Risk Neutral Valuation(2)

Risk-neutral Valuation: A Gentle Introduction (2)

Joseph Tham

Abstract

This teaching note is a continuation of the previous teaching note on risk-neutral valuation. In Section One, we estimate the value of levered equity in a levered company in an M & M world with risk-free debt and without taxes. The structure of the presentation will facilitate the subsequent analysis with taxes. We find the value of the levered firm in two ways. First, we use the replicating portfolio method. Second, we use the risk-neutral approach, which is equivalent to the replicating portfolio method.

In Section Two, we analyze risky debt in a world without taxes. In Section Three, we introduce taxes. However, we continue to assume risk-free debt. The risk of the tax shield is indeterminate. In Section Four, we analyze risky debt in the presence of taxes and derive the relevant expressions for the returns to the equity and debt holders. JEL codes D61: Cost-Benefit Analysis G31: Capital Budgeting H43: Project evaluation Key words or phrases Risk-neutral valuation, Cost of Capital Currently, Joseph Tham (in collaboration with Ignacio Vélez-Pareja) is writing a book on cash flow valuation. Previously, he taught at the Fulbright Economics Teaching Program (FETP) in HCMC, Vietnam and worked with the Program on Investment Appraisal and Management (PIAM) at the Harvard Institute for International Development (HIID). Email address: [email protected].

Again, this teaching note is dedicated to the proverbial grandmother who is diligent, well read and intelligent but has not taken any course in finance. We have erred on the side of over-explanation and repetition rather than brevity, conscious of the risk of boredom and unavoidable loss of the soul in wit. Critical comments and constructive feedback for clearer explanations and further clarification on obscurities are welcome.

mailto:[email protected]

J.Tham, December 3, 2001 2

Introduction

This teaching note is a continuation of the previous teaching note on risk-neutral

valuation. We use the same framework but we change the context. We assume that the

reader has understood the material and examples presented in the previous teaching note,

and in this teaching note, we refer frequently to the results of the numerical examples in

the previous teaching note. Without reading the previous teaching note, this teaching note

may be difficult to follow.

In Section One, we estimate the value of the levered equity in a levered company

in an M & M world with risk-free debt and without taxes. The structure of the

presentation will facilitate the subsequent analysis with taxes. We find the value of the

levered firm in two ways. First, we use the replicating portfolio method. Second, we use

the risk-neutral approach, which is equivalent to the replicating portfolio method. In

Section Two, we analyze risky debt in a world without taxes.

In Section Three, we introduce taxes and continue to assume that the debt is risk-

free. However, we assume that the tax shield is risky. In Section Four, we analyze risky

debt in the presence of taxes and derive the relevant expressions for the returns to the

equity and debt holders.

Section One

Consider a simple economy with two states of nature (an up state and a down

state), no taxes and two investment opportunities: a risky investment and a risk-free

investment. We can either invest in an unlevered company or invest in units of

government bonds. The return on government bonds is the risk-free rate rf. The individual


investor can borrow and lend government bonds without limit. At the end of year 0, let

VUn(0,1) be the value of the unlevered firm and let D(0,1) be the price of a government

bond.

We calculate the value of the equity in a levered firm in two equivalent ways.

First, we construct a replicating portfolio that consists of α percent of the unlevered

company and β percent of cash (represented by purchase and sale of government bonds).

The portfolio “replicates” the payoff structure for the equity in the levered company and

therefore the value of the portfolio must be equal to the value of the equity in the levered

company. Second, we use risk-neutral valuation to find the value of the equity in a

levered company. As expected both methods yield the same answer.

Unlevered company

The cash flow process for the unlevered company is shown below.

Figure 1: Process (or tree) for the free cash flow of the unlevered company Year 0 1

FCF(1,1) = 1,200 VUn(0,1) = ? FCF(1,2) = 800

At the end of year 1, the free cash flow (FCF) for the unlevered company is

uncertain.1 Let FCF(1,1) be the free cash flow in the up state of nature and let FCF(1,2)

1. Since there are no taxes, it is obvious that the free cash flow (FCF) is also the net-of-tax FCF. In the

presence of taxes, the distinction between gross of tax and net of tax FCF will be important.


be the free cash flow in the down state of nature. The value of FCF(1,1) is $1,200 and the

value of FCF(1,2) is $800.

Government bond

The process for the cash flow to the holder of the government bond is shown

below.

