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A B C D E F G H I J K L M 1 13model 7/7/2010 18:02 12/2/2002 2 3 Chapter 13. Model for Capital Structure and Leverage 4 5 In Chapter 5, we introduced the idea that risk has two principal components, market risk and stand-alone risk. Market 6 risk is measured by beta, while stand-alone risk consists of both market risk plus an element of risk that can be eliminated 7 through diversification. In this chapter, we introduce two new dimensions of risk, business risk and financial risk. 8 Business risk is the risk inherent in the firm's operations, and it would be there even if the firm used no debt. Financial risk 9 is the additional risk borne by the stockholders as a result of the use of debt. 10 11 12 BUSINESS RISK AND OPERATING LEVERAGE 13 14 Operating Leverage reflects the amount of fixed costs embedded in a firm's operations. Thus, if a high percentage of a firm's 15 costs are fixed, hence continue even if sales decline, then the firm is said to have high operating leverage. High operating 16 leverage produces a situation where a small change in sales can result in a large change in operating income. The following 17 example compares two operational plans with different degrees of operating leverage. Using the input data given below, we 18 examine the firm's profitability under two operating plans, in different states of the economy. The probabilities of the 19 economic states are also given in the example. 20 21 Input Data Plan A Plan B 22 Low FC High FC 23 Price $2.00 $2.00 Product sells at same price regardless of how it is produced. 24 Variable costs $1.50 $1.00 Plan A has high variable costs; it doesn't use labor-saving equipment. 25 Fixed costs $20,000 $60,000 Plan B has high fixed costs (depreciation) due to the use of labor-saving equipment. 26 Total assets $200,000 $200,000 27 Tax Rate 40% 40% 28 A's break-even units = 40,000. See table. B's breakeven units = 60,000. See calculation below. 29 Operating Performance 30 Plan A: Low Fixed, High Variable Costs Plan B: High Fixed, Low Variable Costs 31 Data Applicable to Both Plans Net Op Profit Return on Net Op Profit Return on 32 Units Dollar Operating Operating After Taxes Equity Operating Operating After Taxes Equity 33 Demand Probability Sold Sales Costs Profit (EBIT) (NOPAT) (ROE) Costs Profit (EBIT) (NOPAT) (ROE) 34 Terrible 0.05 0 $0 $20,000 ($20,000) ($12,000) -6.0% $60,000 ($60,000) ($36,000) -18.0% 35 Poor 0.2 40,000 $80,000 $80,000 $0 $0 0.0% $100,000 ($20,000) ($12,000) -6.0% 36 Average 0.5 100,000 $200,000 $170,000 $30,000 $18,000 9.0% $160,000 $40,000 $24,000 12.0% 37 Good 0.2 160,000 $320,000 $260,000 $60,000 $36,000 18.0% $220,000 $100,000 $60,000 30.0% 38 Wonderful 0.05 200,000 $400,000 $320,000 $80,000 $48,000 24.0% $260,000 $140,000 $84,000 42.0% 39 Expected Values: 100,000 $200,000 $170,000 $30,000 $18,000 9.0% $160,000 $40,000 $24,000 12.0% 40 Standard Deviation (SD):* 49,396 $98,793 $24,698 7.41% $49,396 14.82% 41 Coefficient of Variation (CV): 0.49 0.49 0.82 0.82 1.23 1.23 42 A B C D E F G H I J K L M 43 *We used the function wizard rather than the formula to calculate the standard deviation. Click fx>Statistical>STDEVP. You must scroll down the dialog box to make all the entries. 44 Then click on each entry of the item whose SD we seek some number of times that is consistent with the item's probability 45 weight. Thus, for units sold, 0 has a probability of 0.05, so we click on it once. 40,000 units has a probability of 0.2, which 46 is 4 times as large as 0.05, so we click on it 4 times. Continuing, we click on 100,000 10 times, 160,000 4 times, and 47 200,000 one time. We have a total of 20 entries. We show a picture of the dialog box on the screen to the right. 