The Assumptions and Math
Behind WACC and APV Calculations
Richard Stanton
U.C. Berkeley
Mark S. Seasholes
U.C. Berkeley
This Version October 27, 2005∗
Abstract
We outline the math and assumptions behind weighted average cost of capital (WACC) and
adjusted present value (APV) calculations. We first derive a general formula for the discount
rate of equity and beta of equity under minimal assumptions. We then take into consideration:
i) The existence or non-existence of taxes; ii) Whether a firm has a constant amount of debt in
dollar terms; iii) Whether a firm targets a constant proportion of debt in its capital structure;
and iv) The frequency of debt rebalancing. These considerations give rise to well known results
such as the Miles and Ezzell (1980) formula for WACC and differnt methods (formulae) for
unlevering and re-levering betas. This document is intended for those who wish to understand
the motivation behind valuing a firm with WACC and/or APV. Doctoral students and professors
may find the document useful when teaching WACC and APV. In particular, this document
helps answer questions like: Why does my firm use this formula to unlever beta, but you have
taught us another formula? Understanding the assumptions behind both formulae turns out to
be the key to answering such questions.
Keywords: WACC, APV, Cost of Capital
JEL Classification number: G32, A22, A23
∗
This document is based on work titled “WACC/APV—Mathematical Details” by Richard Stanton. Contact
information: Mark S. Seasholes, U.C. Berkeley Haas School of Business, 545 Student Services Bldg., Berkeley CA
c
94720; Tel: 510-642-3421; Fax: 510-643-1420; email: [email protected]. 2005.
1
1
Introduction
This document is intended for students who wish to delve into the math behind WACC
and APV calculations. The derivations in this document highlight many of the assumptions
behind these valuation methods. This document can be used by students ranging from
advanced undergraduates to advanced MBAs. However, only the most curious students
will find it interesting and benefit from it. Doctoral students and professors may find the
document useful when teaching WACC and APV. In particular, this document helps answer
questions like: Why does my firm use this formula to unlever beta, but you have taught us
another formula? Understanding the assumptions behind both formulae turns out to be the
key to answering such questions.
2
Definitions
We lay out the notation used throughout the document. There are a myriad of possible ways
to denote the same economic quantity. Many are self explanatory.
2.1
Unlevered Firms
We use the following notation to denote the market value of a firm (and its equity) without
leverage in its capital structure:
VU = M VF,unlev
= Market (enterprise) value of an unlevered firm
EU = M VE,unlev
= Market value of the equity of an unlevered firm
Notice that VU = EU and we address this in Section 3. We also use the following definitions:
F CFt = After-tax cashflow of the unlevered firm at time t
βA = Beta of the FCFs (could be written βU or βEU , but typically isn’t)
rA = Expected return of the unlevered firm’s equity (could be written rU or rEU )
2
None of the results in this paper rely on a specific model of asset returns (such as the CAPM).
Section 3 only relies on the fact that the beta (and return) of a portfolio is the weighted
average of its components.
2.2
Levered Firms
For firms with leverage, we use the following notation:
VL = M VF,lev
= Market (enterprise) value of a levered firm
EL = M VE,lev
= Equity value of a levered firm
DL = M VD,lev
= Market value of a levered firm’s debt
We use βE to denote the beta of the levered firm’s equity (therefore it could be written
βEL ). We use rE to denote the expected return on the levered firm’s equity (could be rEL ).
For the levered firm’s debt, βD is always used to denote the beta (could be βDL ), and rD is
always used to denote the expected return of the debt (could be rDL ). If the levered firm is
profitable, the existance of debt creates an interest tax shield. τ is the corporate tax rate;
IT St is the interest tax shield for year t; and S is the present value of all IT S. We discuss
the correct discount rate rs,t later. For now, we simply note that it may not be constant
over time.