Figure 2: Process (or tree) for the cash flow to the holder of the government bond Year 0 1

CFD(1,1) D(0,1) CFD(1,2)

At the end of year 1, let CFD(1,1) represent the cash flow to the debt holder in the

up state and let CFD(1,2) represent the cash flow to the debt holder in the down state. In

this special case, because we have assumed that the debt is risk-free, CFD(1,1) =

CFD(1,2). In other words, under both states of nature, the debt holder will receive the

same cash flow.

Return to investment in the unlevered company

Let ρ be the return to the equity holder for the investment in the unlevered

company. We have to make an assumption about the value of ρ based on CAPM or a

similar asset-pricing model. We assume that the return to unlevered equity ρ is 20% and

the risk-free rate rf is 10%. In addition, we assume that at the end of year 0, the value of

the unlevered company VUn(0,1) is $950. To be consistent with ρ = 20%, the objective


(or actual) probability for the up state of nature is 85% and the objective probability for

the down state of nature is 15%. The set of objective probabilities is P = {pU, (1 – pU)} =

{85%, 15%} where pU is the probability for the up state of nature at the end of year 1.

For the unlevered company, the expectation of the payoffs at the end of year 1

with respect to the objective probabilities P = {pU, (1 – pU)} = {85%, 15%} discounted

by ρ, equals the no-arbitrage value at the end of year 0. Let E(FCF(1,1:2)) be the

expectation of the free cash flows under the two states of nature at the end of 1, with

respect to the set of objective probabilities P. We can verify that the present value of

E(FCF(1,1:2)), discounted with ρ, is equal to V(0,1), the unlevered value at the end of

year 0.

E(FCF(1,1:2)) = pU*FCF(1,1) + (1 – pU)*FCF(1,2)

= 85%*1,200 + (1 – 85%)*800

= 1,140.00 (1a)

VUn(0,1) = E(FCF(1,1:2)) 1 + ρ = 1,140 = 950.00 (1b) 1 + 20%

We can easily verify that, based on the state prices for the two states of nature at

the end of year 1, the set of risk-neutral probabilities is Q = {qU, (1 – qU)} = {61.25%,

38.75%} where qU is the risk-neutral probability for the up state of nature at the end of

year 1. See discussion on state prices in the previous teaching note.

qU = VUn(0,1)*(1 + rf) – FCF(1,2) FCF(1,1) – FCF(1,2) = 950*(1 + 10%) – 800 = 61.25% (2) 1,200 - 800


Risk-neutral valuation

Equivalently, we can calculate the no-arbitrage value at the end of year 0 by

taking the expectation of the payoffs at the end of year 1 with respect to the “risk-neutral”

probabilities Q = {qU, (1 – qU)} = {61.25%, 38.75%}, discounted by the risk-free rate rf.

Let EQ(FCF(1,1:2)) be the expectation of the free cash flows under the two states of

nature at the end of 1, with respect to the set of risk-neutral probabilities Q. Then we can

verify that the present value of EQ(FCF(1,1:2)), discounted with rf, is equal to the

unlevered value at the end of year 0, VUn(0,1).

EQ(FCF(1,1:2)) = qU*FCF(1,1) + (1 – qU)*FCF(1,2)

= 61.25%*1,200 + (1 – 61.25%)*800

= 1,045.00 (3a)

VUn(0,1) = EQ(FCF(1,1:2)) 1 + rf = 1,045 = 950.00 (3b) 1 + 10%

Combining line 1b with line 3b, we can express the unlevered return ρ in terms of

the risk-free rate rf.

1 + ρ = E(FCF(1,1:2))*(1 + rf) (4) EQ(FCF(1,1:2))

Value of levered company

We assume that VUn(0,1), the market value of the unlevered company at the end

of year 0 is $950. Recall that we assume that there are no taxes and therefore no tax

savings from debt financing. In other words, if we were to invest in the unlevered


company without any debt financing, the equity contribution would be $950. Suppose at

the end of year 0 the company finances the investment with risk-free debt equal in value

to $300.2 Then under both states of nature, the cash flow to the debt holder is $330 and

the value of debt at the end of year 0 D(0,1) is $300, which is the cash flow to debt

discounted by the risk-free rate rf.

CFD(1,1) = CFD(1,2) = D(0,1)*(1 + rf)

= 300*(1 + 10%) = 330.00 (5)

With the debt financing, what is the value of the equity in the levered company?

In a world without taxes, the question seems silly and self-evident because the answer is

obviously $650. Since in competitive markets the value of the levered company is equal

to the sum of the (levered) equity and the debt, the value of the levered equity EL(0,1) is

simply the difference between the unlevered value of $950 and the debt of $300.

