Schmalenbach Business Review ◆ Vol. 52 ◆ October 2000 ◆ pp. 363 – 373

Joachim Henkel*

THE RISK-RETURN FALLACY ABSTRACT We assume that in the world of business, higher risks are only taken when rewarded with higher expected returns. This supposition has been confirmed empirically using capital market data. However, accounting measures have yielded puzzling results: using the mean and variance of ROE as measures of return and risk, respectively, Bowman (1980) and other researchers find a negative relationship between the measures. There are many suggested explanations of this seemingly paradoxical result, some of which relate to model misspecifications. Surprisingly, one obvious source of an artificial risk-return “paradox” has been neglected. This is the skewness of each firm’s ROE distribution. Using data from German firms, my study finds a significantly negative skewness. This skewness alone, even if firms were otherwise identical, brings about a negative relationship between mean and variance that is comparable in size to that found in the data. Thus, it is not clear if the empirical result really stems from a negative relationship between risk and return. It could be an artifact resulting from the inadmissible ex post measurement of risk. Hence, the “paradox” is highly questionable.

1 INTRODUCTION The corporate passion for diversification has passed, and today, strategic management groups activities around core businesses. This leaves diversification mainly to the investor, who must attain his most preferred risk-return profile by constructing a suitable portfolio mix. Nonetheless, both the return and risk of investments as well as their relationship to one another continue to be important issues for strategic management. There is little disagreement that of two investments with equal expected returns, investors will prefer the one with lower risk (variance). Although capital markets confirm this risk-aversion, accounting measures have yielded seemingly paradoxical results: Bowman (1980, 1982, 1984) finds a negative relationship between risk and return for most of the industries he analyzes. In his papers, as in many of the succeeding studies, “return” is measured as the average return (on assets, equity, or sales) over some periods, while “risk” has been operationalized as the variance or standard deviation of this return over the same periods. Other studies refine Bowman’s result, restricting the negative riskreturn association to declining industries 1, to the below-target performers of an industry 2, and to time periods in which there was a generally more uncertain, less * Dr. Joachim Henkel, Ludwig-Maximilians-Universität, Institut für Innovationsforschung und Technologiemanagement, Ludwigstr. 28 RG, D-80539 Munich, [email protected]. I thank Dietmar Harhoff for helpful discussions and comments. 1 See Fiegenbaum/Thomas (1985, 1986). 2 See Fiegenbaum/Thomas (1988); Chang/Thomas (1989). sbr 52 (4/2000)

363

J. Henkel

predictable environment 3. Bettis/Hall (1982) and Bettis/Mahajan (1985) find that the risk-return relationship is contingent on the firms’ diversification pattern. Several possible explanations of the “paradox” have been advanced, some of which imply some form of model misspecifications. Surprisingly, one possible source of a misspecification has not been analyzed so far, namely, the skewness of the return distribution. Bowman (1980) does mention skewness, although in a simplified formulation: “If there is some maximum ROE feasible in an industry, then perhaps most variance is really variance down from this upper bound (asymptote). The larger variance is then automatically associated with a lower mean.” While there has been no evidence of an upper bound or an asymptote, it appears plausible that in a bad year, the firm’s return can take a deeper dive than the rise it will show in a good year. To express this idea mathematically, the distribution of returns might be left-skewed: asymmetric, with a flat tail to the left, the mode on the RHS, and a relatively steep decline to the right. In this paper, I analyze theoretically and empirically how a model misspecification is caused by this left-skewness, and what implications this has for measuring the risk-return relationship. The paper proceeds as follows: After a brief review of the literature in Section 2, Section 3 derives the theory that underlies the argument and formulates hypotheses. Section 4 describes the data and presents the results. Section 5 discusses and summarizes the findings. 2 REVIEW

OF

PREVIOUS RESEARCH

The possible explanations of the “risk-return paradox” that have so far been advanced can be categorized into those based on prospect theory, strategic and organizational factors, and on model misspecifications. The prospect theory 4 approaches postulate a target level of returns above which individuals are riskaverse, but are risk-seeking below this target level 5. This induces badly performing firms to take greater risks to improve their situation, which leads ex post to a negative risk-return relationship. The second group of explanations comprises quality of management 6, organizational structure and decision processes 7, firm size 8, and diversification pattern 9. For a discussion of these explanations, see Wiemann/Mellewigt (1998). In the third category, explanations attribute the “risk-return paradox” to model misspecifications. The fact that the commonly used measure of risk is an ex post concept was criticized early on. Alternative measures have been proposed by 3 4 5 6 7 8 9

