CAPITAL STRUCTURE
2
Two primary funding sources: debt and equity
4
Each source has different risk level
2
Each funding source must be compensated for opportunity
cost of what suppliers of funds can earn elsewhere, on
investments of equivalent risk
2
In order to be accepted, projects must increase owners’
expected utility of wealth
Each project must provide, on a risk adjusted basis, enough
CF to pay required returns to equity and debt holders, pay
back original investments, and leave something extra to
increase owners’ (equity holders’) EU of wealth.
2
Cost of capital: minimum risk adjusted return to
shareholders
2
With no debt and CAPM:
4
2
Cost of Capital = E (kj) ' k0 % E (km & k0)
j
We need to consider effect of debt on the cost of capital
Also does capital structure really matter?
That is, does a firm’s capital structure impact the firm’s
value?
VALUE OF THE FIRM WITH CORPORATE TAXES
2
Value of a Levered Firm
4
Modigliani and Miller (1958, 63)
3
3
3
3
3
3
3
3
3
2
Many of the assumptions can be relaxed without changing
major conclusions
4
2
Capital markets are frictionless
Borrow and lend at the risk-free rate
No cost to bankruptcy
Two types of claims: risk-free debt and (risky)
equity
All firm’s in the same risk class
Only corporate taxes
All CFs are perpetuities (no growth)
No signaling opportunities
Managers maximize shareholder wealth
e.g., risk-free assumption on debt can be relaxed
without impacting results
Bankruptcy and personal tax assumptions do have critical
impact on results
2
Clarification: All firms in the same risk class
2
Implication:
˜ '
CF
i
Risky CFs vary only by a scale factor
˜ where = constant scale factor
CF
j
In other words, CFs are perfectly correlated
Consider gross returns
CFit & CFit&1
rit '
CFit&1
CF i '
CF j Y rit '
CFjt &
CFjt&1
CFjt&1
CFjt & CFjt&1
'
' rjt
CFjt&1
So if CFs differ only by scale factor, they will have the
same distribution of returns, the same risk, and will require
the same expected return.
2
Assume the firm’s assets generate the same distribution of
operating CFs each period after-taxes forever.
Thus value of the firm without any debt is
Vu '
E (c̃)
ku
where
Vu
=
present value of an unlevered firm
E (c̃) = expected after-tax cash flow in perpetuity
ku
=
discount rate for all equity firm
Remember stuff on estimating CF
˜
˜ % Dep
ATCF
' NIAT
' (R̃ & Ṽ c & F c & Dep) (1 & tc) % Dep
4
4
4
no other accruals
no interest cost because no debt
no growth
We've assumed c is generated in perpetuity.
This implies depreciation each year must be replaced by
investment in order to keep the same amount of capital in
place.
Thus we're assuming Dep = I where I is capital investment
each year.
c̃
˜ % Dep & I
= NIAT
=
=
=
˜ & Ṽ & F & Dep) (1 & t ) % Dep & I
(Rev
c
c
c
˜ & Ṽ & F & Dep) (1 & t )
(Rev
c
c
c
˜
˜ (1 & t ) ' NIAT
(EBIT)
c
When all CFs are perpetuities, CFs to investors is the same
as NIAT, so
V
u
˜ (1 & t )
E (EBIT)
E (c̃)
c
'
'
ku
ku
2
Now assume the firm issues debt
After-tax CFs must be split between debt and equity
holders
2
˜ % Dep & I
Equity holders get: NIAT
Debt holders get:
where
2
kd D
kd = interest rate on debt
D = face value of debt
Thus to total CF to debt and equity holders is
˜ % Dep & I) % k D '
(NIAT
d
˜ & Ṽ & F & Dep & k D) (1 & t ) % Dep & I % k D
(Rev
c
c
d
c
d
2
Assuming no growth (Dep = I), the total CF is
˜
˜ % k D = EBIT(1
& tc)
NIAT
d
8
CF to unlevered
firm (c̃) . Will have
same risk level.
%
kd D t c
8
Tax shield from
using debt (risk-free
by assumption).
