**COST OF CAPITAL**

**COST OF CAPITAL**

• OBJECTIVE 1

**Compute the cost of debt.**

The method of computing the yield to maturity for bonds will be used how to compute the cost of debt. Because interest payments are tax deductible, only after-tax costs are used.

• OBJECTIVE 2

**Compute the cost of preferred stock.**

The preferred stock is priced using the formula for the value of

perpetuity. The formula for finding the price of perpetuity is used to estimate the component cost of preferred stock. One important point to remember is that unlike bonds, preferred dividend payments are not tax deductible. Therefore, no adjustment is necessary to convert the cost of preferred to an after-tax basis.

**COST OF CAPITAL**

**COST OF CAPITAL**

• OBJECTIVE 3

**Compute the cost of equity.**

There are several different models discussed that can be used to compute the cost of equity financing. The reason for using multiple models is that the inputs to each model are uncertain. By combining the results, a better estimate of cost can be achieved.

• OBJECTIVE 3

**Compute the weighted average cost of capital.**

Once all of the component costs are found, the weighted average cost of capital (WACC) is computed where the weights are the proportion of each source of capital in the firm's target capital structure. The WACC is men used as the discount rate in capital budgeting problems

**COST OF CAPITAL**

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• **Finding the After-tax Cost of Debt**

**The before-tax cost of long-term debt – k**_{d}**. Remember that k**** _{d}** is
also called the yield to maturity on a bond (In most cases, firms
accumulate long-term debt through borrowing in the form of bond
issues with fixed coupon interest payments).

Therefore, the cost of debt will be a function of the coupon payments
*required by the investors is these newly issued bonds.*

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**A) BEFORE-TAX COST OF DEBT**

When the net proceeds from the sale of a bond are equal to its face value, the coupon interest rate can be used for the bond's before-tax cost.

Also, when the yield to maturity (YTM) for other bonds of similar risk is available, the similar bond's YTM can be used for the firm's before- tax cost of long-term debt. This is termed a cost quotation.

When the net proceeds from the sale of the bond differ from the face value of the bond, it is necessary to either calculate the cost or to approximate the cost. The calculation of the before-tax cost of debt for a bond involves finding the yield to maturity for the bond by either the trial and error approach, or by using a hand-held

business/financial calculator.

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*The equation used for approximating the before-tax cost of loaf-term *
debt for a bond is the YTM approximation formula:

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**B) AFTER-TAX COST OF DEBT**

As mentioned earlier, the cost of financing must be stated on
*an after-tax basis. The interest on debt is tax deductible, so it *
reduces the firm's taxable income by the amount of deductible
interest.

The equation used to find the after-tax cost of debt is demonstrated in the following formula:

Where:

K** _{d}** = the before tax cost of debt
T = corporate tax rate

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**Example:**

*Velvet Corporation needs to issue new long-term debt in order to *
*raise capital. Their existing long-term bonds have various coupons, *
*but currently have a YTM of approximately 10%. Therefore Velvet *
*believes they could issue new, 20-year annual-pay bonds with a *
*10% coupon. Velvet will incur a $20 flotation cost an each bond *
*sold, meaning they will net only $980 per bond. Velvet has a *
*corporate tax rate of 40%. What is the after-tax cost of debt for *
*Velvet Corporation?*

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**The first step in solving the problem is to find k**** _{d}**. the before-lax cost
of debt.

In this case the easiest approach is to plug the information into a financial calculator and solve for YTM. However, if you don't have access to a financial calculator, you can use the approximation formula,

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**Once you have obtained the before-tax cost of debt, k**** _{d}**, the next
step is to convert it to the after-tax cost of debt.

Since the interest on the debt is tax deductible. Velvet is only paying 6.14% for the debt and is receiving a 4.10% (10.24%-6.14%) tax deduction for every dollar spent on interest expense.

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**Finding the Cost of Preferred Stock**

Preferred stock is a special form of ownership because its dividends must be distributed before any other dividends are paid to the

common stockholders. Most preferred stock dividends are stated as a dollar amount but sometimes they are stated as a percentage of the stock's par value. In order to calculate the cost of preferred

stock, it is necessary to convert the dividend of any preferred stock that is stated as a percentage into a dollar amount. Once the

dividend is stated as a dollar amount, the cost of the preferred stock can be found by using the following equation:

Where:

kp = cost of preferred stock Dp = annual dollar dividend

Np = net proceeds from the sale of me stock

Preferred stock dividends are paid out of the firm's after-tax cash flows, so no tax adjustments are necessary.

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**Example:**

*Velvet Corporation's outstanding preferred stock currently yields *
*11%. Velvet can issue new $100 par preferred with an 11% *

*dividend, but will incur flotation costs of 5%. Find the cost of newly *
*issued preferred stock.*

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**Solution:**

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**Finding the Cost of Equity**

Some people assume equity has no real cost. This seems logical because dividend payments are not required on common stock, and the firm is not required to pay them.

