# Portfolio decisions and the small firm effect.

ABSTRACTMany empirical studies have found excess returns on the stocks of small firms. This "small firm effect" may cause individual investors to choose to diversify into smaller firms when they make asset allocation decisions. This paper questions whether investors should consider the small firm effect. Monte Carlo simulations cast doubt on the small firm effect. We urge investors to exercise caution when buying small stocks.

The views expressed in this paper are those of the authors and do not necessarily represent the views or policies of the International Monetary Fund. This paper describes research by the authors and is published to elicit further debate. The authors are indebted to Stephen Smith, Peter Dadalt, Jacqueline Garner and an anonymous referee for several useful comments and suggestions.

INTRODUCTION

Many empirical studies have observed excess returns on the stocks of small firms. For example, Ibbotson and Sinquefeld (1999) report that common stocks earned an average annual return of 13.2 percent from 1926 to 1998 while small capitalization stocks earned an average annual return of 17.4 percent. This implies an excess annual return of 4.2 percent before adjusting for risk. Investors often include small stocks in their portfolios because they expect to receive higher returns. The higher returns will enable them to reach their goals or to reach their goals sooner. These investors are usually aware of the increased level of volatility associated with the returns of small stocks but they expect to be generously rewarded for the additional risk. According to this assumption, they will receive more return per unit of risk than if they had invested in the stocks of larger firms. We question whether this assumption is valid and whether the small firm effect (SFE) truly exists in the markets or whether it is an artifact of the datasets that are studied. By investing in small stocks, investors may be accepting even more risk than they believe they are.

Reinganum (1981/1999) finds excess risk-adjusted returns on small firms. Aharony and Falk (1992) find that small banks' returns second-order stochastically dominate returns for large banks. Roll (1981) offers an explanation for the phenomenon. He conjectures that the SFE may be attributable to improper estimation of systematic risk due to non-synchronous trading. According to this hypothesis, infrequent trading of smaller firms' stocks leads to autocorrelation in the returns for portfolios of small stocks. This, in turn, leads to an underestimation in their variances and systematic risk. Reinganum (1982), however, tests Roll's proposition and concludes that the SFE continues to be significant even after correcting for non-synchronous trading. Isberg and Thies (1992) attribute the SFE to the higher direct and indirect transaction costs involved in investing in small firms and the difficulty associated with measuring these. Wei and Stansell (1991) find that benchmark error (measurement error on the market) may explain the SFE. Knez and Ready (1997) find that the risk premium on size completely disappears when they use a regression technique that trims the one percent most extreme data observations each month. They believe that the differences in how various small firms grow determine the higher returns on small firms.

None of the above studies consider survival as a possible problem. Aharony and Falk (1992) test for the presence of survivorship bias arising out of the non-inclusion of failed firms. They do so by testing for differences in results between the entire sample period and a sub sample when no firms fail. However, the bias due to survival is present in the first period even though no firms fail in the sub sample period. On the other hand, survival, as we define it, refers to the fact that "default" states are not observed for the firm. Hence, the time series data on the stock does not comprise a representative sample for drawing inferences. Survival is a problem in the sense that this incompleteness of the dataset leads to biased measurements.

Past studies of the SFE have not corrected for the upward bias in observed mean returns arising out of survival. This upward bias increases proportionally with the probability of default and it is generally acknowledged that the probability of default is higher for small firms than for large firms. For example, consider a security with a true expected return of 12.5 percent. A 2 percent probability of default biases the observed mean returns upward by 2.3 percent. A 3 percent probability of default biases the observed mean returns upward by 3.48 percent. See Best and Smith (1993). The SFE has been noticed in tests of models of mean-variance efficiency (e.g. CAPM), but some of the observed excess return could be explained by a downward bias in the observed betas or insufficient risk adjustment for small firms. Therefore, the excess return may result from an upward bias in the mean and a downward bias in the beta. These excess returns on account of survival would be observed even if CAPM held and markets were always mean-variance efficient.

In this paper, we examine the hypothesis that the SFE is an empirical artifact resulting from differences in the survival rates across firms. Our study's results support the hypothesis that the SFE may be explained by survivorship bias in the sample data and that investors should be more cautious when including small stocks in their portfolios.