Implicitly we assume that VL(0,1) the value of the levered company is equal to VUn(0,1)

the value of the unlevered company. In turn, the levered value is equal to the sum of the

equity and the debt. In symbols,

VL(0,1) = VUn(0,1) (6a)

VL(0,1) = EL(0,1) + D(0,1) (6b)

Nevertheless, for illustration, we use the replicating portfolio method and the risk-neutral

valuation method to value the levered equity EL(0,1) and verify that the answer is $650.3

Based on the debt financing arrangement, we can construct the process for the cash flow

to the levered equity holder.

2. By risk-free debt, we mean that under both states of nature at the end of year 1, the company will be

able to pay back in full the principal and interest on the loan.


Cash Flow to levered equity holder

At the end of year 1, let CFE(1,1) represent the cash flow to the equity holder in

the up state and let CFE(1,2) represent the cash flow to the equity holder in the down

state. Without debt financing, the equity holder would receive all of the FCF under both

states of nature. With debt financing, the debt holder and the equity holder share the FCF.

The free cash flow is equal to the sum of the cash flow to debt and the cash flow to

equity.

FCF(1,j) = CFD(1,j) + CFE(1,j) for j = 1,2 (7)

The process for the cash flow to the levered equity holder is shown below.

Figure 3: Process (or tree) for the cash flow to the levered equity holder Year 0 1

CFE(1,1) = 870 EL(0,1) CFE(1,2) = 470

Rearranging line 7, in the two states of nature at the end of year 1, the cash flow

to the equity holder is the difference between the free cash flow and the cash flow to debt.

CFE(1,1) = FCF(1,1) – CFD(1,1)

= 1,200 – 330 = 870.00 (8a)

CFE(1,2) = FCF(1,2) – CFD(1,2)

= 800 – 330 = 470.00 (8b)

3. The risk-neutral approach is general and a clear understanding of the simplest case will facilitate the

understanding of cash flow processes with more complex payoff structures.


If the up state of nature occurs, the equity holder will receive $870; if the down

state of nature occurs, the equity holder will receive $470.

Replicating portfolio

To calculate the value of the levered equity, we construct a portfolio Z that

consists of α percent of the unlevered company and β percent of the value of the

borrowing, and replicates the payoff structure for the levered equity holder. Let Z(i,j;α,β)

denote the value of the portfolio Z for the jth node in the ith period with α percent of the

unlevered company and β percent of the value of the borrowing. The expression for the

value of portfolio Z at the end of year 0 is given below.

Z(0,1; α,β) = α*VUn(0,1) + β*D(0,1) (9)

The value of the portfolio Z is equal to the sum of the investment in the unlevered

company and the value of the borrowing, where the value of the investment in the

unlevered company is equal to α times VUn(0,1) and the value of the borrowing is equal

to β times D(0,1).

In the up state of nature, the value of the investment in the unlevered company is

α times FCF(1,1) and in the down state of nature, the value of the investment in the

unlevered company is α times FCF(1,2). Similarly, in the up state of nature, the value of

the borrowing is β times CFD(1,1) and in the down state of nature, the value of the

borrowing is β times CFD(1,2). We have two equations and two unknowns, namely the

two coefficients α and β.

α*FCF(1,1) + β*CFD(1,1) = CFE(1,1) (10a)

α*FCF(1,2) + β*CFD(1,2) = CFE(1,2) (10b)


Substituting the appropriate numerical values, we obtain,

α*1,200 + β*330 = 870 (11a)

α*800 + β*330 = 470 (11b)

We can verify that α = 1 and β = -1.

1*1,200 + -1*330 = 870 (12a)

1*800 + -1*330 = 470 (12b)

With α = 1 and β = -1, the value of the portfolio Z at the end of year 0 is $650.

Z(0,1; 1,-1) = α*VUn(0,1) + β*D(0,1)

= 1*950 + -1*300 = 650.00 (13)

Since the payoff of the portfolio Z replicates the payoff structure for the cash flow

to levered equity at the end of year 1, the value of the levered equity is equal to the value

of Z, namely 650.

EL(0,1) = Z(0,1; 1,-1) = 650 (14)

Return to levered equity

Let e be the return to levered equity. To find the return to levered equity, we take

the expectation of the cash flow to equity at the end of year 1 with respect to measure P

and divide by the value of levered equity EL(0,1).

E(CFE(1,1:2)) = pU*CFE(1,1) + (1 – pU)*CFE(1,2)

= 85%*870 + (1 – 85%)*470

= 810.00 (15)

Then 1 + e = E(CFE(1,1:2)) EL(0,1)


= 810 = 1.246154 (16) 650

The return to levered equity is 24.6125%.