See See See See See See See

364

Fiegenbaum/Thomas (1986). Kahneman/Tversky (1979). Bowman (1982); Fiegenbaum/Thomas (1988, 1990); Johnson (1992); Sinha (1994). Bowman (1980); Bettis/Mahajan (1985); Jemison (1987). Jemison (1987); Wiemann/Mellewigt (1998). Perlitz/Löbler (1985); Wiemann/Mellewigt (1998). Bettis/Hall (1982); Bettis/Mahajan (1985); Wiemann/Mellewigt (1998). sbr 52 (4/2000)

Risk-Return Fallacy

Bowman (1982, 1984), who analyzes the content of annual reports for risk-related keywords, and by Bromiley (1991), who uses the variance of analysts’ profit predictions. Another important criticism concerns the fact that the variance of a series of returns does not depend on the sequence of these values: linearly in- or decreasing returns lead to the same variance as any random order of the same values. However, although a time trend could be mistaken for risk, Wiseman/ Bromiley (1991) find that this explanation is not empirically relevant. When they correct for possible time trends, the risk-return relationship in their data remained basically unchanged. Oviatt/Bauernschmidt (1991) argue that, rather than only modeling return as a function of risk and other variables, both return and risk should be modeled as dependent on each other and further variables. Accordingly, they estimate the resulting equation system using a three-stage least squares (3SLS) estimator. While their OLS estimate showed a significantly negative riskreturn relationship, this significance disappears in the 3SLS estimation. This shows that the result obtained from OLS is caused by correlations between error terms and endogenous variables, that OLS does not deal with properly. As will become clear in Section 4, my analysis gives an interpretation to their result. Ruefli (1990) declares that statements about the relationship between risk and return are “inherently unverifiable” with the mean-variance approach. His point is that the riskreturn relationship strongly depends on the length of the time period over which the single observations of “return” extend. He constructs an example in which annual return data, sampled over five years, yield a negative risk-return relationship, but for each year, return data for subperiods (e.g., months), yield a positive relationship. This can happen, but for two reasons his critique is too strong: First, of course it is possible that the variance of some firms’ returns is low over the long term and high over the short term, and vice versa for other firms. However, it appears plausible that this is the exception, and that the majority of firms show a similar degree of volatility over both the long and short terms. Second, the purpose of all the risk-return research is to provide insights for strategic management; and management’s most important time period is a year. Although quarters do play an important role, because of seasonal effects, they are best compared to the corresponding period of the preceding year. Hence, although theoretically measures of return can refer to any time period, the year is presumably the most relevant one. 3 THEORETICAL ASPECTS This section develops the mathematical argument and formulates a number of hypotheses. Following other researchers in this field, I use the return on equity (ROE) as the measure of return. Let Rit denote the ROE of firm i in year t; Ri the mean ROE of firm i over the years t = 1:…T; and S i2 the variance of Rit over this period. I model Rit in a very simple fashion: Rit = R i* + εit .

(1)

That is, I assume firm i ’s ROE to be given by a firm-specific constant Ri*, which one could call the “latent ROE potential”, and a term εit carrying the time-dependence. This term could be considered an error term, catching all non-systematic

sbr 52 (4/2000)

365

J. Henkel

influences on Rit. The expected value of εit, ex ante, is zero for all i and t.10 The most interesting aspect about εit is that its distribution could be asymmetric, possibly with fewer but larger negative deviations from the mean, and more but smaller positive deviations. The following proposition shows mathematically how such an asymmetry would affect the relationship between risk and return (S i2 and Ri ), before hypotheses concerning the empirical analysis are formulated. Proposition 1: Assume that εit in equation (1) is a random variable with mean µ = 0, independent and identically distributed for all i and t. Let the variance be denoted by σ 2, and the third and fourth moment about the mean by α 3 and κ 4, respectively.11 Then firm i’s risk and return, measured by S i2 and Ri respectively, are negatively associated when the ∈its’ distribution is left-skewed, that is, has a negative skewness (α 3 < 0). More precisely, as proven in the Appendix, both the correlation Corr [Ri, S i2 ] and the regression coeffcient b in a regression Ri = a + bS i2 are proportional to α 3: Corr[Ri ,Si2 ] =

b =

α3

(2)

