COST_OF_CAPITAL Excel Book Chapter

COST_OF_CAPITAL Excel Book Chapter - CHAPTER 10 .7716 COST...

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Unformatted text preview: CHAPTER 10 .7716 COST 0f Capital _ After studying this chapteig you should be able to: 1. - Define "hurdle rate ” and Show how it relates to rliefimz is Weighted Average Cost of Capital (PMCCD. 2. Calculate the WACC using both boolc- and market—value weights. 3. Calculate component costs of capital with flotation costs and taxes. 4. Eaplain how and why afirm. is WXCC changes as total capital require- ments change. 5. Use Excel to calculate the "br-ealgaoints” in a firm is marginal W400 came, and graph this came in Excel. Suppose that you are offered an investment opportunity that you believe will earn a return of 8%. If your required rate of return is 10%, would you make this invesunent? Clearly not. Even though you would. earn a profit, in the accounting sense of the word, you wouldn’t be making as much as required to make the investment attractive. Presumably, you have other investment alternatives, with similar risk, that will earn your required return. So, 10% is the opportunity cost of your funds, and you will reject investments that earn less than this rate. This rate is also known as your cost of capital, and corporations use the concept every day to make investment decisions. 303 CHAPTER 10: The Cost of Capital f capital is vital if managers are to make appropriate decisions regarding the use of the iirrn’s fiinds. Without this loiowledge, poor investmentsmayr be made that actually reduce shareholder Wealth. In this chapter ' . * ' you will learn what the cost of capital is, and how to calculate it. Knowledge of a. firm‘s cost 0 ‘ ' ' often referred to as its hurdle rate because all projects must earn a rate of return high enough to clear this rate. st of financing, thereby reducing Otherwise, a project will not cover its co shareholder wealth. But what is the appropriate rate to Me? Let’s look at an example. The managers of Rocky Mountain Motors (RMM) are considering the pur— that will he held for one year. The purchase price chase of a new tract of land of the land is $10,000. RMM‘S capital structure is currently made up of 40% debt, 10% preferred stock, and 50% common equity. Because this capital structure is considered to be optimal, any new financing will he raised in the same proportions. RMM must raise the new funds as indicated in Table 104. TABLE 1 0—1 FUNDING FOR RMM’S LAND PURCHASE mm RNLM’S managers must determine what required sly satisfy all of their capital providers. What is ill accomplish this goal? Before making the decision, rate of return will simultaneou the minimum rate of return that w Looking at the third colunni of Table 10-1, it is clear that the total financing cost is ate at least $930 in excess of its cost in order to $980. So, the project must gener nts a minimum required return of 9.8% on the cover the financing costs. This represe investment of $10,000. Table 10—2 shows what would happen under three alternative rate of return scenarios. / 304 The Appropriate “Hurdle” Rate TABLE 10-2 ALTERNAT-WE SCENARIOS r011 RMM Total FundsAvailable s 10,800 $10,980 ' $11,100 Less: Debt Costs 4,280 4,230 Less: Preferred Costs 1,100 1,100 Available to Common 5,420 5,600 5,720 Shareholders Recall that the common shareholders” required rate of return is 12% 011 the $5,000 that they provided. This means that the shareholders expect to get back at least $5,600. If RMM earns only 8%, the common shareholders will receive only $5,420; $180 less than required. We assume that the common shareholders have alternative investment opportunities (with equal risk} that would return 12%. Therefore, if the project can return only 8%, the best decision that the managers could make would be to allow the common shareholders to hold on to their money. In other words, the project should be rejected On the other hand, if the project is expected to return 9.8% the common shareholders will receive exactly the amount that they require. If the project returns 11%, ‘d1ey will be more than satisfied. Under these latter two scenarios the project should be accepted because shareholder wealth will either be increased by the amount required ($600) or increased by more than required ($7120).1 The Weighted Average Cost of Capital It still remains to determine, in- a general way, what required rate of return will simultaneously satisfy all .of the film’s stakeholders. Recall that 40% of RMIM’s funds were provided by the debt holders. Therefore, 40% of this minimum required rate of return must go to satisfy the debt holders. For the same reason, 10% of this minimum required rate of return must go to satisfy the preferred stockholders, and 50% will be required for the common stockholders. 