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An investment-based explanation for the dispersion anomaly
Journal Pre-proof An investment-based explanation for the dispersion anomaly Byoung-Kyu Min, Tai-Yong Roh
PII: DOI: Reference:
S0165-1765(19)30421-5 https://doi.org/10.1016/j.econlet.2019.108832 ECOLET 108832
To appear in:
Economics Letters
Received date : 11 August 2019 Revised date : 5 November 2019 Accepted date : 5 November 2019 Please cite this article as: B.-K. Min and T.-Y. Roh, An investment-based explanation for the dispersion anomaly. Economics Letters (2019), doi: https://doi.org/10.1016/j.econlet.2019.108832. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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An Investment-Based Explanation for the Dispersion Anomaly
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Byoung-Kyu Min* Tai-Yong Roh†
Abstract
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We provide an investment-based explanation for the dispersion anomaly. The firms’ optimality condition predicts that expected stock returns equal investment returns (the ratio of expected marginal benefits of investment to marginal costs of investment). We show that the investment model does a good job in explaining the dispersion portfolios. Firms with high forecast dispersion have low expected profitability, which is a key component of expected marginal benefit of investment. Consequently, high forecast dispersion portfolio earns lower expected returns. Our results suggest that the dispersion anomaly could be consistent with the firms’ value maximization.
*
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JEL classification: G12; G14 Keywords: Dispersion anomaly; Investment-based asset pricing; Structural estimation
University of Sydney Business School, Codrington Building, The University of Sydney, NSW, 2006, Australia. Phone: +61-2-9036-6356; Fax: +61-2-9351-6461; E-mail: [email protected]. † Corresponding author. Advanced Institute of Finance and Economics, Liaoning University, Shenyang, 110036, China. Phone: +86-24-62602445; Fax: +86-24-62602447; E-mail: [email protected]. 1
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1. Introduction Diether, Malloy, and Scherbina (2002) find that stocks with high analysts’ earnings forecast dispersion earn lower future returns than stocks with low forecast dispersion. If forecast dispersion is a measure of
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future earnings uncertainty, investors should expect higher returns as compensation for bearing high uncertainty. Thus, the negative cross-sectional relation between forecast dispersion and future returns is puzzling. The dispersion anomaly has been confirmed and refined by many following studies. Leippold and Lohre (2014) document the dispersion effect in some European markets. Bali, Bodnaruk, Scherbina, and Tang (2018) show that unusual news flow, which temporarily raises forecast dispersion, predicts negatively stock returns in the cross-section. Most of the literature has adopted the behavioral
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interpretation of the dispersion effect. In particular, Diether, Malloy, and Scherbina (2002) propose that forecast dispersion proxies for divergence of investor opinions and the observed negative dispersionreturns relation is consistent with Miller’s (1997) prediction that optimistic investors dominate the market
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when pessimistic investors cannot short due to short sale constraints.1 Consequently, stocks with greater divergence of opinions are to be initially overpriced and would realize lower future returns as the overpricing is eventually corrected over time.
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Departing from the existing literature, we use the investment-based expected return model from Liu, Whited, and Zhang (2009) and Liu and Zhang (2014) to investigate whether the dispersion-return relation is correctly tied to economic fundamentals via firms’ optimality conditions. The neoclassical theory of investment predicts that, under constant returns to scale, stock returns equal investment returns. The latter returns, determined as the ratio of expected marginal benefits of investment to marginal costs of
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investment, are linked to firm characteristics through the first order condition of firms’ value maximization. Intuitively, firms with high forecast dispersion are likely to have low expected profitability,
1
Using an alternative measure of dispersion of opinions based on individual investor’s trades, Goetzmann and Massa (2005) show high dispersion to predict lower future returns. Boehme, Danielsen, and Sorescu (2006) measure divergence of opinions using trading volume and find that firms with short-sale constraint and high turnover earn lower future returns. 2
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which is a key component of expected marginal benefit of investment.2 Thus, high forecast dispersion portfolio earns lower expected returns. We conduct structural estimation using GMM to examine whether expected investment returns
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predicted from the model can match average realized returns from the data. We find that the investment model explains well the dispersion portfolios. The low-minus-high dispersion decile hedge portfolio has a small and insignificant pricing error (alpha) of 0.32% (t-value = 0.07) per annum, which is remarkably small compared to the Fama-French three-factor alpha of 10.41% (t-value = 4.63). Further, the alphas of ten individual dispersion portfolios do not vary systematically with forecast dispersion and are all insignificantly different from zero. Finally, we find that the expected marginal product of capital
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(profitability) is the key driver in explaining the dispersion anomaly. In the absence of its cross-sectional variation, the low-minus-high dispersion hedge portfolio alpha significantly increases from 0.32% in the benchmark estimation to 13.14%.
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Our work contributes to the existing literature by providing an investment-based explanation for the dispersion anomaly. Differing from Diether, Malloy, and Scherbina (2002) and many other subsequent studies, we suggest that the dispersion effect could be consistent with the firms’ value maximization. Our
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findings do not preclude alternative behavioral explanations. Our study instead suggests that irrationality and market friction are not necessary to explain the dispersion anomaly.
2. The Investment Model
The investment model of Liu, Whited, and Zhang (2009) considers optimal investment decision problem of a levered firm in the economy. At each time t, firm i makes an investment decision and generates
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operating profits, Π 𝐾, 𝑋 . Π 𝐾 , , 𝑋 ,
is a function of capital 𝐾 , and exogeneous state vector 𝑋 ,
(either aggregate or firm-specific). Firms have a Cobb-Douglas production function with constant returns
2
Min, Qiu, and Roh (2019) propose the disclosure manipulation as a potential explanation for the negative relation between forecast dispersion and expected profitability. 3
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to scale. This assumption implies Π 𝐾 , , 𝑋 ,
, ,
,
𝐾,
,
and
,
, ,
𝑘
,
, ,
, in which 𝑘 is
capital’s share. In every period t, capital 𝐾 , depreciates at rate 𝛿 , and is increased (or decreased) by 1
𝛿, 𝐾,
𝐼 , ). Capital investment incurs adjustment costs. The
adjustment cost function is modeled as follows: ,
Φ 𝐼 , ,𝐾,
,
Note that 𝑘 and 𝑎 are(?) homogeneous across firms.