2
Discounting each CF by the appropriate discount rate for
its risk class, we find the value of the levered firm to be
V
L
'
˜ (1 & t )
E (EBIT)
c
ku
%
kd D t c
k0
where
VL = value of levered firm
k0 = risk-free rate
Note:
kdD is the perpetual stream of risk-free payments
to debt holders
This implies the market value of the risk-free
debt is
B=
2
kd D
k0
= market value of debt (bonds)
Rewriting we find,
VL = Vu + Btc
Value of a levered firm is equal to the value of the
unlevered firm plus the present value of the tax shield
provided by debt.
Note that in the absence of any market imperfections
(i.e. tc = 0), the value of the firm is not dependent on
the capital structure of the firm
VL ' Vu
(if tc ' 0)
This famous result is known as Modigliani-Miller
Proposition I (MMI).
MMI (Arbitrage argument)
(tc = 0)
2 firms, identical except for capital structures
Unlevered Company
E(EBIT) = $200
Vu = $1000
Bu = 0
Eu = 1000
ku = .20
Lever Company
E(EBIT) = $200
VL = ?
BL = 500
EL = ?
kd = .10
Vu '
E (EBIT)
ku
Strategy I:
'
200
' $1000
.20
Buy 10% of Unlevered
Investment
Cash Flow
.10($1000) = $100
= .10(Vu)
˜
.10 (EBIT)
2
Strategy II: Buy 10% of Levered Company’s Equity
Cash Flow
˜ & k B )
.10 (EBIT
d L
Investment
.10(EL) = .10(VL - BL)
Common argument:
2
VL (and EL/share) should be lower
because of increased risk
associated with leverage.
Strategy III: Buy 10% of Unlevered using combination of
borrowed funds and equity funds
Borrow 10% of BL and combine with equity funds to buy
10% of Vu.
Cash Flow
Investment
Borrow 10% of BL
-.10BL
10% of Vu
.10Vu
Buy
Total
.10(Vu - BL)
-.10 kd BL
˜
.10 (EBIT)
˜ - kdBL)
.10 (EBIT
Important Point:
Cash flows from buying levered firm are
identical to borrowing and buying unlevered
firm.
(CF to Strategy II = CF to Strategy III)
Therefore:
Cost of each strategy must be the same.
Cost of Strategy II
Cost of Strategy III
.10 (VL - BL)
.10(Vu - BL)
Thus rational investors will require
VL = Vu
(MMI)
2
Critically important result in finance
2
Before MMI, effects of leverage misunderstood
MMI shows if levered firms are priced too high, individuals will
simply borrow on their own accounts and buy shares in
unlevered firms.
Y
Leverage doesn’t effect value of firm!
Example
Consider decision to use debt in BigAg Company.
Financial Structure (tc = 0)
Current
Assets
$8,000,000
Debt
0
Equity
8,000,000
Interest Rate
10%
Shares
400,000
Value/Share
$20
Proposal
$8,000,000
4,000,000
4,000,000
10%
200,000
$20
IMPACT OF CAPITAL STRUCTURE ON RETURNS
No Debt
Recession Expected
ROA
EBIT
5%
15%
Debt = $4,000,000 @ 10%
Expansion
Recession
25%
5%
$400,000 $1,200,000 $2,000,000
Expected Expansion
15%
25%
$400,000 $1,200,000 $1,200,000
0
0
400,000
400,000
400,000
400,000
1,200,000
2,000,000
0
800,000
1,600,000
ROE
5%
15%
25%
0
20%
40%
EPS
$1.00
$3.00
$5.00
0
$4.00
$8.00
Int
0
NIAT
Analysis: Effect of financial leverage depends on company’s income.
Possible Argument:
Expected income is $1,200,000 so the
firm should take on additional debt.
Argument is flawed. Shareholders can
borrow on personal accounts and
duplicate effects of company’s leverage.
2
Invest $2,000 in levered
Leverage Plan
Recession
Expected
Expansion
EPS
0
$4
$8
EPS x 100 shares
0
400
800
Initial Cost = 100 Shares @ $20/sh. = $2000
2
Invest $2,000 in unlevered
Homemade Leverage
(Borrow $2,000 – Buy 200 shares in Unlevered)
EPS x 200 sh.
