provided a total return of approximately 4.3 percent. During the same time, the total of the daily price changes for this fund was measured as 1.362 percent. When 4.3 percent is divided by 1.362 percent, the result is 0.32. Over the same time period, the American Funds Capital Income Builder had a total return of 108 percent compared to a total percentage price change of 561 percent. When total return is divided by total volatility, it results in a ratio of 19 percent--almost 60 times the ratio of the Vanguard S&P 500 fund. This result suggests that, for every dollar of price fluctuation, less than one cent of gain was realized in the Vanguard fund, while the Capital Income Builder returned 19 cents of gain for every dollar of price fluctuation. Clearly, the latter produced better returns with less volatility.
The data in "Percentage Returns & SDs," at lower left illustrates the Total Return, Total Volatility, TRV index, and annualized SDs and returns for several funds for three years ending December 29, 2006. I have also included the SDs from the Morningstar and Smartmoney Web sites for the same period. This latter data illustrates the complexity of using annualized SD. While the data definitions for the SD from both Web sites are identical, there are dramatic differences in the results. The table ranks the funds from the least desirable TRV (VFINX) to the most desirable (CAIBX).
In summary, the TRV provides a rather simple and intuitive approach to measuring historical return relative to volatility through readily accessible data. The method obviates the need for tenuous assumptions, complex equations and often conflicting SDs, while concurrently providing salient information related to expected investment performance--a simple tool for the comparison of a myriad of investment choices.
Patrick Chitwood, ph.D., CPA/PFS, ChFC, is president of the Chitwood Advisory Group in Birmingham, Ala.