Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net A Theory of the Firm’s Cost of Capital How Debt Affects the Firm’s Risk, Value, Tax Rate and the Government’s Tax Claim Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net This page intentionally left blank Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net A Theory of the Firm’s Cost of Capital How Debt Affects the Firm’s Risk, Value, Tax Rate and the Government’s Tax Claim Ramesh K S Rao University of Texas at Austin, USA Eric C Stevens USA World Scientific Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TA I P E I CHENNAI • • • • • • • Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Rao, Ramesh K S A theory of the firm’s cost of capital : how debt affects the firm’s risk, value, tax rate, and the government’s tax claim / by Ramesh K.S Rao & Eric C Stevens p cm Includes bibliographical references ISBN-13 978-981-256-949-3 ISBN-10 981-256-949-9 Corporations Finance Capital costs Corporate debt Capital assets pricing model Financial leverage I Stevens, Eric C., 1962– II Title HG4026.R366 2007 338.6'04101 dc22 2006052555 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Copyright © 2007 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net CheanChian - A theory of the filrm;s.pmd 4/17/2007, 5:13 PM December 12, 2006 15:11 spi-b456 A Theory of the Firm’s Cost of Capital 9in x 6in fm 3rd Reading Preface Modigliani and Miller’s (MM) seminal analyses spawned two broad research strands in corporate finance, the first relating to the effects of leverage on the firm risk and cost of capital, and the second to the firm’s optimal capital structure (mix of debt and equity) This book is concerned with the first, and it is a slightly expanded version of our paper that was published by the Berkeley Electronic Journals in Economic Analysis and Policy.* Our original motivation for this research was the “pie-slicing” analogy that is the core intuition of modern corporate finance theory In essence, the firm’s investment decision determines the size of the economic pie that the firm creates, and debt and equity are simply two different claims on this pie Thus, as MM argued, it does not matter, in frictionless capital markets, how this pie is sliced; the firm’s capital structure is unimportant When this intuition is extended to include corporate taxes, the size of the pie is determined by the firm’s after-tax cash flows and, in this case, thanks to the government’s ∗ Rao, RKS and EC Stevens (2006) The firm’s cost of capital, its effective marginal tax rate, and the value of the government’s tax claim Topics in Economic Analysis & Policy, 6(1), Article 3, published by Berkeley Electronic Press, available at http://www.bepress.com/bejeap/topics/vol6/iss1/art3 This article is adapted here with permission v Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net December 12, 2006 vi 15:11 spi-b456 A Theory of the Firm’s Cost of Capital 9in x 6in fm 3rd Reading A Theory of the Firm’s Cost of Capital subsidy of the firm’s interest payments, maximizing debt becomes optimal With taxes, there are now three claimants to the economic pie— stockholders, bondholders and the tax authority Thus, one should, in principle, be able to value the firm as the sum of the values of three claims Although this intuition was well known, we did not see a satisfactory formal analysis of the “three