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investment, 𝐼 , (i.e, 𝐾 ,
𝐾, .
(1)
At the investment decision time t, firm i also makes a capital structure decision by issuing a certain amount of one-period debt, 𝐵, which must be repaid at next time period t+1. The firm borrows
𝐷, ≡ 1
Π 𝐾, ,𝑋 ,
𝜏
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with interest rate, 𝑟 , . Total firm payout 𝐷 at period t is given by 𝛿, 𝐾,
Φ 𝐼 , ,𝐾,
𝑟,
1 𝐵,
The firm chooses the investment level 𝐼 , and capital stock level 𝐾 ,
𝐼,
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,
,
𝐸 ∑
max
subject to a transversality condition lim 𝐸 𝑀
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→
, ,
𝐵,
The first-order conditions for 𝐼 , and 𝐾 , standard asset-pricing equation 𝐸 𝑀
𝑟,
,
,
,
,
𝐷,
value
⁄ 𝑃,
𝐵,
of
equity,
𝑟,
,
(3)
0. in the value maximization problem yield the
1, in which 𝑟 , ,
is the investment return defined as ,
,
,
.
,
(4)
,
implies 𝐸 𝑀
≡ 𝑃,
(2)
as given,
𝑟,
1 , where 𝑟 ,
is defined as the after-tax corporate bond return. Define 𝑃 , ≡ 𝑉 ,
market 𝐵,
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The first-order condition for 𝐵 , 1 𝜏
𝑟,
𝑀
𝑟, 𝐵 , .
in each period to maximize its
cum-dividend market value 𝑉 , by taking the stochastic discount factor 𝑀 𝑉, ≡
𝐵,
𝐷,
⁄𝑃 ,
as
the
stock
≡ 𝑟,
𝑟,
𝐷 , as the ex-dividend return,
and
𝑤, ≡
as the market leverage. As shown by Liu, Whited, and Zhang (2009), the 4
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investment return equals the weighted average of the stock return and the after-tax corporate bond return, 𝑟,
𝑤 , 𝑟,
1
𝑤 , 𝑟,
𝑟,
,
,
return equation (4) with 𝑟 ,
, ,
,
,
,
,
, stock return 𝑟 , ,
3. Econometric Methodology
,
,
,
,
,
.
(5)
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3.1 GMM Estimation
. Finally, combining investment
,
,
, and RHS
is expressed as
,
,
, ,
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of this equation is equivalent to the levered investment return, 𝑟 ,
,
,
. Solving for the stock return yields 𝑟 ,
We construct annual levered investment returns and match with annual realized stock returns. Following Liu and Zhang (2014), we estimate the moment condition implied by Equation (5) via one-stage GMM
𝑔
a 𝑁
1 vector whose ith component is calculated as ,
𝐸
,
𝑟,
,
,
,
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𝑔
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with the identity weighting matrix. Let 𝑁 be the number of testing portfolios and let moment condition
, , ,
,
,
, ,
,
The parameters {𝑘, 𝑎} are obtained by minimizing the objective function Ω
3.2 Data
,
.
(6)
𝑔 𝐼 𝑔 .
Following Diether, Malloy, and Scherbina (2002), we compute dispersion as the standard deviation of
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analysts’ earnings forecasts in a month divided by the absolute value of the mean forecast in that month. Monthly analysts’ annual earnings forecast data are obtained from the unadjusted file in I/B/E/S. Monthly returns for all common stocks listed on NYSE, AMEX, and NASDAQ are from CRSP. Firm-level accounting data is from Compustat annual. Following Diether, Malloy, and Scherbina (2002), we exclude stocks with a closing price below $5 at the end of each month. We also exclude financial firms and 5
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regulated firms which are not applicable to the investment-based model framework. Our sample period is from 1976 to 2014. Following Liu and Zhang (2014), we set firms’ pre-tax corporate bond returns as one
crediting ratings.3
3.3 Variable Measurement
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of Barclays U.S. aggregate corporate bond returns (Source: Datastream) that match firms’ Compustat
A procedure that measures and matches various portfolio-level characteristics such as 𝑌 , 𝐼,
⁄𝐾 ,
⁄𝐾 ,
,
, 𝑤 , in equation (5) is required to calculate sample moments in equation (6). We calculate
the portfolio-level characteristics as the sum of numerator variable (𝑌 , , 𝑃,
𝐵,
, 𝐼,
) divided by the
), using all stocks in a given portfolio. The detailed
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sum of denominator variable (𝐾 ,
, 𝐵,
construction of the variables used in the paper is provided in Appendix A. To match levered investment returns with stock returns, we use the same timing alignment scheme applied for monthly rebalanced SUE
4. Empirical Results
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portfolios in Liu, Whited, and Zhang (2009).
Table 1 shows that average realized returns decrease from low dispersion to high dispersion portfolio. The
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low-minus-high dispersion decile hedge portfolio has an equal-weighted average return of 10.94% per annum (t-value = 3.82). The Fama-French model performs poorly in explaining the dispersion portfolios: the Fama-French alpha of the hedge portfolio is large and statistically significant at 10.41% (t-value = 4.63). In sharp contrast, the investment-based model performs remarkably well in accounting for the average returns on the dispersion portfolios. The investment model produces alphas for the dispersion
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deciles from 0.16% to 1.23% per annum. The pricing errors are all statistically insignificant, and do not show a systematic pattern across dispersion portfolios. The mean absolute pricing error obtained from the investment model is 0.59%, which is lower than that from the Fama-French model (1.42%). The low-
3 When crediting ratings data are not available for certain firm from Compustat, we set the firms’ credit rating as Baa. 6
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minus-high dispersion decile has a small and insignificant alpha of 0.32% (t-value = 0.07), which is substantially lower than the Fama-French alpha of 10.41%. Figure 1 shows a visual representation of the fit across the different models by plotting the average returns of dispersion deciles predicted by the model
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against their average realized returns. If the predicted and realized average returns for each portfolio are the same, all points should lie on a 45-degree line. The scatter plot from the investment model in Figure 1(a) is fairly aligned with the 45-degree line, while that from the Fama-French model in Figure 1(b) is largely horizontal. Finally, the good fit of the investment model does not come at the cost of having implausible structural parameters. The estimate of the capital’s share (𝑘) is 0.31 (t-value = 2.45), being close to the typical calibration value of 0.30 used in macroeconomic studies (e.g., Rotemberg and
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Woodford, 1992). The estimate for the adjustment cost parameter (𝑎) is 8.76 (t-value = 0.98), comparable to estimates in Liu, Whited, and Zhang (2009). The positive estimate of 𝑎 suggests that the adjustment cost function is increasing and convex in investment.
