The Cost of Debt Because Congress allows you to deduct from your taxable income the

interest you paid, how much does this dollar of debt really cost you? It depends on your marginal tax rate—the tax rate on your next dollar of taxable income. Why the marginal tax rate? Because we are interested in seeing how the interest deduction changes your tax bill in order to see how we will compare your taxes with and without the interest deduction.

Suppose that before considering interest expense you have $2 of taxable income subject to a tax rate of 40%:

Taxes = $2.00 × 0.40 = $0.80

Suppose your interest expense reduces your taxable income by 10, reducing your taxes from 80 cents to 76 cents:

Taxes = $1.90 × 0.40 = $0.76

By deducting the 10-cent interest expense, you have reduced your tax bill by 4 cents. You pay out the 10 cents and get a beneﬁt of 4 cents. In effect, the cost of your debt is not 10 cents, but 6 cents—4 cents is the government’s subsidy of your debt ﬁnancing!

We can generalize this beneﬁt from the tax deductibility of interest. Let r d represent the cost of debt per year before considering the tax deductibility of interest, r * d represent the cost of debt after considering the tax deductibility of interest, and τ be the marginal tax rate. The effective cost of debt for a year is:

(11-1) Suppose that you borrow $100,000 and must repay the amount bor-

rowed plus $10,000. Also suppose that your tax rate is 40%:

$10,000 r d = ------------------------- = 10%

the effective cost of debt is: r * d =

0.06 or 6% per year Creditors require a return of 10% per year on the funds they lend us.

But it only costs us 6% per year.

THE FUNDAMENTALS OF VALUATION

In our example, the required rate of return is easy to figure out: We borrow $100,000, repay $110,000, so your lender’s required rate of return is 10% per year. But your cost of debt capital is 6% per year, less than the required rate of return, thanks to Congress. Most debt financ- ing is not as straightforward, requiring us to figure out the yield on the debt—the lender’s required rate of return—given information about interest payments and maturity value.

Let’s look at an example of the ﬁrm’s cost of a straight coupon bond. Suppose a ﬁrm issues new bonds that have a face value of $1,000, mature in 20 years, and pay interest at a rate of 10% semiannually. If these bonds are issued at face value, the required rate of return of this

new debt capital, r d , is the yield-to-maturity, YTM, of the bonds. The yield-to-maturity for these bonds is the discount rate that causes the present value of the future cash ﬂows—the interest and maturity value— to equal today’s price of the bonds:

Present value of a coupon bond = Present value of interest payments + Present value of maturity value

Let

C indicate the interest payment, t the period, T the number of periods left until maturity, M the maturity value, and r the six-month yield. Then putting together the present values of the interest payments and the maturity value:

C M Present value of a coupon bond =

∑ (11-2)

--------------------- + ----------------------

1 + r t T 1 ( d ) ( d ) or, stated differently,

1 Present value of a coupon bond = C ------------------ t + M -------------------- ∑ 1

↑ Present value

Discount annuity factor

factor Investors are willing to pay today a price for the bond that reﬂects

the present value of its future cash ﬂows, so today’s price is the bond’s present value. Let’s apply this valuation to the bond in our example, solving ﬁrst for the six-month yield, r, and then translating this six-

month yield into an annual yield-to-maturity, r d :

The Cost of Capital

Present value of bond = $1,000

Interest, C = × ------------ $1,000 10% = $50 every six months

Number of periods, T = 20 × 2 =

40 six-month periods

Maturity value, M = $1,000

We solve for r, the six-month yield:

∑ ------------------ + ----------------------

r = 5% per six-month period

This six-month yield is equivalent to an annual yield, r d : r d = 0.05 × 2 = 0.10 or 10%

We can solve for r d in two ways:

■ Trial and error—try different values for r d until the right-hand side of the equation (the discounted value of interest and principal) is equal to the left-hand side (the value of the bond), or

■ Using calculator or spreadsheet programs. As we saw in Chapter 9, the yield obtained from doubling the six-month

yield is called the bond-equivalent yield. Suppose the firm is able to issue the 10% bonds at a price of $900 per bond and interest is not deductible from the firm’s income. The cost of debt would be greater than 10% because we are paying 10% based on the face value of $1,000, but we only get the use of $900. Using the equation for the present value of the bond, we first identify what we know:

Present value of bond = $900

Interest, C = $1,000 × ------------ = $50 every six months

THE FUNDAMENTALS OF VALUATION

Number of periods, T = 20 × 2 =

40 six-month periods

Maturity value, M = $1,000

Again, solve for r

------------------ $50 + ---------------------- ∑ $1,000

r = 5.6342% per period

which we convert into an annual yield: r d = 0.056342 × 2 = 0.112685 or 11.2685%

In this case, the return expected on the bond (the lender’s [the investor’s] required rate of return) and the cost of funds for the ﬁrm (the cost of debt) are 11.2685% since there is no other cost associated with raising funds from debt. Any costs associated with the issuance of debt— borrowing—are incorporated directly into the calculation of the cost of debt to the issuer since the present value of the bond is the proceeds of the bond issuer—the price of the bond less costs of issuance.

Now let’s consider the costs of issuance, called the ﬂotation costs, which are the payments to lawyers, accountants, and investment bank- ers who assist the ﬁrm in issuing debt securities (as well as preferred stock and common stock). There are also SEC registration fees. If these bonds are sold at $900 per bond, investors will require the rate of return of 11.2685%, as we just determined.