The rest of the paper is organized as follows. The literature on survival bias for stocks is discussed in the second section. The motivation and the simulations methodology used in this study are described in the third section. Hypothesis tests and results from the simulations are reported in the fourth section. The last section provides the conclusions.

PRIOR STUDIES ON SURVIVAL

Almost all securities in the market have some positive probability of default. This concept is modeled in Best and Smith (1993) and Brown, Goetzmann and Ross (1995). Best and Smith (1993) study the impact of survivorship bias in a setting where investors are risk neutral. They find that when the probability of default is included in a regression, the CAPM does not have any significant explanatory power over expected returns. Brown, Goetzmann, and Ross (1995) find that survival creates an upward bias in the expected returns of equity time-series data. They urge caution in interpreting the results of equity premium puzzle studies, event studies, and stock split studies.

Some studies model survivorship bias as the bias arising out of the non-inclusion of failed firms in a sample. They attempt to determine whether it is possible to predict future performance using past performance. They argue that the sample drawn for such a study suffers from survivorship bias owing to the non-inclusion of failed firms. The exclusion of failed firms from the sample biases the performance of the sample over the second time period and leads to an illusion of persistent good performance. Brown, Goetzmann, Ibbotson, and Ross (1992) consider samples that are truncated by survivorship bias. They suggest that an adjustment for survivorship bias by including failed firms in the sample should solve this seeming market imperfection. The empirical evidence on this is mixed. Blake, Elton and Gruber (1993) find no evidence of predictability using past performance for bond mutual funds after adjusting for survivorship bias. On the other hand, Hendricks, Patel and Zeckhauser (1993) find persistence in performance for no-load growth oriented mutual funds even after adjusting for the bias. Carpenter and Lynch (1999) simulate tests of mutual fund performance introducing survivorship bias. They reinforce the conclusion from previous studies that, even adjusting for survival bias, mutual fund performance is truly persistent. Kothari, Shanken and Sloan (1995) isolate and analyze the difference in returns between data from the Center for Research in Security Prices (CRSP) and data from Standard and Poor's Compustat Service. They argue that there is significant survivorship bias on the Compustat tapes. This implies that the results of previous empirical studies using Compustat data (e.g. Fama & French, 1992) may be partially explained by the existence of these sample selection biases.

In a more recent paper, ter Horst et al, 2001, show that standard methods of analysis that correct for survivorship bias are subject to look-ahead biases. They show that correcting for survivorship by including failed firms induces a spurious U-shaped pattern in performance persistence. They also show how one can correct for look-ahead bias by using weights based on probit regressions.

MONTE CARLO SIMULATIONS METHODOLOGY

CRSP tapes report stock prices and returns for all stocks listed on the NYSE, AMEX and NASDAQ exchanges. Securities of firms that default fail to meet the exchange's listing criteria at some point and are delisted. Viewed in another manner, default/failure states are not observed for the stocks that are listed on these exchanges. Therefore, not only are the failed securities dropped from a sample, but also the surviving securities' returns are inaccurately measured. This is the source of a "survival" bias in studies using CRSP data. The observed moments of the returns distributions are the moments of the distribution conditional on survival. This entails that: 1) the observed mean is higher than the true mean, 2) the observed variances are downward biased estimates of the true variance, and 3) the observed beta is biased. For example, Best and Smith(1993) show that:

E[R] = E[R/N](1- [gamma]) - [gamma]

where R = the returns on the security, N are the survival states, [gamma] is the probability of failure/default.

The above assumes that realized returns to stocks in failed firms are a negative 100 percent. As expected, for a given E[R], E[R/N] is increasing in [gamma], so that the difference between true and observed returns is also increasing in [gamma]. Note that correcting for survivorship bias merely by including failed firms is an inadequate correction for the bias arising out of the survival of a security. Parameter estimates for surviving firms will continue to be biased even though the sample includes failed firms, since it is the distribution of the individual security that needs to be corrected for the non-observance of failure states.

