Investments

Suppose that the CFO of Textron, in anticipation of future capital requirements, decided in

December 2001 to float a AAA $100 million, 20-year bond at an annual interest rate of 9

percent.By April 2002, interest rates on 20-year AAA bonds had increased to 10 percent.

What is Textron’s pretax cost of debt capital?

Answer:In April 2002, Textron would not want to take a risk-free project yielding 9.5 per-

cent even though it had previously borrowed at 9 percent.It could do better by repurchas-

ing its outstanding bonds.Given the increase in interest rates, the bonds would have fallen

to a level that provides investors with a 10 percent return, which is Textron’s pretax cost of

debt capital.

¹²In

contrast, investment-grade debt typically has a beta of about .2.

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The Effect of Leverage on a Firm’s WACC When There Are No Taxes

In determining the relevant discount rate for a firm’s expected unlevered cash flows, a

manager needs to know how leverage affects the firm’s WACC. Anaive manager might

note that the cost of debt is typically less than the cost of equity, so that an increase in the

proportion of debt financing would reduce its weighted average cost of capital. However,

this logic ignores the fact that an increase in a firm’s debt level also increases the risk of

its equity and (usually also its) debt and therefore raises the required return of each source

of financing. In the absence of taxes, the increase in the risk of the two sources of capi-

tal is exactly offset by a shift in the WACC formula’s weights toward the cheaper source

of financing, leaving the WACC unchanged. Result 13.3 states this formally.

Result 13.3

In the absence of taxes and other market frictions, the WACC of a firm is independent ofhow it is financed.

Result 13.3 implies what this chapter suggested earlier: in the absence of taxes, the

WACC is the same as the unlevered cost of capital as a consequence of both being

identical to the expected return of assets (in the no-tax case). Thus, Result 13.3, despite

being at the heart of modern corporate finance, is an insight derived from portfolio the-

ory. Because the assets of the firm are identical to a portfolio of equity and debt, their

expected return

ED
r r r	(13.10)
AD EED ED

The WACC equation (13.8), with T0, is identical to the right-hand side of equation

c

(13.10). Hence, as long as the firm’s leverage ratio, D/E, does not affect r, it will not

A

affect the WACC.

It should not be surprising that early versions of this theorem were presented in

the language of Result 13.3. The leverage shift will not alter the firm’s WACC.

The “Modigliani-Miller Theorem” (examined in detail in Chapter 14), says that, in

the absence of taxes and other frictions that might alter eitherror expected future cash

A

flows, the financing mix is irrelevant for valuation.

Example 13.11:The Effect of Debt on the WACC without Corporate Taxes

Divided Technologies has no debt financing and has an equity beta of .5.Assume that the risk-

free rate is 8 percent, the CAPM holds, the expected rate of return of the market portfolio is 14

percent, and there are no corporate taxes.If the firm can repurchase one-third of its outstand-

ing shares and finance the repurchase by issuing risk-free debt carrying an 8 percent interest

rate, what will be the effect of a debt-financed share repurchase on its WACC and cost of equity?

Answer:The cost of capital of Divided Technologies before issuing risk-free debt is its

cost of equity:

8% .5(14%8%)11%

After the repurchase, Divided Technologies has a 1 to 2 debt to equity ratio, but the same

WACC D/E1.5.The WACC’s (2/3, 1/3) weighted average of the cost of equity and the

8 percent cost of debt can only be 11 percent if the cost of equity increased to 12.5 per-

cent.(Equation (13.5) could also have been used here.)

Exhibit 13.4 graphs the WACC, the cost of equity capital, and the cost of debt

capital as a function of the leverage ratio, D/E,based on the figures given for Divided

Technologies in Example 13.11. The WACC is a weighted average of the cost of equity,

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D

EXHIBIT13.4WACC, Cost of Equity, and Cost of Debt vs. with No Taxes

E

	Divided Technologies
Return

Cost of

equity after

repurchase

r E

WACC before

repurchase

12.5%

11%WACC

WACC after

^repurchaser D

8%

	D/E
.5

the upwardly sloping line beginning at 11 percent, and the cost of debt, the horizontal

line at 8 percent. The WACC line is horizontal because, as D/Eincreases, the WACC

weight D/(D E) on the 8 percent horizontal line rincreases while the complemen-

D

tary weight on the upward sloping cost of equity line decreases.