claims view of the firm” with risky debt and corporate tax effects The literature’s focus was on “two claims models” of the firm Our primary goal, thus, was to develop a theoretical framework that can identify how the value of the government’s tax claim varies with corporate borrowing In the analysis that is presented here, the value of the firm is consistent with the standard perspective that the firm’s after-tax output is distributed between the debt and the equity, and also with the view that the pre-tax output is apportioned among three risky claims, with the tax authority being the third claimant As we worked on this research, it became clear that with risky debt and corporate taxes it is critical to understand how the risks of the firm’s depreciation and the debt tax shields change with leverage To our knowledge, the risk of the tax shields had not been adequately formalized in the research, and authors have relied on various ad hoc assumptions about the tax shields’ risks A second research goal, therefore, was to model how the tax shields’ risks are affected by leverage The outcome of this effort, which is presented here, is a framework for better measuring the firm’s cost of capital while, at the same time, identifying the marginal effects of debt policy on market values, risks, and expected rates of return The ability of our model to capture several important economic interdependencies (e.g., between the borrowing rate and the tax shields) within a simple analytical framework allows us to illustrate the model with numerical examples and graphical illustration As we discuss, the model can be used to generate better estimates of the firm’s cost of capital and marginal Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net December 12, 2006 15:11 Preface spi-b456 A Theory of the Firm’s Cost of Capital 9in x 6in fm 3rd Reading vii corporate tax rates In addition, it provides a conceptual framework for evaluating the implications of exogenous market forces (e.g., interest rates, tax laws, the market price of risk) on the firm’s economic balance sheet and on the value of the government’s claim on output, and thus may be useful for studies of tax and public policies We are grateful to the Berkeley Electronic Press for permission to reproduce our earlier paper in modified form We also thank our spouses for their support, and the colleagues that have provided feedback on various drafts of the manuscript Finally, we thank the staff of World Scientific, namely Juliet Lee, Venkatesh Sandhya, Chean Chian Cheong and Hooi-Yean Lee for their efforts at bringing this book into the present form Ramesh K S Rao Austin, Texas, USA Eric C Stevens Salt Lake City, Utah, USA September 2006 Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net December 12, 2006 15:11 spi-b456 A Theory of the Firm’s Cost of Capital 9in x 6in fm 3rd Reading This page intentionally left blank Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net December 12, 2006 15:11 spi-b456 A Theory of the Firm’s Cost of Capital 9in x 6in fm 3rd Reading Contents Preface v List of Figures xi List of Tables xiii I Introduction II Model Setting III Distributional Assumptions 19 IV Model Solution Procedure 23 V Discussion of Results 33 VI Extension to s × s states 45 VII Numerical Illustration 47 VIII Conclusion 57 Appendix A 63 Appendix B 77 References 85 Index 89 