Part 5
Chapter 20
The valuation of a project under different
financing strategies.
Marc B.J. Schauten
Example
Consider a project to produce product SMART. I = 4000.
Cash flow is $ 1600 pre-tax per year for 5 years. RU = 10%. Tax rate = 35%.
NPV base case = -I +  [EBIT x (1-C)] /(1+RU)t
Example
Ru =
Tax rate =
Year
1
1
2
3
4
5
10%
35%
EBIT EBIT(1-tc) disc.factor PV of 3
2
3
4
5
1600
1040
0,9090909 945,5
1600
1040
0,8264463 859,5
1600
1040
0,7513148 781,4
1600
1040
0,6830135 710,3
1600
1040
0,6209213 645,8
3942
NPV base case = -I +  [EBIT x (1-C)] /(1+RU)t = -4000 + 3942 = -58
Financing rule 1: Equal principal repayments
Suppose that because of the value of the expected CF’s of this project, the firm can
borrow 2000 more, the loan is repaid in equal installments during 5 years
and Rd = 8%.
Question: PV tax shield?
Example
Rd =
Tax rate =
Year
1
1
2
3
4
5
8%
35%
Debt Interest * tc
3
2
56
2000
44,8
1600
33,6
1200
22,4
800
11,2
400
APV = -58 + 141 = 83 > 0!
disc.factor PV of 3
5
4
0,9259259 51,9
0,8573388 38,4
0,7938322 26,7
0,7350299 16,5
7,6
0,6805832
141,0
Financing rule 2: Balloon repayment
Suppose that because of the value of the expected CF’s of this project, the firm can
borrow 2000 more, the loan is repaid at the end of year 5 and Rd = 8%.
Question: PV tax shield?
Example
Rd =
Tax rate =
Year
1
1
2
3
4
5
8%
35%
Debt Interest * tc
3
2
56
2000
56
2000
56
2000
56
2000
56
2000
disc.factor PV of 3
5
4
0,9259259 51,9
0,8573388 48,0
0,7938322 44,5
0,7350299 41,2
0,6805832 38,1
223,6
APV = -58 + 223.6 = 165.58 > 0!
Financing rule 3: Target capital structure
Assume that the project will be financed for 50% with debt, Rd = 8%.
Use the WACC MM?
Assumptions (a.o.) of WACC MM:
-
PVTS = tc D
Alternatives are: 3a) Miles Ezzell WACC and 3b) Harris Pringle / Ruback WACC
Ad 3a) Assumptions WACC Miles Ezzell:
-
debt as a proportion of the total market value is remains constant during
the life of the project;
-
the project generates stable/unstable CFs that could be finite and/or
variable.
-
ME do not discount the tax shield with RD only. As long as future tax
shields are tied to uncertain future cash flows, discount with Ru.
WACCME  RU 
 1 RU 
D

RD xTc 
DE
 1 RD 
WACC method (textbook)
NPVWACC ME = -I + EBITt (1-tc)/(1+WACCME)t
WACCME
Year
1
1
2
3
4
5
 1 RU 
D
 1.10 
  0.10  0.5x0.08x0 .35
 RU 
RD xTc 
  8.57407%
DE
1

R
1.08


D 

EBIT
2
1600
1600
1600
1600
1600
EBIT(1-tc)
3
1040
1040
1040
1040
1040
NPV = 4,090.34 – 4,000 = 90.34
disc.factor
4
0.9210302
0.8482966
0.7813068
0.7196071
0.6627799
PV of 3
5
957.871
882.228
812.559
748.391
689.291
4090.34
Inselbag, I. and H. Kaufold, 1997, Two DFF approaches
for Valuing Companies under alternative financing
strategies (and how to choose between them), Journal of
Applied Corporate Finance, 114-122.
Check with APV method; using Ru and Rd!
Vl WACC ME
Debt at start year
at the start of year
= 0,5 x Vl
1
2
4,090.34
2,045.17
3,401.05
1,700.53
2,652.66
1,326.33
1,840.10
920.05
957.87
478.94
NB explanation column 5:
53.02 = 57.26 / (1.08)
40.08 = 47.61 / [(1.10)(1.08)]
28.42 = 37.14 / [(1.10)2(1.08)]
17.92 = 25.76 / [(1.10)3(1.08)]
8.48 = 13.41 / [(1.10)4(1.08)]
RdxD
3
163.61
136.04
106.11
73.60
38.31
tcxRdxD
PV of 4
(= tc x 3)
4
5
57.26
53.02
47.61
40.08
37.14
28.42
25.76
17.92
13.41
8.48
PV tax shield
147.92
Base case
3,942.42
APV
4,090.34
CFE method