1. Note that the difihrence between the amount that is available to the common shareholders and the amount required is known as the net present value (NP V ). This concept will be explored in Chapter 11. M— 305 CHAPTER 1D:The Cost of Capital In general, the minimum required rate of return must be a weighted average of the n of capital provided. Therefore, we individual required rates of return on each for1 refer'to this minimum required rate of return as the weighted average cost ofcapital (WACC). The Weighted average cost of capital can be found as follows: - “.4 WA CC = wdkd + wpkp + wast“ (10-1) each source of capital, and the k’s are the costs f capital. In the case of RM, the WACC is: 9.80% where the w’s are the weights. of (required returns) for each source 0 WACC = 0.40(0.07)+0.10(0.10)+0.50(0.12) = 0.098 = which is exactly the required return that we found above. Determining the Weights The weights that one uses in result. Therefore, an import Actually, there are two possible answers to this-qua answer is to find the weights on the balance sheet. the calculation of the WACC will obviously affect the ant question is, “where do the weights come from?” stion. Perhaps the most obvious 5 (usually referred to as the book-value weights) can be procedure. Find the total long-term debt, total preferred Add together each of these to arrive at the f capital. Finally, divide each component by age that each source is of total capital. for RMM. The balance sheet weight obtained by the folloan equity, and the total common equity. grand total of the long-term sources 0 the grand total to detennine the percent Table 10—3 summarizes these calculations TABLE 10-3 CALCULATION on BOOK-VALUE WEIGHTS FOR RMM weights is that they represent the Weights as they originally sold. That is, the book-value weights culated WACC would better represent current The problem with book-value were when the securities were represent historical weights. The cal ___—___—____—_—-————————-—-—n——— WACC Calculations in Excel .t reality if we used flie present weights. Since the market constantly revalues the firm’s securities, and we assume that the capital markets are efficient, we can find the weights by using the current marketvaiues of the securities. The procedure for determining the market~value weights is similar to that used to find the book—value weights. First, determine the total market value of each type of security. Total the results and then divide the market value of each source of capital by the total to determine the weights. TABLE 10—4 CALCULATION OF MAMCET—VALUE WEIGHTS FOR RMM -..«— 1. ~.—.=-_.-d. 1.75.5.1ie'ik‘iS—‘5xmw 5.5:: -— Table 10-4 shows RMM’s current capital structure in marlcet~value terms. Note that, in market value terms, the percentage of common equity has risen considerably, while the percentages of debt and preferred equity have fallen. Using these weights _ we can see that their WACC is: E}; WACC = 0.3114(0.07) + 0.0861(010) + 0.6025(012) = 0.1027 = 10.27% I In this example, the book-value WACC and the market-value WACC are quite close together. This is not always the case. Whenever possible, use the market values of the firm’s securities to determine the WACC. ll WelCC Calculations m Excel We can easily set up a worksheet to do the calculations for the WACC as in Table 10-4. To do this, first copy the data from Table 10—4 into a new worksheet, starting with the headings in A1. ' In column D we want to calculate the total market value of the securities, which is the price times the number of units outstanding. So, in D2 enter: =B2* CZ and GOP}! 307 W CHAPTER 10: The Cost of Capital ' W the formula down to D3 and D4. Cell D5 should have the total market value of the securities, so enter: =Sum (D2 : D4 ). In Column B we need the percentage that each security represents of the total market value. These are the weights that we will use to calculate the WACC. In E2 enter: =D2 / D$ 5 and. copy down to B3 and E4. As a check, calculate the total in 135. Next, we want a column for the after-tax costs of each source of capital, and the ' weighted—average cost of capital. In Fl enter the label: After—tax Cost. Now, 1 in F2:F4 enter the after—tax coat of each component from Table 10-1. We could i - calculate the WACC in F5 with the formula: =EZ* F2+E 3* F3 +E 4* F4. Even easier would be to use the array formula: =SUM (E2 : E4* F2 : F4 ); just remember to press Ctrl-I-Shiit—l-Enter when entering this formula. The completed worksheet appears in Exhibit lO-l. Note that the WACC is exactly as we calculated earlier.2 You are encouraged to experiment by changing the market prices of the securities to see how the weights, and the WACC, change. 312?.- \._.. zen. . 1'. J. EXHIBIT 10—1 WORKSEEET T0 CALCULATE RMM’S H5400 _ource i i I $904.53 a 361,812 31.14% 7.00% s 100.00 a 100,000 8.61% 10.00% a , . . ' D ‘.’ ‘ 1 ill-“1:11 Total Market Value Percentage of Total j. a 70.00 10,000 700 000 50 25% 12 00% —— 11 1.151812 101.00% 111.21% Calculating the Component Costs Up to this point, we have taken the component costs of capital as a given. In reality, these costs are anything but given, and, in fact, change continuously. How we calculate these costs is the subject ofthis section. To begin, note that the obvious way of determining the required rates of return is to simply ask each capital provider what her required rate of return is for the particular 2. Note that this is a simplified example. In reality, most companies will have multiple debt issues outstanding, and many have more than one class of cornmon and preferred stock outstanding as well. The calculations will work in exactly the same way, regardless of the number of issues outstanding. However, you will first have to calculate a weighted average cost for each source of capital (e.g., a weighted average afier—tax cost of debt). 308 I Calculating the Component Costs _'_____________,___.__.——n—————-—-— security that she owns. For all but the most closely held of firms, this would be exceedingly impractical and you would likely get some outlandish responses. However, there leeway by which weaan accomplish the Same end result. Recall from Chapter 8 that the market value of a security is equal to the hittinsic value of the marginal investor. Further, if investors are rational, they will buy (sell) securities as the expected return rises above (falls below) their required return, Therefore, we can say that the investors in the firm “vote with their dollars" on the issue of the firm’s cost of capital. This force operates in all markets.3 So at any given moment, the price of a security will reflect the overall required rate of return for that security. All we need, then, is a method of converting the observed market prices of securities into required. rates of return. Since we have already discussed the valuation of securities (common stock, preferred stock, and bonds) you should recall that a major input was the investor’s required rate of return. As we will see, we can simply invert the valuation equations to solve for the required rate of return. The Cost of Common Equity Because of complexities in the real world, finding a company’s cost of common equity is not always straightforward. In this section we will look at two approaches to this problem, both of which we have seen previously in other guises. Using the Dividend Discount Model Recall that a share of common stock is a pelpetual security, which we assume will periodically pay a cash flow that grows over time. We have previously demonstrated that the present value of such a stream of cash flows is given by equation (8-3): ' D0(1+g) D1 _ V = ____. = C5 kcs ‘ 8 lies — g assuming an infinite holding period and a constant rate of growth for the cash flows. 3. Anybody who isn‘t convinced should check the history of bond and stock prices for companies such as Enron and WorldCom. They were falling dramatically long before those firms filed for bankruptcy. ______r____________”-—————— 309 W CHAPTER 10:The Cost of Capital 310 If we know the current market price of the stock, we can use this knowledge to solve for the common shareholder’s required rate of return. Simple algebraic ' manipulation will reveal that this rate of return is given by: 00(1'i‘g) Vcs D1 ices = +3 = — +g (10-2) Vcs Note that this equation says that the required rate of return on common equity is equal to the sum of the dividend yield and the growth rate of the dividend stream. We could also use anyr of the other common stock valuation models, though solving for the required return is slightly more complicated. Using the CAPM Not all common stocks will meet the assumptions of the Dividend Discount Model. In particular, many companies do not pay dividends. An alternative approach to determining the cost of equity is to use the CapitalAsser Pricing Model (CAPM). The CAPM gives the expected rate of return for a security if we Imow the risk-free rate of interest, the market risk premium, and the 1iskiness of the security relative to the market portfolio (i.e., the security’s beta). The CAPM, you will recall, is the equation for the security market line: Em.) = Rf+ Bi(E(Rm)'—Rf) Assuming that the stockholders are all price-takers, their expected return is the same as the firm’s required rate of return.4 Therefore, we can use the CAPM to determine the required rate of return on equity. The Cost of Preferred Equity Preferred stock, for valuation purposes, can be viewed as a special case of the common stock with the growth rate of dividends equal to zero. We can carry this idea to the process of solving for the preferred stockholders’ required rate of 4. A price—taker casmot materially attect the price of an asset through individual buying or selling. This situation generally exists in the stock market because most investors are small when compared to the market value of the firm’s common stock. _#_n_—u__ Calculating the Component Costs return, First, recall that the value of a share of preferred stock was given by equation (8-19): D I z _ IF ftp As with common stock, we can algebraically manipulate this equation to. solve for the required return if the market price is known: 13 lip = 7,; (10-3) The Cost of Debt Finding the cost of debt is more difficult than finding the cost of either preferred or common equity. The process is similar: determine the market price of the security, and then find the discount rate which makes the present value of the expected mun-e cash flows equal to this price. This rate is the same as the yield to maturity (see page 276). However, we cannot directly solve for this discount rate. Instead, we must use an iterative trial~and—error process. Recall that the value of a bond is given by'equation (9-1): 1i_.