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Which is the Correct Discount Rate? Arithmetic Versus Geometric Mean

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  2. Economics, Finance, Business & Management
  3. Credit and Capital Markets – Kredit und Kapital
  4. Vol. 53, Issue 3
  5. pp. 355–381

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Kipp, M., Koziol, C. Which is the Correct Discount Rate? Arithmetic Versus Geometric Mean. Credit and Capital Markets – Kredit und Kapital, 53(3), 355-381. https://doi.org/10.3790/ccm.53.3.355
Kipp, Martin and Koziol, Christian "Which is the Correct Discount Rate? Arithmetic Versus Geometric Mean" Credit and Capital Markets – Kredit und Kapital 53.3, 2020, 355-381. https://doi.org/10.3790/ccm.53.3.355
Kipp, Martin/Koziol, Christian (2020): Which is the Correct Discount Rate? Arithmetic Versus Geometric Mean, in: Credit and Capital Markets – Kredit und Kapital, vol. 53, iss. 3, 355-381, [online] https://doi.org/10.3790/ccm.53.3.355
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Which is the Correct Discount Rate? Arithmetic Versus Geometric Mean

Kipp, Martin | Koziol, Christian

Credit and Capital Markets – Kredit und Kapital, Vol. 53 (2020), Iss. 3 : pp. 355–381

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Article Details

Publisher Name
Duncker & Humblot
DOI
https://doi.org/10.3790/ccm.53.3.355
Language
English
Pages
27
Keywords

Author Details

Martin Kipp, M.Sc., University of Tuebingen, Department of Finance, Nauklerstr. 47, 72074 Tuebingen, Phone: +49 7071 29 78176

Prof. Dr. Christian Koziol, University of Tuebingen, Department of Finance, Nauklerstr. 47, 72074 Tuebingen

