Seminar 5
Corporate Finance
Equivalence of APV, FTE, and WACC
Consider the following setting.
The firm can make an investment I that generates constant EBIT in perpetuity and has a
perpetual debt D with equal annual interest payments. In this setup both the value of debt and the
debt to equity ratio remain constant over time. We assume that MM Proposition II holds and the
formula for rwacc holds too: rwacc = E/V rE + D/V rD (1-t)
If financed fully with equity:
NPV = - I + EBIT (1-t) / ru
(1)
where t is corporate tax rate, ru is the discount rate (cost of equity of the unlevered firm).
If part of the investment is financed with debt D at the interest rate rD, then tax shield generates
additional value
APV = NPV + rDDt / rD = - I + EBIT (1-t) / ru + Dt
(2)
The firm’s value is equal to
V = I + APV = EBIT (1-t) / ru + Dt
(3)
Let’s get this value using FTE, looking at cash flows to stockholders:
V-I = - (I-D) + (EBIT - (EBIT - rDD) t - rDD)/rE = - (I-D) + (EBIT-rDD)(1-t)/rE (4)
From (3) and (4) one can get
EBIT (1-t) / ru + Dt =
D + (EBIT-rDD)(1-t)/rE
EBIT / ru
=
D + (EBIT-rDD) /rE
rE = ru(EBIT-rDD) / (EBIT-ruD) = ru + (ru-rD)D / (EBIT/ru-D))
(5)
Substituting EBIT/ru from (3)
rs = ru+ (ru-rD)D / ((V-Dt)/(1-t) -D)) = ru + (ru-rD)D / ((V-Dt)/(1-t) -D) =
=ru + (ru-rD)D / ((V-Dt)/(1-t) -D) = ru + (ru-rD)(1-t)D/(V-D)
Since V-D is E we have
rE = ru + (ru-rD)(1-t)D/E
(6)
which is true (MM Proposition II), hence, if it holds FTE and APV methods are equivalent
Now let’s get the firm’s value using WACC
V = EBIT (1-t)/rwacc
Equating with V from (4) we get
V = D + (EBIT-rDD)(1-t)/rE =
(7)
EBIT (1-t)/rwacc
rwacc = EBIT (1-t)/V = [(V-D)rE + rDD(1-t)]/ V = E/V rE + D/V rD (1-t)
which is true, hence WACC and APV methods are equivalent
(8)
Numerical example on using APV, FTE, and WACC
I = 1000
EBIT = 200
t=40%
ru=10%
D = 600
rD=8%
NPV = -I + EBIT (1-t)/ru = -1000 + 200 x 0.6/0.1 = -1000 +1200 = 200
APV = NPV + DT = 200 + 600 x 0.4 = 440
V = I + APV = 1440
E = V- D = 1440- 600 = 840
rE = ru + (ru-rD)(1-t)D/E = 10% + (10%-8%)(1-0.4)600/840 = 10.857%
FTE = - (I-D) + (EBIT-rDD)(1-t)/rE = - (1000-600) + (200 – 0.08x600)(1-0.4)/0.10857 =
= -400 + 91.2/0.10857 = -400 + 840 = 440
rwacc = E/V rE + D/V rD (1-t) = 840/1440 x 10.857% + 600/1440 x (1-0.4) 8% = 8.333%
NPV (using WACC) = -I + EBIT (1-t)/rwacc = -1000 + 200 (1-0.4)/0.08333 = 440
All three methods are equivalent.
2.