T − 3 σ κ − σ   T − 1 2

4

4

α3

(3)

T −3 κ −σ T −1 4

4

This proposition can be illustrated with the following example: Assume Rit can only take on the values 10% and zero, with probabilities 0.95 and 0.05, respectively. This two-point distribution is clearly left-skewed. For T = 10 time periods, Table 1 gives the expected relative frequencies of the possible risk-return combinations. The relationship between Ri and S i2 in this example is negative: With average return decreasing, risk goes up12. Hence, the data show a “risk-return paradox”. But there is no paradox at all: the firms are (ex ante) symmetric; the negative association between risk and return stems solely from the left-skewness of the Rit s’ distribution, together with the ex post measurement of risk. Table 1: Simulation of mean-variance relationship with two-point distribution Average Return (Ri, in %) Risk (S i2 , in (%)2) Expected Relative Frequency

10 0 0.6

9 10.0 0.32

8 17.8 0.07

7 23.3 0.01

smaller values … < 0.001

10 This assumption is not tenable if there is a time trend in the data. However, Wiseman/Bromiley (1991) exclude time trends as a source of a false risk-return paradox. Hence, for the purpose of my model, this assumption is justified. 11 That is, α 3 = E [(εit − µ)3] = E [εi 3t ] and κ 4 = E [(εit − µ)4 ] = E [ε 4it ], where E […] stands for “expected value of …”. For ease of comparison, α 3 and κ 4 are usually normalized to obtain the coefficients of skewness (a3 ) and kurtosis (a4 ): a3 = α 3/σ 3, a4 = κ 4/σ 4 − 3. 12 This relationship is reversed for values of Ri below 5. However, because of their low probability, these combinations of risk and return are irrelevant.

366

sbr 52 (4/2000)

Risk-Return Fallacy

Figure 1 presents another example, depicting the result of a simulation with 1000 firms over 10 time periods. The return of each firm in each period is given by a random variable with a triangular density distribution between zero and 10%. That is, the density equals 2r (10%) − 1 for 0% ≤ r ≤ 10%, zero otherwise. As Figure 1 clearly shows, this left-skewed distribution leads to a negative relationship between mean return and empirical variance, although all underlying return distributions are identical. The slope of the regression line obtains as bsim = − 0.151. This compares well to equation (2), which yields btheor = − 0.149. Now I have set the stage to formulate hypotheses concerning the effect of skewness in actual data. To start with, it must be verified that the “risk-return paradox” is present in my data at all. Figure 1: Simulation of mean-variance relationship with a triangular density ρ(r) of the return distribution: ρ(r) = 2r (10%)− 1 for r ∈ [0%; 10%].

H1: The sample shows a negative relationship between average returns (Ri ) and variances (S i2 ). H2: The average empirical skewness of the firms’ return distribution is negative. H3: The distribution’s skewness alone brings about a negative relationship between Ri and S i2 that is in size comparable to the empirical one. 4 EMPIRICAL RESULTS My analysis uses annual report data from West German firms for the ten-year period from 1988 through 1997. The original data set originates with Hoppenstedt AG and is the most comprehensive commercially available set of financial data for German firms. It includes firms from all industries, except the financial services sector. To restrict the influence of outliers, I discard the top and bottom percentile of single year ROE observations (Rits) and the top percentile of firms with extreme sbr 52 (4/2000)

367

J. Henkel

variances. However, including these observations would leave the findings qualitatively unaltered. Finally, I keep in the sample only those firms for which there are observations for at least eight years. The resulting total number of firms in the sample equals N = 1250. Table 2 summarizes the mean-variance analysis of the data set. The column titled “empirical” shows various measures of mean-variance relationships, all of which are negative and highly significant. Hence, whichever measure is prefered, the data clearly show a “risk return paradox”, which verifies Hypothesis H1. Table 2: Relationship between mean and variance of ROE

correlation Corr [Ri, S i2 ] regression coefficient b Spearman’s rank correlation ρ

empirical − 0.60 − 0.54 − 0.28

calculated − 0.66 − 0.46 –

Significance levels for empirical values: p < 0.01%.