48 49 Which plan is better? Based on expected profits and the return on equity (ROE), Plan B looks better. However, Plan B 50 is also riskier as measured by the standard deviation (SD) and the coefficient of variation (CV). So, we face a tradeoff 51 between risk and return--B is more profitable, but A is less risky. Someone will have to choose between the two plans, but 52 at this point we have no basis for making the choice. 53 54 Note also that Plan A will break even at sales as low as 40,000 units, while Plan B would have a $20,000 loss at that level. 55 B's breakeven point is 50% higher, at 60,000 units. 56 57 The results generated above are graphed here: 58 59 60 61 Plan A: Low Fixed Costs, Plan B: High Fixed Costs, 62 Low Operating Leverage High Operating Leverage 63 64 $200,000 Revenues and costs Revenues and costs 65 $200,000 Revenues 66 $150,000 $150,000 Revenues 67 VC $100,000 $100,000 VC 68 FC 69 $50,000 FC $50,000 70 $0 71 $0 0 50000 100000 72 0 50000 100000 Sales (units) 73 Sales (units) 74 75 76 (1) B has a much higher breakeven point, and (2) B has more operating leverage in the sense that a given change in sales 77 leads to a larger change in profits than for A. 78 79 We can see from the table that A's breakeven point is at 40,000 units. We can see from the table and also from the graph 80 that B's breakeven point is between 40,000 and 100,000 units, but we cannot tell the exact point. However, we can use the 81 following formula to find the exact breakeven point: 82 FC 83 Q BE = / (P - VC) In words, the quantity at which a firm breaks even is found as Fixed costs divided by 84 the difference between Price and Variable costs. 85 Plan A 86 Q BE = FC / (P - VC) 87 Q BE = $20,000 / $2.00 - $1.50 88 Q BE = 40,000 Units. 89 90 Plan B 91 BE B = FC / (P - VC) 92 BE B = $60,000 / $2.00 - $1.00 A B C D E F G H I J K L M 93 BE B = 60,000 Units. A B C D E F G H I J K L M 94 95 At this point, we know that Plan B has a higher expected rate of return, but it is also more risky. Our analysis is based on 96 stand-alone risk. However, given a positive correlation between the firm's returns and that on the market, Plan B's higher 97 risk will result in a higher "unlevered beta." We do not calculate unlevered betas here, but in the next section we assume 98 that B's beta is 1.5, and we also assume that management believes that choosing Plan B would lead to a higher stock 99 price than Plan A. 100 101 102 FINANCIAL RISK AND LEVERAGE 103 104 Financial Leverage refers to the use of fixed-income securities (preferred stock and debt) in the capital structure. The 105 firm has a certain amount of business risk as discussed above in connection with operating leverage. This business risk is 106 measured by the firm's "unlevered beta," which is the beta it would have if it had no debt. If the firm uses no financial 107 leverage, i.e., no debt or preferred stock, then each stockholder would bear that business risk in proportion to his or her 108 share of the stock. However, if the firm uses debt, then the business risk will be concentrated on its stockholders, and each 109 will have to bear more of that risk than if the firm had remained debt free. 110 111 The risk of the stock is reflected in the stock's beta coefficient, and, as we discuss below, beta rises with the use of debt-- the 112 more debt, the higher the beta. The lowest beta is the one that would exist if no debt were used--this is the "unlevered 113 beta," and it reflects the firm's business risk as discussed above. 