IT St = rD DL,t τ
S = P V (IT S)
∞
X
IT St
=
(1 + rs,t )t
t=1
3
General Derivations
For an unlevered firm, the value of the firm is the value of the equity: VU = EU . For a
levered firm, the value of the firm is the value of the equity plus debt: VL = EL + DL . The
3
value of a levered firm can also be written as the value of an unlevered firm plus the present
value of interest tax shields: VL = VU +S. If we equate these two different ways of expressing
VL , substitute EU for VU , and reorganize we get:
EL + DL = VU + S
EL = VU − DL + S
EL = EU − DL + S
(1)
Equation (1) tells us that we can interpret EL as being equivalent to a portfolio of EU , DL ,
and S. Next, remember the beta of a portfolio P is the weighted average of its component
betas. Weights are determined by market values:
VP = V1 + V2 + V3
V1
V2
V3
βP =
β1 +
β2 +
β3
VP
VP
VP
This fact allows us to write the beta of EL in terms of the betas of EU , a short position in
DL , and S. Following convention, we continue to write βA instead of βU or βEU , and obtain:
EU
DL
S
βE =
βA −
βD +
βS
EL
EL
EL
(2)
DL − S
DL
S
=
1+
βA −
βD +
βS
EL
EL
EL
Equation 2 gives us a general relationship between βE , βA , βD and βS for an arbitrary debt
policy. Later, we make assumptions about target leverage ratios which allows us to calculate
S and βS . Not only is the beta of a portfolio equal to a weighted average of the betas of its
components, but so is the expected return of a portfolio equal to a weighted average of the
expected returns of its components. Following exactly the same logic as above, we obtain:
DL
S
DL − S
rA −
rD +
rS
(3)
rE = 1 +
EL
EL
EL
Note that this result holds regardless of whether or not the CAPM is true.
3.1
No Taxes
In the case of no taxes, the present value of tax shields is zero (S = 0) and Equation (2)
becomes a familiar expression. Thus, we get our first results. In a world with no taxes and
4
arbitrary debt policy:
βE
βA
DL
DL
=
1+
βA −
βD
EL
EL
or
EL
DL
=
βE +
βD
VL
VL
For the returns, Equation (3) gives us a similar expression (remember S = 0):
DL
DL
rE =
1+
rA −
rD
EL
EL
or
EL
DL
rA =
rE +
rD
VL
VL
3.2
(4)
(5)
Constant Amount of Debt
Having a constant dollar amount of debt is the usual assumption in many APV calculations.
If there is a constant dollar amount of debt outstanding (DL ), constant interest rate (rD ),
and constant tax rate (τ ), the interest tax shield in every year is constant and we have:
IT St = IT S
= rD DL τ
The interest tax shields are known in advance (in expectation) every year, forever. Thus,
we can calculate their present value using the perpetuity formula. The chance (risk) of
not getting the interest tax shield in any given year is related to the debt rate. Therefore,
we discount future interest tax shields to the present using rD . Discounting the interest
tax shields by rD is based on the structure of cash flows when one assumes a firm issues a
constant dollar amount of debt. The cash flow structure implies βS = βD as the tax shield
each period is proportional to the cash flow to bond holders.
rS = rD
βS = βD
∞
X
S =
t=1
IT S
rD DL τ
=
(1 + rD )t
rD
= DL τ
5
Substituting into Equation (2) gives:
DL − S
DL
S
βE =
1+
βA −
βD +
βS
EL
EL
EL
DL − DL τ
DL
DL τ
=
1+
βA −
βD +
βD
EL
EL
EL
DL (1 − τ )
DL (1 − τ )
= 1+
βA −
βD
EL
EL
Substituting into Equation (3) gives:
DL (1 − τ )
DL (1 − τ )
rE = 1 +
rA −
rD
EL
EL
(6)
(7)
Notice, under the following assumptions: i) A constant amount of debt; ii) A debt beta of
zero; iii) βS = βD = 0 and iv) rS = rD , we get a common formula used to unlever (relever)
betas:
DL (1 − τ )
βE = βA 1 +
EL
VL
= βA
(1 − τ )
EL
3.3
Constant Proportion of Debt
Having a constant proportion of debt (constant leverage ratio) is the usual assumption behind
WACC calculations. Suppose a firm changes the amount of debt outstanding each year to
keep the proportion of debt constant. Split S into two pieces. The first piece is the present
value (P V ) of the first year’s (period’s) tax shield (t = 1). The second piece is the P V of all
remaining years’ (periods’) tax shields (t > 1). The first part of the tax shield is a multiple
of the first period’s interest payment and has a beta of βD . The second part of the tax shield
goes up or down in proportion to the value of the firm, and therefore has a beta of βA . Our
assumptions are: βS = βD and rS = rD for the first period (t = 1). After the first period
6
(t > 1) we assume βS = βA . Finally, S = PV(1st tax shield) + PV(Remaining tax shields):
PV(1st tax shield) =
rD DL,t=1 τ
1 + rD
rD DL,t=1 τ
1 + rD
rD DL,t=1 τ
rD DL,t=1 τ
βD + S −
βA
=
1 + rD
1 + rD
PV(Remaining tax shields) = S −
SβS
Substitute the right-hand side of the equation directly above, for the SβS in Equation (2).