Problem 1 from HA3
Eureka Space Techology Group (an all-equity firm) is announcing a $100 million R&D project,
which is expected to drastically improve its satellite launching technology by reducing its annual
launching costs from $500 million to $475 million. The firm can finance the project through
either retained earnings or a new bond issue. The firm’s cost of equity capital is 12.5%. The
prevailing market rate of interest for comparable corporate bonds is 8%. Currently, the firm has
15 million shares of stock outstanding at $32.5 a share. The firm has sufficient earnings to fully
utilize the corporate tax shield if the project is debt financed. Assume that the relevant corporate
tax rate is 35%.
a) Which financing method, retained earnings or external debt financing, would you recommend
to the management and why?
b) What is the resulting PV of the firm from each of these two financing methods?
c) What do you expect will be the stock market stock price response to the two different methods
d) Explain the intuition behind the result in part c).
Solution to problem 1 from HA3
Denote VU – firm value before investment, VU’ – firm value with the investment, if the funds
(100) were raised by selling equity
VU = 15mln*32.5 = 487.5 mln
VU’ = VU + 25(1-t)/rU = 487.5 + 130 = 617.5 mln
By MM I we have: VL’ = VU’ + tD = 617.5 mln + 0.35*100 = 652.5 mln
Hence, PV of the project: PV = VL’ – VU = 625.5 – 487.5 = 165 mln.
The same value can be obtained in the following way:
Denote
Dp – project debt (i.e. incremental increase in debt due to the project)
Ep – project equity (i.e. incremental increase in equity due to the project)
The following is true: Ep = PV – Dp
For the project, from MM II with constant level of debt (NOT constant D/E ratio):
rE = rU + (Dp/(PV-Dp))(rE – rD)(1-t)
For the project rwacc = ((PV-Dp)/PV)rE + (Dp/PV)rD(1-t)
Hence, PV = 25(1-t)/rwacc = 25(1-t)/(((PV-Dp)/PV)rE + (Dp/PV)rD(1-t))
We know: Dp = D = 100, rU = 0.125, rD = 0.08. Hence, we can express rE in terms of PV and
solve the above equation for PV. We will obtain PV = 165 mln (you can check this by
substituting this number instead of PV into the equation).
Hence, the new equity value in the levered firm will be VL’ – D = 652.5 – 100 = 552.5 mln.
Dividing by the number of shares you get the share price.
If the firm is financed through retained earnings, the equity value will be VU’ – 100 = 517.5 mln.
Dividing by the number of shares you get the share price (hence, it will be lower).
3. Problem 18.5, 18.9, 18.10 from Berk-DeMarzo
NOTE: In Berk-DeMarzo they always use enterprise value instead of firm value, EV = E + D –
cash. By Debt they mean Net Debt, i.e. D – cash, where cash is supposed to be the cash asset of
the firm invested in the market at the market rate (e.g. in risk-free bonds).
Suppose Lucent Technologies has an equity cost of capital of 10%, market capitalization of
$10.8 billion, and an enterprise value of $14.4 billion. Suppose Lucent’s debt cost of capital is
6.1% and it’s marginal tax rate is 35%.
a. What is Lucent’s WACC?
b. If Lucent maintains a constant debt-equity ratio, what is the value of a project with
average risk and the following expected free cash flows?
Year 0
1
2
3
FCF -100 50
100 70
c. If Lucent maintains it’s debt-equity ratio, what is the debt capacity of the project in part
(b)?
d. What is Lucent’s unlevered cost of capital?
e. What is the unlevered value of the project?
f. What are the interest tax shields from the project? What is their present value?
g. Show that the APV of Lucent’s project matches the value computed using the WACC
method.
h. What is the free cash flow to equity for this project?
i. What is its NPV computed using the FTE method? How does it compare with the NPV
based on the WACC method?
