capital structure and cost of equity pdf
DESCRIPTION
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers http://awesomefinance.weebly.com/TRANSCRIPT
Capital Structure and Cost of Equity
Learning Objec-ves
¨ Understand basic concepts of corporate finance ¤ Capital structure, cost of equity, dividend policy
¨ Calculate rate cost of equity capital, kE ¨ Calculate unleveraged rate cost of capital, kU
¤ Capital structure assuming no tax advantaged debt
¨ Systemic equity risk ¨ Miller and Modigliani
¤ Assump-ons ¤ Proposi-ons
¨ Demonstrate that under M&M assump-ons the DCF valua-on methods are equivalent
2
Simple Firm Assump-ons
¨ Fairway Corp financial structure plus ¤ T = 0, ∆T = 0, IDI = 0, NOA = 0 ¤ C = IC ¤ τ ≥ 0, EB > 0, DB ≥ 0
¨ M&M Assump-ons ¤ FCF is a perpetuity
n FCF = NOPAT – ∆IC = EBIT(1-‐τ) n ∆ IC = CX – DX -‐ CC + ∆OWC = 0 n CX – DX = 0, CC=0, ΔOWC=0
¤ Debt is constant (a perpetuity) n ∆DB = ∆D = 0
¤ kTS = kD
3
Firm Value
4
Assume: NOA=0, T=0 -----------------------
OA = TA, NOCE = IS = 0 NIBCL NIBCL NIBCL NIBCLIC = EB + DB , LE = IC + NIBCLIC = OWC + NC = CV = PV(FCF) = Fair Value of ICV = IC + MVA
-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Book Value LE & TA -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐CE AP ITP NIBCL NIBCLAR
IC
MVA
VU
VTS
-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Fair Value LE & TA -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐
D
E
INV
NC
STD
LTD
EB
DB
EB
Value of TA
V Value of IC
TA
OWC
NC
APV Valua-on with Constant FCF Growth
0
20
40
60
80
100
120
140
160
Fair Va
lue [$M]
VU
VTS
D
ETS
FCFU
1
TSU
FCF
1
Vgk
FCF
VVVAPVM
gkFCFEDV
FCFM
+−
=
+=
−=+=
5
No assump-on yet on growth of debt, D, tax shield, TS, or present value of tax shield, VTS
Simple Firm Example 6
)τ1(EBITFCF
0OWCΔ0TΔ
0CCDXCX
OWCΔ)CCDXCX(TΔ)τ1(EBIT OWCΔNΔNOPATFCF
−⋅=
=
=
=−−
−−−−+−⋅=
−−=
Dτk
τ)(1EBIT
Vk
τ)(1EBITV
U
TSU
⋅+−⋅
=
+−⋅
=
Without Debt With Debtt 33% 33%(1-‐τ) 67% 67%kD 10% 10%D -‐$ 100,000$ IX -‐$ 10,000$ ΔT -‐$ -‐$ IDI -‐$ -‐$ Δ IC -‐$ -‐$
EBIT 223,881$ 223,881$ τ·∙EBIT 73,881$ 73,881$
EBIT·∙(1-‐τ) 150,000$ 150,000$ IX·∙(1-‐τ) -‐$ 6,700$
NP 150,000$ 143,300$
IX·∙(1-‐τ) -‐$ 6,700$ IDI·∙(1-‐τ) -‐$ -‐$
ΔT -‐$ -‐$ NOPAT 150,000$ 150,000$
Δ IC -‐$ -‐$ FCF 150,000$ 150,000$
M&M assump-ons including kTS = KD
APV Valua-on with No FCF Growth
$0
$20
$40
$60
$80
$100
$120
$140
$160
$180
Fair Va
lue [$M]
VU
VTS
D
E
Dτk
τ)EBIT(1 DτkFCF V :APVM
UU
⋅+−
=⋅+=
kτ)EBIT(1
kFCF V :FCFM
−=
=
Dk
τ)(1Dk-‐τ)EBIT(1
Dk
FCFE V :FCFEM
E
D
E
+−⋅⋅−
=
+=
Rates of Return on Equity 8
Eτ)(1Dk -‐ τ)(1EBIT
Eτ)(1D)k-‐(EBIT
Eτ)](1D)k-‐E[(EBIT
E]E[NPr
D
D
D
0
1E
−⋅⋅−⋅=
−⋅⋅=
−⋅⋅=
=
‘Forward’ (expected) net profit on present equity fair value
