What is behind the magic of O-Score? An alternative interpretation of Dichev’s (1998) bankruptcy risk anomaly


Using Ohlson’s (J Account Res 18(1):109–131, 1980) measure of bankruptcy risk (O-Score), Dichev (J Fin 53(3):1131–1147, 1998) documents a bankruptcy risk anomaly in which firms with high bankruptcy risk earn lower than average returns. This study first demonstrates that the negative association between bankruptcy risk and returns does not generalize to an alternative measure of bankruptcy risk. Then, by examining the nine individual components of O-Score, I find that funds from operations (FFO) is the only component that is associated with returns. Furthermore, I show that the return-predictive power of FFO is due to cash flows from operations. Taken as a whole, this study provides evidence that Dichev’s bankruptcy risk anomaly is a manifestation of investors’ under (over)-pricing of cash flows (accrual) component of earnings, i.e., the accrual anomaly documented by Sloan (Account Rev 71(3):289–316, 1996).

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Fig. 1


  1. 1.

    One exception is Campbell et al. (2008), who find that stocks with high bankruptcy risk have delivered low returns since 1981. Their measure of bankruptcy risk utilizes accounting information as well as market information and lies outside the scope of this study.

  2. 2.

    The results in Houge and Loughran (2000) and Pincus et al. (2007) suggest that there may be an accrual anomaly and a cash flow anomaly and that they do not always occur together, i.e., there is no mechanical relationship going on such that if one is present, the other must also be present. Section 4.4 investigates whether the negative association between O-Score and returns would disappear when Sloan’s original accrual measure is employed.

  3. 3.

    Unlike the Compustat Industrial File, in which all data were displayed in U.S. dollars, the Fundamental File displays data for Canadian companies (dual listed or not) using Canadian dollars.

  4. 4.

    More precisely, Griffin and Lemmon find that the negative association between Ohlson’s (1980) measure of bankruptcy risk and subsequent returns is primarily from growth firms, suggesting that the bankruptcy risk anomaly is likely to occur as a result of a high degree of information asymmetry among growth firms. However, I repeat my analysis using small and large firms as well as growth and value firms because growth firms tend to be small and Kothari (2001) documents that evidence of market inefficiency tends to be more pronounced among small firms.

  5. 5.

    Firms listed in AMEX or NASDAQ tend to be smaller than those in NYSE. Hence using the median of NYSE size distribution leads me to classify less than 20 % of firm-years to the subsample of large firms. Thus I employ the first quintile of NYSE size distribution. In any case, the tenor of results remains unchanged when I use the median as a size cutoff.

  6. 6.

    The data library is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.

  7. 7.

    Studies that examine the bankruptcy-predictive power of Ohlson’s O-Score include Begley et al. (1996) and Hillegeist et al. (2004).

  8. 8.

    The average annual bankruptcy rate during the period of 1980 through 2000 was 0.97 % among the industrial firms in the intersection of the CRSP and Compustat databases (Hillegeist et al. 2004, Table 1, p. 12).

  9. 9.

    A decline of book-to-market from the second highest O-Score decile portfolio to the highest decile portfolio is much milder in Dichev because he winsorizes all the variables at the 1st and 99th percentile, whereas I use the variables without any adjustment. I intentionally do not follow Dichev in this respect due to the concern raised by Kothari et al. (2005), who demonstrate that data deletion induces a spurious association between subsequent returns and ex ante information variables.

  10. 10.

    Note that BSM-Prob is a measure based on market prices. If the market is not adequately anticipating bankruptcy probabilities, then BSM-Prob may not show the bankruptcy anomaly even if the anomaly exists. I cannot confirm or refute this possibility.

  11. 11.

    I include value-weighted portfolio analysis because Fama (1998) finds that, when value-weighted returns are examined, evidence of stock market inefficiency shrinks substantially and typically becomes statistically unreliable.

  12. 12.

    The trading strategy in Table 4 (as well as in Table 8) earns large negative net returns in 1999. This period is believed to be a time in which mispricing was prevalent (see Ritter and Welch 2002; Lamont and Thaler 2003).

