traditional versus theoretical risk measures

CFA Institute

Traditional versus Theoretical Risk MeasuresAuthor(s): Russell J. Fuller and G. Wenchi WongSource: Financial Analysts Journal, Vol. 44, No. 2 (Mar. - Apr., 1988), pp. 52-57+67Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4479102 .

Accessed: 16/06/2014 01:43

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

CFA Institute is collaborating with JSTOR to digitize, preserve and extend access to Financial AnalystsJournal.

http://www.jstor.org

This content downloaded from 193.104.110.48 on Mon, 16 Jun 2014 01:43:19 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=cfa

http://www.jstor.org/stable/4479102?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


by Russell J. Fuller and G. Wenchi Wong

Traditional versus Theoretical Risk Measures

For thousands of years, mankind has understood that there is a tradeoff between risk and return. Yet the struggle to define the exact nature of the risk-return relationship continues. In the investment field, the Capital Asset Pricing Model (CAPM) is the best known traditional description of this relationship. Long before the CAPM came on the scene, however, analysts relied on experience and' common sense to quantify risk.

An examination of the reltation between return and three different measures of risk-one traditional measure, the Value Line Safety Rank, and two theoretical measures, beta and standard deviation-reveals some surprising conclusions. These suggest that theoretical asset pricing models could be improved by considering the factors that contribute to the risk measures investors have trad'itionally used.

Over the 12-year period 1974-85, safety rank exhibited the highest correlation with return, followed by standard deviation, with beta a distant third. This held true for both portfolios and individual stocks. For explaining individual stock returns, safety rank was three to four times more powerful than beta.

"Great deeds are usually wrought at great risk." -Herodotus, 485-425 B.C.

HE BEST KNOWN MO[)EL of risk and return in the investment field to date is the Capital Asset Pricing Model (CAPM)

developed in the mid-1960s by 'Sharpe, Lintner and others.1 In addition to a theory describing the relation between risk and return, the CAPM provides an explicit measure of risk-i.e., the security's beta. Long before the advent of the CAPM, however, investors were attempting to quantify risk, even though they may not have had a well-developed theory to explain the relation between their particular risk measure and return.

Bond rating agencies, for examnple, have no particular theory to go by in constructing their traditional risk measures. Nor is there a theory indicating exactly what the relation between

bond returns and bond ratings should be. Rath- er, it is common sense that suggests that bonds are riskier if the issuing firm has a relatively high level of debt, if the coverage ratios of the bond are low, if the earnings of the firm are erratic and so forth. Common sense also suggests that bonds with low ratings (say, BB) should be priced to provide, on average and over time, higher returns than bonds with high ratings (say, AAA).

Similarly, on the basis of common sense and experience, analysts have constructed traditional measures of risk for common stocks. This article examines the relative explanatory powers of two theoretical risk measures-beta and standard deviation-and one traditional risk measure-the Value Line Safety Rank.2

The Risk Measures Value Line assigns rankings from 1 (highest safety) to 5 (lowest safety) to all the common stocks it follows. Table I describes how, in general, the Value Line Safety Rank is determined.

The safety rank takes into account two primary factors-the stock's price stability index,

1. Footnotes appear at end of article.

Russell Fuller is Professor and Chair, Department of Finance, Washington State University. G. Wenchi Wong is Assistant Professor of Finance at DePaul University.

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 0 52



which is a ranking of the stock's standard deviation of returns compared with the standard deviation of returns for all other stocks followed by Value Line, and the stock's financial strength rating, which is based on such variables as debt coverage ratios, firm size, quick ratio and accounting methods. The safety rank is thus based on both statistical concepts and funda- mental variables.