If the cost of equity did not increase as leverage increases, the WACC would

decline as leverage increases. However, as D/Eincreases, which means a movement

to the right on the graph, the increase in rexactly offsets the effect from placing

E

greater weight on the lower cost of debt line, resulting in a horizontal (blue) line

for the WACC. When the WACC line is horizontal, it means that the WACC is unaf-

fected by leverage.

The Effect of Leverage on a Firm’s WACC with a Debt Interest Corporate Tax Deduction

The picture of the WACC in Exhibit 13.4 changes when a tax gain is associated with

leverage. With corporate taxes, the WACC declines with an increase in debt because,

relative to all-equity financing, part of the cost of financing is borne by the govern-

ment. To analyze this issue, note that equation (13.10) is valid even when there are

taxes. Rearranging this equation to obtain the equity term in the WACC equation,

(E/(D E))r, we get

E

ED

rr r

D E^EAD E^D

Substituting this into the WACC equation (13.8) yields

D
WACCr T r	(13.11)
AD EcD

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482Part IIIValuing Real Assets

Recall, now, that earlier in this chapter we learned that the expected return of assets is

a portfolio-weighted average of the expected returns of the unlevered assets and the

debt tax shield. As leverage increases, the portfolio weight on the debt tax shield

increases. It would be extremely aberrational to have a debt tax shield with higher beta

risk than the unlevered assets. Because of this, it is safe to conclude that the expected

return on assets does not increase as leverage increases.

Equation (13.11)¹³thus implies the following result:

Result 13.4

When debt interest is tax deductible, the WACC will decline as the firm’s leverage ratio,D/E,increases.

The Adjusted Cost of Capital Formula.In the Hamada model, with static

perpetual debt, both debt and its tax shield are risk-free, implying rr, 0,

DfTX

and TXTD.In this case, the portfolio-weighted average of the expected re-

c

turns of the unlevered assets and the debt tax shield, given in equation (13.1b),

simplifies to

TDTD

r cc

1r r

^AD E^UAD E^D

If we substitute this equation into equation (13.11), equation (13.11) reduces to

Modigliani and Miller’s (1963) adjusted cost of capital formula, which gives the

firm’s WACC as a function of its debt to value ratio:

D

WACCr1T

UA^cD E

An application of this formula is provided in Example 13.12.

Example 13.12:The Effect of Leverage on the WACC with Corporate Taxes

Example 13.8 found that United Technologies (UT), with liabilities consisting of 20 percent

debt and 80 percent equity, had a WACC of 13.2 percent when the corporate tax rate was

34 percent.In a financial restructuring designed to raise to 40 percent the proportion of UT

financed with debt, UT issues debt and buys back its equity with the proceeds.Compute the

firm’s new WACC given the assumptions of the Hamada model.

Answer:To calculate the firm’s new cost of capital, first estimate UT’s unlevered cost of

capital.From the adjusted cost of capital formula, a WACC of 13.2 percent with 20 percent

debt financing corresponds to an rthat satisfies

UA

.132r[1(.34)(.20)]

UA

Therefore, r.1416.Plugging this value into the adjusted cost of capital formula at a 40

UA

D

percent ratio leaves a WACC satisfying

D E

WACC .1416[1 (.34)(.4)] .122

Therefore, the new WACC is 12.2 percent.

Graphing the WACC, Cost of Debt, and Cost of Equity with Corporate Taxes.

Exhibit 13.5 graphs United Technologies’WACC, cost of equity capital, and cost of

¹³Equation

(13.11) applies to risky debt as long as one adjusts Tto account for nonuse of the tax

c

shields, as discussed earlier in the chapter.