__1_, (1 + ltd)” F]! 7 =P I ._._. TB m k, +(1+k,)N The problem is to find kd such that the equality holds between flie left and right sides of the equation. Suppose that, as in Exhibit 10—1, the current price of RMM’s bonds is $904.53, the coupon rate is 10%, the face value of the bonds is $1,000, and the bonds will mature in 10 years. lfflie bonds pay interest annually, our equation looks as follows: 1,; (1 +165)” 1,000 = 100 — 904.53 kd + (1 + ltd)“, We must make an initial, but intelligent, guess as to the value of kd. Since the bond is selling at a discount to its face value, we lcnow that the yield to maturity (kd) must be greater than the coup0n rate. Therefore, our first guess should be something 311 _ _____,___u--m__._..__ - :HAPTER 10:The Cost of Capital / 312 greater than 10%. If we choose 12% we will find that the price would be $886.99, which is lower than the actual price. Our first guess was incorrect, but we now know that the answer must lie between 10% and 12%. The next logical guess is 11%, which is the halfway point. Inserting this for ltd we get a price of $941.11, which is too high, but not by much. Further, we have narrowed the range of possible answers to those between 11% and 12%. Again, we choose the halfway point, 11.5%, as our next guess. This results in an answer of $913.48. Continuing this process, we will eventually find the correct answer to be 11.67%.5 Making an Adjustment for Taxes Notice that the answer that we foimd for the cost of debt, 11.67%, is not the same as that listed in Exhibit 10-1. Because interest is a tax-deductible expense, interest payments actually cost less than the full amount of the payment. In this case, if Rh/IM were to make an interest payment of $116.70, and the marginal tax rate is 40%, it would only cost them $70.02 (= 116.70 X (1 — 0.40) ). Notice that 7002/ 1,000 m 0.07 , or 7%, which is the after—tax cost of debt listed in Exhibit 9-1. In general, we need to adjust the cost of debt to account for the deducfibility of the interest expense by multiplying the before~tax cost of debt (i.e., the yield to matmity) by 1 — t, where tis the marginal tax rate. Note that we do not make the same adjustment for the cost of common or preferred equity, because dividends are not tax deductible.6 Using Excel to Calculate the Component Costs A general principle that we have relied on in constructing our worksheet models is that we should make Excel do the calculations wheneVer possible. We will now make changes to our worksheet in Exhibit 10-1 to allow Excel to calculate the component costs of capital. _____,__————-" 5. The method presented here is lcnouui as the bisection method. Briefly, the idea is to quickly bracket the solution and to then choose as the next approximation 1he answer that is exactly halfway betwoen the previous possibilities. This method can lead to very rapid convergence on the solution if a good begiiming guess is used. ' 6. This is just a close approximation, but close enough for most purposes since the cost of capital is just an estimate anyway. It would be more accurate to use the after-tax cash flows in the equation. This will result in the after—tax cost of debt with no additional adjustment required, and will differ slightly from that given above W. Usmg Excel to Calculate the Component Costs W The After-Tax Cost of Debt We cannot calculate any of the component costs on our worksheet without adding some additional information. We will first add information which will be used to calculate the afler-tax cost of debt. Beginning in A7 with the label: Additional Bond Data, add the information from Table 10-5 into your worksheet. For simplicity, we assume that the bonds pay interest annually. TABLE 10-5 ‘ ADDITIONAL DATA FOR CALCULATmG THE COST OF DEBT FOR RMM Coupon Rate I Face Value $1,000 Mme With this information entered, we now need a function to find the cost of debt. Excel provides two built—in fiinctions that will do the job: RATE and YIELD. We have ah'eady seen both of these functions. Since YIELD (defined on page 277) requires more information than we have supplied, we will use RATE. Recall that RATE, which works only on a payment date, will solve for the yield for an annuity— type stream of cash flows and allows for a different present value and future value. Specifically, RATE is defined as: RATE(NPER, PMT, PI’, FV, Tree, Guess) The only unusual aspect of our usage of this function is that we will be supplying _ both a PV and an FV. Specifically, PV will be the negative of the current bond price, and FV is the face value of the bond. In F2 enter the RATE fimction as: =RATE (1311, 39* B10, -B2, 1310). The result is 11.67%, which we found to be the pretax cost of debt. Remember that we must also make an adjustment for taxes, so we need to multiply by 1 — 1:. The final form of the formula in F2 then is: =RATE (Bil , 139*310, —BZ , B10)* (1—338 ) , and the result is 7.00%. With the new bond information, your worksheet should resemble Exhibit 10—2. 