References

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  2. Brealey, R./Myers, S. C./Allen, F. (2011): Principles of Corporate Finance, 10th ed., McGraw-Hill.  Google Scholar
  3. Cochrane, J. H. (2009): Asset pricing: Revised Edition, Princeton University Press.  Google Scholar
  4. Cornish, E. A./Fisher, R. A. (1938): Moments and Cumulants in the Specification of Distributions, Revue de l’Institut international de Statistique, 307–320.  Google Scholar
  5. Damodaran, A. (2013): Equity Risk Premiums (ERP): Determinants, Estimation and Implications – the 2012 edition’, Managing and Measuring Risk: Emerging Global Standards and Regulations after the Financial Crisis, 343–455.  Google Scholar
  6. Dorfleitner, G. (2002): Stetige versus diskrete Renditen: Überlegungen zur richtigen Verwendung beider Begriffe in Theorie und Praxis, Kredit und Kapital, Vol. 35, 216–241.  Google Scholar
  7. Dorfleitner, G. (2003): Why the return notion matters, International Journal of Theoretical and Applied Finance, Vol. 6(1), 73–86.  Google Scholar
  8. European Commission (2014): Regulation (EU) no 1286/2014 of the European Parliament and of the Council of 26 November 2014 on Key Information Documents for Packaged Retail and Insurance-based Investment Products (PRIIPs). URL: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:32014R1286  Google Scholar
  9. European Commission (2017): Commission delegated Regulation (EU) 2017/653 of 8 March 2017 Supplementing Regulation (EU) no 1286/2014 of the European Parliament and of the Council on Key Information Documents for Packaged Retail and Insurance-based Investment Products (PRIIPs) by laying down Regulatory Technical Standards with regard to the Presentation, Content, Review and Revision of Key Information Documents and the Conditions for fulfilling the Requirement to provide such Documents. URL: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:32017R0653  Google Scholar
  10. Graf, S. (2019): PRIIP-KID: Providing Retail Investors with Inappropriate Product Information?, European Actuarial Journal, Vol. 9(2), 361–385.  Google Scholar
  11. Huang, C.-f./Litzenberger, R. H. (1988): Foundations for Financial Economics, Prentice Hall.  Google Scholar
  12. Hull, J. C. (2017): Options, Futures, and Other Derivatives, 9th ed., Pearson.  Google Scholar
  13. Koller, T./Goedhart, M./Wessels, D. (2015): Valuation: Measuring and Managing the Value of Companies, 6th edition, John Wiley & Sons.  Google Scholar
  14. KPMG (2019): Cost of Capital Study 2019, 14th ed.  Google Scholar
  15. May, S. (2019): Arithmetische und geometrische versus diskrete und stetige Renditen, Working Papers, Technische Hochschule Ingolstadt.  Google Scholar
  16. Stehle, R. (2004): Die Festlegung der Risikoprämie von Aktien im Rahmen der Schätzung des Wertes von börsennotierten Kapitalgesellschaften, Die Wirtschaftsprüfung, Vol. 57(17), 906–927.  Google Scholar
  17. Tsay, R. S. (2005): Analysis of Financial Time Series, Vol. 543, John Wiley & Sons.  Google Scholar
  18. Black, F./Scholes, M. (1973): The Pricing of Options and Corporate Liabilities, Journal of Political Economy, Vol. 81(3), 637–654.  Google Scholar
  19. Brealey, R./Myers, S. C./Allen, F. (2011): Principles of Corporate Finance, 10th ed., McGraw-Hill.  Google Scholar
  20. Cochrane, J. H. (2009): Asset pricing: Revised Edition, Princeton University Press.  Google Scholar
  21. Cornish, E. A./Fisher, R. A. (1938): Moments and Cumulants in the Speci?cation of Distributions, Revue de l’Institut international de Statistique, 307–320.  Google Scholar
  22. Damodaran, A. (2013): Equity Risk Premiums (ERP): Determinants, Estimation and Implications – the 2012 edition’, Managing and Measuring Risk: Emerging Global Standards and Regulations after the Financial Crisis, 343–455.  Google Scholar
  23. Dor?eitner, G. (2002): Stetige versus diskrete Renditen: Überlegungen zur richtigen Verwendung beider Begri?e in Theorie und Praxis, Kredit und Kapital, Vol. 35, 216–241.  Google Scholar
  24. Dor?eitner, G. (2003): Why the return notion matters, International Journal of Theoretical and Applied Finance, Vol. 6(1), 73–86.  Google Scholar
  25. European Commission (2014): Regulation (EU) no 1286/2014 of the European Parliament and of the Council of 26 November 2014 on Key Information Documents for Packaged Retail and Insurance-based Investment Products (PRIIPs). URL: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:32014R1286  Google Scholar
  26. European Commission (2017): Commission delegated Regulation (EU) 2017/653 of 8 March 2017 Supplementing Regulation (EU) no 1286/2014 of the European Parliament and of the Council on Key Information Documents for Packaged Retail and Insurance-based Investment Products (PRIIPs) by laying down Regulatory Technical Standards with regard to the Presentation, Content, Review and Revision of Key Information Documents and the Conditions for ful?lling the Requirement to provide such Documents. URL: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:32017R0653  Google Scholar
  27. Graf, S. (2019): PRIIP-KID: Providing Retail Investors with Inappropriate Product Information?, European Actuarial Journal, Vol. 9(2), 361–385.  Google Scholar
  28. Huang, C.-f./Litzenberger, R. H. (1988): Foundations for Financial Economics, Prentice Hall.  Google Scholar
  29. Hull, J. C. (2017): Options, Futures, and Other Derivatives, 9th ed., Pearson.  Google Scholar
  30. Koller, T./Goedhart, M./Wessels, D. (2015): Valuation: Measuring and Managing the Value of Companies, 6th edition, John Wiley & Sons.  Google Scholar
  31. KPMG (2019): Cost of Capital Study 2019, 14th ed.  Google Scholar
  32. May, S. (2019): Arithmetische und geometrische versus diskrete und stetige Renditen, Working Papers, Technische Hochschule Ingolstadt.  Google Scholar
  33. Stehle, R. (2004): Die Festlegung der Risikoprämie von Aktien im Rahmen der Schätzung des Wertes von börsennotierten Kapitalgesellschaften, Die Wirtschaftsprüfung, Vol. 57(17), 906–927.  Google Scholar
  34. Tsay, R. S. (2005): Analysis of Financial Time Series, Vol. 543, John Wiley & Sons.  Google Scholar

Abstract

The paper revisits the two major concepts for average historical returns, i.?e., the arithmetic mean and the geometric mean, in order to clarify which approach must be used for which application. Conducting a rigorous derivation with a geometric Brownian motion, we can explain that the appropriate discount rate refers to the mean discrete return and, therefore, to the arithmetic mean rather than the often wrongly applied geometric mean. Likewise, the prominent CAPM relationship between the expected asset return and the expected market return is only valid for the arithmetic mean rather than the geometric mean. Using historical data for the German stock index, we illustrate that an inconsistent application can cause severe deviations from the meaningful ex-ante expected performance of an asset, the true discount rate, the true CAPM risk-adjusted return, and the intended performance scenarios of packaged retail and insurance-based investment products (PRIIPs) within the key information documents (KIDs).

Table of Contents

Section Title Page Action Price
Martin Kipp / Christian Koziol: Which is the Correct Discount Rate? Arithmetic Versus Geometric Mean 1
Abstract 1
I. Introduction 1
II. Theoretical Background 3
1. Basic Principles of Asset Returns 3
2. Geometric Brownian Motion as Stock Price Process 6
III. Arithmetic and Geometric Mean of Discrete Returns 9
1. Geometric Mean of Discrete Returns 9
2. Arithmetic Mean of Discrete Returns 1
IV. The Proper Discount Rate 1
V. Capital Asset Pricing Model 1
1. Proof of the CAPM 1
2. Analysis of a Potential Inaccuracy 2
VI. PRIIPs 2
VII. Conclusion 2
References 2

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