ix Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net December 12, 2006 11:15 spi-b456 A Theory of the Firm’s Cost of Capital 9in x 6in App-B 3rd Reading Appendix B Examples Illustrating the Firm’s Balance Sheet Using the Data in Table 77 Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net December 12, 2006 78 11:15 Table B.1 Results for the × example, for seven debt levels The debt is risky when D > $46,296.30 Debt amount (D) 80,000 100,000 105,000 DMAX = 110,661.75 2 1 9 Debt ΦD,o ΦD,p E(Φ˜D ) βD kD VD Par yield, r Promised pmt, D(1 + r) 0 Null 0.08 0.08 43,200.00 43,200.00 43,200.00 0.08 40,000 0.08 43,200.00 50,000.00 50,000.00 50,000.00 0.08 46,296.30 0.08 50,000.01 99,970.39 50,000.00 92,474.83 1.898 0.1559 80,000 0.2496 99,970.39 129,623.15 50,000.00 117,679.68 2.420 0.1768 100,000 0.2962 129,623.15 137,036.34 50,000.00 123,980.89 2.519 0.1808 105,000 0.3051 137,036.34 145,430.66 50,000.00 131,116.06 2.621 0.1848 110,661.75 0.3142 145,430.66 Debt tax shield ΦDTS ,o ΦDT S,p ˜ ) E(ΦDTS βDT S kDT S VDT S 0 Null 0.08 960.00 816.00 4.506 0.2602 647.49 1,111.11 944.44 4.506 0.2602 749.42 5,991.12 5,092.45 4.506 0.2602 4,040.85 8,886.94 7,553.90 4.506 0.2602 5,994.01 9,610.90 8,169.27 4.506 0.2602 6,482.30 10,430.67 8,866.07 4.506 0.2602 7,035.21 (Continued) Pricing case (Table 3) App-B 3rd Reading Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net 9in x 6in 46,296.30 spi-b456 A Theory of the Firm’s Cost of Capital 40,000 A Theory of the Firm’s Cost of Capital 0 40,000 46,296.30 100,000 105,000 DMAX = 110,661.75 spi-b456 A Theory of the Firm’s Cost of Capital 80,000 Depreciation tax shield (for any feasible D) ΦNT S,o 30,000.00 ΦNT S,p 15,000.00 E(ΦNT 27,750.00 ˜ S) βNTS 1.899 kNTS 0.1560 VNTS 24,005.99 Total tax shield ΦT T S,o 30,000.00 ΦT T S,p 15,000.00 ˜ E(Φ) 27,750.00 βT T S 1.899 kT T S 0.1560 VT T S 24,005.99 35,991.12 15,000.00 32,842.45 2.275 0.1710 28,046.84 38,886.94 15,000.00 35,303.90 2.420 0.1768 30,000.00 39,610.90 15,000.00 35,919.27 2.453 0.1781 30,488.29 40,430.67 15,000.00 36,616.07 2.490 0.1796 31,041.20 14,040.00 11,934.00 4.506 0.2602 9,469.61 13,888.89 11,805.56 4.506 0.2602 9,367.69 9,008.88 7,657.55 4.506 0.2602 6,076.25 6,113.06 5,196.10 4.506 0.2602 4,123.09 5,389.10 4,580.73 4.506 0.2602 3,634.80 4,569.33 3,883.93 4.506 0.2602 3,081.89 (Continued) App-B βT kT VT 15,000.00 12,750.00 4.506 0.2602 10,117.10 31,111.11 15,000.00 28,694.44 1.978 0.1591 24,755.41 9in x 6in Taxes ΦT,o ΦT,p ˜ E(Φ) 30,960.00 15,000.00 28,566.00 1.968 0.1587 24,653.49 11:15 Debt amount (D) December 12, 2006 (Continued) Appendix B Table B.1 79 3rd Reading Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net (Continued) December 12, 2006 80 Table B.1 Debt amount (D) 80,000 100,000 105,000 DMAX = 110,661.75 135,000.00 50,0000.00 ˜U) 122, 250.00 = E(Φ 2.493 = β U 0.1797 = k U 103, 626.54 = V U 100,000.00 3,626.54 92,760.00 6,800.00 79,866.00 4.065 0.2426 64,274.04 60,000.00 4,274.04 86,111.11 73,194.44 4.506 0.2602 58,079.66 53,703.70 4,375.96 41,020.73 34,867.62 4.506 0.2602 27,667.39 20,000.00 7,667.39 14,263.80 12,124.23 4.506 0.2602 9,620.55 9,620.55 7,574.56 6,438.38 4.506 0.2602 5,108.84 10,108.84 5,000.00 0.01 0.01 4.506 0.2602 0.01 10,661.76 10,661.75 135,000.00 50,0000.00 122,250.00 2.493 0.1797 103,626.54 135,960.00 50,000.00 123,066.00 2.505 0.1802 104,274.04 136,111.11 50,000.00 123,194.44 2.507 0.1803 104,375.96 