Int = .10(2000)
Recession
Expected
Expansion
$1 x 200 = 200
3 x 200 =
600
5 x 200 =
1000
200
200
200
0
400
800
Initial Cost = 200($20) - 2000 = $2000
2
This is another illustration of MMI. In a world without
transaction costs, capital structure doesn’t matter.
Effectively, increases in expected returns from leverage are
offset by additional risk (more later).
2
Doesn’t match reality
2
Letting tc > 0,
V L ' V u % B tc
giving debt preferential tax treatment (allowing a tax
deduction for interest payments) increases the value of the
firm as the firm takes on more and more debt.
Y
2
firms should use almost all debt financing.
Doesn’t match reality
WEIGHTED AVERAGE COST OF CAPITAL (WACC)
Suppose project is funded with
B = $ by debt holders
E = $ by equity holders
I = $ of initial investment
I=B+E
The WACC is defined so that suppliers of capital receive their
respective required return given the risk they must bear.
Debt holders require kd. After-tax cost = kd(1 - tc)
Equity holders require ke.
I kw = B kd(1 - tc) + E ke
kw = WACC
kw '
Let wd '
B
k (1 & tc) %
I d
E
k
I e
B
E
and we '
I
I
kw ' wd kd (1 & tc) % we ke
(WACC)
2
Could allow multiple debt and equity types
2
wd and we often assumed set at some "target level" (more
later)
2
kw supplies required return to each contributor of capital
Relationship between ke and debt
VL = Vu + Btc
2
Expected ATCF into levered firm
Vu (ku) + Btc (kd)
8
8
same risk
same risk
as c̃
as kd D
2
Expected ATCF to debt and equity holders
E ke % B kd
2
Cash inflows = cash outflows (no growth)
V u ku % t c B kd ' E ke % B kd
Vu
B
ke '
ku & (1 & tc) kd
E
E
Remember
V L ' V u % tc B ' B % E
so V u ' E % B (1 & tc)
Substituting,
ke '
2
E % B (1 & tc)
E
ku & (1 & tc)
ke ' ku % (1 & tc) (ku & kd)
B
E
B
kd
E
(MMII)
4
Opportunity cost of equity capital increases linearly
B
with a change in
.
E
4
With no debt, ke = ku.
Example:
AGFIRM is currently unlevered. It is considering restructuring
to allow $200 in debt. Company expects to generate $151.52 in
EBIT (perpetuity). Corporate tax rate = 34%. Cost of debt is
10%. Unlevered firms in the industry require a 20% return.
2
What will AGFIRM’S value be if it restructures?
˜ u) ' E (EBIT)
˜ (1 & t )
E (ATCF
c
' $151.52(1 & .34)
' $100
V
u
'
E (ATCF u)
ku
'
100
' $500
.20
V L ' V u % tc B ' $500 % 200 (.34) ' $500 % $68
' $568
Suppose AGFIRM started with 500 shares
Vu
$500
Share price =
' $1
'
500
shares
u
VL & B
$568&$200
Share price =
'
' $1.23
shares
300
L
Value of equityL = EL = VL - B = $568 - $200 = $368
2
What is the required return on AGFIRM’s equity?
ke ' ku % (1 & tc) (ku & kd)
B
E
' .20 % (1 & .34) (.20 & .10)
200
368
' .2359
Use of debt increased required return on equity from
ku = .20 to ke = .2359
Check
EL '
'
2
(E (EBIT) & kd B) (1 & tc)
ke
(151.52 & .10 (200)) (1 & .34)
= $368
.2359
Find AGFIRM’s WACC
wd '
kw
B
VL
'
200
' .352
568
we '
368
' .648
568
= weke + wdkd (1- tc)
= (.648)(.2359) + (.352) (.10)(1 - .34)
= .15286 + .02323
= .1761
The firms WACC decreased from kw = ku = 20% to
kw = 17.61% as a result of the debt use.
Check value of the firm
VL '
E (EBIT) (1 & tc)
kw
'
151.52 (1 & .34)
' $568
.1761
Suppose the firm holds a $100 in assets. Required payment
to capital suppliers = I · kw
= 100 · .1761
= $17.61
Funding
Source
Investment
Required
Return
Cash
Flow
Equity
.648(100) = $64.88
Debt
Govt.