114 115 In the following example we illustrate all this, continuing with the situation described in our operating leverage example. We 116 assume that the company decided on Plan B and thus requires $200,000 of capital. We also assume that the firm must 117 borrow in increments of $20,000, and that its debt cannot exceed $120,000 due to restrictions in its corporate charter. 118 Further, we assume that Bigbee Inc currently has 10,000 shares of common stock outstanding, and that the company will 119 use debt to repurchase stock at the current market price. 120 121 Discussions with its bankers indicate that Bigbee can borrow different amounts, but the more it borrows, the higher the cost 122 of its debt as shown in the table below: 123 124 Debt Cost Schedule 125 Amount 126 D/A ratio borrowed Cost of debt 127 0% $0 8.0% 128 10% $20,000 8.0% 129 20% $40,000 8.3% 130 30% $60,000 9.0% 131 40% $80,000 10.0% 132 50% $100,000 12.0% 133 60% $120,000 15.0% 134 A B C D E F G H I J K L M 135 OPTION 1: Finance Plan B entirely with common equity (Equity = 100%, Debt = 0%) 136 137 Assets $200,000 138 Debt ratio 0% 139 Equity Ratio 100% 140 Debt $0 141 Equity $200,000 142 Interest rate 8.00% Interest rate from debt cost schedule above. 143 Tax rate 40% 144 Shares outstanding 10,000 145 146 147 Pre-tax Taxes Net 148 Demand Probability EBIT Interest Income 40% Income ROE EPS 149 Terrible 0.05 ($60,000) $0 ($60,000) ($24,000) ($36,000) -18.00% ($3.60) 150 Poor 0.20 ($20,000) $0 ($20,000) ($8,000) ($12,000) -6.00% ($1.20) 151 Normal 0.50 $40,000 $0 $40,000 $16,000 $24,000 12.00% $2.40 152 Good 0.20 $100,000 $0 $100,000 $40,000 $60,000 30.00% $6.00 153 Wonderful 0.05 $140,000 $0 $140,000 $56,000 $84,000 42.00% $8.40 154 155 Expected values: $40,000 $0 $40,000 $16,000 $24,000 12.00% $2.40 156 Standard Deviation: 14.82% $2.96 157 Coefficient of Variation: 1.23 1.23 158 159 OPTION 2: Finance Plan B with $100,000 of debt and $100,000 of equity (50% equity, 50% debt) 160 161 Assets $200,000 162 Debt ratio 50% 163 Equity Ratio 50% 164 Debt $100,000 165 Equity $100,000 166 Interest rate 12% Interest rate from debt cost schedule above. 167 Tax Rate 40% 168 Shares outstanding 5,000 Assumes stock can be repurchased at the current price. 169 170 171 Pre-tax Taxes Net 172 Demand Probability EBIT Interest Income 40% Income ROE EPS 173 Terrible 0.05 ($60,000) $12,000 ($72,000) ($28,800) ($43,200) -43.20% ($8.64) 174 Poor 0.20 ($20,000) $12,000 ($32,000) ($12,800) ($19,200) -19.20% ($3.84) 175 Normal 0.50 $40,000 $12,000 $28,000 $11,200 $16,800 16.80% $3.36 176 Good 0.20 $100,000 $12,000 $88,000 $35,200 $52,800 52.80% $10.56 177 Wonderful 0.05 $140,000 $12,000 $128,000 $51,200 $76,800 76.80% $15.36 178 179 Expected values: $40,000 $12,000 $28,000 $11,200 $16,800 16.80% $3.36 180 Standard Deviation: 29.64% $5.93 181 Coefficient of Variation: 1.76 1.76 182 A B C D E F G H I J K L M 183 Typically, financial leverage increases the expected rate of return on equity. In this case, the return on equity rises from 184 12% to 18%. However, this higher return comes at a price--the higher the debt ratio, the greater the risk as indicated by 185 the coefficient of variation, which rises from 1.23 to 1.76. 186 187 In the tables just above, we calculated the expected EPS and risk measures at zero debt and at 50% debt. We can use an 188 Excel Data Table to calculate these values at a number of different debt ratios, as shown below. 