We agree with Altman, Haldeman, and Narayanan (1977), who show that the probability of failure is greater for smaller firms. This implies that observed mean returns are overstated for small firms. In addition, if observed betas are lower than true betas for small firms, the SFE may be explained by an understatement in required returns. This in fact, is the central hypothesis of our paper. Here, we model the difference between observed and true betas. Other studies have also hypothesized an understatement in computed betas relative to true betas for small firms. However, the understatement in those studies arises out of sources other than survival. We show that this difference is related to the probability of failure and potentially explains the SFE.

We use Monte Carlo simulations to isolate the misspecification arising out of survival from the market efficiency hypothesis. As a first step in our simulations and to mimic the CAPM framework, we generate a finite set of securities with their complete "true" distributions. The "market" we generate is a mean variance efficient portfolio of the original securities. Thereafter, we compute "true" betas of the securities based on the complete true distributions. Thus, our simulation creates a situation where the market is indeed a mean-variance efficient portfolio and all securities earn true returns in accordance with their betas.

Our next step is to compute "observed" betas based only on the "survival" states in the distribution. It is important to realize that for all stocks in the market, at any point in time, the observed multivariate distribution contains only survival states for all securities. For example, one cannot observe returns for IBM where Microsoft has been in default and vice versa. One can only observe returns where both IBM and Microsoft have survived. Therefore, in this study, we incorporate the effect of survival, by dropping realizations of the distribution where any security is in default. ter Horst, et al, 2001, show that correcting for survivorship bias by including failed firms is subject to look-ahead bias (spurious U-shaped pattern in performance persistence), when analyzing the performance of mutual funds. We then compute "observed" betas. This approach does not incorporate time-varying betas.

The Monte Carlo simulations of securities and market returns enable us to isolate the bias as the difference between true and observed betas. We then test the hypothesis that survival causes measurement error in returns and betas causing the illusion of a SFE.

We report summary statistics for these variables in Table 1. The beta of the market will always be 1, regardless of survival bias and by construction. Therefore, survival bias will imply that for some stocks, [??] is upward biased, and is downward biased for other stocks. True betas are distributed between 0.7039 and 1.2117 and have an unweighted mean of 0.9182. Observed betas are distributed between 0.6832 and 1.2113 and have an unweighted mean of 0.9137. On average the observed betas are lower than true betas. The difference ([??] - [[beta].sub.i]) has a mean close to zero and is distributed between -0.0619 and 0.0286. Given that the equity risk premium in the market is approximately 8.5 percent, the security that has the biggest downward biased beta ([??] - [[beta].sub.i] = -0.0619) will have its expected return biased downwards by approximately 53 basis points, relative to its true expected return (8.5 x .0619 = 0.525). In this case, realized returns will appear to have "excess returns". The stock that has the most upward biased beta ([??] - [[beta].sub.i] = 0.0286) will have its expected return biased upwards by approximately 24 basis points, relative to its true expected return (8.5 x .0286 = .243). In this case, realized returns will appear to not compensate the investor for risk. This creates a bias of approximately 77 basis points in expected returns between the two stocks. The question, of course, arises whether betas are likely to be biased downwards for smaller stocks and upwards for larger stocks.

