Subdominant Eigenvalue Location and the Robustness of Dividend Policy Irrelevance

  • Adam J. OstaszewskiEmail author


This paper, on subdominant eigenvalue location of a bordered diagonal matrix, is the mathematical sequel to an accounting paper by Gao et al. (J Bus Financ Acc 40:673–694, 2013). We explore the following characterization of dividend-policy irrelevance (DPI) to equity valuation in a multi-dimensional linear dynamics framework L: DPI occurs under L when discounting the expected dividend stream by a constant interest rate iff that rate is equal to the dominant eigenvalue of the canonical principal submatrix A of L. This is justifiably the ‘latent’ (or gross) rate of return, since the principal submatrix relates the state variables to each other but with dividend retention. We find that DPI reduces to the placement of the maximum eigenvalue of L between the dominant and subdominant eigenvalues of A. We identify a special role, and a lower bound, for the coefficient measuring the year-on-year dividend-on-dividend sensitivity in achieving robust equity valuation (independence of small variations in the dividend policy).


Dividend irrelevance Dominant eigenvalue Bordered diagonal matrix Performance stability Dividend-on-dividend sensitivity 

Mathematics Subject Classification (2010)

Primary 91B32 91B38; Secondary 91G80 49J55 49K40 


  1. 1.
    Ashton, D.: The cost of equity capital and a generalization of the dividend growth model. Account. Bus. Res. 26, 34–18 (1995)CrossRefGoogle Scholar
  2. 2.
    Ashton, D., Cooke, T., Tippett, M., Wang, P.: Linear information dynamics, aggregation, dividends and “dirty surplus”. Account. Bus. Res. 34, 277–299 (2004)CrossRefGoogle Scholar
  3. 3.
    Bellman, R.: Introduction to matrix analysis. Reprint of the second (1970) edition. In: Classics in Applied Mathematics, vol. 19 (SIAM, Philadelphia, 1997)Google Scholar
  4. 4.
    Bhattacharya, S.: Imperfect information, dividend policy, and “the bird in the hand” fallacy. Bell J. Econ. 10, 259–270 (1979)CrossRefGoogle Scholar
  5. 5.
    Davies, R.O., Ostaszewski, A.J.: Optimal forward contract design for inventory: a value-of-waiting analysis. In: Brzdek, J., Popa, D., Rassias, T.M. (eds.) Ulam Type Stability, pp. 73–96. Springer, Cham (2019)Google Scholar
  6. 6.
    Dybvig, P.H., Zender, J.F.: Capital structure and dividend irrelevance with asymmetric information. Rev. Financ. Stud. 4, 201–219 (1991)CrossRefGoogle Scholar
  7. 7.
    Gao, Z., Ohlson, J.A., Ostaszewski, A.J.: Dividend policy irrelevancy and the construct of earnings. J. Bus. Financ. Acc. 40, 673–694 (2013)CrossRefGoogle Scholar
  8. 8.
    Gietzmann, M.B., Ostaszewski, A.J.: Predicting firm value: the superiority of q-theory over residual income. Account. Bus. Res. 34, 349–377 (2004)CrossRefGoogle Scholar
  9. 9.
    Henrici, P.: Applied and Computational Complex Analysis. Power Series-Integration-Conformal Mapping-Location of Zeros, vol. I, Reprinted 1988. (Wiley, Hoboken, 1974)Google Scholar
  10. 10.
    Hinrichsen, D., Kelb, B.: Stability radii and spectral value sets for real matrix perturbations. In: Systems and Networks: Mathematical Theory and Applications, vol. II, pp. 217–220. Invited and Contributed Papers (Akademie-Verlag, Berlin, 1994)Google Scholar
  11. 11.
    Horn, R.A., Johnson, C.R.: Matrix Analysis (Cambridge University Press, Cambridge, 1985)CrossRefGoogle Scholar
  12. 12.
    Hwang, S.-K.: Cauchy’s interlace theorem for eigenvalues of Hermitian matrices. Am. Math. Mon. 111, 157–159 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jack, A., Johnson, T., Zervos, M.: A singular control model with application to the goodwill problem. Stoch. Process. Appl. 118, 2098–2124 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Klinger, A.: The Vandermonde matrix. Am. Math. Mon. 74, 571–574 (1967)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lo, K., Lys, T.: The Ohlson model: contribution to valuation theory, limitations, and empirical applications. J. Acc. Audit. Financ. 15, 337–367 (2000)Google Scholar
  16. 16.
    Marden, M.: The Geometry of the Zeros of a Polynomial in a Complex Variable (American Mathematical Society, New York, 1949)zbMATHGoogle Scholar
  17. 17.
    Miller, M.H., Modigliani, F.: The cost of capital, corporation finance and the theory of investment. Am. Econ. Rev. 48, 261–297 (1958)zbMATHGoogle Scholar
  18. 18.
    Miller, M.H., Modigliani, F.: Dividend policy, growth, and the valuation of shares. J. Bus. 34, 411–433 (1961)CrossRefGoogle Scholar
  19. 19.
    Miller, M.H., Modigliani, F.: Corporate income taxes and the cost of capital: a correction. Am. Econ. Rev. 53, 433–443 (1963)Google Scholar
  20. 20.
    Noble, B.: Applied Linear Algebra (Prentice-Hall, Upper Saddle River, 1969)zbMATHGoogle Scholar
  21. 21.
    Ohlson, J.: Earnings, book values, and dividends in equity valuation. Contemp. Account. Res. 11, 661–687 (1995)CrossRefGoogle Scholar
  22. 22.
    Ohlson, J.: Accounting earnings, book value, and dividends: the theory of the clean surplus equation (part I). In: Brief, R.P., Peasnell, K.V. (eds.) Clean Surplus: A Link Between Accounting and Finance, pp. 165–230 (Garland Publishing, Princeton, 1996)Google Scholar
  23. 23.
    Ohlson, J.A., Gao, Z.: Earnings growth and equity value. Found. Trends Acc. 1, 1–70 (1981)zbMATHGoogle Scholar
  24. 24.
    Preinreich, G.: The fair value and yield of common stock. Acc. Rev. 11, 130–140 (1936)Google Scholar
  25. 25.
    Rugh, W.J.: Linear System Theory (Prentice-Hall, Upper Saddle River, 1996)zbMATHGoogle Scholar
  26. 26.
    Seneta, E.: Non-negative Matrices and Markov Chains, 2nd edn. Revised reprint (1st ed. 1973) (Springer, New York, 1981)Google Scholar
  27. 27.
    Tippett, M., Warnock, T.: The Garman-Ohlson structural system. J. Bus. Financ. Acc. 24, 1075–1099 (1997)CrossRefGoogle Scholar
  28. 28.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem (Oxford University Press, Oxford, 1965)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsLondon School of EconomicsLondonUK

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