Simplify and do the same with Equation (3) to obtain the following key formulae. Since the
firm targets a constant proportion of debt, it is not necessary to carry the t = 1 subscripts
DL,t=1
L
around. In other words, D
= EL,t=1
.
EL
βE
rD τ
DL
rD τ
DL
1−
βA −
1−
βD
= 1+
EL
1 + rD
EL
1 + rD
rE
DL
rD τ
DL
rD τ
= 1+
1−
rA −
1−
rD
EL
1 + rD
EL
1 + rD
(8)
Note: These formulae do not explicitly depend on S, which cancels out of the calculations.
They depend only on the relative values of DL and EL , so they are valid for any pattern
of cash flows from the firm. In addition, the constant proportional debt assumption plays
only one role after the first period—the tax shields have a beta of βA . The exact rule for
how the debt is adjusted each period is not important. The same result will, therefore, hold
for any rule that yields this beta. For example, if a firm adjusts its debt each period to be
proportional to that period’s all-equity cash flow (constant “debt-coverage”) the same result
holds.
3.4
Rebalancing Frequency
Suppose the firm makes interest payments and adjusts its debt level more than once a year.
As the adjustment frequency increases, the relationship between equity and asset betas
eventually converges to Equation (4). To see this, consider a rebalancing frequency of k
periods per year. If the debt rate (rD ) is given as an effective annual rate, then we discount
7
the first term by (1 + rD )1/k . More importantly, the tax shield over the first fractional period
of the year depends on (1 + rD )1/k − 1 τ . As the rebalancing frequency becomes higher
and higher, the value of βE becomes the closer and closer to value we obtain in the absence
of taxes!
"
βE
lim βE
k→∞
"
!#
(1 + rD )1/k − 1 τ
DL
=
1−
βA −
EL
(1 + rD )1/k
DL
DL
=
1+
βA −
βD
EL
EL
DL
1+
EL
!#
(1 + rD )1/k − 1 τ
1−
βD
(1 + rD )1/k
(9)
(10)
Equation (10) makes it clear that continuous rebalancing in a world with taxes gives the
same result as Equation (4) which comes from a world without taxes! Thus, students who
believe that Equation (10) does not consider taxes (because there is no τ term) are not
correct.
We show the effect of different rebalancing periodicity in Table 1. In practice, using Equation (4) rather than Equation (9) usually gives economically similar results. In the example
in Table 1, the tax rate is 35% per year, the asset beta is 0.8571, the company has $2,000 of
debt outstanding, the costs of debt is 7.00%, the beta of debt is 0.2857, and the market value
of equity is $6,760.40. Equation (9) gives a equity beta of 1.0223 with annual rebalancing.
The equity beta increases to 1.0262 if the firm rebalances daily. The equity beta as calculated from Equation (4) is 1.0262 which is the same—to four decimals—as daily rebalancing.
The economic difference between using 1.0223 and 1.0262 is small. Any estimation error (of
equity betas) is typically orders of magnitude larger than this difference.
4
Weighted Average Cost of Capital (WACC)
Assume that a firm maintains a constant leverage ratio:
DL
DL
= Constant
=
VL
(DL + EL )
Remember F CFt refers to the all-equity firm’s expected cash flow at time t (i.e. the cash
flow that would have occurred had the firm been all-equity financed.) From earlier, we can
write: VL = EL + DL . The levered firm is a portfolio with two components (EL and DL ).
8
The expected return on VL is the weighted average return of its components:
EL
DL
rV =
rE +
rD
EL + DL
EL + DL
or
EL
DL
rV =
rE +
rD
VL
VL
(11)
Consider buying both the equity and debt of a levered firm (EL and DL ) today, holding
them for one period, then selling both positions. In exchange for the initial cost of this
investment, VL , we receive next period’s free cash flow (F CFt+1 ), next period’s interest tax
shield (rD DL τ ), and next period’s value of the equity plus debt (VL,t+1 ). If F CFt+1 is the
cash flow of an all-equity firm at time t+1, then we can define the cash flow of the levered
firm at t+1 (CL,t+1 ) as the all equity cash flow plus the interest tax shield (rD DL τ ):
CL,t+1 ≡ F CFt+1 + rD DL τ
Here CL,t+1 is the total cash flow paid to debt and equity holders combined. The price we
pay today must equal the present value of the future cash flows, so:
VL =
F CFt+1 + rD DL τ + VL,t+1
1 + rV
(12)
=
CL,t+1 + VL,t+1
1 + rV
Substituting repeatedly for VL,t+1 , VL,t+2 , etc., we obtain:
VL =
CL,t+2
CL,t+3
CL,t+1
+
+ ...