The details of the simulation are as follows: First, we generate 100 independent normal return distributions with 5000 observations each. The mean of the independent normal distributions is picked randomly from a uniform distribution (over the range 4 to 19 percent). The mean for the independent normal distributions is bounded below 4 percent because of the requirement that the riskless asset should dominate no security. We assume a return of 4 percent for the riskless asset. The standard deviation is also picked randomly from a uniform distribution (over the range 0 to 300 percent). "Securities" are created using weighted combinations of the independent normal distributions. The weights are also picked randomly from a uniform distribution (over the range 0 to 1). Using weights from a distribution over the range 0 to 1 ensures two things. First, all securities will have an "expected return" greater than 4 percent, and second, they will have mostly positive covariances. In practice, most stocks are observed to have positive covariances. For every security return that is less than -100 percent, the security return is set to -100 percent. In this manner, the security's distribution is truncated at -100 percent in order to be consistent with limited liability. The securities, thus modified, have expected returns distributed between 9.98 percent and 14.3 percent and standard deviations distributed between 16.4 percent and 45.1 percent. It is worth pointing out that expected returns and standard deviations are randomly picked in this simulation, and some of the securities that are associated with high risk do not come with a higher expected return. Therefore, they may be stochastically dominated by other securities in the set and will not be optimally included in an efficient portfolio. The riskless security is assumed to have a rate of return equal to 4 percent. We compute the "market" as a mean variance efficient portfolio of the 101 securities (100 risky securities and one riskless security) through a standard mean variance optimization routine with constraints; the constraints imposed are that the market must yield an expected return of 12.5 percent, and that all securities must have non-negative weights in the market portfolio. These parameters are consistent with statistics for the history of the CRSP tapes. (See Ibbotson & Sinquefeld, 1999). The optimization routine reports the weights of the individual securities in the resulting market portfolio. On average, only about 28 securities have a positive weight in the market portfolio, and the rest have a zero weight. The resulting market portfolio has a standard deviation of approximately 20 percent. We drop securities that have zero weight in the market portfolio from subsequent computations. Including these securities will, in fact, bias results since they have been optimally excluded from investors' portfolios and their behavior can not carry any implications for efficient markets or asset pricing. Indeed, using an equally weighted portfolio of all 100 securities faces a similar problem. In simulations, it is first necessary to establish that the securities included in a market portfolio are "optimal" so that investors would choose to hold them. We now compute the "true" beta ([beta]) for each security by regressing individual security returns against the market portfolio. We compute the probability of default ([gamma]) for each security by counting the number of observations for which the security's return is equal to -100 percent and dividing that number by the total number of observations. The probability of default ([gamma]) across securities in the simulations ranges from 0 to about 0.86 percent. We introduce a survival bias into the simulations by dropping those observations where any security has a return of -100 percent and we retain only those observations where none of the securities is in default. The survival bias introduced here would pertain primarily to the states when the security is finally delisted and it becomes worthless. Clearly, most of the loss in value happens prior to delisting; these are states where the returns are greater than -100 percent, although negative. On average, approximately 1 percent of the stocks on the CRSP tapes disappear each year because of delisting. Biased standard deviations for the simulations range between 15.7 and 44.3 percent while biased expected returns range between 11 and 16 percent. We compute the "observed " beta ([??]) for each security using the incomplete distribution of the security and market returns. The variables generated in the simulation are: [[bar.R].sub.i] = "true" expected return on security i, i = 1 ... n. [[??].sub.i] = "observed" expected return on security i, i = 1 ... n. [[beta].sub.i] = "true" beta of security i, i = 1 ... n. [[??].sub.i] = "observed" beta of security i, i = 1 ... n. [[gamma].sub.i] = probability of default/failure of security i, i = 1 ... n.

HYPOTHESES AND TESTS AND RESULTS

In order to address the question of how bias in betas is related to firm size, we make the reasonable and well-established assumption that small stocks have a greater probability of default than large stocks. Accordingly, our first hypothesis is that stocks with [[??].sub.i] > [[beta].sub.i] have a low probability of default, and stocks with [[??].sub.i] < [[beta].sub.i] have a high probability of default. We define a dummy variable, [delta], such that [delta] = 0 if the bias, [[??].sub.i]-[[beta].sub.i] > 0, 1 otherwise. Therefore, according to our hypothesis, [[gamma].sub.1]([delta] = 0) < [[gamma].sub.i] ([delta] = 1). We test this hypothesis with a t-test, using all observations for all simulations. We report the results in Table 2. The results provide strong support for the aforementioned hypothesis.

We also conduct this t-test for the stocks generated in each simulation individually, and find that results are qualitatively consistent with those in Table 2.

To provide further evidence, we next test the hypothesis that the bias in beta, [[??].sub.i - [[beta].sub.i], is related to the firm's probability of default, [[gamma].sub.i]. For this purpose, we run a regression with [[??].sub.i] - [[beta].sub.i] as a dependent variable and [[gamma].sub.i] as an independent variable. We expect to find that the greater the [[gamma].sub.i], the lower the [[??].sub.i]. More formally stated, we predict a negative relationship between [[??].sub.i] - [[beta].sub.i] and [[gamma].sub.i]. We conduct the test for each of the 100 simulations to ensure consistency. The results are reported in Table 3. In an overwhelming number of the market simulations (79 out of 100), we find, as expected, a negative and significant relationship between [[??].sub.i] - [[beta].sub.i] and [[gamma].sub.i]. In a few simulations (4 out of 100), the relationship is positive, but not significant. It is noteworthy that in no case, is the relationship between [[??].sub.i] - [[beta].sub.i] and [[gamma].sub.i] positive and significant. These simulation regressions and t-tests provide strong support for the hypothesis that betas of small firms are biased downward while those of large firms are biased upward, hence contributing to the empirical observation of the so-called SFE.