EBτ)IX)(1-‐(EBIT
EBNP
roe1-‐
0
−=
=
‘Trailing’ net profit on present equity book value
Cost of Equity: M&M Assump-ons 9
( ) U
U
U
kED)τ-‐(1 kD)τE(Dτ)(1EBIT
EDDτk
τ)(1EBIT V
⋅+⋅=
⋅⋅−+=−⋅
+=⋅+−⋅
=
( )E
τ)(1DkkED)-‐(1r
Eτ)(1Dk -‐ τ)(1EBIT r
DUE
DE
−⋅⋅−⋅+⋅τ=
−⋅⋅−⋅=
ED)kk()1(rr
ED)kk()1(kr
DUUE
DUUE
⋅−⋅τ−+=
⋅−⋅τ−+=
ED)kk()1(kk DUUE ⋅−⋅τ−+=
But we s-ll don’t know kU
M&M Assumptions FCF and Debt are perpetuities
Cost of Equity: General 10
¨ Most common model is Capital Asset Pricing Model (CAPM) ¤ Defines a measure of risk as a single parameter ¤ Remember: kE ≡ E[rE] = rE
¨ rE is a func-on of the ¤ Risk free rate of return, rF ¤ Investor’s addi2onal expected return rate for the expected risk on
equity investment n The investor’s return rate is rela-ve to equity market value – not the firm’s
equity book value
¨ kE ≡ rE = rF + f( risk[rE] )
Risk Free Rate of Return, rF
¨ Return rate is risk free (known) over some planning period and in some currency ¤ Full return of principal ¤ ‘Nominal rate’ not real
n Real rate of return may not be known
n Future purchasing power of return and principal may not be known
¨ In the U.S. the risk free rate of return is the treasury debt zero coupon bond yield ¤ 12 mo. treasury bill yield for 1 yr investment horizon ¤ 10 year zero coupon treasury strip yield might be used for a long
term investment horizon
11
Capital Asset Pricing Model (CAPM)
]rr[Er ]r[E FMFE −⋅β+=
12
E[rM-‐rF] is the expected, excess risky return rate on the ‘market’ over some investment horizon (Market risk premium, MRP)
]rr[E ]rr[E FMFE −⋅β=−β is a risk parameter for an equity’s expected excess return rate rela-ve to the market’s expected excess return rate (Equity risk premium)
Return rate is a random variable with expected value rE and rM Risk is an measure of return rate variance – actually the standard devia-on and usually annualized Beta for the market, βM = 1 Firm’s equity beta almost always 0.25 < β < 2 Examples: SO GG AAPL BIDU
-5.0%
-4.0%
-3.0%
-2.0%
-1.0%
0.0%
1.0%
2.0%
3.0%
4.0%
-3% -2% -1% 0% 1% 2% 3%
• Plot historical excess return pairs
• i is index for historical sample pairs
• weekly or monthly historical samples are typical β calcs
• Linear (OLS) regression • Excess returns normally distributed about trend line
• Trend line slope is β
Capital Asset Pricing Model (CAPM) 13
( )iiii FMFE rr, rr −−
β=.7
ii FM rr −
ii FE rr −
More About Beta
¨ Calcula-on ¤ Stock: i ¤ Market: M ¤ Correla-on of returns : ρiM (1 ≥ ρiM ≥ -‐1) ¤ Standard devia-on of return rates: σi , σM (σi , σM > 0)
n Annualized standard devia-on of return rate is called ‘vola-lity’
¨ Insights ¤ Is β = +2 more risky than -‐2 ? ¤ Is ρ = +1 more risky than ρ = -‐1 ? ¤ Is a larger more risky ?