Algebraic formulation for the WACC

Here we show the algebraic derivation of the standard weighted average cost of

capital (WACC). Let w be the WACC. The expectation of the free cash flow at the end of

year 1 with the set of probabilities P divided by the levered value is equal to the one plus

the weighted average cost of capital.

1 + w = E(FCF(1,1:2)) (17) VL(0,1)

First, we know that

E(FCF(1,1:2)) = pU*FCF(1,1) + (1 – pU)*FCF(1,2) (18a)

Rewriting the free cash flow as the sum of the cash flow to equity and the cash

flow to debt, we obtain the expectation of the FCF in terms of the sum of the expectation

of the CFE and the CFD.

E(FCF(1,1:2)) = pU*[CFE(1,1) + CFD(1,1)

+ (1 – pU)*[CFE(1,2) + CFD(1,2)]

= E(CFE(1,1:2)) + E(CFD(1,1:2)) (18b)

We know that without taxes, the unlevered value is equal to the levered value.

And in turn, the levered value is equal to the sum of the levered equity and the debt.

VUn(0,1) = VL(0,1) (19a)

VL(0,1) = EL(0,1) + D(0,1) (19b)


We can rewrite the expectations of the cash flows in line 19b in terms of the

values by using the expressions for the cash flows in line 1b and line 16.

(1 + ρ)*VUn(0,1) = (1 + e)*EL(0,1) + (1 + d)*D(0,1) (20a)

ρ*VUn(0,1) = e*EL(0,1) + d*D(0,1) (20b)

In addition, we apply line 19a, which says that the unlevered value is equal to the

levered value and using line 17 in line 20b, we solve for the weighted average cost of

capital w.

w*VL(0,1) = e*EL(0,1) + d*D(0,1) (21a)

w = %EL(0,1)*e + %D(0,1)*d (21b)

where %EL(0,1) = EL(0,1) is the levered equity as a percent of the levered value and VL(0,1) %D(0,1) = D(0,1) is the debt as a percent of the levered value. VL(0,1)

Algebraic formulation for e

Now we can derive an algebraic expression for e, the return to levered equity.

From above, we know that

ρ*VUn(0,1) = e*EL(0,1) + d*D(0,1) (22a)

ρ*[EL(0,1) + D(0,1)] = e*EL(0,1) + d*D(0,1) (22b)

Solving for e, we obtain that

e*EL(0,1) = ρ*EL(0,1) + (ρ - d)*D(0,1) (23a)

e = ρ* + (ρ - d)*D(0,1) (23b) EL(0,1)

The debt-equity ratio is

D(0,1) = 300 = 0.46154 (24) EL(0,1) 650


e = ρ* + (ρ - d)*D(0,1) EL(0,1)

= 20% + (20% - 10%)*0.46154 = 24.6154% (25)

Using the expression for e, we can verify the calculations for the WACC.

VL(0,1) = EL(0,1) + D(0,1)

= 650 + 300 = 950.00 (26)

%EL(0,1) = EL(0,1) = 650 = 68.42% (27) VL(0,1) 950 %D(0,1) = D(0,1) = 300 = 31.58% (28) VL(0,1) 950

w = %EL(0,1)*e + %D(0,1)*d

= 68.42%*24.6154% + 31.58%*10%

= 16.84% + 3.16% = 20.00% (29)

Section Two

In this section, we introduce risky debt and continue to assume that there are no

taxes. Suppose we borrow some amount D(0,1) and “promise” to pay $825 to the debt

holder at the end of year 1. Let X be the amount that we promise to pay back.4 There is

no guarantee that we can actually pay $825 to the debt holder at the end of year 1. The

amount that we can pay will depend on the state of nature that occurs at the end of year 1.

If the up state of nature occurs, we can easily pay back the X amount. However, if the

down state occurs, then we simply pay the debt holder the cash flow that occurs, namely

4. In the language of option pricing, X is the exercise price.


FCF(1,2) = $800, which is less than the amount that we had promised. Thus, the debt is

risky. At the end of year 0, what is the value of the debt D(0,1)?

First, we calculate the value of the levered equity EL(0,1) because it is simpler.

Second, we calculate the value of the debt D(0,1).

Payoff for the levered equity holder

With risky debt, what is the payoff to the levered equity holder? The structure is

equivalent to a call option. If the “up” state occurs, the project can pay the debt holder

$825, and the levered equity holder receives the remaining $375; if the “down” state

occurs, the project cannot pay the debt holder $825. The project simply pays the debt

holder the free cash flow, namely $800, and the equity holder receives nothing.