In addition to our central results discussed above, we point out some interesting patterns that appear in these simulations. One of these is the weak relationship between [[gamma].sub.i] and [[beta].sub.i]. Table 4 contains the results of the cross sectional regressions of [[beta].sub.i] on [[gamma].sub.i] and it turns out that there is not much of a relationship between true betas and the probability of default. A similar pattern appears in Table 5 where we report the results of cross sectional regressions of observed betas ([[??].sub.i]) against the probability of default. A majority of these regressions have coefficients that are not significant, which tells us that neither [[beta].sub.i] nor [[??].sub.i] are related to the probability of default. Yet, the bias in beta measurement does significantly depend on probability of default ([[gamma].sub.i]) as discussed in Table 3. This is an interesting result and is consistent with a number of empirical studies that find that size and beta are not significantly related to one another.

Finally, we also mimic the cross-sectional studies as in Fama and French (1992) by conducting cross-sectional regressions of [[??].sub.i] and [[gamma].sub.i] on [[??].sub.i]. The results for these regressions are presented in Table 6. As the academic literature has been finding with stock-market data, our simulations indicate that, in the presence of survival bias, an inclusion of [[gamma].sub.i] as one of the independent variables improves the explanatory power of the regressions significantly (see Table 7). In our results, [[gamma].sub.i] is positively related to expected returns on stocks in 94 regressions out of a 100 regressions.

These results have been obtained in the context of a CAPM mean-variance efficient world and can be explained by the existence of survival bias alone. Therefore, according to our results, the SFE could be an artifact of the way we observe the returns on stocks in the market, and small firms may actually be earning no excess returns. The risk of small stocks may actually be fairly priced in the market, and investors should not invest in small stocks if they are expecting high excess returns.

CONCLUSION AND IMPLICATIONS

Certainly the simulations above are not an exact replication of the market. However, we have made an attempt to match the moments of the simulation to the moments of the market. An advantage of our approach is that we have been able to isolate the issue of survival bias from the issue of market mean-variance efficiency. In this context, our study casts doubt on the SFE. The simulation results provide strong support for the hypothesis that the SFE could be due to a bias introduced in the dataset by the survival of stocks.

For investors this means they should exercise caution when buying small stocks. It may be true that some portfolios can benefit from this diversification but there is no guarantee that small stocks will outperform larger stocks in the market. The larger excess returns observed on small stocks in various studies may simply be attributed to a measurement error in the dataset. Investors in small stocks will usually experience considerable volatility in returns. They may not, however, be rewarded by higher returns.

REFERENCES

Aharony, J. & Falk, H (1992). Small Firm Effect: The Case for Banks. Journal of Financial Services Research, 6(2), 157-68.

Altman, E., Haldeman R.G. & Narayanan P. (1977). Zeta Analysis: A New Model to Identify Bankruptcy Risk of Corporations. Journal of Banking and Finance, 1, 29-54.

Best, R. W. & Smith, S. D. (1993). Some Implications of Risk Neutrality for Time Variation in Stock Returns. Federal Reserve Bank of Atlanta Working Paper 93-4.

Blake, C., Elton, E. J. & Gruber, M. J. (1993). The Performance of Bond Mutual Funds. Journal of Business, 66(3), 370-403.

Brown, S. J., Goetzmann, W., Ibbotson, R.G. & Ross, S. A. (1992) Survivorship Bias in Performance Studies. Review of Financial Studies, 5(4), 553-80.

Brown, S. J., Goetzmann, W. N. & Ross, S. A. (1995). Survival. Journal of Finance, 50(3), 853-873.

Carpenter, J. N. & Lynch, A. W. (1999). Survivorship Bias and Attrition Effects in Measures of Performance Persistence. Journal of Financial Economics, 54, 337-374.