Total
.352(100) = 35.20
tax shield = $3.52(.34)
100
.2359
.10
.1761
$15.
28
3.52
(1.20)
$17.60
GRAPHICAL LOOK AT COST OF CAPITAL
2
No taxes
%
ke=ku+(ku-kd) B/S
ku
kw = ku
kd
B/E
Note:
k w ' we k e % wd k d
'
E
B
ke %
kd
B%E
B % E
'
E
B
B
(k u % (k u & kd) ) %
kd
B%E
E
B%E
'
E
B
B
ku %
(k u & kd) %
kd
B%E
B%E
B%E
' ku
Corporate taxes (tc > 0)
Note:
ke=ku+(1-t c)(ku-kd) B/E
%
ku
kw=ku(1-tcB/B+E)
ku(1-tc)
kd(1-tc)
B/E
kw =
E
B
B
(k u % (1 & tc) (k u & kd) ) %
kd (1 & tc)
B % E
E
B % E
=
E
B
ku %
(1 & tc) k u
B % E
B % E
=
E
B
B
ku %
ku &
tc k u
B % E
B % E
B % E
= ku &
B
tc k u
B % E
6 = k u 1 & tc
B
B % E
B
) ' 1
B64 B % E
lim (
Y
lim kw ' k u (1 & tc)
B64
With or without taxes, increasing debt increases ke because of higher
risk (also higher expected return).
However, value of firm and equity claims only change as a result of
tax shield.
V
L
'
E (EBIT) (1 & tc)
kw
% B tc
Without taxes reduces to
VL '
EBIT
ku
Results so far:
Without taxes — Capital structure irrelevant
Corporate taxes — Firms should use almost all debt
CAPITAL STRUCTURE — LIMITS TO DEBT USE
4
Remember: world without personal taxes
V L ' V u % tc B
4
Firms should use all debt financing
4
Inconsistent with real world
4
MM theory help point out possible answers
3
Financial distress costs
3
Personal taxes
2
Bankruptcy:
Risk or Cost?
4
Debt provides tax benefits
4
Debt also puts financial pressure on firm
4
Ultimate financial distress 6 bankruptcy
4
Bankruptcy costs can offset benefits of debt
Example (no taxes)
Day Company and Knight Company both plan to operate one more
year.
Knight Company: $49 principle and interest obligation
Day Company:
$60 principle and interest obligation
Both companies have the same operating CFs.
Knight Company
Boom
Day
Company
Rec
Boom
Rec
Prob.
(.50)
(.50)
(.50)
(.50)
CF
100
50
100
50
Debt
49
49
60
Distribution to
Stockholders
51
1
40
60
50
0
Day Company bankrupt in regression
Assume: Risk neutral investors
Interest rate ke = kd = 10%
Ek '
(.5) (51) % (.5) (1)
' 23.64
1.10
ED '
(.5) (40) % (.5) (0)
' $18.18
1.10
Bk '
(.5) 49 % (.5) (49)
' 44.54
1.10
BD '
(.5) (60) % (.5) (50)
' 50
1.10
Vk
68.18
VD
2
Firms have same value
2
Debt holders recognize bankruptcy impacts
68.18
Promised payment $60, but debt holders only willing to
pay $50.
Required payment without bankruptcy
60
= $54.55
(1 % .10)
Actual yield on Day’s debt
60
& 1 ' .20 or 20%
50
(junk bond)
Example is unrealistic
Reality: lawsuits, liquidation, other cost increase
realized bankruptcy costs
Suppose these costs = $15
Day Company
(.5)
(.5)
Boom
Rec
CF
100
Debt
60
Distribution
to Equity
40
50 - 15
ED '
60 50 35
BD =
0
(.5) (40) % (.5) (0)
' 18.18
1.10
(.5)(60) % (.5) (35)
= 43.18
1.10
VD
61.36
2
Bankruptcy "costs" reduce value of firm
2
Bankruptcy "risk" did not impact firm value
Without bankruptcy costs
$50
$100
$60
Debt
$40
Eq.
$50
Debt
With bankruptcy costs
$100
$60
Debt
$40
Eq.