The CAPM beta, by contrast, is based solely on statistical concepts, although there is an economic interpretation of beta as a measure of relative market risk. The beta for the ith stock (pi) is measured as the covariance of the stock's returns with the market index's returns (Covim) divided by the variance of the market's returns ((X2):

Covim i =

-1 (1)

Beta can also be expressed as the correlation coefficient of the ith stock's returns with the market's return (pim) multiplied by the ratio of the standard deviation of returns for the ith stock divided by the standard deviation of returns for the market index:

i= Pim (2)

Both beta and the Value Line Safety Rank are at least partly determined by a common variable-the stock's standard deviation of returns. This suggests that the two risk measures should be correlated. But the two risk measures differ with respect to other variables. They are thus not perfect proxies for each other.

As both safety rank and beta are partly deter-

mined by the stock's standard deviation of returns (sigma), we also considered this risk measure. Sigma (frequently called total risk) can be shown theoretically to be the appropriate measure of risk for individual securities in cir- cumstances where only limited diversification is possible.3

Data and Method To examine the relation between realized stock returns and the three risk measures, we collect- ed data for three four-year subperiods-1974- 77, 1978-81 and 1982-85. We used Value Line Safety Ranks and beta estimates for each available stock as of the beginning of 1974, 1978 and 1982.4 Monthly returns for each stock were taken from the CRSP tapes for New York and American stock exchange stocks. The sample consisted of all stocks for which complete data sets (Value Line Safety Rank, Value Line betas and monthly returns) were available.

For each stock in a particular four-year (48- month) subperiod, we calculated the arithmetic mean monthly return.5 Sigma was measured as the standard deviation of returns based on the 48 months prior to the start of the subperiod in question. By using the prior 48 months of returns, we based sigma on data that would have been available to investors at the start of the subperiod, just as the Value Line Safety Ranks and beta estimates would have been available.

As noted, Value Line assigns each stock to one of five safety ranks, rank 1 being the least risky and 5 being the most risky. The distribution of stocks across the five ranks is not uni- form, and the distribution varies somewhat over time. Of the approximately 1,700 stocks currently followed by Value Line, approximately 140 (about 8 per cent) are assigned a safety rank of 1; approximately 240 (about 15 per cent) are assigned a rank of 2; 140 or so are assigned a rank of 5; about 240 are ranked 4; and the balance (approximately 900 stocks currently) are assigned what is considered an average safety rank of 3. Value Line used this same general type of distribution at the beginning of our first subperiod (i.e., 1974).

Two factors caused the number of sample stocks in each rank to vary by subperiod. First, the number of stocks followed by Value Line varied slightly over the entire time period of the study, and at any point in time Value Line might not assign a safety rank or beta estimate to every stock. Second, our criteria for inclusion

Table I The Value Line Safety Rank

The Value Line Safety Ranks range from 1 to 5, with a rank of 1 indicating the lowest risk and a rank of 5 indicating the greatest risk. The safety rank is computed by averaging two other Value Line indexes-the price stability index and the financial strength index.

The price stability index is a ranking based on the standard deviation of weekly stock price changes over the most recent five years. The 5 per cent with the lowest standard deviations receive a ranking of 95 and so on down to a ranking of 5 for the highest standard deviations.

The financial strength ratings range from A++ for those firms deemed to have the best relative financial strength to a low of C. The primary variables used to determine this rating are equity coverage of intangibles, quick ratio, accounting methods, variability of return, fixed charge coverage, stock price stability and company size.

Source: A. Bernhard, "How to Use the Value Line Investment Survey, A Subscriber's Guide" (Value Line Inc., New York).

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 O 53



in the sample required that a stock must have a complete record of return data, as well as a safety rank and beta estimate; this meant some stocks were eliminated. The number of stocks in rank 1 for subperiod 1974-77, for example, was reduced from 102 to 72. Table II lists the number of sample stocks in each safety rank for each of the three subperiods.6

For the purpose of comparing the three risk measures (safety rank, beta and sigma), we

assigned beta ranks and sigma ranks to stocks to duplicate the distribution for the Value Line Safety Rank, shown in Table II. That is, for subperiod 1974-77, we assigned the 72 stocks with the lowest betas (sigmas) a beta (sigma) rank of 1, the next 225 lowest betas (sigmas) a rank of 2, and so on.7

Results We would expect the three risk measures to be positively correlated, because sigma is one of the variables involved in determining both the safety rank and beta. This turned out to be the case for our sample. Table III reports correlation coefficients for the three risk measures by subperiod.

