Lessons from the small firm effects.

Chatrath, Arjun ; Adrangi, Bahram ; Anderson, Robin 等

INTRODUCTION

The initial public offering (IPO) market for common stock has been both active and extremely cyclical over the past few decades. For instance, a study by Ibbotson, Sindelar, and Ritter (1993) notes that, between 1970 and 1995, more than 8000 firms went public, raising more than $130 billion. During this period, the number of IPOs have ranged from only 9 in 1974 to over 900 in 1986, and the proceeds from these IPOs ranged from $50 million to almost $25 billion.

There are both, demand-side and supply side explanations for the cyclical nature of the IPO market in the US. Choe, Masulis, and Nanda (1993), providing a demand side explanation for the cyclical behavior, suggest that there are periods when exceptionally large number of firms have capital needs which are unlikely to be met by private funding. The internet startups in the mid 1990s are good examples. On the supply side, Loughran and Ritter (1995) suggest that there might be periods when investors that traditionally invest in IPOs have money and the urge to invest. The liquidity surge into internet startups in the late 1990s is a good representation.

A firm considering an IPO or a public offering should be interested in knowing whether a particular hot issue period is demand driven or supply driven. For instance, if the hot issue period is demand driven, entrepreneurs may wish to avoid going public during that time as competition for funds could drive up the cost of capital. On the other hand, the entrepreneur could benefit by timing his IPO to the hot issue periods that are supply driven. Such periods are characterized by high price to earnings ratios, a symptom of lower cost of equity for a new issue.

In this paper we provide further insight on how an entrepreneur might time his IPO to avoid high costs of capital. We suggest that small capitalization stocks have several inherent characteristics that at least temporarily distinguish them from medium and large capitalization stocks. The paper presents evidence on these tendencies, or small firm effects, and goes on to elaborate on how they might impact cost of capital for new issues. The ideas generated in this paper can be employed together with those developed in Choe, Masulis, and Nanda (1993) and Loughran and Ritter (1995) to form a checklist of sorts that could ultimately aid in the effective timing of an IPO.

The main results of our paper may be summarized as follows. First, we demonstrate that small stock prices are more sensitive to the general market when the market is falling than when it is rising. We rule out the possible asymmetry in variance of small stocks across falling and rising markets as a cause for this anomaly. Second, we update prior research and demonstrate a relationship between the sensitivity of small stocks to the general market and the level of business risk: the betas of small stocks are found to be positively related to the spread between Baa-rated and Aaa-rated (default-free) bonds. In other words, we demonstrate that small capitalization stocks are especially sensitive to movements in the overall market when there is a great deal of risk in the market. For IPOs, this translates to an especially high cost of equity when the markets are falling or when the yield spread is high. However, this paper also finds that the yield spread holds little in terms of predictive capacity for the return behavior of small stocks.1

The remainder of the paper is organized as follows. In section II we discuss the nature and significance of the small firm effects and further motivate our empirical section. Section III presents empirical evidence of the small firm effects. Section IV provides some concluding thoughts.

SMALL STOCKS EFFECTS

Small capitalization stocks have several behavioral characteristics that distinguish them from medium and large capitalization stocks. For instance, it is well noted that small capitalization stocks are more market-sensitive (have a larger beta) than large stocks, and as they progress to become medium and large capitalization stocks, their beta approaches that of the market. It is also well documented that small stock returns have been higher than those predicted by the Capital Asset Pricing Model. This size-effect was first noted for the US by Banz (1981), and since then, there has been a great deal of international evidence on the subject. Notably, small stocks have been found to have larger returns than the overall market despite their higher market sensitivities. (2)

A prominent explanation to this size effect has been forwarded by Jagannathan and Wang (1996). The authors suggest that the effect arises because human capital, more specifically the present value of an individuals future wages, is not explicitly included in the benchmark market portfolio. According to the authors, since individuals typically want to insure against job-loss, they as investors will be willing to accept lower rates of return on stocks that do relatively well during high layoff periods. If investors believe that large firms are likely to outperform smaller stocks during economic events that lead to increased layoffs or wage reductions, they would prefer investing in larger stocks, even at the expense of lower expected returns. (3) Similarly, under adverse economic conditions, investors may avoid small stocks even though expected returns are higher.