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We next examine which firm fundamental is a key driver for the investment model to explain the dispersion anomaly. The investment equation (5) proposes several economic forces that determine the cross-section of stock returns. The first driver is the marginal product of capital, 𝑌 ,
; the second
⁄𝑞 , ; the third is investment-to-capital, 𝐼 , ⁄𝐾 , ; the fourth is the
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is the growth rate of investment, 𝑞 ,
⁄𝐾 ,
depreciation rate, 𝛿 ,
; and the fifth is the market leverage, 𝑤 , . All other things being equal, firms
with high expected marginal product of capital, high expected investment growth, low investment-tocapital, low expected depreciation rate, and high market leverage should earn higher average stock returns. Table 2 presents the averages of five drivers for each dispersion portfolio.4 The results show that
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expected marginal product of capital (profitability) is the only component going in the right direction to explain the dispersion portfolios. The high dispersion portfolio has significantly lower sales-to-capital than low dispersion portfolio (1.09 vs 1.75), with a spread of 0.66 (t-value = 7.72). In contrast, other
4 The investment growth contains the adjustment cost parameter, which is not directly observable from the data. We therefore report the growth rate of investment-to-capital instead. 7
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components go in the wrong direction to explain the expected returns. Intuitively, firms with high forecast dispersion are likely to have low expected profitability. As such, they should earn lower expected returns as predicted by the neoclassical theory of investment.
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To evaluate quantitatively how much each component contributes in matching the dispersion portfolios, we exercise comparative static experiments. We remove the cross-sectional variation of each firm characteristic by assigning it to its cross-sectional mean in each year, while maintaining all the other characteristics unchanged. We then recalculate investment returns using the estimated parameters of 𝑘 and 𝑎. Table 3 presents the predicted returns and pricing errors obtained from the comparative static experiment. Without the cross-sectional variation in sales-to-capital, the pricing error of low-minus-high
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dispersion decile substantially increases to 13.14% per annum from the level of 0.32% in the benchmark estimation, suggesting that expected marginal product of capital is quantitatively important. Further, in the absence of the cross-sectional variation in sales-to-capital, the model predicts high dispersion portfolio to
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earns higher returns, which is at odds with data. In contrast, removing the cross-sectional variation of other drivers predicts high dispersion portfolio to have lower returns. As other characteristics (except sales-to-capital) go in the wrong direction to explain the dispersion portfolio, eliminating their cross-
5. Conclusion
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sectional variation is helpful to match the negative relation between dispersion and expected returns.
We provide an investment-based explanation for the dispersion anomaly. The first order condition of firms’ value maximization predicts that stock returns equal investment returns (the ratio of expected marginal benefits of investment to marginal costs of investment). Using GMM, we show that the
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investment model goes a long way in explaining the dispersion portfolios. Firms with high forecast dispersion have low expected profitability, which is a key component of expected marginal benefit of investment. Consequently, high forecast dispersion portfolio earns lower expected returns. Our findings suggest that the dispersion anomaly could be consistent with the firms’ value maximization. 8
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References Bali, T., A. Bodnaruk, A. Scherbina, and Y. Tang, 2018, Unusual news flow and the cross section of stock returns, Management Science 64 (9): 4137–4155.
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Boehme, R., B. Danielsen and S. Sorescu, 2006, Short-sale constraints, differences of opinion, and overvaluation, Journal of Financial and Quantitative Analysis 41: 455–487 Diether, K., C. Malloy, and A. Scherbina, 2002, Differences of opinion and the cross section of stock returns, Journal of Finance 57(5): 2113–2141. Goetzmann, W., and M. Massa, 2005, Dispersion of opinion and stock returns, Journal of Financial Markets 8: 324–349. Leippold, M., and H. Lohre, 2014, The dispersion effect in international stock returns, Journal of Empirical Finance 29: 331–342.
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Liu, L.X., T. Whited, and L. Zhang, 2009. Investment-based expected stock returns, Journal of Political Economy 117: 1105–1139. Liu, L.X., and L. Zhang, 2014. A neoclassical interpretation of momentum. Journal of Monetary Economics 67: 109–128.
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Miller, E., 1977, Risk, uncertainty, and divergence of opinion. Journal of Finance 32: 1151–1168. Min, B., B. Qiu and T. Roh, 2019, What Drives the Dispersion Anomaly? Working paper, University of Sydney.
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Rotemberg, J., and M. Woodford, 1992, Oligopolistic pricing and the effects of aggregate demand on economic activity, Journal of Political Economy 100: 1153–1207.
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Appendix A. Variable Description Variable
Definition
Gross property, plant, and equipment (Compustat annual item PPEGT)
𝐼 , (Investment)
Capital expenditures (item CAPX) minus sales of property, plant, and
pro of
𝐾 , (Capital Stock)
equipment (item SPPE, set to zero if missing) 𝑌 , (Output)
Sales (item SALE)
𝐵 , (Total Debt)
Long-term debt (item DLTT) plus short-term debt (item DLC)
𝑃 , (Market Equity)
The stock price at the fiscal yearend (item PRCC_F) multiplied by common shares outstanding (item CSHO)
The amount of depreciation (item DP) divided by capital stock 𝐾 ,
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𝛿 , (Depreciation rate)
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Table 1 Average Realized Excess Returns and Pricing Errors for Dispersion Decile Portfolios The table reports average realized excess returns ( 𝑅 ) and the pricing errors from the Fama-French and the Investment model. The testing portfolios are ten dispersion decile portfolios. L-H denotes the low-minus-high
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dispersion hedge portfolio. The Fama-French alpha is obtained from regressing annual excess returns of dispersion decile portfolios on annual returns of the market factor, a size factor, and a book-to-market factor. Numbers in parentheses are t-statistics. The sample period is from 1976 to 2014.