  13. 13.

    The Pearson (Spearman) correlation coefficient is −0.74 (−0.85).

  14. 14.

    Note that cash flows from operations could be derived from reported funds from operations for observations on pre-SFAS No. 95 regime, but it is standard practice to adopt Sloan’s definitions of accruals (and cash flows from operations, if applicable) among studies that examine the accrual anomaly.

  15. 15.

    I want to express my gratitude to an anonymous referee for raising this question.

  16. 16.

    There are cases where the iterative process cannot solve the two equations. This occurs because of the boundary condition, which requires solutions of the equations to be positive. Accordingly, a few observations with valid data are lost; the number of lost observations is less than 0.01 % of the total number of firm-years in my sample. Thus I do not expect that this deletion will make any change in the tenor of the results.


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This study is based on a chapter of my dissertation at the University of British Columbia. I am grateful to members of my dissertation committee: Sandra Chamberlain (co-chair), Kin Lo (co-chair), and Kai Li for their guidance and direction. Also, I gratefully acknowledge Jim Ohlson (editor), an anonymous referee, Darlene Bay, Joy Begley, Larry Brown, Duane Kennedy, James Moore, Michael Shih, Dan Simunic, Samir Trabelsi, and seminar participants at Brock University, Georgia State University, McMaster University, National University of Singapore, University of Alberta, University of British Columbia, the 2010 Canadian Academic Accounting Association annual meeting, and the 2011 American Accounting Association annual meeting for helpful comments and suggestions. Any errors are my own.

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Correspondence to Sohyung Kim.


Appendix 1

Estimation of Hillegeist et al.’s (2004) BSM-Prob

Hillegeist et al. (2004) modify the original option-pricing equation as follows:

$$ V_{E} = V_{A} e^{ - \delta T} N(d_{ 1} ) - Xe^{ - rT} N(d_{ 2} ) + ( 1- e^{ - \delta T} )V_{A} , $$

where N(d 1 ) and N(d 2 ) are the standard cumulative normal distribution of d 1 and d 2, respectively, and

$$ d_{ 1} = \frac{{\ln [V_{A} /X] + (r - \delta + (\sigma_{A}^{ 2} / 2))T}}{{\sigma_{A} \sqrt T }}, $$


$$ d_{ 2} = d_{ 1} - \sigma_{A} \sqrt T , $$

V E is the current market value of equity; V A is the current market value of assets; X is the face value of debt maturing at time T; r is the continuously compounded risk-free rate; δ is the dividend rate expressed in terms of V A ; and σ A is the standard deviation of assets returns. They make two modifications to the traditional option-pricing model. The V A e δT term accounts for the reduction in the value of the assets due to the dividends that are distributed before time T. The addition of the (1 − e δT )V A term is necessary because it is the equity holders who receive the dividends.

Under the BSM model, the probability of bankruptcy is simply the probability that the market value of assets, V A , is less than the face value of the liabilities, X, at time T. The BSM model assumes that the natural log of future asset values is distributed normally as follows:

$$ \ln V_{A} (t) \sim N\left[ {\ln V_{A} + \left( {\mu - \delta - \frac{{\sigma_{A}^{2} }}{ 2}} \right)t,\sigma_{A}^{ 2} t} \right], $$

where μ is the continuously compounded expected return on assets. Then, as shown in McDonald (2002, p. 604), the probability that V A (T) < X is, as follows:

$$ N\left( { - \frac{{\ln (V_{A} /X) + (\mu - \delta - (\sigma_{A}^{ 2} / 2))T}}{{\sigma_{A} \sqrt T }}} \right) = BSM{ - }Prob. $$

Equation (5) shows that the probability of bankruptcy is a function of the distance between the current value of the firm’s assets and the face value of its liabilities ln(V A /X) adjusted for the expected growth in asset values (μ − δ) relative to asset volatility (σ A ). Note that the value of the call option in Eq. (1) is not a function of μ because Eq. (1) is derived under the assumption of risk-neutrality. However, the probability of bankruptcy depends upon the actual distribution of future asset values, which is a function of μ.