r T  D 
 0.08  0.35  0.5 
re  ru  (ru  rd )1 d    0.1  (0.1  0.08)1

  0.11948148
1.08  0.5 

 1 rd  E 
Year
1
1
2
3
4
5
Vl WACC ME
Debt at start year
at the start of year
= 0,5 x Vl
2
3
4,090.34
2,045.17
3,401.05
1,700.53
2,652.66
1,326.33
1,840.10
920.05
957.87
478.94
RdxD
EBT
4
163.61
136.04
106.11
73.60
38.31
5
1,436.39
1,463.96
1,493.89
1,526.40
1,561.69
Tax
(= tc x 5)
6
502.74
512.39
522.86
534.24
546.59
Redemption
CFE
PV(CFE)
7
344.65
374.20
406.28
441.11
478.94
8
589.01
577.38
564.75
551.04
536.16
9
526.14
460.71
402.54
350.85
304.94
2045.17
CFE1 = (EBIT-Rd D)(1-tc) – redemption = (1600 – 163.61)(1-0.35) – 344.65 = 589.01
Market value of E at t=0: 2,045.17
Equity holders invested: 4,000 – D0 = 4,000 – 2,045.17 = 1,954.83
NPV = 2,045.17 – 1,954.83 = 90.34
Financing rule 3: Target capital structure
Assume that the project will be financed for 50% with debt, Rd = 8%.
Ad 3b) Harris and Pringle (1985) and Ruback (2002) assume tax shields are
discounted at RU (see Part 5 note B)
WACCRuback  RU 
D
RD  Tc
DE
WACCRuback  RU 
D
RD  Tc  0.10  0.5  0.08  0.35  0.08600
DE
NPVWACC Ruback = -I + EBITt (1-tc)/(1+WACCRuback)t
WACC method (textbook)
Year
1
1
2
3
4
5
EBIT
2
1600
1600
1600
1600
1600
EBIT(1-tc)
3
1040
1040
1040
1040
1040
NPV = 4,087.57 - 4,000 = 87.57
disc.factor PV of 3
4
5
0.9208103
957.64
0.8478916
881.81
0.7807474
811.98
0.7189202
747.68
0.6619892
688.47
4087.57
Check with APV method; using Ru only!
Year
1
1
2
3
4
5
Vl WACC Ruback
at the start of year
2
$4,087.57
$3,399.10
$2,651.43
$1,839.45
$957.64
D=0,5*Vl
3
$2,043.79
$1,699.55
$1,325.71
$919.73
$478.82
NPV = 4,087.57 - 4,000 = 87.57
RdxD
4
$163.50
$135.96
$106.06
$73.58
$38.31
tcxRdxD
(=tcx(4))
5
$57.23
$47.59
$37.12
$25.75
$13.41
PV tax shields
Base case
PV of (5)
6
$52.02
$39.33
$27.89
$17.59
$8.32
$145.15
$3,942.42
$4,087.57
CFE method
RE Ruback  RU 
Year
1
1
2
3
4
5
Dt
(RU  RD )  0.1  1(0.1- .08)  0.12
Et
Vl at the
start of year
2
$4,087.57
$3,399.10
$2,651.43
$1,839.45
$957.64
D=0,5*Vl
RdxD
3
$2,043.79
$1,699.55
$1,325.71
$919.73
$478.82
4
$163.50
$135.96
$106.06
$73.58
$38.31
EBT
5
$1,436.50
$1,464.04
$1,493.94
$1,526.42
$1,561.69
Tax
6
$502.77
$512.41
$522.88
$534.25
$546.59
Redemption
7
$344.23
$373.84
$405.99
$440.90
$478.82
CFE1 = (EBIT-Rd D)(1-tc) – redemption = (1600 – 163.50)(1-0.35) – 344.23 = 589.49
Market value of E at t=0: 2,043.79
Equity holders invested: 4,000 – D0 = 4,000 – 2,043.79 = 1,956.21
NPV = 2,043.79 – 1,956.21 = 87.57
CFE
PV(CFE)
8
9
$589.49
$526.33
$577.78
$460.61
$565.07
$402.21
$551.27
$350.34
$536.28
$304.30
Total
$2,043.79
Summary example
Financing Rule
1
APV
- Base case
- tax shield
83,4
-57,6
141,0
2
166,0
-57,6
223,6
WACC ME, CFE ME
3a
3b
Miles
Ezzell
Harris Pringle/
Ruback
90,3
- 57,6
147,9
87,6
- 57,6
145,2
90,3
87,6
Financing rule
1
equal principlal repayments (2,000; 1,600; 1,200; 800; 400)
2
balloon repayment (2,000; 2,000; 2000; 2000; 2000)
3
target capital structure / debt rebalanced (50% of project value)
a) ME: (2,045; 1,701; 1,326; 920; 479)
b) HP/R:(2,044; 1,700; 1,326; 920; 479)
Levering and Unlevering betas, some formulas
MM1963, PVTS discounted at RD, g = 0
D
βE = βU + (βU - βD ) (1- t c )
E
and
βU 
β E  β D 1 τ c 
1 1 τ c 
HP1985, PVTS discounted at RU
βE  βU  (βU - βD )
D
E
D
βD
E
βU 
 D
1  
 E
βE 
and
D
E
D
E
MM1963: Discount rate tax shield is rd, g = 0
The expected economic income for the providers of capital is: VU (RU ) + τ c (RD )D = E(RE ) + D(RD )
Rewriting gives: RE 
VURU   c RDD  RDD
E
MM63 tells us that: VL  VU   c D
→
RE 
VU
D
RU  (1   c ) RD (1)
E
E
and VL  D  E , this results in: VU  E  (1   c )D
If we insert (2) in (1) we find: R E  R U 
D
(1   c )(R U  R D )
E
(3)
Remark: (3) = Proposition II MM63!
Relation WACC and rd
WACC 
D
E
(1   c )R D 
RE
DE
DE
(4)
If we insert (3) in (4) we find:
WACC =
Since
D
E
E
D
(1- τc )(RU - RD ) = E RU + D (1- τc )RU
(1 - τ c )RD +
RU +
D+E
D+E
D+E E
D +E
D+E
E
D
+
=1
D+E D+E
(2)
D