Figure 4: Process (or tree) for the cash flow to the levered equity holder Year 0 1

Max{[FCF(1,1) – X],0} EL(0,1) Max{[FCF(1,2) – X],0}

Algebraically, we represent the payoff to the equity holder as the maximum of the

difference between the FCF and the exercise price X and zero.

Up state: CFE(1,1) = max{[FCF(1,1) – X], 0)}

= max{[1,200 – 825],0)}

= max{ 375,0} = $375 (30a)

Down state: CFE(1,2) = max{[FCF(1,2) – X], 0)}


= max{[800 – 825],0)}

= max{-25,0} = 0 (30b)

In the up state, the maximum of the difference (between the FCF and X) is $375

and in the down state, the maximum of the difference (between the FCF and X) is $0.

Value of the levered equity and return to levered equity

To find the value of the levered equity EL(0,1), we take the expectation of the

cash flow to equity at the end of year 1 with respect to the risk-neutral probabilities and

discount with the risk-free rate.

EQ(CFE(1,1:2)) = qU*CFE(1,1) + (1 – qU)*CFE(1,2)

= 61.25%*375 + (1 – 61.25%)*0

= 229.6875 (31a)

EL(0,1) = EQ(CFE(1,1:2)) 1 + rf = 229.6875 = 208.807 (31b)

1 + 10%

To find the return to levered equity, we take the expectation of the cash flow to

equity at the end of year 1 with respect to measure P and divide by the value of levered

equity EL(0,1).

E(CFE(1,1:2)) = pU*CFE(1,1) + (1 – pU)*CFE(1,2)

= 85%*375 + (1 – 85%)*0

= 318.750 (32a)

Then 1 + e = E(CFE(1,1:2)) EL(0,1)


= 318.75 = 1.52653 (32b) 208.807

With risky debt, the return to levered equity is 52.653%.

Payoff for the Debt Holder

Now, let us consider the payoff structure for the debt holder. If the “up” state

occurs, the project pays the debt holder $825. If the “down” state occurs, the project pays

the debt holder the cash flow, namely $800. Algebraically, we represent the payoff to the

debt holder as the difference between the exercise price X, and the maximum of the

difference between the exercise price X and FCF and zero.

Up state: CFD(1,1) = X - max{[X – FCF(1,1)], 0)}

= 825 - max{[825 - 1,200],0)}

= 825 - max{-375,0} = $825 (33a)

Down state: CFE(1,2) = X - max{[X - FCF(1,2)], 0)}

= 825 - max{[825 – 800,0)}

= 825 - max{ 25,0} = 800 (33b)

In the up state, the maximum of the difference (between X and the FCF) is $0 and

thus the debt holder simply receives the amount X. In the down state, the maximum of

the difference (between X and the FCF) is $25 and the debt holder receives the amount X

less $25, which is the same as the cash flow in the down state of nature.


Value of the debt and return to debt

To find the value of the debt D(0,1), we take the expectation of the cash flow to

debt at the end of year 1 with respect to the risk-neutral probabilities and discount with

the risk-free rate.

EQ(CFD(1,1:2)) = qU*CFD(1,1) + (1 – qU)*CFD(1,2)

= 61.25%*825 + (1 – 61.25%)*800

= 815.3125 (34a)

D(0,1) = EQ(CFD(1,1:2)) 1 + rf = 815.3125 = 741.193 (34b)

1 + 10%

Since the repayment of the amount of $825 is uncertain, at the end of year 0, the

debt holder will provide $741.19.5

To find the return to the debt holder, we take the expectation of the cash flow to

debt at the end of year 1 with respect to measure P and divide by the value of debt D(0,1).

E(CFD(1,1:2) = pU*CFD(1,1) + (1 – pU)*CFD(1,2)

= 85%*825 + (1 – 85%)*800

= 821.250 (35a)

Then 1 + d = E(CFD(1,1:2)) D(0,1)

= 821.25 = 1.10801 (35b) 741.193

5. Note that this amount of $741.19 is less than the $750 that the debt holder would have provided at the

end of year 0 if the debt were risk-free. $750*(1 + 10%) = 825.00


As expected, with risky debt, the return to the debt holder is 10.80%, which is

higher than the risk-free rate of 10%.

The WACC

We can verify that the returns to the equity and debt holders give the correct value

for the WACC. With no taxes, again, we verify that the WACC is equal to the unlevered

return ρ.