But if the firm only gets $890 per bond (the flotation costs are $10 per bond), this means the cost to the firm is more than 11.2685% per year:

------------------ + ∑ ----------------------

r = 5.7040% per period

which we convert into an annual yield: r d = 0.057040 × 2 = 0.11408 or 11.408%

The Cost of Capital

Flotation costs of $10 per bond increase the cost of the bond to the ﬁrm from 11.2685% to 11.408%. This rate is referred to as the all-in-cost

of debt. But investors pay $900 a bond, which reﬂects their required

rate of return of 11.2685%. Next we consider the tax deductibility of interest. If a dollar of interest is paid, is the interest cost to the ﬁrm one dollar? No, because interest on debt is deductible for tax purposes. Since interest expense reduces income,

a dollar of interest reduces taxable income by one dollar. If the ﬁrm issues the 10% bonds at par, with interest paid annually and no ﬂotation costs, and has a 40% marginal tax rate, the after-tax cost of debt is 6% per year:

0.06 or 6% per year If the ﬁrm issues the 10% bonds at 90 (this is the way prices are quoted

in the bond market), meaning 90% of the bond’s face value, or $900 per bond with a $1,000 par value, with no ﬂotation costs, the before-tax cost of debt is 11.2685% and the after-tax cost of debt is:

r * d = 0.112685 1 0.40 ( – ) = 0.06761 or 6.761% per year But to be more complete we must include ﬂotation costs, therefore

using the all-in-cost of debt. If the ﬁrm issues the 10% bonds at 90, receiving only $890 per bond after ﬂotation costs, the after-tax cost of

debt, , r d * is:

r d * = 0.11408 1 0.40 ( – ) = 0.6845 or 6.845% per year Therefore, the tax deductibility of interest reduces the cost of debt to

the borrower. The greater the marginal tax rate, the greater this beneﬁt from deductibility and hence the lower the cost of debt. For example, the cost of the 10% bonds for different marginal tax rates, with annual interest and no ﬂotation costs, is shown in Exhibit 11.1.

EXHIBIT 11.1 Cost of Debt at Varying Tax Rates

Marginal Tax Rate After-Tax Cost of Debt () r * d

THE FUNDAMENTALS OF VALUATION

Not all bonds are straight coupon bonds. Suppose you issue a zero- coupon bond. Though no interest is paid in cash each year, there is implicit interest, which you are allowed to deduct each period for tax purposes.

Suppose you issue a zero-coupon bond at the beginning of 1999 that matures in ﬁve years and has a face value of $1,000. If this bond does not pay interest—not explicitly at least—no one will buy it at its face value. Instead, investors will pay some amount less than the face value, and their reported return will be based on the difference between what they pay for the bond and (assuming they hold it to maturity) the face value. If the bonds are issued at 60% of the $1,000 face value or $600, what is the yield-to-maturity? We compare the present value of the bond, which is $600, with the maturity value, which is $1,000. We can start with the basic valuation relation:

FV = PV(1 + r) t

where FV = future value

PV = present value r

= interest per period T = number of periods

Modifying this to ﬁt our needs:

Maturity value = Present value(1 + r ) T d

First, let’s identify the known values: Maturity value, M

Present value, PV

Number of periods, T=5 We then insert these known values into this equation and solving for

the one unknown, r d : $1,000 5 = $600 1 ( + r

------------------ = ( 1 + r 5 d )

1.6667 5 = ( 1 + r

d = 1.6667 – 1 = 0.1076 or 10.76% per year

The Cost of Capital

EXHIBIT 11.2 Implicit Interest on a Zero-Coupon Bond

Implicit Year

End-of-Period

Previous-Period

Accrued Value

97.01 The implicit interest over the life of the bond is the difference

between the face value and the issue price, $600. The implicit interest for a given year is the growth in the value of the bond during the year

that is expected at the time the bond is issued. 1 For example, the implicit interest for the ﬁrst year is the difference between the $600 issue price and $600 grown one year—accrued—at 10.76% per year:

Implicit interest in ﬁrst year = $600(1 + 0.1076) – $600 = $64.56

↑ Value of the bond

Issue at the end of the

price

ﬁrst year

We can see the growth in the implicit interest (due to compounding) for each year, comparing the end of period accrued value of the bond with the previous period’s accrued value, as shown in Exhibit 11.2.

Each period, we deduct the implicit interest on our tax return to arrive at taxable income. There is a cost of raising this capital, just as there was with a coupon bond. For example, in year 5, the accrued interest is $64.56, which is 10.76% of the beginning of the year value of $600.00. The issuer deducts $64.56 interest from its income for tax purposes. If the marginal tax rate is 40%, the effective cost per bond is:

r * d = 0.1076 1 0.40 ( – ) = 0.646 or 6.46% per year As you can see from our calculations with a straight coupon bond

and a zero-coupon bond, the starting point is the investor’s required 1 The implicit interest does not depend on what happens to the actual price of the

bond, rather it depends only on the price of the bond at issuance and its time path, assuming the bond is held to maturity.

THE FUNDAMENTALS OF VALUATION

rate of return. Once we have that rate, we adjust it for ﬂotation costs and the issuer’s tax-beneﬁt from interest deductibility.

The Cost of Debt Because Congress allows you to deduct from your taxable income the