Fama, E. F. & French, K. R. (1992). The cross-section of expected returns. Journal of Finance, 47, 427-465.

Fortune, P. (1991). Stock Market Efficiency: an Autopsy? New England Economic Review, 17-40.

Hendricks, D., Patel, J. & Zeckhauser, R. (1993). Hot hands in Mutual Funds: Short-Run Persistence of Relative Performance, 1974-1988. Journal of Finance, 48(1), 93-130.

Ibbotson, R.G. & Sinquefeld, R.A. (1999). Stocks, Bonds, Bills and Inflation: Market Results for 1926-1998. Chicago, IL: Ibbotson Associates.

Isberg, S.C. & Thies, C.F. (1992). Small Firm Mutual Funds: Additional Evidence on the small firm effect. Small Business Economics, 4(3), 211-219.

Knez, P.J. & Ready, M. J. (1997). On the Robustness of Size and Book-to-Market in Cross-Sectional Regression. Journal of Finance, 52 (4), 1355-1382.

Kothari, S.P., Shanken, J. & Sloan, R.G. (1995). Another Look at the Cross-section of Expected Stock Returns. Journal of Finance, 50(1), 185-224.

Reinganum, M. (1981). Mis-specification of Capital Asset Pricing: Empirical Anomalies Based on Earnings' Yields and Market Values. Journal of Financial Economics, 9, 9-46.

Reinganum, M. (1982). A Direct Test of Roll's Conjecture on the Firm Size Effect. Journal of Finance, 37, 27-35.

Reinganum, M. (1999). The Significance of Market Capitalization in Portfolio Management over Time. Journal of Portfolio Management, 25(4), 39-50.

Roll, R. (1981). A Possible Explanation of the Small Firm Effect. Journal Of Finance, 36(4), 879-888.

ter Horst, J.R. , Nijman, T.E. & Verbeek, M. (2001). Eliminating Look-Ahead Bias in Evaluating Persistence in Mutual Fund Performance. Journal of Empirical Finance, 8 (4), 345-73.

Wei, K.C. & Stansell, S. R. (1991). Benchmark Error and the Small Firm Effect: a Revisit. Journal of Financial Research, 14 (4), 359-369.