I calculate the average of the single firms’ higher moments about the mean for the returns Rit, that is, σ2 = Σi σ i2 /N etc.: σ2 = 0.059, α 3 = − 0.055, and κ 4 = 0.123. The average of the third moments, α 3, is significantly negative (p < 0.01%). Hence, Hypothesis H2 is valid: on average, the firms’ ROE distributions are left-skewed. Dividing the sample in quartiles by the mean return Ri and performing separate skewness analyses yields a negative skewness for all quartiles. It is highly significant for the two bottom quartiles, insignificant for the other two. Like the meanvariance relationship, this is at least partly due to the ex post measurement: some firms do have ROE values in the negative tail of the distribution, others do not; those that do show both a larger negative empirical skewness and a lower mean. According to Proposition 1, the left-skewness alone creates a negative relationship between mean Ri and variance S i2 even when firms have ex ante identical return expectations. It would be ideal if one could quantify which fraction of, say, the negative correlation Corr [Ri , S i2 ] = − 0.60 is to be attributed to skewness, but this would require a very complex maximum likelihood estimation with a high number of parameters. Instead, I look at it the other way round: assuming all firms are ex ante identical, and their ROEs Rit are independent, identically distributed random variables with mean µ (which is irrelevant in this context) and higher moments σ 2, α 3, and κ 4 (as observed empirically): what will be the relationship between mean Ri and variance S i2 , I evaluated equations (2) and (3) using the empirically determined average moments to calculate these expected values (see the column “calculated” in Table 2). Comparing their size to that of the empirical results yields ratios of 1.08 and 0.85 for correlation and regression coefficient b, respectively. The difference between these ratios is due to the simplifying assumption of identical ROE distributions for all firms. Nevertheless, the analysis clearly shows the following: by the mechanism described in the last section, the empirically found skewness of the return distribution alone can generate a “risk-return paradox” that is comparable in size to that found in the data. This verifies Hypothesis H3. It is hence misleading to infer a negative correlation between mean µ and 368

sbr 52 (4/2000)

Risk-Return Fallacy

variance σ 2 of the return distribution – which would constitute a real risk-return paradox – from a negative correlation between mean Ri and variance S i2 of the empirical series of returns. The reasoning can be taken a step further to include the impact of the fourth moment of the return distribution: Assume the firms’ ex ante return distribution shows flat but long tails to both sides. I.e., outliers, both negative and positive, are rare, but “far out”, such that the distribution shows a high kurtosis. One would expect that, ex post, both the top and the bottom firms with respect to mean return Ri show a high variance S i2 , since their extreme mean is likely to be caused by outliers, either positive or negative. The result would be a U-shaped dependence of empirical variance (“risk”) on the mean. If, in addition, the return distribution is left-skewed, then this U-shape will exhibit a more pronounced left (falling) branch. In fact, this is what both Fiegenbaum/Thomas (1988) and Wiemann/Mellewigt (1998) observe. They interpret the U-shape as a confirmation of prospect theory: troubled firms show a high variance because their behaviour is risk-loving, while top firms enjoy the fruits of more risky but also more promising projects. However, my analysis questions this result: the U-shaped (quadratic) component of the risk-return relationship could, at least partly, be caused by a high kurtosis. 5 DISCUSSION The goals of this paper are three-fold: to calculate theoretically the impact of skewness of the return distribution on various measures of the risk-return relationship; to demonstrate the existence of skewness in real data; and to give a quantitative idea of the effect of this skewness on the risk-return relationship. These goals have been accomplished by proving Proposition 1 and Hypotheses H1 – H3. Furthermore, the study also shows that the U-shaped risk-return relationship observed by Fiegenbaum/Thomas (1988) and Wiemann/Mellewigt (1998) is perhaps erroneous, and, at least partly, attributable to a high kurtosis of the return distribution. My intention was not to replicate the numerous investigations concerning the influences of industry, size, or average performance. I did in fact check some of them, but found my results unchanged. This work relates to that of Oviatt/Bauernschmidt (1991). Their comparison of OLS to 3SLS estimates shows that the “risk-return paradox” found using OLS was false and attributable to a correlation between error terms and endogenous variables. My results give what I believe is a lucid explanation of this correlation: because of skewness, bad years for firm i lead to both a larger negative error term εit in Ri and a higher variance S i2 . Why is skewness likely to appear? One explanation might be capacity constraints that prevent firms from taking full advantage of upward demand shifts, while downward shifts hit them with full force. Another possible explanation relates to smoothing behavior of managers who prefer a series of continuously good results followed by one very bad year to a series of more volatile results.