Solution
a. rwacc 
10.8
14.4  10.8
10% 
6.1%(1  0.35)  8.49%
14.4
14.4
b. Using the WACC method, the levered value of the project at date 0 is
VL 
50
100
70


 185.86
2
1.0849 1.0849 1.08493
Given a cost of 100 to initiate, the project’s NPV is 185.86 – 100 = 85.86.
c. Lucent’s debt-to-value ratio is d = (14.4 – 10.8) / 14.4 = 0.25. The project’s debt
capacity is equal to d times the levered value of its remaining cash flows at each date:
Year
FCF
VL
D = d*VL
d. rU 
0
–100
185.86
46.47
10.8
14.4  10.8
10% 
6.1%  9.025%
14.4
14.4
1
50
151.64
37.91
2
100
64.52
16.13
3
70
0
0.00
(Note that this formula (pretax WACC for rU) is true
only when we maintain the constant debt-equity ratio, see lecture slides)
e. V U 
50
100
70


 184.01
1.09025 1.090252 1.090253
Year
FCF
VL
D = d*VL
Interest
Tax Shield
0
–100
185.86
46.47
1
50
151.64
37.91
2.83
0.99
2
100
64.52
16.13
2.31
0.81
3
70
0
0.00
0.98
0.34
f. Using the results from problem (c) (and interest = debt cost of capital = 6.1%):
The present value of the interest tax shield is
PV(ITS) 
0.99
0.81
0.34


 1.85
2
1.09025 1.09025 1.090253
g. Using the debt capacity calculated in problem 5, we can compute FCFE by adjusting FCF
for after-tax interest expense (D*rD*(1 – tc)) and net increases in debt (Dt – Dt-1):
Year
D
FCF
After-tax Interest Exp.
Inc. in Debt
FCFE
h. NPV  53.53 
0
46.47
1
37.91
2
16.13
3
0.00
-$100.00
$0.00
$46.47
-$53.53
$50.00
-$1.84
-$8.55
$39.60
$100.00
-$1.50
-$21.78
$76.72
$70.00
-$0.64
-$16.13
$53.23
39.60 76.72 53.23


 $85.86
1.10 1.102 1.103
4. Problem 18.12 from Berk-DeMarzo
Amarindo, Inc. (AMR), is a newly public firm with 10 million shares outstanding. You are doing
a valuation analysis of AMR. You estimate its free cash flow in the coming year to be $15
million, and you expect the firm’s free cash flows to grow by 4% per year in subsequent years.
Because the firm has only been listed on the stock exchange for a short time, you do not have an
accurate assessment of AMR’s equity beta. However, you do have beta data for UAL, another
firm in the same industry:
UAL: Equity Beta = 1.5, Debt Beta = 0.30, Debt-Equity Ratio =1.
AMR has a much lower debt-equity ratio of 0.30, which is expected to remain stable, and its debt
is risk free. AMR’s corporate tax rate is 40%, the risk-free rate is 5%, and the expected return on
the market portfolio is 11%.
a) Estimate AMR’s equity cost of capital
b) Estimate AMR’s share price
Solution
a) Using the same expression for beta as for rU (see previous problem, part (d), again
remember that we are in the situation of maintaining constant debt-equity ratio, hence we
can use unlevered WACC to estimate rU), UAL Asset beta = (1/2) 1.5 + (1/2) 0.3 = 0.90
We can use this for AMR’s asset beta.
To derive the equity beta, since AMR’s debt is risk free we have:
Equity Beta = Asset Beta  (1 + D/E) = 0.9  1.30 = 1.17
From CAPM
re = 5% + 1.17(11% – 5%) = 12.02%
Alternatively, given an asset or unlevered beta of 0.90 for AMR, we have (from CAPM):
ru = 5% + 0.90(11% – 5%) =10.4%
Then we can solve for re:
re = 10.4% + 0.30 (10.4% – 5%) = 12.02%
b) Since D/E ratio is stable, we can value AMR using the WACC approach.
1. WACC = (1/1.3) 12.02% + (.3/1.3) 5% (1 – 40%) = 9.94%
2. Levered value of AMR (as a constant growth perpetuity):
3. D  E  VL = FCF/(rwacc – g) = 15 / (9.94% – 4%) = $252.52
million
4. E  E D E  VL  252.52 1.3  $194.25 million
5. Share price = 194.25 / 10 = $19.43