189 190 D/A Ratio Exp. EPS Std dev of EPS CV of EPS 191 50% $3.36 $5.93 1.76 192 0% $2.40 $2.96 1.23 Minimum risk 193 10% $2.56 $3.29 1.29 194 20% $2.75 $3.70 1.35 195 30% $2.97 $4.23 1.43 196 40% $3.20 $4.94 1.54 197 50% $3.36 $5.93 1.76 Maximum EPS 198 60% $3.30 $7.41 2.25 199 200 There's a conflict between risk and return so we must decide on a tradeoff, i.e., must decide the optimal capital structure. 201 202 203 DETERMINING THE OPTIMAL CAPITAL STRUCTURE 204 205 The optimal capital structure is the one that maximizes the stock price. Also, that same capital structure minimizes the 206 WACC. To find--or, really, estimate--the optimal capital structure, we need information on how capital structure affects 207 the costs of debt and equity. The effects on debt are usually estimated by talking with bankers and investment 208 bankers--Bigbee's debt cost schedule as shown above was determined in this way. The effects on the cost of equity are 209 determined in various ways, but one logical starting point is the Hamada Equation, which is explained below. 210 211 THE HAMADA EQUATION 212 213 Hamada developed his equation by merging the CAPM with the Modigliani-Miller model. We use the model to determine 214 beta at different amounts of financial leverage, and then use the betas associated with different debt ratios to find the cost of 215 equity associated with each of those debt ratios. Here is the Hamada equation: 216 217 bL = bU x [1 + (1-T) x (D/E)] 218 219 Here bL is the leveraged beta, bU is the beta that the firm would have if it used no debt, T is the marginal tax rate, D is the 220 market value of the debt, and E is the market value of the equity. 221 222 In the table below, we apply the Hamada equation to Bigbee Electronics, given its unlevered beta and tax rate. 223 224 BU 1.5 225 Tax rate 40% 226 227 D/A D/E BL 228 0.0 0.00 1.50 229 0.1 0.11 1.60 230 0.2 0.25 1.73 231 0.3 0.43 1.89 A B C D E F G H I J K L M 232 0.4 0.67 2.10 233 0.5 1.00 2.40 234 0.6 1.50 2.85 A B C D E F G H I J K L M 235 236 As the table shows, beta rises with financial leverage. With beta specified, we can determine the effects of leverage on the 237 cost of equity and then on the WACC. Here we assume that the risk-free rate is 6% and the required return on the market 238 is 10%. We also assume that Bigbee pays out all of its earnings as dividends, hence its earnings and dividends are not 239 expected to grow. Therefore, its stock price can be found by using the perpetuity equation, Price = Dividend/k s. 240 241 Risk-free rate, kRF: 6% 242 Market return, kM: 10% 243 244 D/A ratio D/E ratio A-T kD EPS = DPS Estimated beta Cost of equity Est. Price P/E Ratio WACC 245 0% 0.00% 4.8% $2.40 1.50 12.0% $20.00 8.33 12.00% 246 10% 11.11% 4.8% $2.56 1.60 12.4% $20.65 8.06 11.64% 247 20% 25.00% 5.0% $2.75 1.73 12.9% $21.33 7.75 11.32% 248 30% 42.86% 5.4% $2.97 1.89 13.5% $21.90 7.38 11.10% 249 40% 66.67% 6.0% $3.20 2.10 14.4% $22.22 6.94 11.04% 250 50% 100.00% 7.2% $3.36 2.40 15.6% $21.54 6.41 11.40% 251 60% 150.00% 9.0% $3.30 2.85 17.4% $18.97 5.75 12.36% 252 253 We see that the stock price is maximized, and the WACC is minimized, if the firm finances with 40% debt and 60% 254 equity. This is the optimal capital structure. 255 256 Below, we graph the key data from the table above. 257 258 Cost of Capital Graph 259 260 261 20.0% 18.0% 262 16.0% 263 14.0% Debt 264 12.0% 265 10.0% Equity 8.0% 266 6.0% WACC 267 4.0% 268 2.0% 269 0.0% 270 0% 20% 40% 60% 80% 271 Debt Ratio 272 273 274

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