sbr 52 (4/2000)

369

J. Henkel

To estimate the impact of the measured skewness on the mean-variance relationship, I assumed time-independent and identical return expectations for all firms. Time-dependence could be introduced by modeling the latent ROE potential R i* (1) as a function of time-dependent environmental and firm-specific variables. However, although this would be an interesting approach in its own right, none of the studies I refer to used such a model. Hence, for the sake of comparability, I use a time-independent model as well. The assumption of identical return expectations for all firms is more heroic: different firms certainly do have expected return distributions that differ both in mean and variance. These differences interact with the distributions’ skewness to yield the empirically measured mean-variance associations. As Table 2 shows, these empirical values can be both higher and lower than the values calculated solely from skewness (see column “calculated”), depending on the measure used. However, the important point is that the calculated values are comparable in size to the empirical values. Hence, the observed “risk-return paradox” might as well be a complete fallacy. To calculate the risk-return relationship correctly, the effects of left-skewness and the real risk-return relationship have to be disentangled. For this, higher moments of the return distribution must be taken into account. The question how this is to be done is subject of ongoing research.

370

sbr 52 (4/2000)

Risk-Return Fallacy

APPENDIX Proof of Proposition 1: For Ri and S i2 , the empirical variance of the series of observations Ri 1,…, Rit, we have: Ri

=

Si2

=

1 ∑ Rit T t 2 1 1   ∑  Rit − ∑ Rit ′   T −1 t  T t′

(4) (5)

2

= =

1 1   ∑  εit − ∑ εit ′   T −1 t  T t′ 2  1    2 T ∑ εit −  ∑ εit   T (T − 1)  t t 

(6) (7)

For the covariance of Ri and S i2 , this leads to: 13 Cov[Ri ,Si2 ] = E[(Ri − E[Ri ]) ⋅ (Si2 − E[Si2 ])] 2 2      1  ∑ εit ⋅ T ∑ εi2t ′ −  ∑ εit ′  −T E  ∑ εi2t ′  + E  ∑ εit ′    E  t ′   t′      t′ T 2 (T − 1)  t  t ′       1 = 2 E T ∑ εit εi2t ′ − ∑ εit εit ′ εit ′′ −T (T − 1) σ 2 ∑ εit  t ,t ′,t ′′ t T (T − 1)  t ,t ′  1   = 2 T (T − 1)E  ∑ εit3  T (T − 1) t 

=

=

α3 T

(8)

For calculation of Corr[Ri,S i2 ] and the regression coefficient b the variances Var [Ri ] and Var [S i2 ] are required:

13 E […] stands for “expected value of …”. sbr 52 (4/2000)

371

J. Henkel

Var [Ri ] = E[(Ri − E[Ri ]) 2 ] 2 1    E ε   ∑   it   T 2  i  1   = 2 E  ∑ εit2  t T   σ2 = T Var [Si2 ] = E[(Si2 − E[Si2 ]) 2 ]

=

(9)

= E[(Si2 ) 2 ] − σ 4   1 E T 2 ∑ εit2 εi2t ′ − 2T ∑ εit2 εit ′ εit ′′ + ∑ εit εit ′ εit ′′ εit ′′′  − σ 4 2 t ,t ′,t ′′,t ′′′ T (T − 1)  t ,t ′ t ,t ′,t ′′  1 = 2 ( κ 4T (T − 1) 2 + σ 4T (T − 1)(T 2 − 2T + 3)) − σ 4 T (T − 1) 2 =

=

2

1 4 4 T − 1 κ − σ  T  T − 3

(10)

Taking together equations (8), (9), and (10) leads to the sought for expressions: Corr[Ri ,Si2 ] = =