The correlation coefficients fall in the range of 0.4 to 0.8 and are all highly significant statistically. For example, the correlation between safety rank and beta rank is 0.705 for the 1974-77 subperiod. While the correlations are high, however, the risk measures are not perfectly correlated and therefore not perfect proxies for each other. It is of interest to see which risk measure is most strongly associated with return.

Table II Number of Sample Stocks in Each Rank by Subperiod

Subperiod Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Total 1974-1977 72 225 413 184 60 954 1978-1981 63 235 486 181 49 1014 1982-1985 105 182 520 127 60 994

Table III Correlation Coefficients for Risk Measures

Safety Rank Safety Rank Beta Rank and and and

Subperiod Beta Rank Sigma Rank Sigma Rank

1974-1977 0.705 0.820 0.702 1978-1981 0.434 0.703 0.510 1982-1985 0.501 0.760 0.616 Averages 0.547 0.761 0.609

Table IV Descriptive Statistics

Statistics for Safety Ranks Subperiod Averages Rank 1 Rank 2 Rank 3 Rank 4 Rank 5

1974-1977 Monthly Return 1.104% 1.016% 1.484% 2.263% 2.573% 1978-1981 Mean Return 0.759% 1.125% 1.617% 2.118% 2.700% 1982-1985 Mean Return 1.966% 1.841% 1.771% 1.393% 1.438%

Averages for 1974-1985 Mean Return 1.276% 1.327% 1.624% 1.925% 2.237%

Statistics for Beta Ranks Subperiod Averages Rank 1 Rank 2 Rank 3 Rank 4 Rank 5

1974-1977 Monthly Return 1.382% 1.329% 1.417% 1.980% 2.312% Beta Value 0.585 0.819 1.077 1.401 1.760

1978-1981 Mean Return 1.391% 1.365% 1.567% 1.959% 1.827% Beta Value 0.586 0.733 1.028 1.318 1.599

1982-1985 Mean Return 2.056% 2.054% 1.772% 1.374% 0.596% Beta Value 0.597 0.773 1.021 1.306 1.552

Averages for Mean Return 1.610% 1.583% 1.585% 1.771% 1.578% 1974-1985 Beta Value 0.589 0.788 1.042 1.342 1.637

Statistics for Sigma Ranks Subperiod Averages Rank 1 Rank 2 Rank 3 Rank 4 Rank 5

1974-1977 Monthly Return 1.129% 1.001% 1.540% 2.235% 2.297% Sigma Value 4.657% 6.990% 10.034% 13.386% 16.986%

1978-1981 Mean Return 1.002% 1.143% 1.604% 2.098% 2.496% Sigma Value 5.704% 7.703% 10.618% 15.351% 21.546%

1982-1985 Mean Return 2.091% 1.984% 1.749% 1.271% 1.230% Sigma Value 4.987% 6.663% 9.796% 14.114% 19.003%

Averages for Mean Return 1.407% 1.376% 1.631% 1.868% 2.008% 1974-1985 Sigma Value 5.116% 7.119% 10.149% 14.284% 19.178%

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 O 54



Figure A Return and Three Risk Measures

2.6

2.4- 5

2.2-

5 2.0

Iu 1.6 an Rank ~ ~ ~ ~ ~ ~ Rn

1.4 2 2~~~~~~~~~~~

1.2

1.02

0.

cu 0.8 . .

> 0.6

0.4

0.2 .. .

0.0. .

Safety Rank Beta Rank Sig,ma Ranik

Table IV presents simple descriptive statistics for each of the risk measures. Over the first two subperiods (1974-77 and 1978-81), there is a very strong, positive relation between risk and return. That is, as the risk ranking increases, mean monthly return also generally increases, regardless of the risk measure being considered.