140,991.12 50,000.00 127,342.45 2.568 0.1827 107,667.39 143,886.94 50,000.00 129,803.90 2.603 0.1841 109,620.55 144,610.90 50,000.00 130,419.27 2.611 0.1845 110,108.84 145,430.67 50,000.00 131,116.07 2.621 0.1848 110,661.76 App-B 3rd Reading Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net 9in x 6in Debt + equity + tax (for any feasible D) ΦD+E+T,o 150,000.00 ΦD+E+T,p 50,0000.00 ˜ D+E+T ) E(Φ 135,000.00 βD+E+T 2.672 kD+E+T 0.1869 VD+E+T 113,743.65 spi-b456 A Theory of the Firm’s Cost of Capital 46,296.30 A Theory of the Firm’s Cost of Capital Debt + equity ΦD+E,o ΦD+E,p ˜ D+E ) E(Φ βD+E kD+E = WACC VD+E = V L 40,000 11:15 Equity ΦE,o ΦE,p ˜E) E(Φ βE kE VE E0 Wealth gain, ∆W DIV0 105,000 DMAX = 113,108.40 0 0 0 Null 0.08 0.08 43,200.00 43,200.00 43,200.00 43,200.00 43,200.00 43,200.00 0.08 40,000 0.08 43,200.00 82,853.60 82,853.60 82,853.60 82,853.60 82,853.60 82,853.60 0.08 76,716.29 0.08 82,853.60 88,173.63 88,173.63 88,173.63 88,173.63 82,853.60 86,961.95 0.176 0.0870 80,000 0.1022 88,173.63 120,576.27 120,576.27 120,576.27 120,576.27 82,853.60 111,984.59 0.996 0.1198 100,000 0.2058 120,576.27 130,728.00 130,728.00 130,728.00 124,994.98 82,853.60 118,783.66 1.282 0.1313 105,000 0.2450 130,728.00 181,528.35 145,005.02 135,000.00 124,994.98 82,853.60 133,720.44 2.556 0.1822 113,108.40 0.6049 181,528.35 0 0 0 Null Null 960.00 960.00 960.00 960.00 71.35 4.278 0.2511 592.54 1,841.19 1,841.19 1,841.19 1,841.19 1,421.84 4.278 0.2511 1,136.45 2,452.09 2,452.09 2,452.09 2,452.09 1,893.60 4.278 0.2511 1,513.51 6,172.88 6,172.88 6,172.88 6,172.88 4,766.95 4.278 0.2511 3,810.11 7,718.40 7,718.40 7,718.40 7,498.49 8,169.27 4.389 0.2556 4,715.50 20,525.99 13,501.51 10,500.00 7,498.49 10,392.04 8.599 0.4240 7,297.92 (Continued) 3rd Reading Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net App-B 100,000 81 80,000 9in x 6in βDTS kDTS VDTS 76,716.29 spi-b456 A Theory of the Firm’s Cost of Capital Debt tax shield φDT S,1 ΦDTS ,2 ΦDTS ,3 ΦDTS ,4 ΦDTS ` ,5 ´ ˜ DTS E Φ 40,000 11:15 Debt ΦD,1 ΦD,2 ΦD,3 ΦD,4 ΦD,5 ` ´ ˜D E Φ βD kD VD Par yield, r Promised pmt, D(1 + r) December 12, 2006 Debt amount (D) Appendix B Table B.2 Results for the × example, for seven debt levels The debt is risky when D >$76,716.29 (Continued) 11:15 Debt amount (D) 40,000 76,716.29 DMAX = 113,108.40 32,452.09 32,452.09 32,452.09 32,452.09 24,856.08 30,722.03 0.724 0.1090 27,703.40 36,172.88 36,172.88 36,172.88 36,172.88 24,856.08 33,595.38 0.996 0.1198 30,000.00 37,718.40 37,718.40 37,718.40 37,498.49 24,856.08 34,748.98 1.109 0.1244 30,905.39 50,525.99 43,501.51 40,500.00 37,498.49 24,856.08 39,220.46 2.280 0.1712 33,487.81 (Continued) Depreciation tax shield (for any feasible D) ΦNT S,1 30,000.00 ΦNT S,2 30,000.00 ΦNT S,3 30,000.00 ΦNT S,4 30,000.00 ΦNT 24,856.08 S,5 ` ´ ˜ NT S E Φ 28,828.43 βNT S 0.519 kNT S 0.1007 VNT S 26,189.89 Total tax shield ΦT T S,1 30,000.00 ΦT T S,2 30,000.00 ΦT T S,3 30,000.00 ΦT T S,4 30,000.00 ΦT` T S,5 ´ 24,856.08 ˜TTS E Φ 28,828.43 βT T S 0.519 kT T S 0.1007 VT T S 26,189.89 30,960.00 30,960.00 30,960.00 30,960.00 24,856.08 29,569.78 0.602 0.1041 26,782.43 31,841.19 31,841.19 31,841.19 31,841.19 24,856.08 30,250.27 0.675 0.1070 27,326.34 3rd Reading Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net App-B 105,000 9in x 6in 100,000 A Theory of the Firm’s Cost of Capital 80,000 spi-b456 A Theory of the Firm’s Cost of Capital