$50
$35
$15
B. cost Debt
Debt and Equity
holders no longer
split earnings
Bondholders account for bankruptcy "costs" in returns
$60
& 1 ' 39.0%
$43.18
Bond holders pay fair price after accounting for prob. and
costs of bankruptcy
Suppose
Rec Prob
0
.5
.5
Bankruptcy Cost
0
0
$15
Repayment
Promise
60
60
60
Amount
Borrowed
$54.55
50.00
43.18
2
Bankruptcy costs increase cost of debt
2
Financial distress causes same impact as bankruptcy
costs
TYPES OF COSTS
Direct Costs of Financial Distress
2
Legal and Administration Costs of Liquidation and
Reorganization
4
4
4
4
Legal fees
Administrative fees
Accounting fees
Court fees (expert witnesses)
Estimated Costs (publically traded firms) . 3% of market
value
2
Indirect Costs of Financial Distress
4
Impaired ability to conduct business
3
impacts relations with customers and suppliers
Estimated direct and indirect costs . 20%+ of market value
AGENCY COSTS
Conflict of interest between shareholders and debt holders
1.
Incentives to take large risks
4
Firms near bankruptcy take larger risks
Suppose promised debt payment is $100
Rec.
Boom
Low Risk Project
Prob. Value = Equity + Debt
.5
$100 =
0 + 100
.5
200 = 100 + 100
V L' (.5) (100) % (.5) (200) ' $150
Rec.
Boom
High Risk Project
Prob. Value = Equity + Debt
.5
$ 50 =
0 + 50
.5
240 =
140 + 100
V H ' (.5) (50) % (.5) (240) ' $145
(bankrupt)
All equity firm would select low risk project
VL > VH
However equity value is
EL= (.5)(0) + (.5)(100) = 50
EH = (.5)(0) + (.5)(140) = 70
Equity holders will want high risk project when debt is
used.
2.
Incentive toward under investment
New investment may help debt holders at shareholders
expense
Consider firm
$4,000 debt payment due at end of year
Bankrupt in recession
Project cost $1,000 and brings in $1,700
Firm without Project
(.5)
(.5)
Boom
Rec
CFs
5,000
2,400
Debt
4,000
2,400
1,000
0
Firm with Project
(.5)
(.5)
Boom
Rec
6,700
4,100
4,000
4,000
2,750
100
Debt holders prefer project
Suppose $1,000 cost of project comes from equity
Ew/o = (.5) 1,000 + (.5) 0 = $500
Ew = (.5) (2,700) + (.5)(100) - 1,000 = $400
Equity holders prefer to reject project.
Equity holders contribute entire investment but must share
rewards.
3.
Milk Property — like strategy 2 except instead of
deciding not to invest you actually
divest through increasing dividends.
4
These strategies only occur when there is a possibility
of bankruptcy
Again — increases the cost of debt
4
May be reduced through "protective covenants" in
loon documents
3
Negative Covenants (limit actions)
restrict dividends, collateral, mergers, asset
sales, debt levels
3
Positive covenant (specify action)
NWC, financial statements
3
Debt Consolidation also reduces bankruptcy
costs
3
Combining the tax effects and financial distress
costs
Value of PV of tax
Firm
shield
VL = Vu + tcB
PV of financial
distress cost
Maximum
Value
Vu
B*
(Optional debt level)
Tax shield
Y 8 value of firm
Distress costs Y 9 value of firm
Debt
VALUATION UNDER PERSONAL
AND CORPORATE TAXES
2
Earlier we saw gains in leverage with only corporate taxes:
G ' V L & V u ' tc B
2
Now consider value of firm in a world with both corporate
and personal taxes
Equity holders receive (EBIT - kdB)(1 - tc)(1 - te)
Debt holders get
kdD(1 - td)
Total CF to all stakeholders:
(EBIT & kd D) (1 & tc) (1 & te) % kd D (1 & td)
or
EBIT (1 - tc)(1 - te) + kdD(1 - td) 1 &
8
CF to unlevered
firm after taxes
8
CF to debt
holders after
taxes
(1 & tc) (1 & te)
(1 & td)
So
VL ' Vu % B 1 &
(1 & tc) (1 & te)
(1 & td)
where B = kdD (1 - td) / kd
Gain from leverage in world with personal taxes
G ' B1 &
(1 & tc) (1 & te)
(1 & td)
Notes:
2
When te = td, gain is same as in world without personal
taxes
2
When te < td, gain from leverage is reduced from world
without personal taxes
4
Reasons te may be less than td
3
3
3
3
capital gains tax break
capital gains may be delayed by reinvestment
gains and losses in a portfolio may offset each other
dividend exclusions for corporations
when te < td, more taxes get paid in a levered firm, than
an unlevered firm at the personal level
2
If (1 & tc) (1 & te) ' (1 & td) , gain from leverage is zero.