2 +
1 + rV
(1 + rV )
(1 + rV )3
(13)
i.e. VL equals the present value of all future cash flows (including tax shields), discounted
at rV . This is not very surprising. However, there is a problem using Equation (13) in
practice. We need to know the size of the tax shields each period. These depend on the
amount of debt, which in turn (and by assumption) is a constant multiple of VL . However,
VL is what we are trying to calculate. This is a way around this problem. Instead of adding
the tax shields to the all-equity cash flows, we can value the firm instead by using only
the all-equity cash flows (whose value we do know in expectation), and discounting by an
adjusted discount rate called the WACC (or Weighted Average Cost of Capital). To see this,
9
rewrite Equation (12) using the fact that DL = VL DVLL to obtain:
VL =
F CFt+1 + rD VL DVLL τ + VL,t+1
1 + rV
Collecting terms in VL , this becomes:
VL =
F CFt+1 + VL,t+1
1 + rV − DVLL rD τ
(14)
From Equation (11) we can substitute for rV to get:
VL =
F CFt+1 + VL,t+1
1+
+ DVLL rD − DVLL rD τ
EL
r
VL E
(15)
=
F CFt+1 + VL,t+1
1 + rW ACC
where:
rW ACC ≡
EL
EL + DL
rE +
DL
EL + DL
rD (1 − τ )
(16)
=
EL
VL
rE +
DL
VL
rD (1 − τ )
Substituting repeatedly for VL,t+1 , VL,t+2 , . . . etc. gives:
VL =
F CFt+3
F CFt+2
F CFt+1
+
+ ...
2 +
1 + rW ACC (1 + rW ACC )
(1 + rW ACC )3
Note: Using WACC works for any pattern of cash flows, as long as the firm maintains a
constant leverage ratio. Equation (16) is the usual way WACC is defined, and is the most
convenient way to calculate it given rE and rD . We can also (and equivalently) express
WACC in terms of rA instead. To do so, start the same way as above, writing VL as the
present value of the next period’s payoff, but now write the present value of each of the
pieces separately, as 1
VL =
1
D
F CFt+1 rD VL VLL τ
VL,t+1
+
+
1 + rA
1 + rD
1 + rA
rA is the correct discount rate for VL,t+1 , since VL is always a constant multiple of VU .
10
(17)
This can be rewritten as:
VL =
F CFt+1 + VL,t+1
DL
r τ
VL D
(1 + rA ) 1 − 1+rD
Comparing this equation with Equation (15), we see that:
"
D
L
1 + rW ACC = (1 + rA ) 1 −
VL
rD τ
#
1 + rD
(18)
rW ACC = rA −
DL rD τ
VL
1 + rA
1 + rD
Equation (18) is the Miles and Ezzell (1980) formula for WACC.
5
Example: Valuing a Firm with Taxes
In this section, we shall value one firm in a number of different ways. To highlight the
robustness of our derivations, growth has been included. All results from this section are
shown in Table 2. The firm has the following details: i) The expected operating income
(EBIT) is $1,000 next year; ii) The growth rate of EBIT is constant g = 3.00%; iii) The
tax rate is τ = 35%; iv) rA = 11.00%; and v) The firm rebalances its capital structure, if
necessary, once a year. The firm’s all-equity cash flow next year is: $1, 000 × (1 − 0.35) =
$650.00 and using the formula for a growing perpetuity, we get:
650
VU =
0.1100 − 0.0300
= $8, 125.00
Now assume the firm currently has $2,000 of riskless debt (rD = 5.00%), and will keep a
constant proportion of debt outstanding in future.
5.1
APV
Using APV, we calculate the PV of the tax shields, then add this to VU . To calculate the
PV of the tax shields, note that each tax shield is discounted back one year at rD , and
11
remaining years at rA . Why? The first year’s tax shield is known today. However, all future
tax shields depend on the value of the firm. Since we have assumed the firm will keep a
constant proportion of debt outstanding in the future, the amount of debt outstanding scales
with firm value. The growth in firm value is related to rA . Thus, the first year’s tax shield
is:
IT St=1 = rD DL,t=1 τ
$35.00 = 0.05 × $2, 000 × 0.35
The present value of all tax shields is:
S = P V (IT S) =
=
35
35 × 1.03
35 × 1.032
+
+
+ ...