If Jagannathan and Wang's (1996) understanding of investor behavior is correct, the behavior of small stocks in the IPO and secondary markets may also depend on the investors' perceptions of general business risk rather than market risk alone. In support of such an explanation for small stock performance, Chan, Chen, and Hsieh (1985) and Jagannathan and Wang (1996) make a number of interesting observations on small stock price behavior. First, the authors find that small company stock returns seem to covary more with per capita income than do large company stocks. Second, they indicate that small stock returns appear positively correlated with the changes in the spread between the Baa-rated and Aaa rated bonds. Third, the authors find that the small companies have higher market betas when this spread is higher. Thus, small stocks tend to be especially sensitive to the market when the risk in the market is at its highest.

Another important implication of Jagannathan and Wang's (1996) claims on investor behavior, is the likelihood of asymmetries in the sensitivity of small stocks across rising and falling markets. Given that there is considerable overlap between business- and market-risk factors, periods of high market risk are also likely to be periods of high business risk. In other words, it is implicit that if investors systematically avoid small stocks in periods of high business risk, they systematically avoid small stocks during periods of market downturns.

There is already some evidence of such a tendency in real estate investment trusts (REITs), noted to be fairly small in capitalization. Goldstein and Nelling (1999) and Chatrath, Liang and McIntosh (2000) find that REITs tend to be more market sensitive when markets are falling than when they are rising. Chatrath, Liang and McIntosh (2000) regress the excess returns of equity REITs on the product of the S&P 500 index returns and a dummy representing a rising market, and on the product of the S&P 500 index returns and a dummy representing a declining market. The authors find highly significant differences in the betas corresponding to the two series. However, it is notable that the findings for REITs may not be easily applicable to the remainder of the small-stock universe. Unlike most small stocks, REITs have relatively low betas and relatively high dividend yields. Moreover, their correlation with the general market has been decaying of late (for instance, see Chatrath, Liang & McIntosh, 2000).

Thus, while we already know much about small stock price behavior, some important questions remain unanswered. Namely, is there a pattern of asymmetry in the sensitivity of small stocks across rising and falling markets? Does the nature of the variance of small stock returns play a role in these tendencies (4) What information does the yield spread have on the behavior of small stocks? The next section develops some testable hypotheses on these and other questions and presents empirical evidence from the US stock market.

EVIDENCE FROM THE U.S. MARKET

Summary Statistics

Our study employs monthly total returns of the S&P 500, Small Stocks, Russell 1000, and Russell 2000 indices, and the 30-day treasury bill. The Small Stocks total returns are prepared by Ibbotson Associates, which are published in Stocks, Bonds, Bills and Inflation (Ibbotson and Sinquefeld (1999)). The series are from the U.S. Investment Benchmark data module supplied by Ibbotson Associates. Returns are given by [([P.sub.t]+[d.sub.t])/[P.sub.t-1]]*100, where [P.sub.t] and [d.sub.t] represent the index value and dividend at time t, respectively. The data for S&P 500 index, the Small Stocks index, and treasury bills span the interval 1/1972 - 12/1998. The Russell index returns are available only from 1/1979. It is notable that the Russell 2000 and the Small Stock Indices represent the smallest stocks among the four, and the Russell 1000 returns are expected to more closely mimic the S&P 500. We find that the four stock index returns series are stationary employing standard unit root tests: the Dickey-Fuller and Phillips-Perron tests reject the null for nonstationarity in the return series at the 1 percent level. (5)

Table 1 presents summary statistics and diagnostics for the four return series. It is apparent from comparing the standard deviations that the small stock returns have had larger variations. Further, the Q statistics indicate significant autocorrelation in the monthly returns for small stocks (at least for 6 lags), but no autocorrelation for large stocks. Finally, all four series suggest considerable variance persistence. The solution of the conditional variance equation from Bollerslev's (1986) Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model yielded [??] and [beta] coefficients that would suggest that the impact of volatility shocks persist over long periods of time. In summary, Table 1 captures some notable similarities and differences in small and large capitalization stocks.