L
2
3
4
5
6
7
8
9
H
L-H
Panel A: Average Excess Returns 𝑅
14.29
11.46
10.95
10.70
10.40
9.85
9.60
8.27
6.62
3.35
10.94
(t-stat)
(4.13)
(3.41)
(3.65)
(3.15)
(3.14)
(2.97)
(2.68)
(2.24)
(1.74)
(0.83)
(3.82)
Panel B: Alphas from the Fama-French three-factor model
(t-stat)
2.41
0.16
1.08
0.04
0.48
-0.05
-0.19
-2.52
-4.53
-8.00
10.41
(1.32)
(0.08)
(0.70)
(0.03)
(0.54)
(-0.04)
(-0.14)
(-2.11)
(-3.21)
(-5.99)
(4.63)
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𝛼
Panel C: Alphas from the Investment Model 0.44
-0.30
0.07
-0.69
0.41
0.01
1.23
-1.08
0.32
(-0.25)
(0.43)
(0.49)
(-0.41)
(0.06)
(-0.90)
(0.28)
(0.01)
(0.87)
(-0.59)
(0.07)
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0.86
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(t-stat)
-0.76
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𝛼
11
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(depreciation rate), and
𝑰𝒊,𝒕 𝟏
𝑰𝒊,𝒕
𝑲𝒊,𝒕 𝟏 𝑲𝒊,𝒕
𝒀𝒊,𝒕 𝟏
(sale-to-capital),
𝑰𝒊,𝒕 𝑲𝒊,𝒕
(investment-to-
(investment growth). The average of the difference in the components between the
𝑲𝒊,𝒕 𝟏
(depreciation rate)
𝑤 , (market leverage)
𝛿,
,
(investment growth)
(investment-to-capital)
,
,
,
,
(sale-to-capital)
,
,
,
Characteristics
0.20
0.08
0.12
0.99
1.75
L
0.20
0.08
0.11
0.99
1.50
2 1.47
4
0.21
0.08
0.11
0.99
0.23
0.08
0.11
1.00
5
0.25
0.27
0.07
0.11
1.01
1.36
6
re0.08
0.11
1.00
1.41
12
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1.48
3
0.34
0.07
0.11
0.99
1.20
8
0.41
0.08
0.10
0.97
1.13
9
0.46
0.08
0.09
0.95
1.09
H
pro of
0.29
0.07
0.11
1.00
1.27
7
-0.26
0.01
0.02
0.04
0.66
L-H
-16.69
2.00
5.32
1.63
7.72
t_[L-H]
lowest dispersion decile and the highest dispersion decile (L-H) and its corresponding t-statistic are also reported. The sample period is from 1976 to 2014.
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capital), 𝒘𝒊,𝒕 (market leverage), 𝜹𝒊,𝒕 𝟏
This table reports the averages of the following 5 firm characteristics that drive the expected investment returns:
Firm Fundamentals across Dispersion Portfolios
Table 2
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𝒒𝒊,𝒕 𝟏 𝒒𝒊,𝒕
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(investment-to-capital), 𝒘𝒊,𝒕 (market leverage), and
𝟏 𝟏
𝒊,𝒕 𝟏
𝟏 𝝉𝒕 𝒂 𝒊,𝒕 𝑲𝒊,𝒕
𝑰
𝑰 𝟏 𝝉𝒕 𝟏 𝒂 𝒊,𝒕 𝟏 𝑲
(investment-to-capital)
(sale-to-capital)
,
,
-2.06
Pricing errors
Pricing errors
-0.76
14.49
15.79
-4.27
Pricing errors
Predicted average returns
18.00
5.26
8.47
0.86
10.55
-0.18
11.59
-0.55
11.96
2.94
8.47
2 9.24
4
0.44
10.46
-0.57
11.47
-1.68
12.58
2.17
-0.30
10.95
-1.08
11.73
-2.64
13.29
1.41
0.66
9.69
5
11.33
-0.91
10.71
6
8.59
-1.23
10.78
7
6.97
-2.99
11.21
8
0.07
10.28
-0.55
10.90
-1.52
-0.69
10.49
-0.90
10.70
-1.53
0.41
9.14
0.44
9.11
0.96
0.01
8.21
0.83
7.39
1.25
9
1.27
5.31
1.76
4.82
5.62
0.96
-3.75
10.33
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11.87
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8.73
3
13
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Predicted average returns
Pricing errors
Predicted average returns
(growth of investment) Predicted average returns
𝑤 , (market leverage)
,
,
,
,
L
(sale-to-capital),
-1.08
4.39
-0.28
3.59
10.98
-7.67
-7.88
11.19
H
0.32
10.11
-1.78
12.21
-15.25
25.68
13.14
-2.71
L-H
(the capital gain component of the investment return). We also report pricing
𝑲𝒊,𝒕 𝟏
errors which are defined as difference between average realized stock return and the investment returns. The sample period is from 1976 to 2014.
𝑲𝒊,𝒕
𝑰𝒊,𝒕
assigning it to its cross-sectional mean in each year, while maintaining all the other characteristics unchanged. We consider 4 components:
𝒀𝒊,𝒕 𝟏
This table reports the predicted returns and pricing errors obtained under a setting that removes the cross-sectional variation of each firm characteristic by
Predicted Returns and Pricing Errors from Comparative Static Experiments
Table 3
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(a) The Investment model
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the Investment and the Fama-French model, respectively.