1.05 1.05 × 1.1100 1.05 × 1.11002
1.1100
35
×
0.1100 − 0.0300 1.0500
= $462.50
From Equation (1), the value of equity (in this levered firm) is the value of unlevered firm,
minus the value of debt, plus the present value of the interest tax shields:
EL = VU − DL + S
= 8, 125.00 − 2, 000.00 + 462.50
= $6, 587.50
Also from the derivation of Equation (1), the value of the levered firm (VL ) is greater than
the unlevered firm (VU ) by exactly the present value of interest tax shields:
VL =
EL + DL
= VU + S
= 6, 587.50 + 2, 000.00 = 8, 125.00 + 462.50
=
$8, 587.50
= $8, 587.50
Using Equation (9) with k = 1, we can calculate rE , which will be useful later:
rE
=
rD τ
DL
rD τ
DL
1−
rA −
1−
rD
1+
EL
1 + rD
EL
1 + rD
=
2, 000
0.05 × 0.35
2, 000
0.05 × 0.35
1+
1−
0.1100 −
1−
0.05
6, 587.50
1.05
6, 587.50
1.05
=
12.7913%
12
Note: APV is easy to use in this case. Everything is growing at a constant rate which makes
it easy to calculate the value of the future tax shields. APV also works well in cases where we
assume the amount of debt remains constant or when the amount changes deterministically
over time. However, in cases where the firm’s cash flows are changing over time, it is harder
to calculate the size of future tax shields.
5.2
WACC
Given the results above, we can calculate WACC:
rD DL τ 1 + rA
rW ACC = rA −
VL
1 + rD
1.1100
0.05 × 2, 000 × 0.35
×
= 0.1100 −
8, 587.50
1.05
= 10.5691%.
The value of the entire levered firm (VL ) is the growing perpetuity value of the free caseflows
using rW ACC as the discount value:
F CFt+1
VL =
rW ACC − g
VL =
650
0.105691 − 0.0300
= $8, 587.50
The value of equity (EL ) in this levered firm is, again, the total firm value (VL ) minus the
value of debt (DL ):
EL = 8, 587.50 − 2, 000 = $6, 587.50
Note: To calculate WACC based on a known amount of debt (DL ), one needs the value of
EL to calculate VL . This is fine if we are valuing a marginal project, but less acceptable if
we are trying to value the whole firm. We can get around this problem by solving iteratively
using tools such as Excel’s Solver. The idea is to: 1) get the value for EL ; 2) calculate
WACC, discount the cash flows, and calculate (a new value for) EL , 3) Have a solver try
different values for EL until the calculated value in step 2 equals the input value in step 1.
13
5.3
Discounting Total After Tax Cash Flow
The total after-tax cash flow in year 1, including the tax shield, is $650 + $35 = $685. This
is expected to grow at 3% per year. The pretax weighted average cost of capital is:
DL
EL
rE +
rD
rV =
EL + DL
EL + DL
EL
DL
=
rE +
rD
VL
VL
2, 000.0
6, 587.50
0.127913 +
0.05
=
8, 587.50
8, 587.50
= 10.9767%.
Note that this is (slightly) lower than rA since the first tax shield has been discounted at rD
not rA . So, from Equation 13 we use the growing perpetuity formula to get:
VL =
=
CL,t+1
rv − g
685
0.109767 − 0.0300
= $8, 587.50
Note: Discounting total, after-tax cashflows shares the disadvantages of both APV and
WACC discussed above. We need to forecast future tax shields (like APV), and we also need
rE and EL before we can calculate rV (like WACC).
5.4
Discounting Cash Flows to Equity (Only)
A common valuation methodology discounts the cash flows to equity holders (only) using
the equity rate rE . This method has the advantage of not having to calculate VL and then
subtracting the value of the debt. It does work, but you have to be very careful. We already
know that rE = 12.7913%. The total after tax cash flow to equity and debt holders combined
in year 1 is $685, of which $2, 000 × 0.05 = $100 goes to debt holders, leaving $585 for equity
14
holders. But:
585
= $5, 974.69 =???