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EXHIBIT13.5	D WACC, Cost of Equity, and Cost of Debt vs. with Corporate
	E Taxes

	United Technologies
Return

r E

14.2%
13.2%
12.2%
	WACC

8%(1 – .34)

r D (1 – T ) c

		D/E
.25	.67
Before	After

debt capital as a function of the leverage ratio, D/E,when the corporate tax rate is 34

percent, based on the figures in Example 13.12. Note, as suggested earlier, that as D/E

increases, more weight is placed on the bottom horizontal line, r(1T), in the com-

Dc

putation of the WACC. However, in contrast to the no-tax case, the increase in ras

E

D/Eincreases is insufficient to offset the additional weight on the lower debt cost

r(1T). Hence, unlike the pattern shown in Exhibit 13.3, the WACC in Exhibit 13.4

Dc

declines from the starting point of 14.16 percent (r)as D/Eincreases. The WACC

UA

continues to decline as D/Eincreases, up to the point where the firm eliminates all cor-

porate taxes. Thereafter the WACC is flat.

Dynamic Perpetual Risk-Free Debt.Up to this point we have assumed that the firm

has a fixed amount of debt outstanding. Alternatively, if the firm wants to maintain a

given ratio of debt to equity, it will need to issue new debt and repurchase equity as

the firm’s (or project’s) value rises and issue equity to retire debt as the value of the

firm (or project) falls.

In the Miles and Ezzell (1980, 1985) model, Dis perfectly correlated with the value

of the unlevered assets of the firm and thus the tax savings from debt issuance are per-

fectly correlated with the prior period’s value of the unlevered assets. This implies that,

at least approximately,

TXUA

implying that the expected returns of assets, unlevered assets, and the debt tax shield

are about the same. As periods become arbitrarily short, this equality between the

expected returns of the assets and the unlevered assets holds exactly, rather than approx-

imately, in which case equation (13.11) reduces to

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484Part IIIValuing Real Assets

D

WACCr Tr

^UAD E^cD

Dynamic updating of debt to maintain a constant leverage ratio has implications

for both the WACC and the APVmethods because it affects asset betas and, thus, the

formulas for leveraging and unleveraging equity betas. The Miles and Ezzell model,

for example, which assumes risk-free debt, implies that only the tax shield associated

with the first interest payment, which has a present value of TDr/(1 r) is certain,

cff

and the remainder of the tax shield has the same beta risk as the unlevered assets. Thus,

the beta of all the assets is a portfolio weighted average of 0 and with the portfo-

UA

lio weight on 0 beingTDr/(1 r) and the portfolio weight on being

cffUA

(1T)Dr/1 r. This implies that the of the assets can be expressed as

cff

TDr

1 c f

^AD E 1 r^UA

f

When the updating interval is short, the term in brackets is very close to one, and thus

this formula says that with dynamic updating of debt, the beta of the assets and the beta

of the unlevered assets are approximately the same. As periods become arbitrarily short,

the value of the first debt interest payment becomes infinitesimal and the two betas

become exactly the same. As suggested earlier, this implies that the no tax versions of

equations (13.6) and (13.7) can be used to adjust equity betas for leverage when taxes

exist, provided that the firm dynamically maintains a constant ratio of DtoE.

One-Period Projects.If a project lasts only one period, there is only a single inter-

est payment. In this case, the present value of the tax shield, TDr/(1 r), is the same

cff

as the present value of the riskless component of the tax shield in the Miles and Ezzell

model. Thus, the beta of the assets when there is debt that lasts only a single period

has the same formula as that in the Miles and Ezzell model. In general, the Miles and

Ezzell formulas for the WACC and the beta of equity apply to one-period projects. This

is because the fraction of the assets that is risk-free is identical in the two cases.

Which Set of Formulas Should Be Used?If the unlevered assets of the firm have a

significant risk premium, the models of Hamada and Miles/Ezzell generate different

adjusted cost of capital formulas, as well as different formulas for leverage adjustments of

equity betas. Dynamic updating of debt and equity to maintain a constant leverage ratio,

as in Miles and Ezzell, tends to generate larger WACC changes for a given leverage change

than the Hamada model, which assumes that the amount of debt does not change.