Beta Asymmetry

A central issue in our study relates to the potential differences in small stock behavior across rising and falling markets. We first examine the nature of the small stock betas by estimating variations of the regression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [R.sub.t,i] and [R.sub.t,SP] represent, respectively, the monthly returns for the ith small-firm index and the S&P 500 index, and D- is a dummy variable equal to 1 if [R.sub.t,SP] is below the return on the 30-day t-bill (i.e., when excess market returns are positive), and 0 otherwise. A positive and significant [[beta].sub.2] coefficient would indicate that small stock sensitivities are higher when markets are falling than when they are rising. Table 2 reports the results from these regressions.

The results from the regression of small stock returns on the S&P 500 index returns (Model 1 in Table 2) indicate that the small stock betas are generally larger than 1 (Panel A and C), while the beta of the Russell 1000 is similar to that of the market (Panel B). It is also notable that the R (2) is almost equal to 1 for the Russell 1000 index and much smaller for the Small Stock and Russell 2000 indices, again suggesting that the former more closely mimics the market. The table also provides strong evidence of asymmetry in the betas. The estimation of equation 1 (Model 2 in Table 2) resulted in highly significant [beta]2 coefficients for the Small Stock and Russell 2000 index. (6) On the other hand, the [beta]2 coefficient was barely statistically significant for the larger-cap Russell 1000 index.

Further testing is conducted to consider the possibility that the asymmetric beta pattern found in Table 2 is a reflection of a positive relationship between small stock returns and their variances. Specifically, conditional return variances have been noted to be higher in falling markets (e.g., Glosten, Jagannathan, and Runkel (1993)), and as already noted, small stocks have been found to be especially sensitive when risk levels are high. If it is true that the asymmetry in small stock betas noted in Table 2 result from dependencies between returns and variances, adjustments for such variance effects in the return data should subsequently remove the asymmetries that have been noted. Such an exploration could also be considered a test of the robustness of the regressions we have undertaken so far: if return-variance effects are substantial, heteroskedasticity may be leading us to false inferences regarding beta behavior.

Consider, first, a generalization of the noted asymmetric relationship between small stocks and the market. The greater sensitivity between small stocks and the market in falling markets can be represented as

Cov[([R.sub.i],[R.sub.m]).sup.-] > Cov[([R.sub.i,[R.sub.m]).sup.+] (2)

where the superscript signifies market performance. We can be rewrite the above as

[E([R.sub.t],[R.sub.m])] - E([R.sub.m])E[([R.sub.t],[R.sub.m]).sup.-] > [[E([R.sub.t][R.sub.m]) - E([R.sub.m])E([R.sub.t])].sup.+]. (3)

Assuming conditional means to be time-invariant, we have

[[R.sub.t], [R.sub.m]].sup.-] > [[[R.sub.t],[R.sub.m]].sup.+], (4)

which can be extended to residual returns, without loss of generality, as in,

[[[epsilon].sub.i],[[epsilon].sub.m]].sup.-] > [[[epsilon].sub.i],[[epsilon].sub.m].sup.+]. (5)

Given the notion that the return-dependence in betas may actually be the result of return-variance relationships, one would like to see this inequality dissipate when adjustments for variance-effects are made.

To evaluate the impact of accounting for the possible return-variance relationship in our tests for beta asymmetry, we undertake the regression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [[??].sub.t,i]/[??][h.sub.t,i] is the standard residuals of the ith small stock index, and [[??].sub.t,SP]/[??][h.sub.t,SP] is the standard residuals of the S&P 500 index; [[??].sub.t,i] represents the return shock (error) at t, and [??][h.sub.t,i] is the conditional standard deviation at t. The standard residuals are obtained from the GARCH model with controls for variance asymmetries as suggested in Glosten et al. (1993). (7) In effect, if we find that [[beta].sub.2] in (6) is positive and significant, the asymmetry in the small stock betas would not be explained by variance effects in the data.

