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.
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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.
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
J.Tham, December 3, 2001 13
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.
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)
J.Tham, December 3, 2001 21
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
J.Tham, December 3, 2001 22
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
J.Tham, December 3, 2001 23
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
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
J.Tham, December 3, 2001 26
at the end of year 1 with respect to the risk-neutral probabilities and discount with the