December 12, 2006 82 Table B.2 βT kT VT 80,000 100,000 105,000 DMAX = 113,108.40 26,143.92 13,501.51 10,500.00 7,498.49 11,671.57 9.781 0.4712 7,933.22 25,183.92 12,541.51 9,540.00 6,538.49 10,930.22 10.225 0.4890 7,340.68 24,302.73 11,660.32 8,658.81 5,657.30 10,249.73 10.701 0.5080 6,796.78 23,691.83 11,049.42 8,047.91 5,046.40 9,777.97 11.078 0.5231 6,419.71 19,971.04 7,328.63 4,327.12 1,325.61 6,904.62 14.865 0.6746 4,123.11 18,425.52 5,783.11 2,781.60 0 5,751.02 17.682 0.7873 3,217.73 5,617.93 0 0 1,279.54 23.352 1.0141 635.30 161,002.48 131,503.51 124,500.00 117,496.49 82,853.60 ` U´ ˜ 123,328.43= E Φ 2.139 = β U 0.1656 = k U 105, 810.48 = V U 100,000.00 5,810.48 118,762.48 89,263.51 82,260.00 75,256.49 39,653.60 80,869.78 3.447 0.2179 66,403.03 60,000.00 6,403.03 79,990.08 50,491.11 43,487.59 36,484.08 41,896.67 7.648 0.3859 30,230.64 23,283.71 6,946.93 75,280.94 45,781.97 38,778.46 31,774.94 34,867.62 8.006 0.4002 27,324.00 20,000.00 7,324.00 46,599.10 17,100.13 10,096.61 3,093.10 16,110.79 14.865 0.6746 9,620.59 9,620.59 37,992.88 8,493.91 1,490.40 0 10,465.32 20.346 0.8938 5,525.98 10,525.98 5,000.00 0.12 0 0 0.03 23.352 1.0141 0.01 13,108.41 13,108.40 (Continued) App-B 76,716.29 9in x 6in Equity ΦE,1 ΦE,2 ΦE,3 ΦE,4 ΦE,5 ` ´ ˜E E Φ βE kE VE E0 Wealth gain, ∆W DIV0 40,000 spi-b456 A Theory of the Firm’s Cost of Capital Taxes ΦT,1 ΦT,2 ΦT,3 ΦT,4 ΦT,5 ` ´ ˜T E Φ 11:15 Debt amount (D) December 12, 2006 (Continued) Appendix B Table B.2 83 3rd Reading Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net 11:15 (Continued) Debt amount (D) 40,000 76,716.29 80,000 100,000 105,000 DMAX = 113,108.40 161,002.48 131,503.51 124,500.00 117,496.49 82,853.60 123,328.43 2.139 0.1656 105,810.48 161,962.48 132,463.51 125,460.00 118,456.49 82,853.60 124,069.78 2.151 0.1660 106,403.03 162,843.67 133,344.71 126,341.19 119,337.68 82,853.60 124,750.27 2.162 0.1665 106,946.93 163,454.57 133,955.60 126,952.09 119,948.58 82,853.60 125,222.03 2.169 0.1668 107,324.00 167,175.36 137,676.39 130,672.88 123,669.37 82,853.60 128,095.38 2.213 0.1685 109,620.59 168,720.88 139,221.91 132,218.40 124,994.98 82,853.60 130,419.27 2.235 0.1694 110,525.98 181,528.47 145,005.02 135,000.00 124,994.98 82,853.60 133,720.46 2.556 0.1822 113,108.41 App-B 3rd Reading Trắc nghiệm kiến thức chứng khoán Mỹ : www.sachchungkhoan.net 9in x 6in Debt + equity + tax (for any feasible D) ΦD+E+T,1 187,146.40 ΦD+E+T,2 145,005.20 ΦD+E+T,3 135,000.00 ΦD+E+T,4 124,994.98 ΦD+E+T,5 82,853.60 ` ´ ˜ D+E+T E Φ 135,000.00 βD+E+T 2.672 kD+E+T 0.1869 VD+E+T 113,743.71 A Theory of the Firm’s Cost of Capital βD+E kD+E = WACC VD+E = V L spi-b456 A Theory of the Firm’s Cost of Capital Debt + equity ˜ D+E,1 Φ ΦD+E,2 ΦD+E,3 ΦD+E,4 ΦD+E,5 ` ´ ˜ D+E E Φ December 12, 2006 84 Table B.2 December 12, 2006 11:15 spi-b456 A Theory of the Firm’s Cost of Capital 9in x 6in Refs 2nd Reading References Auerbach, AJ (1983) Taxation, corporate financial policy and the cost of capital Journal of Economic Literature, 21(3), 905–940 Auerbach, AJ and K Hassett (1992) Tax policy and business fixed investment in the United States Journal of Public Economics, 47(2), 141–170 Baron, DP (1975) Firm valuation, corporate taxes and default risk Journal of Finance, 30(5), 1251–1264 Bierman, HJ (1993) Capital budgeting in 1992: A survey Financial Management, 22, 22–24 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