For the 1982-85 subperiod, however, there is a strong but negative relation between risk and return. This could be easily explained if the 1982-85 period had been a bear market. During bear markets, we would expect a negative relation between risk and return; it is the nature of risk that the most risky securities should provide the poorest returns in poor markets. But the 1982-85 period would have to be classified as a bull market.8 We can offer no explanation for this peculiar result, other than the usual caveat: During relatively short periods of time, any- thing can happen in the securities markets.

Averaging results over all three subperiods provides an interesting picture. The bottom row for each risk measure in Table IV presents

simple averages, which are plotted in Figure A. For each of the three risk measures, five sets of bars represent the return for each rank; that is, the first of the five bars represents the return for rank 1 stocks (lowest risk) and the fifth bar represents the return for rank 5 stocks (highest risk).

For the safety ranks there is a strongly positive relation between risk and return; for the entire 1974-85 periQd, the average return increases as the safety rank increases. This is also true for the sigma ranks, with the one exception that the average return for sigma rank 1 was slightly greater than the return for sigma rank 2. For the beta ranks, however, the relation between risk and return is not so clear. In fact, simple observation of Figure A suggests that there is no relation between risk, as measured by beta, and realized return.

The process of averaging over time periods and across securities by forming portfolios (as is the case for the results reported in Figure A) can sometimes obscure a true underlying relation-

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 U 55



Table V Regression Analysis of Risk Measures (t-statistics in parentheses)

Safety Beta Sigma F Subperiod Regression Intercept Rank Rank Rank R2 Ratio** 1974-1977 #1 0.0019 0.0047 0.146 162.88

(1.70) (12.76)** #2 0.0080 0.0026 0.044 44.69

(6.67)** (6.69)** #3 0.0031 0.0043 0.122 132.58

(6.75)** (11.51)**

1978-1981 #1 0.0017 0.0049 0.136 160.29 (1.41) (12.66)**

#2 0.0106 0.0019 0.019 21.07 (8.71)** (4.59)**

#3 0.0037 0.0042 0.100 113.40 (3.05)** (10.65)**

1982-1985 #1 0.0218 -0.0016 0.011 12.42 (16.27)** (-3.52)**

#2 0.0264 -0.0031 0.047 49.84 (19.56)** (-7.06)**

#3 0.0246 -0.0025 0.032 33.23 (18.51)** (-5.77)**

Total #1 0.0088 0.0025 0.138 102.26 Period (11.90)** (10.11)**

1974-1985 #2 0.0121 0.0013 0.036 24.51 (15.39)** (4.95)**

#3 0.0090 0.0024 0.129 95.28 (12.13)** (9.76)**

* Significant at the 5 per cent level. ** Significant at the 1 per cent level; all F ratios are significant at the 1 per cent level.

ship. To provide a more formal test of the relation between return and the three risk measures, we regressed the average return for individual stocks against each stock's three risk rankings over each of the three subperiods. Table V reports the results.

For the first subperiod, 1974-77, the first regression (denoted #1) lists the results of regressing individual stock mean returns for the 48-month subperiod against safety rank at the beginning of the subperiod. The regression coefficient for safety rank is a positive 0.0047 and is highly significant in the statistical sense; the t- statistic of 12.76 is significant at the 1 per cent level. The R2 is 0.146, which means that roughly 15 per cent of the variation in stock returns can be explained by safety rank.9

The second regression (#2) lists the results of regressing individual stock mean returns against beta rank at the beginning of the subperiod. While the coefficient for beta rank is positive (0.0026) and statistically significant (t- statistic of 6.69), the relation between return and beta rank is not nearly as strong as the return-safety rank relationship. The beta rank

R2 is only 0.044-about one-third the size of the safety rank R2.

The third regression lists the results uising the sigma ranks as the risk measure. The relation between return and sigma rank is positive and statistically significant. It is stronger than the result for beta rank, but not quite as strong as that for safety rank.