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(b) The Fama-French model
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Figure 1 plots the average returns of dispersion deciles predicted by the model against their average realized returns. Figures 1(a) and 1(b) plot the results from
Average Predicted Returns versus Average Realized Returns
Figure 1
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PII: DOI: Reference:
S0165-1765(19)30421-5 https://doi.org/10.1016/j.econlet.2019.108832 ECOLET 108832
To appear in:
Economics Letters
Received date : 11 August 2019 Revised date : 5 November 2019 Accepted date : 5 November 2019 Please cite this article as: B.-K. Min and T.-Y. Roh, An investment-based explanation for the dispersion anomaly. Economics Letters (2019), doi: https://doi.org/10.1016/j.econlet.2019.108832. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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An Investment-Based Explanation for the Dispersion Anomaly
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Byoung-Kyu Min* Tai-Yong Roh†
Abstract
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We provide an investment-based explanation for the dispersion anomaly. The firms’ optimality condition predicts that expected stock returns equal investment returns (the ratio of expected marginal benefits of investment to marginal costs of investment). We show that the investment model does a good job in explaining the dispersion portfolios. Firms with high forecast dispersion have low expected profitability, which is a key component of expected marginal benefit of investment. Consequently, high forecast dispersion portfolio earns lower expected returns. Our results suggest that the dispersion anomaly could be consistent with the firms’ value maximization.
*
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JEL classification: G12; G14 Keywords: Dispersion anomaly; Investment-based asset pricing; Structural estimation
University of Sydney Business School, Codrington Building, The University of Sydney, NSW, 2006, Australia. Phone: +61-2-9036-6356; Fax: +61-2-9351-6461; E-mail: [email protected]. † Corresponding author. Advanced Institute of Finance and Economics, Liaoning University, Shenyang, 110036, China. Phone: +86-24-62602445; Fax: +86-24-62602447; E-mail: [email protected]. 1
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1. Introduction Diether, Malloy, and Scherbina (2002) find that stocks with high analysts’ earnings forecast dispersion earn lower future returns than stocks with low forecast dispersion. If forecast dispersion is a measure of
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future earnings uncertainty, investors should expect higher returns as compensation for bearing high uncertainty. Thus, the negative cross-sectional relation between forecast dispersion and future returns is puzzling. The dispersion anomaly has been confirmed and refined by many following studies. Leippold and Lohre (2014) document the dispersion effect in some European markets. Bali, Bodnaruk, Scherbina, and Tang (2018) show that unusual news flow, which temporarily raises forecast dispersion, predicts negatively stock returns in the cross-section. Most of the literature has adopted the behavioral
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interpretation of the dispersion effect. In particular, Diether, Malloy, and Scherbina (2002) propose that forecast dispersion proxies for divergence of investor opinions and the observed negative dispersionreturns relation is consistent with Miller’s (1997) prediction that optimistic investors dominate the market
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when pessimistic investors cannot short due to short sale constraints.1 Consequently, stocks with greater divergence of opinions are to be initially overpriced and would realize lower future returns as the overpricing is eventually corrected over time.
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Departing from the existing literature, we use the investment-based expected return model from Liu, Whited, and Zhang (2009) and Liu and Zhang (2014) to investigate whether the dispersion-return relation is correctly tied to economic fundamentals via firms’ optimality conditions. The neoclassical theory of investment predicts that, under constant returns to scale, stock returns equal investment returns. The latter returns, determined as the ratio of expected marginal benefits of investment to marginal costs of
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investment, are linked to firm characteristics through the first order condition of firms’ value maximization. Intuitively, firms with high forecast dispersion are likely to have low expected profitability,
1
Using an alternative measure of dispersion of opinions based on individual investor’s trades, Goetzmann and Massa (2005) show high dispersion to predict lower future returns. Boehme, Danielsen, and Sorescu (2006) measure divergence of opinions using trading volume and find that firms with short-sale constraint and high turnover earn lower future returns. 2
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which is a key component of expected marginal benefit of investment.2 Thus, high forecast dispersion portfolio earns lower expected returns. We conduct structural estimation using GMM to examine whether expected investment returns
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predicted from the model can match average realized returns from the data. We find that the investment model explains well the dispersion portfolios. The low-minus-high dispersion decile hedge portfolio has a small and insignificant pricing error (alpha) of 0.32% (t-value = 0.07) per annum, which is remarkably small compared to the Fama-French three-factor alpha of 10.41% (t-value = 4.63). Further, the alphas of ten individual dispersion portfolios do not vary systematically with forecast dispersion and are all insignificantly different from zero. Finally, we find that the expected marginal product of capital
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(profitability) is the key driver in explaining the dispersion anomaly. In the absence of its cross-sectional variation, the low-minus-high dispersion hedge portfolio alpha significantly increases from 0.32% in the benchmark estimation to 13.14%.
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Our work contributes to the existing literature by providing an investment-based explanation for the dispersion anomaly. Differing from Diether, Malloy, and Scherbina (2002) and many other subsequent studies, we suggest that the dispersion effect could be consistent with the firms’ value maximization. Our
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findings do not preclude alternative behavioral explanations. Our study instead suggests that irrationality and market friction are not necessary to explain the dispersion anomaly.
2. The Investment Model
The investment model of Liu, Whited, and Zhang (2009) considers optimal investment decision problem of a levered firm in the economy. At each time t, firm i makes an investment decision and generates
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operating profits, Π 𝐾, 𝑋 . Π 𝐾 , , 𝑋 ,
is a function of capital 𝐾 , and exogeneous state vector 𝑋 ,
(either aggregate or firm-specific). Firms have a Cobb-Douglas production function with constant returns
2
Min, Qiu, and Roh (2019) propose the disclosure manipulation as a potential explanation for the negative relation between forecast dispersion and expected profitability. 3
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to scale. This assumption implies Π 𝐾 , , 𝑋 ,
, ,
,
𝐾,
,
and
,
, ,
𝑘
,
, ,
, in which 𝑘 is
capital’s share. In every period t, capital 𝐾 , depreciates at rate 𝛿 , and is increased (or decreased) by 1
𝛿, 𝐾,
𝐼 , ). Capital investment incurs adjustment costs. The
adjustment cost function is modeled as follows: ,
Φ 𝐼 , ,𝐾,
,
Note that 𝑘 and 𝑎 are(?) homogeneous across firms.