Many firms, particularly those with low to moderate amounts of leverage, try to

maintain a target debt to equity ratio, but update rather slowly, perhaps because the cost

of frequently issuing and repurchasing debt and equity is prohibitive. For such firms,

truth lies somewhere between the models of Hamada and Miles/Ezzell. The weighting

of the two models depends on which behavioral assumption the firm conforms to bet-

ter. Large firms, firms with existing shelf registrations, and those with a history of

repurchasing equity are likely to be more active in dynamically updating. Also, for proj-

ects with short lives, truth is probably closer to that given by the Miles and Ezzell for-

mulas (with the caveat that comparison firms are perpetual).

However, we noted earlier that for many highly leveraged firms, it is possible that

the debt tax shield has a negative beta. For firms and projects with this property, it is

important to use even a lower WACC and lower equity beta adjustments for leverage

than those suggested by the Hamada model.

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Finally, it is important to recognize that the field of corporate finance has yet to

develop formulas for how leverage changes affect the WACC, equity betas, and the

expected returns of assets, unlevered assets, debt tax shields, and equity in many real-

istic situations. Foremost among these is the case of the growing firm with debt tied

to its growth and reinvestment. This kind of problem, however, can often be analyzed

by a skilled practitioner using the PVmethod. For this reason, we are still puzzled by

the overwhelming popularity of the WACC method as a tool for valuation.

Evaluating Individual Projects with the WACC Method

The appropriate discount rate for a particular project must reflect the risk and debt

capacity of the project rather than the risk and debt capacity of the firm as a whole.

While the tracking portfolio analysis in Chapter 11 emphasized this in great detail, some

financial managers believe that they are losing money whenever a project returns less

than the firm’s WACC. Examples 13.13 and 13.14 provide an additional perspective on

why this line of thinking is fallacious by examining an extreme case where a risky firm

is evaluating a project that has no risk.

Riskless Project, Riskless Financing.In Example 13.13, the project has riskless cash

flows. The project’s financing also is riskless, consisting of debt with tax-deductible

interest payments. In this case, analysts can evaluate the project directly by comparing

the project’s proceeds with its financing costs. In contrast to the analysis of riskless

projects in Chapter 10, however, the analyst now has to account for the debt tax shield.

Example 13.13:Adopting Projects That Have Rates of Return Below Your

Cost of Capital

GECC, a subsidiary of General Electric with a 20% CAPM-based weighted average cost of

capital, leases private corporate jets.For a Gulfstream jet, it charges $500,000 per year on

its 10-year lease payable at the end of each year.The cash flows associated with such leases

are risk-free and once the lease is signed, it is not possible to get out of the lease.GECC

is in the process of closing a deal with the management of AMD (Advanced Micro Devices)

on one of these jets.AMD, which has an equity beta of one and a cost of capital equal to

12%, suddenly makes an intriguing offer.AMD offers to lease the jet for 10 years, but with

an upfront payment of $3 million in lieu of the 10 separate annual payments of $500,000.

What discount rate should GECC use to decide whether to accept the offer if the corporate

tax rate is 50 percent and the risk-free rate is 6 percent?

Answer:This is clearly a project that can be evaluated with the discounted cash flow

method.If GECC maintains the status quo, it is, in essence, forgoing a cash flow of $3 mil-

lion at date 0 in exchange for $500,000 per year in annual pretax cash flows at the end of

years 1–10.Note that if GECC uses its 20 percent cost of capital to discount the after-tax

cash flows of 10 annual payments, it computes the present value of the after-tax lease pay-

ments as $1.048 million, which would make it think that AMD was offering it a good deal at

$3 million (even though half of it must be paid in taxes to the government).However, using

the risk-free rate of 6 percent generates a present value for the ten payments (after tax) equal

to $1.84 million, which exceeds the $1.5 million they would otherwise receive after taxes from

the initial $3 million payment.In this case, AMD would be obtaining a good deal but GECC

would be obtaining a bad deal if it accepts the up-front payment.It should be clear that the

risk-free rate of 6%, rather than GECC’s 20% cost of capital, is the appropriate discount rate

here and that they should reject the offer.If GECC were to take the upfront payment, pay

taxes on it, and buy a 10-year annuity with the remaining cash, the payments on the annu-

ity would be computed at the 6 percent rate.It would generate only $203,801 a year, less

than the $250,000 it would earn after taxes from the $500,000 per year lease.