To empirically estimate BSM-Prob from Eq. (5), the market value of assets, V A , asset volatility, σ A , and the expected return on assets, μ, need to be estimated because these values are not directly observable. In the first step, Hillegeist et al. (2004) estimate the values of V A and σ A by simultaneously solving Eq. (1) and the following optimal hedge equationFootnote 16:

$$ N(d_{ 1} ) = \frac{{V_{E} \sigma_{E} }}{{V_{A} e^{ - \delta T} \sigma_{A} }}. $$

V E is set equal to the total market value of equity at the end of the firm’s fiscal year ending in calendar year t−1. σ E is computed using daily return data from the CRSP over the entire fiscal year ending in calendar year t−1. The strike price X is set equal to the book value of total liabilities, T equals one year, and r is the one-year Treasury bill rate. The dividend rate, δ, is the sum of common and preferred dividends divided by the sum of the market value of equity plus total liabilities at the end of the fiscal year. Note that all applicable accounting data are from financial statements with fiscal years ending calendar year t−1 so that BSM-Prob is estimated on an ex ante basis.

In the second step, the expected market return on assets, μ, is estimated based on the actual return on assets during the previous year as follows:

$$ \mu (t) = \frac{{V_{A} (t) + \, dividends - V_{A} (t - 1)}}{{V_{A} (t - 1)}}, $$

where dividends is the sum of the common and preferred dividends declared during the year. Note that this process is based on the estimates of V A that were computed in the first step. When expected returns are below the risk-free rate (above one), Hillegeist et al. (2004) set them equal to the risk-free rate (one).

Appendix 2

An association of a variable with a score: an econometric interpretation

Without a loss of generality, suppose that there are n firms and each firm has its score S, which is a linear combination of k components (e.g., accounting variables). Using matrices, I can express scores for n firms as:

$$ {\mathbf{S}} = {\mathbf{X\alpha }}, $$

where S = n × 1, X = n × k, and α = k × 1. Then, an association of a variable Y (e.g., returns) with the score S can be written as:

$$ \beta = ({\mathbf{S^{\prime}S}})^{{{\mathbf{ - 1}}}} {\mathbf{S^{\prime}Y}} \ne 0, $$

where Y is n × 1. Substituting (8) into (9), I obtain:

$$ \beta = ({\mathbf{\alpha^{\prime}X^{\prime}X\alpha }})^{{{\mathbf{ - 1}}}} {\mathbf{\alpha^{\prime}X^{\prime}Y}} \ne 0. $$

With respect to the numerator (αXY), since α ≠ 0, the following holds:

$$ {\mathbf{X^{\prime}Y}} \ne {\mathbf{0}}. $$

The denominator (αXXα) is nonzero as long as XX is positive definite (i.e., rank (X) = k). This condition is satisfied as long as one component of the score is not a linear combination of the other components. Thus it follows that, in a regression of variable Y on X (the component of the score), the resulting coefficients is a nonzero vector:

$$ ({\mathbf{X^{\prime}X}})^{{{\mathbf{ - 1}}}} {\mathbf{X^{\prime}Y}} = {\varvec{\gamma}} = \left[ {\begin{array}{*{20}c} {\gamma 1} \\ {\gamma 2} \\ \vdots \\ {\gamma k} \\ \end{array} } \right] \ne {\mathbf{0}} . $$

Therefore, when there is an association between a variable and a score, it must be true that one or more components of the score have nontrivial associations with the variable, holding all other components constant.

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Kim, S. What is behind the magic of O-Score? An alternative interpretation of Dichev’s (1998) bankruptcy risk anomaly. Rev Account Stud 18, 291–323 (2013). https://doi.org/10.1007/s11142-012-9206-7

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  • Market efficiency
  • Bankruptcy risk
  • Funds from operations
  • Cash flows
  • Accruals

JEL Classification

  • G14
  • G33
  • M41