313 CHAPTER 1D:The Cost of Capital ________4_4____!__————————— 314 EXHIBIT 10—2 RMM WORKSHEET WITH BOND DATA w flit-‘1? £3.21? r-tnx Cost H Toialh’inrketValuc ‘Perce go cf'l‘otal' Afte $ 36l,812 31.14% m The Cost of Preferred Stock Compared to calculating the after-tax cost of debt, finding the cost of preferred stock is easy. We need only add one piece of information: the preferred dividend. In C? type: Additional Preferred Data. In CS type: Dividend and in D8 enter: 1 D. We know from equation (10-3) that we need to divide the preferred dividend by the current price of the stock. Therefore, the equation in F3 is: =D8 /BB. The Cost of Common Stock To calculate the cost of common stock, we need to know the most recent dividend and the dividend growth rate in addition to the current market price of the stock. In E7 type: Additional Common Data. In E8 type: Dividend U and in F8 enter: 3 . 9 6. In E9 enter the label: Growth Rate and in F9 enter: 6%. Finally, we will use equation (10—2) to calculate the cost of common stock in F4. Since we lcoow the most recent dividend (Do) we need to multiply that by 1 + g. The formula in F4 is: = (F8* (1+F9) ) /B4+F9, and the result is 12% as we found earlier. As you will see, we have not yet completed the calculation of the component costs for RMM. We have left out one crucial piece, which we will discuss in the next section. At this point, your worksheet should resemble fliat in Exhibit 10-3. The Role of Flotation Costs EXHIBIT 10-3 RMM COST or CAPITAL WOIUCSHEET. I TotnlMaricct Value Percentage ot'TolaI .2...Dein 090453 301512 31.14% " 9 . s , ' Preferred 1; 100.00 mm a; 100,000 10.00% Common s moo $ 700000 a: s- warm mm ——_—— ; Auditionamonanm 1; won m—— m——_— - .E—___ The Role of Flotation Costs Any action that a corporation takes has costs associated with it. Up to this point we have implicitly assumed that securities can be issued without cost, but this is not the case. Selling securities directly to the public is a complicated procedure, generally requiring a lot of management time as well as the services of an z'mzesnnent banker. An investment bank is a firm that serves as an intermediary between the issuing firm and the public. In addition to forming the underwriting syndicate to sell the securities, the investment banker also fimctions as a consultant to the firm. As a consultant, the investment banker usually advises the firm on the pricing of the issue and is responsible for preparing the registration statement for the Securities and Exchange Commission (SEC). The cost of the investment banker’s services, and other costs of issuance, are referred to as flotation costs. (The term derives from the fact that the process of selling anew issue is generally referred to as floating a new issue.) These flotation costs add to the total cost of the new securities to the firm, and we must increase the component cost of capital to account for them. There are two methods for accounting for flotation costs. The most popular method is the cost of capital adjustment. Under this method the market price of new securities is decreased by the per unit flotation costs. This results in the net amount that the company receives from the sale of the securities. The component costs are 31_5 CHAPTER 10:The Cost of Capital ' (ff/(ff— 1hen calculated in the usual way except that the net amount received, not the market price, is used in the equation. The second, less common, method is the investment cost adjustment. Under this al outlay for the project under consideration to methodology we increase the initi account for the total flotation costs. Component costs are then calculated as we did so it assigns all above. The primary disadvantage of this technique is that, becau flotation costs to one project, it implicitly assumes that the securities used to finance a project will be retired. when the project is completed.7 Because it is more commom and its assumptions are more realistic, we will use the cost of capital adjustment technique. When flotation costs are included in the analysis, the equations for the component costs are given in Table 10-6. TABLE 10-6 COST OF CAPITAL EQUATIONS WITH FLOTATI ON Cosr ADJUSTMENT : " Cost of new common equity Cost of preferred equity Pretax cost of debt (solve for kd) * In these equations the flotation costs (flares. dollar amount per unit. It is also common for flotation costs to be stated as a percentage of the unit price. _______——— 7. For more information on both methods, see Brigham and Gapensici, “Flotation Cost Adjustments,” Financial