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investment, 𝐼 , (i.e, 𝐾 ,
𝐾, .
(1)
At the investment decision time t, firm i also makes a capital structure decision by issuing a certain amount of one-period debt, 𝐵, which must be repaid at next time period t+1. The firm borrows
𝐷, ≡ 1
Π 𝐾, ,𝑋 ,
𝜏
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with interest rate, 𝑟 , . Total firm payout 𝐷 at period t is given by 𝛿, 𝐾,
Φ 𝐼 , ,𝐾,
𝑟,
1 𝐵,
The firm chooses the investment level 𝐼 , and capital stock level 𝐾 ,
𝐼,
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,
,
𝐸 ∑
max
subject to a transversality condition lim 𝐸 𝑀
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→
, ,
𝐵,
The first-order conditions for 𝐼 , and 𝐾 , standard asset-pricing equation 𝐸 𝑀
𝑟,
,
,
,
,
𝐷,
value
⁄ 𝑃,
𝐵,
of
equity,
𝑟,
,
(3)
0. in the value maximization problem yield the
1, in which 𝑟 , ,
is the investment return defined as ,
,
,
.
,
(4)
,
implies 𝐸 𝑀
≡ 𝑃,
(2)
as given,
𝑟,
1 , where 𝑟 ,
is defined as the after-tax corporate bond return. Define 𝑃 , ≡ 𝑉 ,
market 𝐵,
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The first-order condition for 𝐵 , 1 𝜏
𝑟,
𝑀
𝑟, 𝐵 , .
in each period to maximize its
cum-dividend market value 𝑉 , by taking the stochastic discount factor 𝑀 𝑉, ≡
𝐵,
𝐷,
⁄𝑃 ,
as
the
stock
≡ 𝑟,
𝑟,
𝐷 , as the ex-dividend return,
and
𝑤, ≡
as the market leverage. As shown by Liu, Whited, and Zhang (2009), the 4
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investment return equals the weighted average of the stock return and the after-tax corporate bond return, 𝑟,
𝑤 , 𝑟,
1
𝑤 , 𝑟,
𝑟,
,
,
return equation (4) with 𝑟 ,
, ,
,
,
,
,
, stock return 𝑟 , ,
3. Econometric Methodology
,
,
,
,
,
.
(5)
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3.1 GMM Estimation
. Finally, combining investment
,
,
, and RHS
is expressed as
,
,
, ,
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of this equation is equivalent to the levered investment return, 𝑟 ,
,
,
. Solving for the stock return yields 𝑟 ,
We construct annual levered investment returns and match with annual realized stock returns. Following Liu and Zhang (2014), we estimate the moment condition implied by Equation (5) via one-stage GMM
𝑔
a 𝑁
1 vector whose ith component is calculated as ,
𝐸
,
𝑟,
,
,
,
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𝑔
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with the identity weighting matrix. Let 𝑁 be the number of testing portfolios and let moment condition
, , ,
,
,
, ,
,
The parameters {𝑘, 𝑎} are obtained by minimizing the objective function Ω
3.2 Data
,
.
(6)
𝑔 𝐼 𝑔 .
Following Diether, Malloy, and Scherbina (2002), we compute dispersion as the standard deviation of
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analysts’ earnings forecasts in a month divided by the absolute value of the mean forecast in that month. Monthly analysts’ annual earnings forecast data are obtained from the unadjusted file in I/B/E/S. Monthly returns for all common stocks listed on NYSE, AMEX, and NASDAQ are from CRSP. Firm-level accounting data is from Compustat annual. Following Diether, Malloy, and Scherbina (2002), we exclude stocks with a closing price below $5 at the end of each month. We also exclude financial firms and 5
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regulated firms which are not applicable to the investment-based model framework. Our sample period is from 1976 to 2014. Following Liu and Zhang (2014), we set firms’ pre-tax corporate bond returns as one
crediting ratings.3
3.3 Variable Measurement
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of Barclays U.S. aggregate corporate bond returns (Source: Datastream) that match firms’ Compustat
A procedure that measures and matches various portfolio-level characteristics such as 𝑌 , 𝐼,
⁄𝐾 ,
⁄𝐾 ,
,
, 𝑤 , in equation (5) is required to calculate sample moments in equation (6). We calculate
the portfolio-level characteristics as the sum of numerator variable (𝑌 , , 𝑃,
𝐵,
, 𝐼,
) divided by the
), using all stocks in a given portfolio. The detailed
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sum of denominator variable (𝐾 ,
, 𝐵,
construction of the variables used in the paper is provided in Appendix A. To match levered investment returns with stock returns, we use the same timing alignment scheme applied for monthly rebalanced SUE
4. Empirical Results
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portfolios in Liu, Whited, and Zhang (2009).
Table 1 shows that average realized returns decrease from low dispersion to high dispersion portfolio. The
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low-minus-high dispersion decile hedge portfolio has an equal-weighted average return of 10.94% per annum (t-value = 3.82). The Fama-French model performs poorly in explaining the dispersion portfolios: the Fama-French alpha of the hedge portfolio is large and statistically significant at 10.41% (t-value = 4.63). In sharp contrast, the investment-based model performs remarkably well in accounting for the average returns on the dispersion portfolios. The investment model produces alphas for the dispersion
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deciles from 0.16% to 1.23% per annum. The pricing errors are all statistically insignificant, and do not show a systematic pattern across dispersion portfolios. The mean absolute pricing error obtained from the investment model is 0.59%, which is lower than that from the Fama-French model (1.42%). The low-
3 When crediting ratings data are not available for certain firm from Compustat, we set the firms’ credit rating as Baa. 6
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minus-high dispersion decile has a small and insignificant alpha of 0.32% (t-value = 0.07), which is substantially lower than the Fama-French alpha of 10.41%. Figure 1 shows a visual representation of the fit across the different models by plotting the average returns of dispersion deciles predicted by the model
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against their average realized returns. If the predicted and realized average returns for each portfolio are the same, all points should lie on a 45-degree line. The scatter plot from the investment model in Figure 1(a) is fairly aligned with the 45-degree line, while that from the Fama-French model in Figure 1(b) is largely horizontal. Finally, the good fit of the investment model does not come at the cost of having implausible structural parameters. The estimate of the capital’s share (𝑘) is 0.31 (t-value = 2.45), being close to the typical calibration value of 0.30 used in macroeconomic studies (e.g., Rotemberg and
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Woodford, 1992). The estimate for the adjustment cost parameter (𝑎) is 8.76 (t-value = 0.98), comparable to estimates in Liu, Whited, and Zhang (2009). The positive estimate of 𝑎 suggests that the adjustment cost function is increasing and convex in investment.