However, it is important to remember that investors require compensation for the risks they take.

Common stock, being riskier than either bonds or preferred stock, carries a higher required rates of return, in other words; investors require compensation in the form of dividends and/or capital gains which exceeds that of the firm's bondholders and preferred

stockholders.

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**A) CAPM**

* The Capital Asset Pricing Model (CAPM) can be used to estimate *
the cost of common stock equity. The CAPM describes the

relationship between the required return, or the cost of common
equity in this case, and the nondiversificable risk of the firm as
*measured by the beta coefficient, (S. The following equation. *

represents the basic CAPM;

Where:

Rf = risk-free rate of return,
*β= beta coefficient,*

k** _{m}** = market required rate of return.

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**B) GORDON GROWTH MODEL**

The constant growth valuation model is presented in the following equation:

Where:

P** _{o}** = value of common stock,

D** _{1}** = per-share dividend expected at the end of year 1,
k

**= required return on common stock,**

_{s}g = constant rate of growth in dividends.

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To find the cost of common equity, simply solve the cost of equity
**(k**_{e}**), which is the same as the required rate of return on equity (k**** _{s}**).

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**C) BOND YIELD PLUS RISK PREMIUM**

Another way to compute the cost of equity involves adding a risk premium to the firm's cost of debt.

This method is quite intuitive since stockholders will always require a premium above the yield earned by the firm's bondholders, in order to compensate for the additional risk of owning equity. This method is specified in equation:

Where:

δ= the equity risk premium.

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We now have three separate methods for finding the cost of equity,
k** _{e}**. The following example requires the utilization of all three

methods. Work through it carefully to make sure you understand each one.

*Skiing Tours Inc. has a beta of 1.2. The risk-free rate is 10 percent, *
*and the market risk premium (k*_{m}*-R*_{f}*) is 5 percent. *

*Skiing Tours is a constant growth firm, which just paid a dividend of *

*$2.04, and its common stock sells for $27.- per share. The firm's *
*before-tax cost of debt, k*_{d}*, is 12% and their policy is to use a risk *
*premium of 4 percentage points when using the bond-yield-plus-risk-*
*premium method to find k*_{e}*. *

*Flotation costs on new common stock total 10 percent, and the firm *
*is expected to grow at a constant 8% indefinitely. *

*Find the cost of equity using all three methods.*

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Using the CAPM approach, the solution is as follows;

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Using the constant growth model, the solution is as follows:

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Finally, the bond yield plus risk premium approach would yield the
following solution for k** _{e}**.

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The previous example was set up so that the cost of equity would be 16% in all three cases. In practice, however, the three estimates will usually be quite different.

Then it is the responsibility of the analyst to decide whether to throw one or more of the estimates out, or whether to use some type of average value.

Reconciling the estimates is totally at the discretion of the analyst,
so it is important to have a good understanding of the problems
associated with each method in order to arrive at the best possible
**estimate of k**** _{e}**.

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**COST OF CAPITAL**

**D) RETAINED EARNINGS VERSUS NEW EQUITY**

The equity financing component of about eighty percent of all new projects comes from retained earnings.

However, when retained earnings are used up, the firm must issue new common equity in order to maintain the same mix of debt, preferred stock and common equity in its capital structure.

* The cost of new common equity is always higher than the cost of *
retained earnings due to the costs associated with issuing the new

**stock (flotation costs).**The Gordon Growth Model is used to calculate the cost of new equity, with the flotation costs being factored into the price of the stock in the numerator of the equation.

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The process is illustrated in the following example.

*Assume Siding Tours Inc. (from the previous example) has flotation *
*costs of $2.00 per share for the issuance of new common equity. *

*What is the cost of this new equity, and haw does this compare with *
*the cost of retained earnings?*

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To solve this problem, let's first recalculate the cost of retained earnings, using the original data.

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Now we can factor flotation costs into the denominator of the equation and solve for the cost of new common equity.

The price of the common stock in the denominator of the equation is now the net proceeds received by the firm from the issuance.

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Note that the flotation costs associated with the issuance of new
**equity drove k**** _{e}** from 16% to 16.8%.

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**Finding the Weighted Average Cost of Capital**

* The weighted average cost of capital after-tax (WACC) is found *
by weighting the cost of each specific type of capital in the firm's
capital structure to find the average cost of funds over the long run.

To determine the WACC, simply add the total dollar amount of all capital in a firm's capital structure and divide each individual capital component by the total to find its proportion.

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*Velvet Corporation has the following capital structure, as determined *
*from the company's balance sheet:*

*Determine each component's weight for use in the WACC.*

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To solve this problem, we must recognize that the total dollar amount of financed capital is $500,000. Then, simply divide the amount of the financed capital being used by $500,000 as is illustrated in the following solution.