In general, the results for the 1978-81 subperiod mirror those for the 1974-77 subperiod. Safety rank is the most powerful explanatory risk measure, sigma rank the second most powerful; beta rank is a distant third. For this subperiod, the R2 for safety rank is over six times as large as the R2 for beta rank.

This is not the case for the third subperiod, 1982-85. For this subperiod, beta rank is the best explanatory variable, even though its R2 is only 0.047. This is also the subperiod for which there was a peculiar negative relation between beta risk and return, despite the fact that it was a period of generally rising stock prices. Further- more, the regression coefficient for each of the risk measures is negative. We are thus inclined to view the results for this subperiod with a

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 0 56



healthy dose of skepticism. As a final check on the risk-return relation-

ships, we calculated the average return for each stock over the entire 12 years of the study, 1974- 85. We then regressed this 12-year average return against each stock's risk measures at the beginning of the period, 1974. These results are reported at the bottom of Table V. The results are similar to those for the first two subperiods. Safety rank is the most powerful explanatory risk measure, followed by sigma rank, with beta rank a distant third.10

Implications Our results suggest that the intuitive notions of risk utilized by investment professionals in constructing risk measures have been closer to the mark than the theoretically more rigorous risk measures developed by academics.

One interpretation of the many recent find-

ings of market inefficiency (such as the small- firm effect, P/E effect and earnings "surprises") is that the theoretical model for risk and return is misspecified, and the risk measure (typically beta) is consequently inadequate or inappropri- ate. For example, a recent study found that, in addition to beta, measures of liquidity risk need to be accounted for in the risk-return model.'1

Liquidity risk has long been one of the intuitive notions of risk considered by investors; in fact, by incorporating such variables as firm size and stock price stability in their safety ranks, Value Line analysts may be trying to account for this risk. The safety rank may also reflect other traditional notions of risk, such as business risk. If so, this would suggest that our theoretical understanding of the relation between risk and return might be improved by carefully observ- ing the practices of investment professionals. 12 U

Footnotes

1. See, for example, W. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance, September 1964 and J. Lintner, "The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review of Economics and Statistics, February 1965, as well as J.L. Treynor, "Toward a Theory of the Market Value of Risky Assets" (unpublished, 1961), J. Mossin, "Equilibrium in a Capital Asset Market," Econometrica, October 1966, and F. Black, "Capital Market Equilibrium with Restricted Borrowing," Journal of Business, July 1972. Another example of the evolutionary struggle to define the risk-return relationship is the Arbitrage Pricing Theory of S. Ross, "The Arbitrage Theory of Capital Asset Pricing," Journal of Economic Theory, December 1976.

2. Other traditional risk measures for common stocks include Standard & Poor's earnings and dividends ranking and Fitch's quality rating. We concentrate on the Value Line Safety Rank for several reasons. First, the data are readily available to investors, as Value Line is the world's largest investment advisory organization. Sec- ond, as Standard & Poor's is careful to point out, its earnings and dividends rankings do not pre- tend to reflect all the factors, tangible or intangi- ble, that bear on stock quality (Standard & Poor's Stock Guide, page 7.)

3. See, for example, H. Levy, "Equilibrium in an Imperfect Market: A Constraint on the Number of Securities in the Portfolio," American Economic Review, September 1972.

4. The choice of the starting date was dictated by the fact that 1972 was the earliest period for which we could get both Value Line Safety Ranks and Value Line beta estimates. We also estimated simple OLS betas using the CRSP data as well as OLS betas adjusted for their tendency to regress toward the mean. The procedures presented in the text using Value Line betas were repeated using simple OLS betas and adjusted OLS betas and the results were all qualitatively the same as those reported in Tables III, IV and V.

5. We also computed the geometric mean for each stock and repeated the procedures reported in the text. All the results based on the geometric mean were qualitatively the same as those based on the arithmetic mean.