The results from (6) are reported in Table 3. They indicate the asymmetry in the relationship between small stocks and the market persists after controls for possible variance effects. This is implied by the significant [[beta].sub.2] coefficients in Panels A and C. Further, as in Table 2, the [[beta].sub.2]] coefficient is weakly significant for the Russell 1000. In sum, the patterns observed for small stocks, vis a vis their relationship to the market cannot be explained by variance effects in the data.

Small Stock betas, Returns, and Business Risk

In this section we evaluate whether the small stock sensitivity is dependent on the level of business risk, as indicated in prior research (Chan, Chen, and Hsieh (1985)). We also evaluate whether small stock returns are predictable by the yield spread, a common proxy for the level of business risk.

First, we conduct 30-month rolling regressions of the Small Stock Index returns on the returns of the S&P 500 index (over the period 1972-1998) to obtain 240 betas. These betas, along with the corresponding yield spread between Baa and default free bonds are presented in Figure 1. We then regress these betas on the difference between the Baa and Aaa yields (Spread). The results are

[[beta].sub.t] = 0.778 + 0.221 [Spread.sub.t] + [[epsilon].sub.t], [R.sup.2] = .17, (21.88) (7.71)

where the figures in parentheses are t-statistics. The Spread coefficient (.221) is highly significant, indicating a strong, positive relationship between small stock sensitivities and the spread. Similar results were obtained when we estimate the above regression for the Russell 2000 Index returns.

To examine the relationship between small stock returns and the yield spread, and to examine whether the relationship is robust to variance effects, we undertake the pair of regressions

[R..sub.t,i] = [alpha] + [[beta].sub.1][Spread.sub.t] + [[epsilon].sub.t] (7)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The results in Panel A of Table 4 suggest that the Small Stock Index returns and Russell 2000 Index returns are highly related to the spread: the [[beta].sub.1] coefficient is significant at the 1 and 5 percent levels respectively. On the other hand, the coefficient for the Russell 1000 is insignificant, indicating a relative lack in the relationship between the yield spread and larger-cap stock returns. However, there is evidence that variance effect do account for some of these patterns. The [[beta].sub.1] in Panel B are significant only for the Small Stock Index.

As the yield spread seems to be related to the performance of small stocks, it behooves us to determine whether the spread holds significant predictive powers vis a vis the performance of small stock returns. Table 5 reports results pertaining to the forecasting performance of the yield spread. Here we compare the mean square errors (MSE) from a simple autoregressive regression of each index against the MSE from an extended model with contemporaneous and past levels of the yield spread. The results indicate that the drop in MSEs are statistically insignificant. In other words, the inclusion of the yield spread into a simple forecasting model for small stocks does not result in an improvement.

SUMMARY AND CONCLUSIONS

The paper outlines some important differences in the behavior of small- and large capitalization stocks and demonstrates two small stock anomalies that should be of particular interest to entrepreneurs considering a public offering. First, we demonstrate that small stock prices are more sensitive to the general market when the market is falling than when it is rising. We rule out the possibility that variance effects in small stock returns are a cause for this anomaly. Second, we demonstrate a relationship between the sensitivity of small stocks to the general market and the level of business risk: the betas of small stocks are found to be highly related to the spread between Baa rated and default-free bonds. In other words, we demonstrate that small capitalization stocks are especially sensitive to movements in the overall market when the market is most risky.

As discussed in the paper, the hot issue periods for IPOs can be either demand or supply driven. If the hot issue period is demand driven (driven by capital needs of firms), entrepreneurs may wish to avoid going public during that time as competition for funds could drive up the cost of capital. As noted in this paper, the entrepreneur may also benefit from examining an easily available proxy for the level of business risk, the yield spread. Given that the yield spread is closely related to the small firm beta, and that the pricing of new issues will be impacted by the performance of other small stocks at that time, it may benefit the entreprenuer to time his issue when the spread is not unduly high. Further, our results on beta asymmetry demonstrate that cost of equity for small firms may be exaggerated relative to the average firm when the overall market is declining. Thus, entrepreneurs ought to be especially deliberate in avoiding going public when the benchmark indices are faltering.

REFERENCES

Banz, R.(1981). The relation between return and market value of common stocks, Journal of Financial Economics, 9, 3-18.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307-327.