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Riskless Project, Risky Equity Financing.Example 13.14 is more difficult to ana-

lyze because the risk-free project is financed by an equity offering. The expected cost

of new equity financing is higher than the expected return of the project. However, as

we will see in this example, the risk-free rate is still the appropriate marginal, or incre-

mental, cost of raising capital for the project.

The relevant measure for the cost of capital of a project is the firm’s marginal cost

of capital, or the amount by which the firm’s total cost of financing will increase if it

raises an additional amount of capital to finance the project. In Example 13.13, this

amount was apparent because even though the firm’s original capital was composed

entirely of equity, risk-free debt could be used to finance the lease payments. However,

Example 13.14 shows that even if the company funds a new investment with equity,

the same concept applies: The marginal cost of capital for the project reflects the risk

of the project and not the risk of the firm as a whole.

Example 13.14:The Marginal Weighted Average Cost of Capital

Assume that United Technologies (UT) is an all-equity firm, has a market value of $1 billion,

and has a beta of 2.Given the expected rate of return on the market of 14 percent and the

risk-free rate of 8 percent, its cost of capital is 20 percent.The firm is considering a project that

costs $1 billion, but is risk free.Since UT generates no taxable income, it finances the project

by issuing additional equity.How does the company determine whether to accept or reject the

project? To simplify the example, assume that both existing projects and the new project have

perpetual cash flows with expected values that do not change with the cash flow horizon.

Answer:Discount the project’s cash flows at the 8 percent return and see whether the

discounted value exceeds $1 billion.This is equivalent to valuing the firm’s cash flows, both

with and without the project, using the appropriate WACC in each case.

To understand this point, note that at a 20 percent return, shareholders expect to earn

$200 million per year from UT’s existing projects on the $1 billion invested in the firm.

Assume, for the moment, that the new project is a zero-NPVproject.If management decides

to go forward with the project, the total risk of UT will decline as its risk falls from a beta of

2 to a beta of 1, which is the average of the betas of the firm’s existing assets and the beta

of the new project.With a beta of 1, UT’s cost of capital would then be 14 percent, the same

as the market portfolio’s expected return.With this return, shareholders expect to earn $280

million per year on the $2 billion invested in the firm.This is indeed what UT’s shareholders

will receive if the incremental cash flows from the new project are $80 million per year.Thus,

UT’s shareholders are indifferent about whether to adopt the project if it provides exactly $80

million per year in incremental expected cash flow.Note that discounting the $80 million per

year at 8 percent results in a $1 billion present value and a zero net present value.Thus,

8 percent is the correct discount rate to use for the project’s incremental cash flows.

If the project’s expected cash flows exceed $80 million per year, implying that UT’s

investors prefer project adoption, then the 8 percent discount rate will indicate that UT’s proj-

ect has a positive NPV.Analogously, if the expected cash flows are less than $80 million

per year, the 8 percent discount rate will indicate a negative NPVproject.

Example 13.14 illustrates that the marginal cost of capital, that is, the project’s

WACC, provides the appropriate hurdle rate for determining whether a project should

be selected. We also know from the value additivity concept, discussed in Chapter 10,

that the value created by an investment project equals the NPVof the project, calcu-

lated here by discounting the project’s cash flows at the project’sWACC and subtract-

ing from this value the initial expenditure on the project.

The Importance of Using a Marginal Weighted Average Cost of Capital.In Exam-

ples 13.13 and 13.14, shareholders gain from selecting risk-free projects whose rates of

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Chapter 13

Corporate Taxes and the Impact of Financing on Real Asset Valuation

487

return exceed the risk-free rate, but which return less than the firm’s weighted average cost

of capital. In Example 13.13, the project could be financed by risk-free borrowing, so that

accepting the project represented an arbitrage gain. The increase in cash flows from the

project exceeded the cash outflow from the financing. In Example 13.14, an all-equity-

financed firm used additional equity financing to fund the project. In this case, the proj-

ect created value for the firm by lowering its risk and thus the required WACC. Managers

who think firms cannot create value by accepting safe projects that yield 8 percent returns

when the firm as a whole has a cost of capital of 12 percent are forgetting to consider

how the firm’s risk, and hence its cost of capital, is affected by adopting the project.