Practice and Education. (FallJWinter 1991): 29—34. ff 316 The Role of Flotation Costs Adding Flotation Costs to Our Worksheet We can easily incolporate the adjushnent for flotation costs into our worksheet. All we need to do is change the references to the current price in each of our formulas to the current price minus the per unit flotation costs. These costs are given in Table 10—7. TABLE 10-7 FLOTATION Cosrs AS A PERCENTAGE on SELLING PRICE non RMM Enter the information train Table 10—7 into your worksheet. For each security, We have added the information at the end of the “Additional infonnation” section. For example, in A12 enter: Flotation and in B12 enter: 1%, which is the flotation cost for bonds‘ Add similar entries for preferred and common stock. 1 To account for flotation costs, change your formulas to the following: F2 =RATE (Bll,BB*BlO,—B2* (1—1312) ,B10)* (l—B8) F3 =D8/ (133* (l-D9)) i F4 =(F8* (1+F9) ) / {134* (l—F10))+F9 Once these changes have been made, you will notice that the cost of each component has risen. Your worksheet should now resemble the One pictured in Exhibit 10-4. 317 CHAPTER 10:The Cost of Capital EXHIBIT 10-4 COST or CAPITAL WORKSHEET WITH FLOTATION Cosrs — rarer-red . Common as 709.000 _—- Adm-ammonium mm Coupon Rate notation ‘o:‘:»racewue _— ‘ Maturity m___— The Cost of Retained Earnings We have shown how to calculate the required returns for purchasers of new common equity, preferred stock, and bonds, but films also have another source of long—term capital: retained earnings. Is there a cost to such internally generated funding, or is it free? Consider that managers generally have two options as to what they do with the firm’s internally generated funds. They can either reinvest them in profitable projects or return them to the shareholders in the form of dividends or a share repurchase. Since these funds belong to the common shareholders alone, the definiu'on of a “profitable proj ect” is one that earns at least the common shareholder’s required rate of return. If these funds will not be mvested to earn at least this return, they should be returned to the common shareholders. So there is a cost (an opportunity cost) to internally generated fimds: the cost of common equity. Note that the only difference between retained earnings (internally generated common equity) and new common equity is that the firm must pay flotation costs on the sale of new common equity. Because no flotation costs are paid for retained earnings, we can find the cost of retained earnings in the same way we did before learning about flotation costs. In other words, I I, Do(1+gl D1 can — VCS +3 = 7,759+c (10-4) This notion of an opportunity cost for retained earnings is hnportantfor a couple of reasons. Most importantly, managers should be disabused of the notion met die 318 The Marginal WACC Curve funds on hand are “free.” As you now know, there is a cost to these funds and it should be accounted for when making decisions. In addition, there may be times when a project that otherwise appears to be profitable is really unprofitable when the cost of retained earnings is correctly accounted for. Accepting such a project is contrary to the principle of shareholder wealth maximization and will result in the film’s stock price falling. T he Marginal WCC Curve A film’s weighted average cost of capital is not constant. Changes can occur in the HHCC for a number of reasons. As a firm raises more and more new capital, its WHCC will likely increase due to an. increase in supply relative to demand for the firm’s securities. Furthermore, total flotation costs may increase as more capital is raised. Additionally, no firm has an unlimited supply of projects that will return more than the cost of capital, so the risk that new funds will be invested unprofitany increases. We will see in the next chapter that these increases in the MCC play an important role in determining the firm’s optimal capital budget. For the remainder of this chapter we will concentrate on determining the WHCC at varying levels of total capital. Finding the Breakpoints We can model a firm’s marginal WMCC curve with a step function. This type of function resembles a staircase when plotted. They are commonly used as a linear (though discontinuous) approximation to noulinear functions. The accuracy of the approximation improves as the number of steps increases. Estimating the marginal HMC‘C (MCC) curve is a. two-step process: 1. Determine the levels of total capital at which the marginal WMCC is expected to increase. These points are referred to as breaigaoints. 2. Determine the marginal HHCC at each breakpoint. Figure 10—1 illustrates what a marginal WACC curve might look like for Rocky Mountain Motors. Notice that the breakpoints are measured in terms of dollars of 319 M CHAPTER 10: The Cost of Capital M total capital. In this section we will estimate where these breakpoints are likely to " ‘_. !» occur and determine the WMCC at the breakpoints. FIGURE 10-1 THE MARGINAL WACC (MCC) CURVE AS A STEP FUNCTION WACC m) Total Capital (31) mAnlun] wAcc .