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Once the weights have been determined, the cost of each component is multiplied by its weight to determine the WACC.

Where:

ω**d** = proportion of debt in the target capital structure,
k** _{d}** = cost of debt,

ω**p** = proportion of preferred stock in the target capital structure,
*k**_{p}* = cost of preferred stock.

ω**e** = proportion of equity in the target capital structure,
*k*** _{e}** = cost of equity,

*T = the firm's tax rate.*

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**The Marginal Cost of Capital Schedule**

The firm's weighted average cost of capital is a key input in the decision making process, so it is important to understand that as new amounts of financing are introduced, the WACC changes.

With increased amounts of financing, the cost of the various types of financing will increase due to the increased risk to the fund's

suppliers.

The weighted marginal cost of capital is simply the WACC at any point in time that has been adjusted to reflect new financing. To calculate the WMCC, the breaking point, which is the level of total new financing at which the cost of one of the financing components rises, thereby causing an upward shift in the WMCC has to be

calculated.

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The breaking point can be calculated by using the following equation;

Where:

BP_{i}*= breaking point for financing source i,*

TC_{i}*= amount of funds available from financing source i at a *
given cost

ω**i** = capital structure weight

### .

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Once the break points have been calculated, the WMCCAT over the various ranges must be determined.

When these new ranges are grouped together, they can be used to prepare a WMCCAT schedule, which is a graph that relates the firm's WACCAT to the level of new financing.

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**THE INVESTMENT OPPORTUNITIES SCHEDULE (IOS)**

The firm's investment opportunities schedule is simply a ranking of

projects from the highest return to the lowest return. The IOS graphically presents the return on the different investment opportunities facing a firm and the marginal cost of capital. If a project's return is above the

WMCCAT, the project should be accepted. If the project falls below the WMCCAT, the project should be rejected.

In the above graph, the solid line is the return on three different

investments available to the firm. It is called the investment opportunity schedule. The dotted line is the cost of capital at various amounts of investment. This is the marginal cost of capital schedule. In the above example, projects A and B would be accepted and project C would be rejected.

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 1:**

**Cost of debt**

*Find the cost of debt for a bond that is selling for $950, has 5 years *
*to maturity and has a coupon interest rate of 10%. Assume the tax *
*rate is 40%.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 1- Solution:**

The cost of debt is found using the procedure shown earlier for computing yield to maturity. The difference is that the market price is adjusted to reflect any discounts, fees, or costs that the firm must pay. In this problem, no costs are mentioned so we assume that there are none. The YTM can be found easily on a financial

calculator or by using the approximation equation

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 1- Solution:**

The final step is to convert the cost of debt to the after-tax cost of debt using equation.

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 2:**

**Cost of long-term debt**

*Laurie's Rose Distributors is contemplating expansion by selling*

*public bonds. Her investment banker advises that the bonds should *
*sell for $1,100 less 3% for fees. *

*If the bonds have a coupon rate of 10% and mature in 10 years, *
*what is Laurie's cost of debt? Assume a 40% tax rate.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 2- Solution:**

The YTM estimation equation can be used. The market price will be

$1,100 - (1.100 x 0.03) = $1,067.

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 3:**

**Cost of preferred stock**

*Compute the cost of preferred stock financing if the annual dividend *
*is $2, the current market price of the stock is $23, and the flotation *
*costs are $1 per share.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 3 - Solution:**

Preferred stock pays a dividend in perpetuity, meaning forever. By rearranging the equation for the value of perpetuity we can find the return (cost) for preferred stock. No tax adjustment is needed since preferred stock dividends are not tax deductible.

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 4:**

**Cost of equity**

*Compute the cost of equity if common stock is selling for $35.-, *
*the growth rate is expected to be 5% and next year's dividend *
*is expected to be $3.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 4 - Solution:**

Given the available information, the only model that can be used is the Gordon Growth Model. By rearranging the term we find that:

Note: The most common mistake made when using this equation is to mix decimals and percentages.

For example, avoid the following: K** _{s}** = 3/35 + 5% = .086 +5 =
5.086%.

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 5:**

**Cost of equity**

*Compute the cost of equity if the firm's beta is 1.12, the risk free rate *
*of interest is 7% and the return on the market is 13%.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 5 - Solution:**

The only method to solve for the cost of equity that is possible, given the above information, is the CAPM,

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 6:**

**Cost of equity**

*Compute the cost of equity if new stock must be sold and the *

*flotation costs are expected to be 5% of the current market price of *
*the stock. The stock is selling for $35.-, the growth rate is expected *
*to be 5% and the next dividend is expected to be $3.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 6 - Solution:**

When flotation costs exist, the cost to the firm rises because, although the firm pays out the same amount, it receives a lesser

amount. In this example, when stock is sold, the public will pay $35.- per share, but the firm will receive only $33.25 (35 X (1-.05)) of the proceeds.