Computing a Project WACC from Comparison Firms.The last two examples illus-

trate that the WACC of a firm is the relevant discount rate for the incrementalcash

flows of one of its projects only when the project has exactly the same risk profile as

the entire firm. In other words, the project must (1) have the same beta and (2) con-

tribute the same proportion as the entire firm to the firm’s debt capacity. If these con-

ditions do not hold, firms can apply the WACC method by finding another firm with

the same risk profile as the project being valued and using the WACC of the compar-

ison firm to discount the expected real asset cash flows of the project. Example 13.15

illustrates how this can be done.

Example 13.15:Adjusting Comparison Firm WACCs forLeverage

This extends Example 13.2, where the unlevered cost of capital for the Marriott restaurant

division was found to be 10.97 percent per year when the corporate tax rate is 34 percent.

Compute the WACC for Marriott’s restaurant division, assuming that the division’s debt capac-

ity implies a target D/E.4 and static perpetual risk-free debt, as in the Hamada model.

Answer:There are two ways to solve this problem.This example uses the Modigliani-

Miller adjusted cost of capital formula.Exercise 13.8 focuses on applying the WACC

DD/E

formula, equation (13.8), directly after releveraging the equity.Note that .

D E1 D/E

D.4

Hence .2857.Substituting this into the Modigliani-Miller adjusted cost of

D E1 4

D

capital formula at the target of .2857 yields a WACC of

D E

9.9% .1097[1 .34(.2857)]

This chapter’s computation of the risk- and tax-adjusted discount rate for Marriott’s

restaurant division, being fairly typical, serves as a blueprint for many of the project

valuations you may do in a practitioner setting. Here is a detailed summary of how we

ended up with the WACC above:

•

We recognized that Marriott’s restaurant division was not traded and thatanalysis of Marriott International, which does have traded stock, would not generate an appropriate discount rate because its risk is affected by the other divisions of Marriott, notably lodging.

•

We identified traded securities for firms that, as a consequence of their line ofbusiness, had unlevered assets that were comparable to those of Marriott’s restaurant division. We recognized that, for an “apples-to-apples comparison,”only the unlevered assets of the comparison firms provide discount rates thatare relevant to the restaurant division. This is because taxes and leverage alterthe risk of the comparison firms’assets and equity.

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488	Part IIIValuing Real Assets

•

To address these points, we estimated the equity betas of three firms with comparable unlevered assets; then, Example 13.1 unlevered the equity betaswith a formula, equation (13.7).14To reduce noise in the estimation process,Example 13.1 used the average unlevered beta as the CAPM input to computean unlevered cost of capital for the restaurant division.

•

To obtain the restaurant division WACC, Example 13.15 converted theunlevered cost of capital, obtained from the comparison firms, into a levered WACC with a formula—the Modigliani-Miller adjusted cost of capital formula,using the division’s target debt ratio for D/(D E). The resulting WACC,used to discount the unlevered cash flows of the restaurant division, generatesthe value of the assets of the restaurant division (including the asset componentgenerated by the debt tax shield).15

13.4	Discounting Cash Flows to Equity Holders

The valuation approaches discussed up to this point value the cash flows of real assets,

which accrue to the debt holders as well as to the equity holders. Because these cash flows

do not account for transfers between debt and equity holders, the decision rules that arise

from their valuation select projects that maximize the total value of the firm’s outstanding

claims; that is, the value of its debt plus the value of its equity. In a number of instances,

this decision rule conflicts with the objective of maximizing the value of the firm’s equity.

Positive NPVProjects Can Reduce Share Prices When Transfers to Debt Holders Occur

The last section examined two risk-free projects and showed that discounting their cash

flows at a risk-free rate was appropriate. This approach was correct as long as the objec-

tive is to maximize firm value. However, maximizing firm value is not always the same

as maximizing the firm’s stock price. The adoption of a positive NPVproject can trans-

fer wealth from equity holders to debt holders, which adversely affects share prices.

Example 13.16 points out that when these conflicts exist, discounting cash flows at a

risk-free rate may not be consistent with maximizing the firm’s stock price.

Example 13.16:When Discounting Riskless Cash Flows at a Risk-Free Rate