Brown, P., Kleidon, A. & T. Marsh. (1983). New evidence on size-related anomalies in stock prices, Journal of Financial Economics, 12, 33-56.

Chatrath, A., Y. Liang & W. McIntosh. (2000). The asymmetric reits beta puzzle, Journal of Real Estate Portfolio Management, 6, Forthcoming.

Chatrath, A., Ramchander, S. & F. Song. (1995). Are market perceptions to layoffs changing? Economics Letters, 47, 335-342.

Chan, K. C., Chen, Nai-Fu & D. Hsieh. (1985). An exploratory investigation of the firm size effect, Journal of Financial Economics, 14, 451-471.

Choe, H., Masulis, R. & V. Nanda. (1993). Common stock offerings across the business cycle: theory and evidence, Journal of Empirical Finance, 3-21.

Glosten, L. R., Jagannathan, R. & D. E. Runkel. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance, 48.

Goldstein, A. & E. F. Nelling. (1999). Reit return behavior in advancing and declining stock markets, Real Estate Finance, 15, 68-77.

Hawawini, G. & D. Keim. (1995). On the predictability of common stock returns: World wide evidence, in Handbooks in Operations Research and Management Science, 9, Chapter 17,

R. Jarrow, V. Maximovic, and W. Ziemba, Editors, Amsterdam: Elsevier Science. Ibbotson, R. & R. Sinquefeld. (1999). Stocks, Bonds, Bills, and Inflation, Chicago: Ibbotson Associates.

Ibbotson, R., Sindelar, J. & J. Ritter. (1993). Initial public offerings, Journal of Applied Corporate Finance, 1, 37-45.

Jagannathan, R. & Z. Wang. (1996). The conditional capm and the cross-section of expected returns, Journal of Finance, 51, 3-53.

Levis, M. (1985). Are small firms big performers? Investment Analyst, 76, 21-27.

Loughran, T. 7 J. Ritter. (1995). The new issues puzzle, Journal of Finance, 50, 23-52.

Ziemba, W. (1991). Japanese security market regularities: monthly, turn-of-the-month, and year, holiday and golden week effects, Japan World Economics, 3, 119-146.

ENDNOTES

(1) Other than being of interest to small firm mangers, the asymmetry in the relationship between small stocks and the market would certainly have some important implications for portfolio managers. In particular, short term estimates of relatedness between small stocks and the market would be considered unreliable, complicating basic questions on hedging small stock returns, or requiring further qualifications to questions like "how much in small stocks?".

(2) For the US, Hawawini and Keim (1995) find that the lowest capitalization stocks (average beta of 1.17) yielded returns of about 9% higher than the largest stocks (average beta of 0.95) over the period of 1951-1989. Ziemba (1991) finds that the smallest Japanese stocks outperformed their larger counterparts by 1.2 percent per month over the 1965-1987 period. Levis (1985) finds that small stocks in the UK market outperformed the large stocks by 0.4 percent over the 1958-1982 period. The most noteworthy evidence comes from the Australian market for which Brown et al. (1983) find a small stocks premium of 5.73 percent from 1958 to 1981. Thus, there is international evidence that small firms provide higher returns despite the tendency to have higher betas.

(3) There is evidence that larger firms perform fairly well during layoffs. For instance, Chatrath and Song (1995) provide evidence that layoffs by large corporations in the early 1990s actually resulted in significant gains in shareholder wealth.

(4) The conditional variances of stocks have been noted to be higher in falling markets (e.g., Glosten, Jagannathan & Runkel, 1993). It has been suspected that inferences on stock return patterns may be biased by such an effect.

(5) Nonstationarity would have implied that the series are not mean-reverting and that the traditional econometric methodologies could lead to false inferences. The stationarity results are available for the authors.

(6) These results are similar to those found for REITs by Goldstein and Nelling (1999) and Chatrath, Liang and McIntosh (2000).