Padamja S. Khandelwal, International Monetary Fund

Natalie Chieffe, Ohio University

Table 1: Summary Statistics [[bar.R].sub.i] = "true" expected return on security i, i = 1 ... n. [[??].sub.i] = "observed" expected return on security i, i = 1 ... n. [[beta].sub.i] = "true" beta of security i, i = 1 ... n. [[??].sub.i] = "observed" beta of security i, i = 1 ... n. [[gamma].sub.i] = probability of default/failure of security i, i = 1 ... n. [[sigma].sup.2.sub.i] = variance of "true" expected returns on security i, i = 1 ... n. 2 [[??].sup.2.sub.i] = variance of "observed" expected returns on security i, i = 1 ... n. Standard Variable Mean Deviation Minimum Maximum [[beta].sub.i] 0.9182 0.0672 0.7039 1.2117 [[??].sub.i] 0.9137 0.0687 0.6832 1.2113 [[??].sub.i] - -0.0045 0.0113 -0.0619 0.0286 [[beta].sub.i] [[gamma].sub.i] 0.0009 0.0016 0.0000 0.0086 [[??].sub.i] - 0.1344 0.0066 0.1129 0.1625 [[bar.R].sub.i] 0.1180 0.0057 0.0998 0.1430 [[sigma].sup.2.sub.i] 0.0972 0.0495 0.0268 0.2031 [[??].sup.2.sub.i] 0.0937 0.0482 0.0247 0.1966 Table 2: Results on t-test for differences in probability of default between the groups where * = 0 and * = 1, using observations for all securities generated in all simulations. Mean value of Assumption [[gamma].sub.i] (metod) t-stat p-value [delta]=0 such that 0.0005 Equal variance -12.55 0.0001 [[??].sub.i] - (pooled) [[beta].sub.i] > 0 [delta]=0 such that 0.0013 Unequal variance -14.71 0.0001 [[??].sub.i] - (Satterthwaite) [[beta].sub.i] [less than or equal to] 0 Test for equality of variance, F-stat = 3.33, pval = 0.0001 Table 3: For each of the 100 simulations, the regression ([??].sub.i] - [[beta].sub.i]) = [a.sub.0] + [b.sub.0] [[gamma].sub.i] + [epsilon] was performed and the results are summarized below. Significance is measured at the 5 percent level. For simulations with ([b.sub.0]) coefficients that [a.sub.0] [b.sub.0] are: N [R.sup.2] (p- value) (p-value) Negative, not significant 17 0.0362 -0.0025 -1.919 (0.4073) (0.2892) Negative, significant 79 0.3116 -0.0008 -4.080 (0.5588) (0.0077) Positive, not significant 4 0.0151 -0.002 2.374 (0.4064) (0.4872) Total 100 0.0151 -0.001 -3.455 (0.5270) (0.0747) Table 4: For each of the 100 simulations, the regression: [[beta].sub.i] = [a.sub.0] + [b.sub.0][[gamma].sub.i] + [epsilon] was performed and the results are summarized below. Significance is measured at the 5 percent level. For simulations with ([b.sub.0]) coefficients that are: N [R.sup.2] Positive, not significant 62 0.0526 Negative, not significant 10 0.0241 Positive, significant 28 0.2752 Total 100 0.1120 For simulations with ([b.sub.0]) [a.sub.0] [b.sub.0] coefficients that are: (p-value) (p-value) Positive, not significant 0.911 6.075 (0.0001) (0.3594) Negative, not significant 0.896 -3.687 (0.0001) (0.0108) Positive, significant 0.918 17.64 (0.0001) (0.0108) Total 0.912 8.338 (0.0001) (0.2881) Table 5: For each of the 100 simulations, the regression: [[??].sub.o] = [a.sub.0] + [b.sub.0][[gamma].sub.i] + [epsilon] and the results are summarized below. Significance is measured at the 5 percent level. For simulations with ([b.sub.0]) coefficients that are: N [R.sup.2] Positive, not significant 50 0.0384 Negative, not significant 32 0.0168 Negative, significant 1 0.2816 Positive, significant 17 0.2515 Total 100 0.1120 For simulations with ([b.sub.0]) [a.sub.0] [b.sub.0] coefficients that are: (p-value) (p- value) Positive, not significant 0.9108 5.3163 (0.0001) (0.4378) Negative, not significant 0.9029 -2.9119 (0.0001) (0.6197) Negative, significant 0.9749 -17.2652 (0.0001) (0.0044) Positive, significant 0.9220 19.5898 (0.0001) (0.0181) Total 0.9121 8.3389 (0.0001) (0.2881) Table 6: For each of the 100 simulations, the regression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] was performed and the results are summarized below. Significance is measured at the 5 percent level. For simulations with ([c.sub.0]) coefficients [a.sub.0] that are: N [R.sup.2] (p-value) Positive, not significant 6 0.86 0.0556 70 (0.0001) Positive, significant 94 0.85 0.0587 23 (0.0005) Total 100 0.85 0.0586 32 (0.0005) For simulations with ([c.sub.0]) coefficients [b.sub.0] [c.sub.0] that are: (p-value) (p-value) Positive, not significant 0.0844 0.4112 (0.0001) (0.4764) Positive, significant 0.08153 1.1806 (0.0001) (0.0016) Total 0.0817 1.134 (0.0001) (0.0301) Coefficient of beta hat is always positive in the above regressions; including probability of default improves the explanatory power and increases the returns. Table 7: For each of the 100 simulations, the regression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] was the performed and the results are summarized below. Significance is measured at the 5 percent level. For simulations with ([b.sub.0]) N [R.sup.2] coefficients that are: Significant 100 0.694 For simulations with ([b.sub.0]) [a.sub.0] [b.sub.0] coefficients that are: (p-value) (p-value) Significant 0.0556 0.0861 (0.0084) (0.0002)

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Title Annotation: | MANUSCRIPTS |
---|---|

Author: | Khandelwal, Padamja S.; Chieffe, Natalie |

Publication: | Academy of Accounting and Financial Studies Journal |

Date: | Sep 1, 2004 |

Words: | 5389 |

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