Lower corporate taxes from leverage are exactly offset by
higher personal taxes.
$
VL = Vu + tcB when te = td
VL = Vu + B[1- (1-tc)(1-te) ]
(1-td)
when (1-td) > (1-tc)(1-te)
VL = Vu when
(1-td) = (1-tc)(1-te)
B
TAX POLICY AND FINANCING INCENTIVES
Before 1986: Corporate Income
– 46% maximum
Interest and Dividends – 50% maximum
Capital Gains
– 20%
Suppose firm pays no dividends and capital gains are deferred
so that the effective tax rates are te = 10%, td = 50% and
tc = 46%.
Income before tax
Less Corporate tax @ 46%
Income after Corporate tax
Less Personal tax (te=.10 and td=.50)
Income after tax
Interest
$1.00
0.00
1.00
0.50
$0.50
Equity Income
$1.00
0.46
0.54
0.054
$0.496
Small advantage to debt
= $.004
Essentially no advantage to debt
1999:
Corporate Income
Interest and Dividend
Capital Gains
– 35% maximum
– 39.6% maximum
– 20%
Suppose effective capital gains rate is 20%/2 = 10% and that no
dividends are paid.
Income before tax
Less Corporate tax (tc = 35)
Income after Corporate tax
Personal tax (td=.396 and te=.10)
Income after tax
Interest
$1.00
0.00
1.00
0.396
$0.604
Equity Income
$1.00
0.35
0.65
0.065
$0.585
Advantage to debt = $0.019
Now suppose the same firm in 1999 pays out 1/2 of equity
income as dividends.
Effective tax rate on equity (.396 + .10)/2 = .248)
Income before tax
Less Corporate tax (tc=.35)
Income after Corporate tax
Personal tax (td=.396 and te=.248)
Income after tax
Interest
$1.00
0.00
1.00
0.396
$.604
Equity Income
$1.00
0.35
0.65
0.161
$0.489
Advantage to debt = $.115
2
Advantage tends to favor debt
2
Magnitude not clear and depends on tax rates of equity and
debt holders as well as dividend payout rates
EXAMPLE
E (EBIT) ' $100,00
tc ' 34%
(perpetuity)
te ' 12%
td ' 28%
ku (1 & tc) ' 15%
Currently all equity, but considering borrowing $120,000
at 10%.
Vu '
$100,000 (1 & .34)(1 & .12)
' $440,000
.15 (1 & .12)
V L ' $440,000 % $120,000 1 &
(1 & .34) (1 & .12)
(1 & .28)
' $463,200
G ' V L & V u ' 463,200 & 440,000 = $23,230
Smaller than tcB = .34(120,000) = $40,800
Extra tax on debt (td > te) at personal level lowers gains
from debt
2
Implications
4
Gains from leverage still positive (probably) but
smaller than thought if te < td
4
"Grossed" up return on debt to equate after-tax returns
(if te < td) offsets same debt advantage
Result
4
Framework also lays out arguments for equilibrium
aggregate debt levels (another day)
2
How firms establish capital structure
2
Practical difficulties —
2
Empirical Evidence
4
no "formula" for optimal
debt structure
Most firms have "low" D/E.