— Siep function approximation After consulting with their investment bankers, the managers of RM have determined that the},r can raise new money at the costs indicated in TableI10~8. Open a new worksheet and enter the data fiom Table 10—8 beginning in cell Al. The percentages in the “% of Total" column should be referenced from the worksheet that was created for Exhibit 10—4. TABLE 10—8 ROCKY MOUNTAJN MOTORS INFORMATION 15.00% Common 60.25% Up to 100,000 100,001 to 500,000 More than 500,000 17.00% Preferred 8.61% Up to 50,000 10.20% Debt 31.14% Up to 250,000 7.10% More than 250,000 8.00% E . 12.31% The Marginal WACC Curve Note that you should enter just the numbers from the “Amomits Which Can Be Sold” column. You can define custom fonnats, if desired, so the numbers are displayed. with the text. This allows us-to have the-text, and still use the numbers for the calculations that follow. For example, you can format the first cell as: “Up to “it. ##0 which will cause the number to be displayed as shown in the table. The second number (500,000) can be formatted with: “ 1 O O , 0 01 to “# , ##0 so that it will display as shown. RMM feels that its current capital structure is optimal, so any new money will be raised in the same percentages. For example, if the firm decides to raise $200,000 in total capital, then $120,500 (60.25% of $200,000) will come from common equity, $62,280 (31.14%) will be debt, and $17,220 (8.61%) will he preferred equity. Using the information in Table 10—8, we can determine the breakpoints in RMM’s marginal 1716400 curve. To do this, first realize that a break will occur wherever the cost of an individual source of capital changes (why?) There will be a hrealqaoint associated with the issuance of $100,000 in common stock, for example. But recall that breakpoints are measured in dollars of total capital. So the question is, “How do we convert this $100,000 in common stock into the amount of total capital?” Since all of the capital will be raised in constant proportion, we can use the following equation: $ Common Stock $ T0131 capital : % Common Stock (10-5) In this case, we can see that if RMM raised $100,000 in new common stock, then they must have raised $165,973 in total capital. Using equation (10-5): $100,000 $165,973 = We can use this information to see that if RM‘M issued $100,000 in new common stock, then they must also have raised $51,684 (= $165,973 X 0.3114 ) in new debt and $14,290 (= $165,973 x 0.0861 ) in new preferred stock. 321 M- CHAPTER 10:The Cost of Capital W To locate all of the breakpoints, all we need to do is find the points at which the cost of each source changes and then convert those into dollars of total capital. Table 10-9, using the information from Table 10—8, shows how to find. these breakpoints. TABLE 10-9 FINDING THE BREAICPOINTS IN RMM’s MARGINAL WA CC CURVE 5 100,000/0.6025 $165 973 500,000/0.6025 s 829.866 Prefeired Stock 503000 “10851 $ 580,906 250,000/03114 s 802.773 In your worksheet enter: Breakpoints in cell E1. The first hreakpoint is associated with the $100,000 level of new common stock. In E2, enter the formula: =C2/BS2. The result is $165,973, exactly as we found. in Table 10-9. Copy this ‘ formula to E3. In E5 the formula is: =C5/B$5. In ET your formula will be: =C'7/ $B$7. The next step is to determine the WACC at each of the breakpoints. To find the WACC, we must convert each breakpoint into its components and then determine the cost of each component. There are a number of ways we might approach this problem in the worksheet. Because we would ultimately like to generate a chart of the marginal WACC, we will set up a table that shows the amount of total capital, the cost of each component, and the WACC at that level of total capital. Begin by entering the labels inA10:E10. In A10 enter: Total Capital. In B10: Cost of Equity. In C10: Cost of Preferred. InDlO: Cost of Debt. In E10: WACC. Now, in A1 I, enter 0. In A12, we want to enter the first breakpomt. We could just reference E2, which has the smallest breakpomt, but that may not be the smallest of the breakpoints if the weights change. To ensure that A12 always has the smallest breakpoint, we should use the SMALL function: SMALL(ARRAI’, K) where ARRAY is a range of numbers and. K is me position that you want to return. In A12, enter: =SMALL {E2 :E7, 1) to get the smallest breakpoint. In A13, enter: =SMALL (E2 :E7 , 2) to get the second smallest breakpoint, and so on. To finish 322 W The MarginatWACC Curve ___—___,___—_._.—_.—...