We can rewrite (5) into: WACC  R U 1 
c 
 DE 
(5)
(6)
Relation E en D
RE  RU 
D
(1   c )(R U  R D )
E
(3)
Following the CAPM:
RE  R F  βE (R M  RF )
(7)
RU  RF  βU (R M  RF )
(8)
RD  RF  βD (R M  RF )
(9)
Inserting (7)-(9) in (3) gives:
D
βE = βU + (βU - βD ) (1- t c )
E
and
βU 
β E  β D 1 τ c 
1 1 τ c 
D
E
D
E
HP1985: Discount rate tax shield is ru
The expected economic income for the providers of capital is: VU (RU ) + PVTS(RU ) = E(RE ) + D(RD )
Substitute VU = E+D - PVTS
→
RU (E + D - PVTS) + PVTS(RU ) = D(RD ) + E(RE )
(RU )E + (RU )D = D(RD ) + E(RE )
E(RE ) = (RU )E + (RU )D - (RD )D
RE = RU +
D
(R - R )
E u D
→
→
→
(1)
Note that (1) is the same as proposition II of Miller and Modigliani (1958)
When substituting (1) in the equation in order to calculate the WACC,
by taking the weighted average of RD after taxes and RE, we find
WACC = Ru -
D
t R
VL c D
(2)
Relation E en D
D
RE = RU + (RU - RD )
E
(3)
βU =
E
D
βE + βD
VL
VL
Following the CAPM:
RE  R F  βE (R M  RF )
(4)
RU  RF  βU (R M  RF )
(5)
RD  RF  βD (R M  RF )
(6)
Inserting (4)-(6) in (3) gives:
R F  β E (R M  R F )  R F  β U (R M  R F ) 
βE  βU  (βU - βD )
D
E
D
R F  βU (R M  R F )  R F  βD (R M  R F )
E
D
βD
E
βU 
 D
1  
 E
βE 
and
→