U.S. average D/E : .3 to .5
Firms pay substantial taxes but clearly don't issue
debt to point where tax shield is exhausted
4
Announced increases (decreases) in anticipated
leverage tend to increase (decrease) firm value
4
Capital structures differ by industry
3
3
3
4
Profitability
Growth
Intangible assets
Firms tend to maintain target levels at D/E
FACTORS TO CONSIDER IN DETERMINING
TARGET D/E
2
Taxes (tax shield)
2
Financial Distress Cost
4
4
4
2
2
Variable income 6 increase probability of financial
distress
Tangible assets 6 less financial distress
Intangible assets 6 more financial distress
Credit Reserves
4
External equity can be expensive to issue relative to
internal equity
4
Maintain credit capacity (low D/E) to allow capital
expenditures without issuing new equity
Industry D/E Ratios
Reconciling M-M and CAPM
Unifies approach to determining discount rate (cost of capital)
Type of
Capital
CAPM
MM
Debt
kd = krf + (km-krf)
d
kd = krf ,
Unlevered Equity
ku = krf + (km-krf)
u
ku = ku
Levered Equity
ke = krf + (km-krf)
L
ke = ku + (ku -kd)(1-tc) B/E
WACC
kw = weke + wdkd(1 - tc)
wd '
B
B%E
wc '
d
=0
kw ' ku 1 & t c
E
B%E
B
B%E
Can easily modify MM risk-free debt assumption:
kd ' krf % (km & krf)
Relationship between
L
+
d
u
ke ' krf % (km & krf)
L
' ku % (ku & kd) (1 & tc)
kd ' krf % (km & krf)
Substitute
ku ' krf % (km & krf)
d
u
Rearrange and simplify
L
'
u
'
u (1 % (1 & t c)
If we observe
% (
'
d) (1 & t c)
, we can estimate
B
L
E
B
1 % (1 & tc)
E
d (1 & t c)
B
E
B
) &
E
L
%
u
u
&
d (1 & t c)
u
B
E
B
E
EXAMPLE
Currently
Considering
wd = .2 (market value)
wd = .35
kd = krf = .07 ( d = 0)
tc = .5
km = .17
(wd = .20)
L = .5
2
Find the current values of ke and kw
ke ' krf % (km & krf)
L
' .07 % (.17 & .07) .5
' .12 or 12%
(at wd ' .2)
kw ' we ke % wd kd (1 & tc)
' (.8) (.12) % (.2) (.07) (1 & .5)
' .103 or 10.3%
(at wd ' .2)
2
Find kw if the new target capital is wd = .35
Remember ke will increase as the debt level rises relative to
equity
MM’s definition of kw says
kw ' ku 1 & t c
B
B % E
which implies (using observed results at wd = .20)
ku '
kw
1 & tc
B
B % E
'
.103
' .1144
1 & (.5)(.2)
So if wd = .35, MM’s definition of kw says
kw = .1144 (1 - (.5)(.35)) = .09438 or 9.438%
2
We could have calculated the new L and ke and preceded
with the standard kw approach directly
At wd = .2,
u
'
L
1 % (1 & tc)
'
B
E
d
=0
.5
' .4444
(1 % (1 & .5)(.25))
So at wd = .35, the new
L
Remember
will be
L
' 1 % (1 & tc)
B
E
u
' [1 % (1 & .5)(.5385)] (.4444)
' .5641
Thus ke = krf + (km - krf)
(at wd ' .35)
L
= .07 + (.17 - .07) .5641
= .1264
and
kw ' wd ke % wd kd (1 & tc)
(at wd ' .35)
' (.65)(.1264) % (.35) (.07) (1 & .5)
' 0.944 or 9.44%
2
Suppose project has same systematic risk as firm, L, and
provides expected return at 9.25%. Should project be
taken?
E (kp) ' .0925 < kw ' .0944
%
ke
12.6%
12%
10.3
9.4
ku(1 - tc)
kd(1 - tc)
kw = ku (1 - tc (B/B+E))
25%
53.85%
B/E
2
Evaluating projects with different risk levels than the firm.
4
Find the required return assuming all equity
investment.
4
Adjust for the firm’s capital structure using the
approach(s) from the previous section.
Example:
4
u
u
j
' 1.2, krf ' .07, km ' .17
kj = krf + (km - krf)
u
j
= .07 + (.17 - .07) 1.2 = .19 or 19%
4
kw = ku (1 - (1 - tc) B/B+E)
= .19 (1 - (.5) (.20)) = .171 or 17.1% at wd = .2
= .19 (1 - (.5) (.35)) = .157 or 15.7% at wd = .35
2
Comment on RTA vs. RTE
2
Comment on business organizations