(7) The Glosten et al. model for each return series is given by

[R.sub.t]=[[b.sub.0] + [[b.sub.1, R.sub.t-1 + b.sub.2, h.sub.t]+ [[epsilon].sub.t] [[epsilon].sub.t][tilde] [[N(0,h.sub.t)] [h.sub.t] = [[alpha.sub.0 + [[alpha.sub.t-1] + [[alpha.sub.2][[member of.sub.2][t.sub.-1]+ [[alpha.sub.3] [[epsilon].sub.t-2][1.sub.n+][u.sub.t],

where [h.sub.t] is the variance of returns, and [[??].sub.t.sup.+] takes on the value of 1 if [[??].sub.t-1] is positive, and 0 otherwise. The variance term in the return equation allows for the returns to be dependent on return variance.

[FIGURE 1 OMITTED]

Arjun Chatrath, University of Portland

Bahram Adrangi, University of Portland

Robin Anderson, University of Portland

Kanwalroop Kathy Dhanda, University of Portland

Table 1

Diagnostics of Monthly Returns

Returns for each index are given by [([P.sub.t]+[d.sub.t])/
[P.sub.t-1]] * 100, where [P.sub.t] and [d.sub.t] represent
the index value and dividend at time t, respectively.

Q is the Ljung-Box-Pierce (Modified Box-Pierce) statistic
(chi-square distributed) for return autocorrelation. The results
from the GARCH(1,1) model are the coefficients from the conditional
variance equation; [[??].sub.1] and [[beta].sub.1] represent the
coefficients of the maximum likelihood estimation of the impact of
past return shocks (error terms) and past variance on the
contemporaneous variance; [[??].sub.1] + [[beta].sub.1] close to 1
suggests the memory in variance persists over long periods of time.

Figures in () are t-statistics. a, b and c represent significance
at the .01, .05 and .10 levels respectively.

 S&P 500 Small
 (1/72-12/98) (1/72-12/98)

Mean 1.183 1.345
Std. Dev 4.425 6.027
Q(Box-Pierce)
lag 1 0.03 8.02 (a)
 2 0.37 9.08 (b)
 3 0.38 9.68 (b)
 4 0.99 11.06 (b)
 5 3.52 11.79 (b)
 6 5.57 12.08 (c)

GARCH(1,1)

 [[??].sub.1] 0.068 (b) 0.049 (a)
 (2.25) (2.72)
 [[beta].sub.1] 0.879 (a) 0.886 (a)
 (20.36) (23.37)
 [[??].sub.1] + 0.947 0.935
 [[beta].sub.1]

 Russel 1000 Russell 2000
 (1/79-12/98) (1/79-12/98)

Mean 1.436 1.27
Std. Dev 4.380 5.488
Q(Box-Pierce)
lag 1 0.13 10.32 (a)
 2 0.29 10.33 (a)
 3 1.10 10.91 (a)
 4 2.69 13.31 (a)
 5 4.24 14.53 (b)
 6 4.56 14.68 (b)

GARCH(1,1)

 [[??].sub.1] 0.056 (c) 0.001
 (1.78)
 [[beta].sub.1] 0.907 (a)
 (17.82)
 [[??].sub.1] + 0.963
 [[beta].sub.1]

Table 2

Evidence of Asymmetry in Small Stock Betas

The results are from the regression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [R.sub.t,i] and [R.sub.t,SP] represent, respectively, the
monthly dividend-inclusive returns for the ith small-firm index
and the S&P 500 index, and D- is a dummy variable equal to 1 if
the S&P 500 excess returns are negative, and 0 otherwise. Returns
for each index are given by [([P.sub.t]+[d.sub.t])/[P.sub.t-1]]*100,
where [P.sub.t] and [d.sub.t] represent the index value and dividend
at time t, respectively. a, b and c represent significance at the
.01, .05 and .10 levels respectively.