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We next examine which firm fundamental is a key driver for the investment model to explain the dispersion anomaly. The investment equation (5) proposes several economic forces that determine the cross-section of stock returns. The first driver is the marginal product of capital, 𝑌 ,
; the second
⁄𝑞 , ; the third is investment-to-capital, 𝐼 , ⁄𝐾 , ; the fourth is the
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is the growth rate of investment, 𝑞 ,
⁄𝐾 ,
depreciation rate, 𝛿 ,
; and the fifth is the market leverage, 𝑤 , . All other things being equal, firms
with high expected marginal product of capital, high expected investment growth, low investment-tocapital, low expected depreciation rate, and high market leverage should earn higher average stock returns. Table 2 presents the averages of five drivers for each dispersion portfolio.4 The results show that
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expected marginal product of capital (profitability) is the only component going in the right direction to explain the dispersion portfolios. The high dispersion portfolio has significantly lower sales-to-capital than low dispersion portfolio (1.09 vs 1.75), with a spread of 0.66 (t-value = 7.72). In contrast, other
4 The investment growth contains the adjustment cost parameter, which is not directly observable from the data. We therefore report the growth rate of investment-to-capital instead. 7
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components go in the wrong direction to explain the expected returns. Intuitively, firms with high forecast dispersion are likely to have low expected profitability. As such, they should earn lower expected returns as predicted by the neoclassical theory of investment.
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To evaluate quantitatively how much each component contributes in matching the dispersion portfolios, we exercise comparative static experiments. We remove the cross-sectional variation of each firm characteristic by assigning it to its cross-sectional mean in each year, while maintaining all the other characteristics unchanged. We then recalculate investment returns using the estimated parameters of 𝑘 and 𝑎. Table 3 presents the predicted returns and pricing errors obtained from the comparative static experiment. Without the cross-sectional variation in sales-to-capital, the pricing error of low-minus-high
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dispersion decile substantially increases to 13.14% per annum from the level of 0.32% in the benchmark estimation, suggesting that expected marginal product of capital is quantitatively important. Further, in the absence of the cross-sectional variation in sales-to-capital, the model predicts high dispersion portfolio to
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earns higher returns, which is at odds with data. In contrast, removing the cross-sectional variation of other drivers predicts high dispersion portfolio to have lower returns. As other characteristics (except sales-to-capital) go in the wrong direction to explain the dispersion portfolio, eliminating their cross-
5. Conclusion
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sectional variation is helpful to match the negative relation between dispersion and expected returns.
We provide an investment-based explanation for the dispersion anomaly. The first order condition of firms’ value maximization predicts that stock returns equal investment returns (the ratio of expected marginal benefits of investment to marginal costs of investment). Using GMM, we show that the
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investment model goes a long way in explaining the dispersion portfolios. Firms with high forecast dispersion have low expected profitability, which is a key component of expected marginal benefit of investment. Consequently, high forecast dispersion portfolio earns lower expected returns. Our findings suggest that the dispersion anomaly could be consistent with the firms’ value maximization. 8
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References Bali, T., A. Bodnaruk, A. Scherbina, and Y. Tang, 2018, Unusual news flow and the cross section of stock returns, Management Science 64 (9): 4137–4155.
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Boehme, R., B. Danielsen and S. Sorescu, 2006, Short-sale constraints, differences of opinion, and overvaluation, Journal of Financial and Quantitative Analysis 41: 455–487 Diether, K., C. Malloy, and A. Scherbina, 2002, Differences of opinion and the cross section of stock returns, Journal of Finance 57(5): 2113–2141. Goetzmann, W., and M. Massa, 2005, Dispersion of opinion and stock returns, Journal of Financial Markets 8: 324–349. Leippold, M., and H. Lohre, 2014, The dispersion effect in international stock returns, Journal of Empirical Finance 29: 331–342.
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Liu, L.X., T. Whited, and L. Zhang, 2009. Investment-based expected stock returns, Journal of Political Economy 117: 1105–1139. Liu, L.X., and L. Zhang, 2014. A neoclassical interpretation of momentum. Journal of Monetary Economics 67: 109–128.
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Miller, E., 1977, Risk, uncertainty, and divergence of opinion. Journal of Finance 32: 1151–1168. Min, B., B. Qiu and T. Roh, 2019, What Drives the Dispersion Anomaly? Working paper, University of Sydney.
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Rotemberg, J., and M. Woodford, 1992, Oligopolistic pricing and the effects of aggregate demand on economic activity, Journal of Political Economy 100: 1153–1207.
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Appendix A. Variable Description Variable
Definition
Gross property, plant, and equipment (Compustat annual item PPEGT)
𝐼 , (Investment)
Capital expenditures (item CAPX) minus sales of property, plant, and
pro of
𝐾 , (Capital Stock)
equipment (item SPPE, set to zero if missing) 𝑌 , (Output)
Sales (item SALE)
𝐵 , (Total Debt)
Long-term debt (item DLTT) plus short-term debt (item DLC)
𝑃 , (Market Equity)
The stock price at the fiscal yearend (item PRCC_F) multiplied by common shares outstanding (item CSHO)
The amount of depreciation (item DP) divided by capital stock 𝐾 ,
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𝛿 , (Depreciation rate)
10
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Table 1 Average Realized Excess Returns and Pricing Errors for Dispersion Decile Portfolios The table reports average realized excess returns ( 𝑅 ) and the pricing errors from the Fama-French and the Investment model. The testing portfolios are ten dispersion decile portfolios. L-H denotes the low-minus-high
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dispersion hedge portfolio. The Fama-French alpha is obtained from regressing annual excess returns of dispersion decile portfolios on annual returns of the market factor, a size factor, and a book-to-market factor. Numbers in parentheses are t-statistics. The sample period is from 1976 to 2014.