0.127913 − 0.03
This is not the right answer. What’s wrong is that we forgot to include part of the cash flow
to equity holders. Next year, we expect the amount of debt outstanding to increase by 3%,
from $2,000 to $2,060. The $60 raised by issuing additional debt is paid as a dividend to
shareholders, so their total cash flow is actually: $685 − $100 + $60 = $645. Thus we get the
same value for EU as before:
EL =
585 + 60
= $6, 587.50
0.127913 − 0.03
Note: Discounting cashflows to equity (only) is generally harder to use in practice than
APV or WACC. Not only do we need to forecast the future tax shields, but we also need to
forecast the amount of new debt issued each period.
5.5
Risky Debt
The valuation example (above) assumes the firm issues $2, 000 of riskless debt. We re-do
the analysis but assume the firm issues $2, 000 of risky debt. The yield on the debt is 7.00%
and it has a 0.2857 beta. The results of the valuation exercise with risky debt are shown
in Table 3. Note the total firm value is now $8,760.40 which is higher than the total firm
value of $8,587.50 from the example in Table 2. The increase in value is due to the higher
tax shields that result from higher interest expense.
6
Conclusion
This paper outlines the math and assumptions behind weighted average cost of capital
(WACC) and adjusted present value (APV) calculations. We first derive a general formula
for the discount rate of equity and beta of equity under minimal assumptions. We then
take into consideration: i) The existence or non-existence of taxes; ii) Whether a firm has a
constant amount of debt in dollar terms; iii) Whether a firm targets a constant proportion
of debt in its capital structure; and iv) The frequency of debt rebalancing.
One result shows a firm’s equity beta is that same in a world with no taxes as it is in a world
with taxes under the assumption the firm continuously rebalances its debt to a constant
15
proportion of capital structure. We highlight how changing assumptions, changes formulas
for equity betas and discount rates (not surprisingly.) We end with a valuation example that
provides consistent results for a firm with growing operating income.
References
[1] Miles, J. and R. Ezzel, 1980, “The Weighted Average Cost of Capital, Perfect Capital Markets and Project Life: A Clarification,” Journal of Financial and Quantitative
Analysis, 15, 719-730.
16
Table 1
Rebalancing Frequency
This table shows the effect of rebalancing on the equity beta ( βE ). The firm rebalances its capital
structure periodically to maintain a constant proportion of debt. The equity beta is calculated using
Equation (10) from the paper. The tax rate ( τ ) is 35%; the asset beta ( βA ) is 0.8571; the company
currently has $2,000 of debt outstanding but this number is expected to change; the cost of debt ( rD ) is
7.00%; the beta of debt ( βD ) is 0.2857; and the value of the firm’s equity ( EL ) is $6,760.40.
Periods
Per Year
Rebalancing
Frequency
βE
1
Annually
1.0223
2
Semi-An
1.0242
4
Quarterly
1.0252
12
Monthly
1.0259
52
Weekly
1.0261
365
Daily
1.0262
8,760
Hourly
1.0262
17
Table 2
Valuation Example with Riskless Debt
This table shows results of a valuation exercise. The firm is expected to have an operating profit (EBIT)
of $1,000 next year. The operating profit is expected to grow at 3% per year after that. The tax rate ( τ )
is 35%; the asset beta ( βA ) is 0.8571; the company has $2,000 of debt outstanding; the cost of debt
( rD ) is 5.00%; the beta of debt ( βD ) is 0.0000 and it is riskless.
Section
of Paper
Calculations
Symbol(s)
Result
5.0
Unlevered Firm
VU
8,125.00
5.1
APV
ITSt=1
PV(ITS)
EL
rE
VL
35.00
462.50
6,587.50
12.79%
8,587.50
5.2
WACC
rWACC
VL
EL
10.57%
8,587.50
6,587.50
5.3
Total A/T CF
rV
VL
10.98%
8,587.50
5.4
CF to Equity
EL
6,587.50
18
Table 3
Valuation Example with Risky Debt
This table shows results of a valuation exercise. The firm is expected to have an operating profit (EBIT)
of $1,000 next year. The operating profit is expected to grow at 3% per year after that. The tax rate ( τ )
is 35%; the asset beta ( βA ) is 0.8571; the company has $2,000 of debt outstanding; the cost of debt
( rD ) is 7.00%; the beta of debt ( βD ) is 0.2857 as this example considers risky debt.
Calculations
Symbol(s)
Result
Unlevered Firm
VU
8,125.00
APV
ITSt=1
PV(ITS)
EL
rE
VL
49.00
635.40
6,760.40
12.16%
8,760.40
WACC
rWACC
VL
EL
10.42%
8,760.40
6,760.40
Total A/T CF
rV
VL
10.98%
8,760.40
CF to Equity
EL
6,760.40
19