VL(0,1) = EL(0,1) + D(0,1)

= 208.807 + 741.193 = 950.00 (36)

%EL(0,1) = EL(0,1) = 208.807 = 21.98% (37a) VL(0,1) 950 %D(0,1) = D(0,1) = 741.193 = 78.02% (37b) VL(0,1) 950

w = %EL(0,1)*e + %D(0,1)*d

= 21.98%*52.65% + 78.02%*10.8%

= 11.57% + 8.43% = 20.00% (38)

Return to levered equity with risky debt

Next, with risky debt, we derive the algebraic expressions for the return to levered

equity and the return to debt. We know that

1 + e = E(CFE(1,1:2))*(1 + rf) (39a) EQ(CFE(1,1:2))

where

E(CFE(1,1:2)) = pU*CFE(1,1) + (1 – pU)*CFE(1,2) and (39b)


EQ(CFE(1,1:2)) = qU*CFE(1,1) + (1 – qU)*CFE(1,2) (39c)

In the case where the debt is risky, the cash flows to equity under the two states of

nature at the end of year 1 are as follows.

CFE(1,1) = max{[FCF(1,1) – X], 0)} = FCF(1,1) – X (40a)

CFE(1,2) = max{[FCF(1,2) – X], 0)} = 0 (40b)

In the up state of nature, the cash flow to equity at the end of year 1 is equal to the

difference between the free cash flow and X. In the down state of nature, the cash flow to

equity at the end of year 1 is zero.

Substituting line 40a and 40b into line 39b and line 39c, we rewrite line 39a as

follows.

1 + e = pU*[FCF(1,1) – X]*(1 + rf) (41a) qU*[FCF(1,1) – X] 1 + e = pU*(1 + rf) (41b) qU

Substituting the numerical values for the parameters in line 41b, we obtain that

the return to levered equity is 52.65%.

1 + e = 85.00%*(1 + 10%) = 1.5265 (42) 61.25%

Note that in this special case, with a simple one period binomial example and

risky debt, the return to levered equity is constant and is not a function of the amount of

debt.

Return to debt with risky debt

We know that


1 + d = E(CFD(1,1:2))*(1 + rf) (43a) EQ(CFD(1,1:2))

where

E(CFD(1,1:2)) = pU*CFD(1,1) + (1 – pU)*CFD(1,2) (43b)

EQ(CFD(1,1:2)) = qU*CFD(1,1) + (1 – qU)*CFD(1,2) (43c)

In the case where the debt is risky, the cash flows to debt under the two states of

nature at the end of year 1 are as follows.

CFD(1,1) = X - max{[X - FCF(1,1)], 0)} = X (44a)

CFD(1,2) = X - max{[X - FCF(1,2) – X], 0)}

= X – [X - FCF(1,2)] = FCF(1,2) (44b)

In the up state of nature, the cash flow to debt at the end of year 1 is equal to X. In

the down state of nature, the cash flow to debt at the end of year 1 is equal to the free

cash flow. Thus,

E(CFD(1,1:2)) = pU*X + (1 – pU)*FCF(1,2) (45a)

E(CFD(1,1:2)) = qU*X + (1 – qU)*FCF(1,2) (45b)

Substituting line 45a and 45b into line 43a, we obtain that

1 + d = pU*X + (1 – pU)*FCF(1,2)*(1 + rf) (46) qU*X + (1 – qU)*FCF(1,2)

Weighted Average Cost of Capital

We know that the value of the levered equity EL(0,1) is equal to the value of the

call option on the free cash flow with an exercise price equal to X. Let J be the value of

the call option. Then

EL(0,1) = J (47a)


We know that the value of the debt D(0,1) is equal to the present value of the

exercise X, with the risk-free rate, less the value of a put option on the free cash flow. Let

G be the value of the call option. Then

D(0,1) = PV(X) - G (47b)

From line 21a, we know that

w*VL(0,1) = e*EL(0,1) + d*D(0,1) (48)

Substituting line 47a and line 47b into line 48, we obtain

w*VL(0,1) = e*J + d*[PV(X) – G] (49a)

w = J *e + [PV(X) – G]*d (49b) VL(0,1) VL(0,1)

Alternative expressions

We know that without taxes, the levered return is equal to the unlevered return.

w = ρ (50)

Also, since the levered value is equal to the sum of the levered equity and debt,

we obtain that,

VL(0,1) = J + PV(X) - G (51)

Combining line 48 and 50 with line 51, we obtain that

ρ*[J + PV(X) - G] = e*J + d*[PV(X) – G] (52)

Rearranging, we obtain that

d*[PV(X) – G] = ρ*[PV(X) - G] + (ρ - e)*J (53a)

d = ρ + (ρ - e)*J (53b) PV(X) - G

Suppose X = 825. Then


d = 20% + (20% - 52.65%)*208.807 741.193

= 20% - 9.20% = 10.80% (54)

Section Three

In this section, we extend the analysis by introducing taxes with risk-free debt.