6. As might be expected, our selection criteria tend- ed to eliminate more stocks ranked 4 or 5 for safety than those ranked 1 or 2. For example, over the three subperiods the rank 1 stocks in our sample averaged 8.1 per cent of the total sample, whereas for all stocks followed by Value Line, those with a safety rank of 1 averaged 6.5 per cent. It is not clear what effect, if any, this bias had on our results.

7. We also assigned beta (sigma) ranks based on equal-size quintiles. Although these quintile rankings are not directly comparable to the method used by Value Line in determining safety ranks, we nevertheless repeated all the tests reported in the text using equal-size quintiles; the results were qualitatively the same.

8. Theoretically, we would expect to see an ex post negative risk-return tradeoff during market peri-

Footnotes concluded on page 67.

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 D 57



assumptions. Now it becomes clear that the formal assumptions are simply a set of techni- cal tools to produce a "valuation" that reveals just enough surplus or deficit to justify the funding contribution chosen.

* Finally, until plan sponsors become familiar with the concept of economic values of pension assets and obligations, decisions will continue to be based on inadequate, and at times misleading, information. U

Footnotes

1. P.O. Dietz, Pension Funds: Measuring Investment Performance (New York: The Free Press, 1966).

2. See Dietz, op.cit., as well as Measuring the Invest-

ment Performance of Pension Funds (Park Ridge, IL: Bank Administration Institute, 1968).

3. See Opinion No. 8, Accountingfor the Cost of Pension Plans (New York: Accounting Principles Board, 1966.

4. Handbook Section 3460, Pension Costs and Obliga- tions (Toronto: Canadian Institute of Chartered Accountants, 1968).

5. The Ontario Pension Benefits Act originally came into force in 1963, but was replaced by a new Act in 1965, on which the legislation of the other Canadian jurisdictions was based.

6. D. Ezra, The Struggle for Pension Fund Wealth (Toronto: Pagurian Press Limited, 1983).

7. Statement of Financial Accounting Standards No. 87: Employers' Accounting for Pensions (Stamford, CT: Financial Accounting Standards Board, 1985) and a new version of CICA Handbook Section 3460.

Fuller and Wong footnotes concluded from page 57.

ods when the return on the risk-free asset ex- ceeds the return on the risky market portfolio. This was not the case during the period 1982-85; for example, the geometric mean return on the S&P 500 was 20.2 per cent during the period, compared with 9.2 per cent for Treasury bills.

9. An R2 of 0.146 may not seem very high, but bear in mind that this is a regression of individual security returns against risk. The R-squares for portfolios are typically much higher.

10. Over this long a time period, individual security risk characteristics can obviously change, which reduces the power of this test. However, the stability of the stocks' risk characteristics over time are perhaps higher than one might think. For example, the correlation coefficient for individual stocks' safety ranks at the beginning of 1974 with their safety ranks at the beginning of 1982 was 0.605. Listed below are the serial correlation coefficients for the three risk measures:

1974 1978 1974 vs. vs. vs.

1978 1982 1982 Safety Rank 0.762 0.710 0.605

Beta Rank 0.691 0.653 0.675 Sigma Rank 0.621 0.597 0.646

11. See Y. Amihud and H. Mendelson, "Asset Pric- ing and the Bid-Ask Spread," Journal of Financial Economics, December 1986.

12. Unfortunately, it does not appear that simply substituting the Value Line Safety Rank for beta will make such anomalies as the small-firm and P/ E effects disappear. We replicated the tests of Banz and Reinganum for the small-firm effect and the low-PE effect for the period of our study (1974-85), using first beta and then safety rank as the risk measure. The size of both the small-firm effect and the low-PE effect was substantially reduced when safety rank was used as the risk measure, but both anomalies remained and were still statistically significant. See R.W. Banz, "The Relationship Between Return and Market Value of Common Stocks," Journal of Financial Econom- ics, September 1981 and M. Reinganum, "Mis- specification of Capital Asset Pricing: Empirical Anomalies Based on Earnings' Yields and Market Values," Journal of Financial Economics, March 1981.

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 C 67



traditional versus theoretical risk measures

Documents