Cov[Ri ,Si2 ] Var [Ri ]Var [Si2 ]

α3

T − 3  σ2κ4 − σ4   T − 1 Cov[Ri ,Si2 ] b= Var [Si2 ] α3 = T −3 κ4 − σ4 T −1

(11)

(12)

REFERENCES Bettis, Richard A./Hall, William K. (1982), Diversification Strategy, Accounting Determined Risk and Accounting Determined Return, in: Academy of Management Journal, Vol. 25, pp. 254 – 264. Bettis, Richard A./Mahajan, Vijay (1985), Risk/Return Performance of Diversified Firms, in: Management Science, Vol. 31, pp. 785 – 799. Bowman, Edward H. (1980), A Risk/Return Paradox for Strategic Management, in: Sloan Management Review, Vol. 21, pp. 17 – 33. Bowman, Edward H. (1982), Risk Seeking by Troubled Firms, in: Sloan Management Review, Vol. 23, pp. 33 – 42.

372

sbr 52 (4/2000)

Risk-Return Fallacy

Bowman, Edward H. (1984), Content Analysis of Annual Reports for Corporate Strategy and Risk, in: Interfaces, Vol. 14, pp. 61 – 71. Bromiley, Philip (1991), Testing a Causal Model of Corporate Risk Taking and Performance, in: Academy of Management Journal, Vol. 34, pp. 37 – 59. Chang, Yegmin/Thomas, Howard (1989), The Impact of Diversification Strategy on Risk-Return Performance, in: Strategic Management Journal, Vol. 10, pp. 271 – 284. Fiegenbaum, Avi/Thomas, Howard (1985), An Examination of Bowman’s Risk-Return Paradox, in: Academy of Management Proceedings, pp. 7 – 11. Fiegenbaum, Avi/Thomas, Howard (1986), Dynamic and Risk Measurement Perspectives on Bowman’s Risk-Return Paradox for Strategic Management: An Empirical Study, in: Strategic Management Journal, Vol. 7, pp. 395 – 407. Fiegenbaum, Avi/Thomas, Howard (1988), Attitudes toward Risk and the Risk/Return Paradox: Prospect Theory Explanations, in: Academy of Management Journal, Vol. 31, pp. 85 – 106. Fiegenbaum, Avi/Thomas, Howard (1990), Stakeholder Risks and Bowman’s Risk/Return Paradox: What Risk Measure is Relevant for Strategists?, in: Bettis, Richard A./Thomas, Howard (eds.), Risk, Strategy, and Management, pp. 111 – 133. Jemison, David B. (1987), Risk and the Relationship among Strategy, Organizational Processes, and Performance, in: Management Science, Vol. 33, pp. 1087 – 1101. Johnson, Hazel J. (1992), The Relationship Between Variability, Distance from Target, and Firm Size: A Test of Prospect Theory in the Commercial Banking Industry, in: Journal of Socio-Economics, Vol. 21, pp. 153 – 171. Oviatt, Benjamin M./Bauerschmidt Alan D. (1991), Business Risk and Return: A Test of Simultaneous Relationships, in: Management Science, Vol. 37, pp. 1405 – 1423. Perlitz, Manfred/Löbler, Helge (1985), Brauchen Unternehmen zum Innovieren Krisen?, in: Zeitschrift für Betriebswirtschaft, Vol. 55, pp. 424 – 450. Ruefli, Timothy W. (1990), Mean-Variance Approaches to Risk-Return Relationships in Strategy: Paradox Lost, in: Management Science, Vol. 36, pp. 368 – 380. Sinha, Tapen (1994), Prospect Theory and the Risk-Return Association: Another Look, in: Journal of Economic Behaviour and Organization, Vol. 24, pp. 225 – 231. Wiemann, Volker/Mellewigt, Thomas (1998), Das Risiko-Rendite Paradoxon. Stand der Forschung und Ergebnisse einer empirischen Untersuchung, in: zfbf, Vol. 50, pp. 551 – 572. Wiseman, Robert M./Bromiley, Philip (1991), Risk-Return Associations: Paradox or Artifact? An Empirically Tested Explanation, in: Strategic Management Journal, Vol. 12, pp. 231 – 241.

sbr 52 (4/2000)

373