The tax rate τ is 34%. We present a new numerical example. The values of some of the

parameters will be similar to the values of the parameters in the numerical examples in

the previous sections. With taxes, we have to construct the income statement to determine

the tax liability and the tax shields under the two states of nature. To ensure that the tax

shield is risky we have set the amount of depreciation equal to the gross-of-tax free cash

flow in the down state of nature.

Unlevered company

The process for the revenues (or gross-of-tax free cash flow) is the same as

before. In the up state of nature, the revenue is $1,200 and in the down state, the revenue

is $800.

The income statements under the two states of nature in year 1 are shown below.

Table 1a: Income Statement, with no debt financing, under the up state of nature, Year 0 1 Revenues 1,200.0 Depreciation 800.0 Gross Income 400.0 Taxes 136.0 Net Income 264.0


Table 1b: Income Statement, with no debt financing, under the down state of nature, Year 0 1 Revenues 800.0 Depreciation 800.0 Gross Income 0.0 Taxes 0.0 Net Income 0.0

In the up state of nature, the net income is $264, and the tax liability is $136. In

the down state of nature, the net income is zero.

We assume that the return to unlevered equity ρ is 15%, the objective (or actual)

probability for the up state of nature is 60% and the objective probability for the down

state of nature is 40%. The set of objective probabilities is P = {pU, (1 – pU)} = {60%,

40%} where pU is the probability for the up state of nature at the end of year 1. The

present value of E(FCF(1,1:2)), discounted with 15%, is equal to V(0,1), the unlevered

value at the end of year 0.

E(FCF(1,1:2)) = pU*FCF(1,1) + (1 – pU)*FCF(1,2)

= 60%*1,064 + (1 – 60%)*800 = 958.40 (55a)

VUn(0,1) = E(FCF(1,1:2)) 1 + ρ = 958.40 = 833.39 (55b) 1 + 15%

The free cash flow statements under the two states of nature are shown below.

Table 2a: Free Cash Flow Statement under the up state of nature, Year 0 1 Revenues 1,200.0 Taxes 136.0 Free Cash Flow 1,064.0


Table 2b: Free Cash Flow Statement under the down state of nature, Year 0 1 Revenues 800.0 Taxes 0.0 Free Cash Flow 800.0

We can easily verify that the set of risk-neutral probabilities is Q = {qU, (1 – qU)}

= {44.2161%, 55.7839%} where qU is the risk-neutral probability for the up state of

nature at the end of year 1. The present value of EQ(FCF(1,1:2)), discounted with rf, is

equal to the unlevered value at the end of year 0, VUn(0,1).

EQ(FCF(1,1:2)) = qU*FCF(1,1) + (1 – qU)*FCF(1,2)

= 44.2161%*1,064 + (1 – 44.2161%)*800

= 916.73 (56a)

VUn(0,1) = EQ(FCF(1,1:2)) 1 + rf

= 916.73 = 833.39 (56b) 1 + 10%

Amount of debt financing

The maximum amount that the project can repay with certainty under both states

of nature in year 1 is equal to the value of the free cash flow under the down state of

nature, namely $800. At the end of year 0, the value of the debt is equal to $727.27,

which is the present value of $800, discounted with the risk-free rate.

D(0,1) = FCF(1,2) = 800 = 727.27 (57) 1 + rf 1 + 10%


Value of tax shield at the end of year 1

The amount of the tax savings at the end of year 1 is equal to the tax rate times the

interest payment.

Tax Savings = τ*d*D(0,1)

= 34%*10%*727.27 = 24.73 (58)

The tax savings is only realized in the up state of nature.

Return to levered equity

Next we construct the cash flow to equity statement under the two states of nature

at the end of year 1. The cash flow to equity is equal to the free cash flow plus the tax

savings less the cash flow to debt.

Table 3a: Cash Flow to Equity Statement under the up state of nature, Year 0 1 Free Cash Flow 1,064.00 Tax Savings 24.73 Capital Cash Flow 1,088.73 Cash Flow to Debt 800.00 Cash Flow to Equity 288.73

Table 3b: Cash Flow to Equity Statement under the down state of nature, Year 0 1 Free Cash Flow 800.00 Tax Savings 00.00 Capital Cash Flow 800.00 Cash Flow to Debt 800.00 Cash Flow to Equity 00.00

In the up state of nature, the cash flow to equity at the end of year 1 is $288.73

and in the down state of nature the cash flow to equity at the end of year 1 is zero. To find

the value of the levered equity EL(0,1), we take the expectation of the cash flow to equity


at the end of year 1 with respect to the risk-neutral probabilities and discount with the

risk-free rate.