**Note: Compare this result to example 4. The only difference was **
* that flotation costs are included here. As a result, the cost of *
equity increases.

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 7:**

**Weighted average cost of capital**

*Compute the weighted average cost of capital for Bob's Crystal Pool *
*Supply given that the target capital structure is 60% common equity, *
*30% debt and 10% preferred stock. K*_{d}*= 10%(after-tax), K*_{p}*s 12%, *
*and K*_{s}*=13.7%.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 7 - Solution:**

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 8:**

**Marginal cost of capital schedule (simple)**

*Suppose that Richard's Pool Basketball Inc. had a target capital*
*structure of 10% preferred, *

*30% debt and *
*60% equity. *

*Its tax rate is 40%. The firm's dividend next period will be $2 and the *
*growth rate is expected to be constant at 5%. The current price of *
*that stock is $20. *

*If new equity is sold the firm expects to get $18 from the sale. The *
*pretax cost of debt is 12%. The cost of preferred is 14%. The firm *
**has $10 in retained earnings. **

*What is the marginal cost of capital schedule?*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 8 - Solution:**

These problems can seem difficult because a number of steps are involved.

First, you must find the cost of each component.

Second you must find breakpoints where the costs change.

Third, you must compute the weighted average cost at each of the breakpoints.

Finally, you can graph the results.

*Step 1: Compute the cost of each component *
a. retained earnings - K** _{e}** = $2/20 + 0.05 = 0.15
b. new equity - K

**= 2/18 + 0.05 = 0.161**

_{e}*Step 2: Compute the break points *

$10/0.6= $16.67

*Step 3: Compute the cost below and above the breakpoint *
a. Cost of capital $0 to $16.67

WACC = 0.6(0.15) + 0.3(0.12)(1-0.4)+ 0.1(0.14) = 0.126 b. Cost of capital above $16.67

WACC = 0.6(0.161) + 0.3(0.12)(0.6) + 0.1(0.14) = 0.132

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 8 - Solution:**

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 9:**

**Marginal cost of capital schedule (complex)**

*Linda's Drive-through Lumber Express has a target capital structure *
*of * *60% common stock,*

*30% debt and *

*10% preferred stock. *

*The cost for retained earnings is 15% and the cost of new or *

*external equity is 16%. Linda's Lumber expects to have $20 million *
*of retained earnings available. Linda can sell $15 million of first-*

*mortgage bonds with an after-tax cost of 9%. The firm's bankers feel *
*the company can sell $10 million of debentures with a cost of 9.5% *

*after-tax. Additional debt would cost 10% after-tax. The cost of *
*preferred stock is 14%. *

*Compute the marginal cost of capital schedule and breakpoints.*

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 9 - Solution:**

*Step 1: Find the breakpoints*

To find the breakpoints divide the amount of money available at a particular cost by the proportion of the target capital structure made up by that component.

a) $20 million/0.60 = $33.33 million b) $15 million/0.30 = $50 million c) $10 million/0.30 = $33.33 million

$20 million can be raised from retained earnings before the cost of retained equity must go up due to flotation costs incurred by selling stocks publicly. $15 million in low cost debt can be sold before more costly debt roust be used. After the $10 million or 9.5% debt is gone, 10% debt must be used.

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 9 - Solution:**

*Step 2: Compute the WACC for each break point*

Look over the breakpoints and find the lowest one. There are two at

$33 million, but the one for debt is after the $50 million breakpoint.

Therefore, the first actual breakpoint is at $33 million due to using up retained earnings. The WACC will be computed using the lowest

cost of each component.

WACC**0 to $33 million** = 0.6(0.15)4. 0.3(0.09) + 0.1(0.14) = 0.1310 =
13.10%

The next breakpoint occurs because Linda's runs out of first- mortgage bonds. A total of $50 million will be raised before this

occurs. In the range between $33 million and $50 million equity will cost more but all other components will slay the same as above.

WACC **$33million to $50million** = 0.6(0.16) + 0.3(.09) + 0.1(0.14) = 0.137

=13.7%

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 9 - Solution:**

The next breakpoint occurs because Linda's runs out of 9.5% debt.

A total of $83 million will have been raised when this happens.

WACC **$50million to $83million** = 0.6(0.16) + 0.3(0.095) + 0.1(0.14) = 0.138

= 13.8%

The last breakpoint is when the firm must begin using 10% debt.

WACC **over $83million** = 0.6(0.16) + 0.3(0.10) + 0.1(0.14) = 0.14 = 14%

**COST OF CAPITAL – Sample Problems and**

**COST OF CAPITAL – Sample Problems and**

**Solutions**

**Solutions**

**Example 9 - Solution:**