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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 "brealgaoints” 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 101, 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 102
ALTERNATWE 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“ (101) 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 thisqua 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 bookvalue weights) can be procedure. Find the total longterm 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 longterm sources 0
the grand total to detennine the percent Table 10—3 summarizes these calculations TABLE 103
CALCULATION on BOOKVALUE WEIGHTS FOR RMM weights is that they represent the Weights as they
originally sold. That is, the bookvalue weights
culated WACC would better represent current The problem with bookvalue
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 104 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 bookvalue WACC and the marketvalue 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 104. 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 aftertax 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 101. 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
CtrlIShiit—lEnter when entering this formula. The completed worksheet appears in
Exhibit lOl. 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 (83): ' 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 (102)
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 riskfree
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 pricetakers, 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 (819): 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,; (103) 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
mune 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 (91): 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 _ _____,___um__._..__  :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 101. Because interest is a taxdeductible 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 91. 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 101 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 aftertax 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 AfterTax 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 aﬂertax cost of debt. Beginning in A7 with the label: Additional
Bond Data, add the information from Table 105 into your worksheet. For
simplicity, we assume that the bonds pay interest annually. TABLE 105
‘ 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?
rtnx Cost H Toialh’inrketValuc ‘Perce go cf'l‘otal' Afte
$ 36l,812 31.14% m The Cost of Preferred Stock Compared to calculating the aftertax 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 (103) 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 103. The Role of Flotation Costs EXHIBIT 103
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 106. TABLE 106
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 107
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* (lD9))
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 104. 317 CHAPTER 10:The Cost of Capital EXHIBIT 104
COST or CAPITAL WORKSHEET WITH FLOTATION Cosrs —
rarerred
. Common as 709.000 _—
Admammonium
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 (104) 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. twostep 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 101
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 usto have thetext, 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 (105) 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 (105): $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 109,
using the information from Table 10—8, shows how to ﬁnd. these breakpoints. TABLE 109
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 109. 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 105. 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 106
THE MARGINAL WACC CURVE FOR RMM 1“ 3.5;; lapﬂ 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 101. 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 107. 
______.__._._————
EXHIBIT 10—7
RMM’S MARGINAL WACC CURVE AS A STEP FUNCTION I .: t'?‘::Cr~;;“=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 1700% 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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 Spring '08
 Olander
 Finance

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