——————m«n—v_—-*m~—m— this series with a round number, in A16 enter: =ROUNDUP (MAX {E2 :E7) , -5) . This will round the largest hreakpoint up to the next $100,000. Next, we will determine the cost of each source for each level of total capital. In B11, we need to find the cost of equity at $0 of total capital. To facilitate later copying, we will set up a nested IF statement. In this case, the formula is: =IF {All* $B$2<=$C$2 , $D$2 , IF (AlP‘ $B$2<=$C$3, $D$3 , $D$4) ) . In words, this formula says: “If the amount of total capital (in Al 1) times the percentage of common stock (B2) is less than or equal to $100,000 (C2), then the cost is 12.31% (D2). Otherwise, if the amount is less than or equal to $500,000 then the cost is 15% (D3). Othelwise, the cost is 17% (D4).” We use similar, but less complicated formulas to determine the cost of preferred stock and debt at each level of total capital. For preferred stock, enter the formula: =IF(A11* $B$5<=$C$5, $D$5, $D$6) into C11. In D11 enter the formula: :IF(A11* $B$7<e$c$7 , $D$ 7, $D$ 8) to detennjne the appropriate cost ofdeht. Finally, we can calculate the marginal weighted average cost of capital (in E11), with the formula: =$B$2* Bl l+$B$ 5* Cl 1+$B$ 7* D1 1. This formula calculates a weighted average of the costs which were calculated in B111D11. Make sure that you have entered the formulas exactly as given, and then copy them down through each row to row 16. Your worksheet should now match the one in Exhibit 10-5. Esmerr 10~5 THE WACC AT EACH BREAKPOJNT xiii"; ' ' "" =2 ._ — — — s 61% Up to 50,000 10.20% 580,906 ' 13.00% Debt 31.14% Up to 250,000 802,773 '1‘ ... ('0 ’3’ E! (I u. — _— ——_—— Cost 01' Preferred Cost of Debt .1 165,973 580.906 = 323 W CHAPTER 10: The Cost of Capital ___________________—_—,,—_—....—_——n—m——-—-m-—~—w— Creating the Marginal WACC Chart I Recall that we want to create a chart of the marginal cost of capital, approximated by a step fimction. To create this chart We need the WACC’s and the brealqioints that were created above. Select A101A16 and then hold down the Ctrl key and select E10:E16. Now use the Chart Wizard to create an XY (Scatter) chart.’3 EXHIBIT 10-6 THE MARGINAL WACC CURVE FOR RMM 1“ 3.5;; lap-fl u r I 2294;“: ifs-:33 {fa-‘2‘ ‘ $5.}; ‘-‘. if: ‘l.fll:_‘_.‘;t‘.‘;, ["2"] 11753505} - t e: m if Marginal WACC Curve for RMIM 14.00% 13.00% 12.00% 11.00% 10.00% 9.00% 800% wacc m] 0 200:000 400,000 600,000 800,000 Total Capital Note that the chart in Exhibit 10—6 does not depict a perfect step function, as shown in Figure 10-1. With a little trick, we can easily change this chart into a perfect step fimction. 8. The most common error in making this type of chart correctly is choosing the wrong type of XY (Scatter) chart. Choose the type illustrated in the lower—right corner of the samples on the Chart Wizard’s Chart Type dialog box. If you choose an XY chart with smoothed lines, the result will be a little too smooth. Try it. Also note that you will not get a good step fisnction using a line chart. ‘ ‘ 324 _——_———_—‘_——n——__‘—_-—-- The MarginaIWACC Curve ' W . First, realize that we want the line to be perfectly verljcal at each hreakpoint. In order to do that, we must have two Y—values (WACC) corresponding to each particular X—value (amount of total capital). However, if we use the exact break point twice, then the WACC will be the same. To get the WACC to increase, we need to increase the hrealcpoint by a very tiny amount. To sac this, select row 13 and insert a new row. Now, in A13 enter the formula: =A12+0 . 0 l, and then copy the other formulas in row 12 down to row 13. Note that the WACC (in E13) is now higher than in E12. Take a look at your chart and notice that you now have a nice step for the first brealqnoint‘ If you zoomed far enough into the chart, you would see that the step is slightly sloped, but at normal size the slope isn’t visible. Repeat these steps with the other three breakpoints, and then your chart should look like the one in Exhibit 10-7. - ______.__._._——-—— EXHIBIT 10—7 RMM’S MARGINAL WACC CURVE AS A STEP FUNCTION I .: t'?‘::C--r~;;“=i".,:1.‘E :‘ mm Cos ofDebt WAC -: 12.31% 7.10% m 12.31% 7.10% 15.00% 7.10% a: 15.00% 7.10% 15.00% " m 15.00% 7.10% - 15.00% 8.00% - 15.00% 8.00% W 17-00% 300% - 17.00% 8.00% Marginal WACC Curve for RMM 14.00% 13.00% 12.00% 11.00% 10.00% 9.00% W'ACC (Mn) 200,000 400,000 600,000 800,000 Total Capital ______—__._....____.__..———--————-—-——— CHAPTER 10:The Cost of Capital W Summary We began this chapter with a discussion of the appropriate required rate of return to use in the evaluation of a company’s scarce capital resources, We demonstrated that a weighted. ayeiage of the cost of each source of capital would be sufficient to simultaneously satisfy the providers of capitaL In addition, we showed. that the costs of the sources of capital can he found by simply inverting the valuation equations from Chapters 8 and 9 and including flotation costs. Finally, we saw that the finn’s marginal weighted average cost of capital changes as the amount of total capital changes. We showed how to determine the location of the breakpoints and how to plot the marginal WHOC curve. FUNCTIONS INTRODUCED IN THIS CHAPTER Determine the yield to maun‘lty for an annuity or bond, 326 ...
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COST_OF_CAPITAL Excel Book Chapter - CHAPTER 10 .7716 COST...

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