EQ(CFE(1,1:2)) = qU*CFE(1,1) + (1 – qU)*CFE(1,2)

= 44.2161%*288.727 + (1 – 44.2161%)*0

= 127.664 (59a)

EL(0,1) = EQ(CFE(1,1:2)) 1 + rf = 127.664 = 116.058 (59b)

1 + 10%

To find the return to levered equity, we take the expectation of the cash flow to

equity with respect to measure P and divide by the value of levered equity EL(0,1).

E(CFE(1,1:2)) = pU*CFE(1,1) + (1 – pU)*CFE(1,2)

= 60%*288.727 + (1 – 60%)*0

= 173.236 (60a)

Then 1 + e = E(CFE(1,1:2)) EL(0,1)

= 173.236 = 1.49267 (60b) 116.058

With risk-free debt and risky tax shield, the return to levered equity is 49.267%.

Value and discount rate for the tax shield

To find the value of the tax shield VTS(0,1), we take the expectation of the tax

shield at the end of year 1 with respect to the risk-neutral probabilities and discount with

the risk-free rate.

EQ(TS(1,1:2)) = qU*TS(1,1) + (1 – qU)*TS(1,2)


= 44.2161%*24.727 + (1 – 44.2161%)*0

= 10.9333 (61a)

VTS(0,1) = EQ(TS(1,1:2)) 1 + rf = 10.9333 = 9.9394 (61b)

1 + 10%

To find the discount rate for the tax shield, we take the expectation of the tax

shield at the end of year 1 with respect to measure P and divide by the value of the tax

shield VTS(0,1).

E(TS(1,1:2)) = pU*TS(1,1) + (1 – pU)*TS(1,2)

= 60%*24.727 + (1 – 60%)*0

= 14.8362 (62a)

Then 1 + ψ = E(TS(1,1:2)) VTS(0,1)

= 14.8362 = 1.49267 (62b) 9.9394

With risk-free debt and risky tax shield, the discount rate for the tax shield is

49.267%. The discount rate for the tax shield is equal to the return to levered equity

because the payoff structure of the tax shield is the same as the payoff structure for the

levered equity holder.

Section Four

In this section, we extend the analysis in Section Three by examining risky debt

and risky tax shields. The value of the maximum amount of risk-free debt at the end of

year 0 is 727.27, with a payment of $800 at the end of year 1.


Suppose we “promise” to pay $835 to the debt holder at the end of year 1. The

debt is risky because under the down state of nature, the free cash flow at the end of year

1 is only $800, which is less than the amount that is due to the debt holder.

In the up state of nature, the cash flow to debt at the end of year 1 is $835. In the

down state of nature, the cash flow to debt at the end of year 1 is equal to the value of the

free cash flow, namely $800. To find the value of the debt D(0,1), we take the

expectation of the cash flow to debt at the end of year 1 with respect to the risk-neutral

probabilities and discount with the risk-free rate.

EQ(CFD(1,1:2)) = qU*CFD(1,1) + (1 – qU)*CFD(1,2)

= 44.2161%*835 + (1 – 44.2161%)*800

= 815.476 (63a)

D(0,1) = EQ(CFD(1,1:2)) 1 + rf = 369.204 = 741.342 (63b)

1 + 10%

To find the return to the debt holder, we take the expectation of the cash flow to

debt at the end of year 1 with respect to measure P and divide by the value of debt D(0,1).

E(CFD(1,1:2)) = pU*CFD(1,1) + (1 – pU)*CFD(1,2)

= 60%*835 + (1 – 60%)*800

= 821.000 (64a)

Then 1 + d = E(CFE(1,1:2)) EL(0,1)

= 821.00 = 1.10745 (64b) 741.342

With risky debt, the return to the debt holder is 10.745%.


Conclusion

We used risk-neutral valuation to determine the value of the levered and

unlevered firms with risk-free and risky debt. In addition, with risk-free and risky debt,

we calculated the returns to the levered equity holder and the debt holder.

REFERENCES Tham, J. (2001) “Risk-neutral Valuation: A Gentle Introduction (1).” Working Paper.

Available on the Social Science Research Network (SSRN) Tham, J & Wonder, N. (2001) “Unconventional Wisdom on psi, the Appropriate

Discount Rate for the Tax Shield.” Working Paper. Available on the Social Science Research Network (SSRN)

Tham, J & Wonder, N. (2001) “The Non-conventional WACC with Risky Debt and

Risky Tax Shield.” Working Paper. Available on the Social Science Research Network (SSRN)

Risk Neutral Valuation(2)

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