 [R.sub.t,SP]
 Constant [R.sub.t,SP] * D [R.sup.2]

A. Small Stock (1/72-12/98)

 Model 1 0.110 1.044 (a) -- .58
 (0.49) (21.43)
 Model 2 0.662 (b) 0.897 (a) 0.337 (b) .59
 (2.05) (11.36) (2.34)

B. Russell 1000 (1/79-12/98)

 Model 1 -0.026 1.009 (a) -- .99
 (-1.07) (190.27)
 Model 2 0.020 0.996 (a) 0.027 (c) .99
 (0.57) (114.97) (1.78)

C. Russell 2000 (1/79-12/98)

 Model 1 -0.260 1.081 (a) -- .72
 (-1.31) (24.63)
 Model 2 0.481 (c) 0.883 (a) 0.452 (a) .74
 (1.68) (12.54) (3.54)

Table 3

The Role of Variance in the Asymmetry in Small Stock Betas

Statistics are from the regression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[??].sub.t,i]/[[??].sub.ht,i] is the standard residuals of the
ith small stock index, and [[??].sub.t,SP]/[[??].sub.ht,SP] is the
standard residuals of the S&P 500 index, and D- is a dummy variable
equal to 1 if the S&P 500 excess returns are negative, and 0 otherwise.
Standard residuals are obtained from the GARCH model with controls for
variance asymmetries as suggested in Glosten et al. (1993). a, b and c
represent significance at the .01, .05 and .10 levels respectively.

 [[??].sub.t,SP]/ [[??].sub.t,SP]/
 Constant [[??].sub.ht,SP] [[??].sub.ht,SP]

A. Small Stock (1/72-12/98)

 Model 1 0.022 0.784 (a) --
 (0.64) (22.89)
 Model 2 0.109 (b) 0.652 (a) 0.239 (b)
 (2.17) (9.90) (2.34)

B. Russell 1000 (1/79-12/98)

 Model 1 -0.061 (a) 1.006 (a) --
 (-9.54) (155.00)
 Model 2 -0.049 (a) 0.988 (a) 0.032 (c)
 (-5.25) (82.56) (1.73)

C. Russell 2000 (1/79-12/98)

 Model 1 -0.061 (c) 0.859 (a) --
 (-1.68) (23.23)
 Model 2 0.053 0.691 (a) 0.316 (a)
 (1.01) (10.25) (2.97)

 Constant [R.sup.2]

A. Small Stock (1/72-12/98)

 Model 1 0.022 .61
 (0.64)
 Model 2 0.109 (b) .63
 (2.17)

B. Russell 1000 (1/79-12/98)

 Model 1 -0.061 (a) .99
 (-9.54)
 Model 2 -0.049 (a) .99
 (-5.25)

C. Russell 2000 (1/79-12/98)

 Model 1 -0.061 (c) .71
 (-1.68)
 Model 2 0.053 .74
 (1.01)

Table 4

Small Stock Returns and Credit Risk

The results are from the regressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [R.sub.t,i] represents the monthly dividend-inclusive returns
for the ith small-firm index, Spread is the difference in the yield
between the Baa and Aaa rated corporate bonds, and [[??].sub.t,i]/
[[??].sub.ht,i] is the standard residuals of the ith small stock
index. a, b and c represent significance at the .01, .05 and .10
levels respectively.

 Small Stock Russell 1000 Russell 2000

Panel A. Returns and the Yield Spread

Constant -0.964 0.944 -0.132
 (-1.10) (1.32) (-0.14)
Spread 2.065 (a) 0.442
 (2.89) (0.77) (1.76)
[R.sup.2] .02 .00 .01

Panel B. Standard Residuals and the Yield Spread

Constant -0.293 (b) -0.286 (c) -0.046
 (-2.02) (-1.75) (-0.27)
Spread 0.282 (b) 0.139 0.183
 (2.34) (1.06) (1.35)
[R.sup.2 .01 .00 .01

Table 5

The Predictive Power of the Yield Spread

The figures are mean square errors from alternative autoregressive
models. MSE-AR represent mean square errors from the model where
index returns are regressed on past returns (3 lags). MSE-AR,Spread
represent mean square errors from the model where index returns are
regressed on past index returns and past and contemporaneous levels
of the spread between Baa and Aaa rated bonds. The figures in ()
are standard deviations. The F statistic tests the equality of the
MSE-AR and MSE-AR,Spread.

 MSE-AR MSE-AR,Spread F

Small Stock Index 35.066 34.095 0.02
 (91.84) (85.83)
Russell 1000 18.575 18.268 0.01
 (44.16) (43.32)
Russell 2000 27.072 26.508 0.01
 (73.54) (72.09)