L
2
3
4
5
6
7
8
9
H
L-H
Panel A: Average Excess Returns 𝑅
14.29
11.46
10.95
10.70
10.40
9.85
9.60
8.27
6.62
3.35
10.94
(t-stat)
(4.13)
(3.41)
(3.65)
(3.15)
(3.14)
(2.97)
(2.68)
(2.24)
(1.74)
(0.83)
(3.82)
Panel B: Alphas from the Fama-French three-factor model
(t-stat)
2.41
0.16
1.08
0.04
0.48
-0.05
-0.19
-2.52
-4.53
-8.00
10.41
(1.32)
(0.08)
(0.70)
(0.03)
(0.54)
(-0.04)
(-0.14)
(-2.11)
(-3.21)
(-5.99)
(4.63)
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𝛼
Panel C: Alphas from the Investment Model 0.44
-0.30
0.07
-0.69
0.41
0.01
1.23
-1.08
0.32
(-0.25)
(0.43)
(0.49)
(-0.41)
(0.06)
(-0.90)
(0.28)
(0.01)
(0.87)
(-0.59)
(0.07)
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0.86
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(t-stat)
-0.76
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𝛼
11
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(depreciation rate), and
𝑰𝒊,𝒕 𝟏
𝑰𝒊,𝒕
𝑲𝒊,𝒕 𝟏 𝑲𝒊,𝒕
𝒀𝒊,𝒕 𝟏
(sale-to-capital),
𝑰𝒊,𝒕 𝑲𝒊,𝒕
(investment-to-
(investment growth). The average of the difference in the components between the
𝑲𝒊,𝒕 𝟏
(depreciation rate)
𝑤 , (market leverage)
𝛿,
,
(investment growth)
(investment-to-capital)
,
,
,
,
(sale-to-capital)
,
,
,
Characteristics
0.20
0.08
0.12
0.99
1.75
L
0.20
0.08
0.11
0.99
1.50
2 1.47
4
0.21
0.08
0.11
0.99
0.23
0.08
0.11
1.00
5
0.25
0.27
0.07
0.11
1.01
1.36
6
re0.08
0.11
1.00
1.41
12
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1.48
3
0.34
0.07
0.11
0.99
1.20
8
0.41
0.08
0.10
0.97
1.13
9
0.46
0.08
0.09
0.95
1.09
H
pro of
0.29
0.07
0.11
1.00
1.27
7
-0.26
0.01
0.02
0.04
0.66
L-H
-16.69
2.00
5.32
1.63
7.72
t_[L-H]
lowest dispersion decile and the highest dispersion decile (L-H) and its corresponding t-statistic are also reported. The sample period is from 1976 to 2014.
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capital), 𝒘𝒊,𝒕 (market leverage), 𝜹𝒊,𝒕 𝟏
This table reports the averages of the following 5 firm characteristics that drive the expected investment returns:
Firm Fundamentals across Dispersion Portfolios
Table 2
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𝒒𝒊,𝒕 𝟏 𝒒𝒊,𝒕
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(investment-to-capital), 𝒘𝒊,𝒕 (market leverage), and
𝟏 𝟏
𝒊,𝒕 𝟏
𝟏 𝝉𝒕 𝒂 𝒊,𝒕 𝑲𝒊,𝒕
𝑰
𝑰 𝟏 𝝉𝒕 𝟏 𝒂 𝒊,𝒕 𝟏 𝑲
(investment-to-capital)
(sale-to-capital)
,
,
-2.06
Pricing errors
Pricing errors
-0.76
14.49
15.79
-4.27
Pricing errors
Predicted average returns
18.00
5.26
8.47
0.86
10.55
-0.18
11.59
-0.55
11.96
2.94
8.47
2 9.24
4
0.44
10.46
-0.57
11.47
-1.68
12.58
2.17
-0.30
10.95
-1.08
11.73
-2.64
13.29
1.41
0.66
9.69
5
11.33
-0.91
10.71
6
8.59
-1.23
10.78
7
6.97
-2.99
11.21
8
0.07
10.28
-0.55
10.90
-1.52
-0.69
10.49
-0.90
10.70
-1.53
0.41
9.14
0.44
9.11
0.96
0.01
8.21
0.83
7.39
1.25
9
1.27
5.31
1.76
4.82
5.62
0.96
-3.75
10.33
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11.87
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8.73
3
13
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Predicted average returns
Pricing errors
Predicted average returns
(growth of investment) Predicted average returns
𝑤 , (market leverage)
,
,
,
,
L
(sale-to-capital),
-1.08
4.39
-0.28
3.59
10.98
-7.67
-7.88
11.19
H
0.32
10.11
-1.78
12.21
-15.25
25.68
13.14
-2.71
L-H
(the capital gain component of the investment return). We also report pricing
𝑲𝒊,𝒕 𝟏
errors which are defined as difference between average realized stock return and the investment returns. The sample period is from 1976 to 2014.
𝑲𝒊,𝒕
𝑰𝒊,𝒕
assigning it to its cross-sectional mean in each year, while maintaining all the other characteristics unchanged. We consider 4 components:
𝒀𝒊,𝒕 𝟏
This table reports the predicted returns and pricing errors obtained under a setting that removes the cross-sectional variation of each firm characteristic by
Predicted Returns and Pricing Errors from Comparative Static Experiments
Table 3
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(a) The Investment model
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the Investment and the Fama-French model, respectively.
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(b) The Fama-French model
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Figure 1 plots the average returns of dispersion deciles predicted by the model against their average realized returns. Figures 1(a) and 1(b) plot the results from
Average Predicted Returns versus Average Realized Returns
Figure 1
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