14
M
iiMi σ
σ⋅ρ=β
Not enough info, investors care about ‘portfolio risk’ Yes M
i
σ
σ
More About Beta
¨ Yahoo ¤ 3 years of monthly returns
¨ Morningstar ¤ 3 years of monthly returns
¨ Bloomberg ¤ “Raw Beta” uses 2 years of weekly returns ¤ “Adjusted Beta” is .67 * Raw Beta + .33 * 1
¨ Ibbotson ¤ 5 years of monthly returns
¨ Value Line ¤ 5 years of weekly returns
¨ Others – Standard and Poors, Barra
15
Cost of Capital in Unleveraged Firm, kU 16
¨ βL for the actual, leveraged firm ¤ from linear regression
¨ βU for unleveraged firm ¤ No tax advantaged debt
)rr(βrrk
rrrr
β
FMUFUU
FM
FUU
−⋅+==
−
−=
)rr(βrrk
rrrrβ
FMLFEE
FM
FEL
−⋅+==
−
−=
Unleveraging and leveraging does not involve ‘-me’ -‐ just transform one scenario to another e.g., ΔDB = 0 Typical to compare firm’s unleveraged β – risk due to business opera-ons
Beta Risk M&M Assump-ons 17
¨ Compute βU from equivalence of
¨ Subs-tute
¨ If firm’s debt is further assumed risk free debt, rD = rF
ED
)rr()rr(
)1(FM
DUUL ⋅
−
−⋅τ−+β=β
⎟⎠⎞
⎜⎝⎛ ⋅τ−+⋅β=β
ED)1(1UL
ED)rr()1(r)rr(r DUUFMLF ⋅−⋅τ−+=−⋅β+
)rr(βrr FMUFU −⋅+≡
)rr(βrr FMLFE −⋅+=ED)kk()1(rr DUUE ⋅−⋅τ−+= M&M
assump-ons General case
M&M Assump-on: Relate k and kU
⎟⎠⎞
⎜⎝⎛ ⋅⋅=
VDτ-‐1kk U
18
VDτ)(1k
VEk k DE ⋅−⋅+⋅=
All firms with
constant D/V
ED)kk()1(kk DUUE ⋅−⋅τ−+=
M&M restric-on of firms with constant D
VDτ)(1k
VD)k-‐(kτ)-‐(1
VEkk DDUU ⋅−⋅+⋅⋅+⋅=
M&M Assump-on: Hamada Equa-on 19
)rr(βrr FMLFE −⋅+=
ED)1()rr()rr(rr FMUFMUFE ⋅τ−⋅−⋅β+−⋅β+=
Risk free rate of return
Business risk
premium
Risk premium due to financial (leverage) risk
[ ]ED)1(1)rr(rr FMUFE ⋅τ−+⋅−⋅β+=
⎟⎠⎞
⎜⎝⎛ ⋅τ−+⋅β=β
ED)1(1UL
Capital Structure Scenario Analysis 20
Sample Problem: ¨ A firm wants to determine its β risk and cost of capital, k, if it
doubles its leverage (D/E ra-o) ¨ Miller & Modigliani
¤ Debt and FCF are constant over -me ¤ But different scenarios may have different levels of debt ¤ But ‘un-‐leveraging’ and ‘re-‐leveraging’ are scenario changes
¨ Given: rM = 12%, τ = 40%, D/E = .33, rF = 5% βL = 1.24 (from linear regression with D/E = .33) Assume rD = rF in this example
Capital Structure Scenario Analysis 21
¨ Calculate kE ¤ kE = rF + 1.24·∙(12% -‐ 5%) = 13.7%
¨ Calculate the unleveraged beta βU
¨ Calculate the unleveraged cost of capital ¤ kU = rF + βU·∙(rM -‐ rF) = 5% + 1.24·∙(12% -‐ 5%) = 12.2%
( )035.1
33.0)40.1(1124.1
ED)τ1(1
1β β LU =⋅−+
⋅=
⎟⎠⎞
⎜⎝⎛ ⋅−+⋅=
Capital Structure Scenario Analysis 22
¨ Calculate a new βL that reflects a D/E of .66
¨ Calculate the new cost of equity kE = rF + 1.445·∙(12%-‐5%) = 15.1
( ) 445.166.04.1035.1 ED)1(1UL =⋅+⋅=⎟⎠⎞
⎜⎝⎛ ⋅τ−+⋅β=β
rM 12%τ 40%rF 5%
Current Unlevered Prospective D/E 33% 0% 66%β 1.240 1.035 1.445kE 13.7% 12.2% 15.1%
The Five Pillars 23
Nobel Prize winner and former Univ. of Chicago professor, Merton Miller, published a paper called the “The History of Finance”
Miller iden-fied five “pillars on which the field of finance rests” These include
1. Miller-‐Modigliani Proposi-ons • Merton Miller 1990 and Franco Modigliani 1985
2. Capital Asset Pricing Model • William Sharpe 1990
3. Efficient Market Hypothesis • (Eugene Fama, Paul Samuelson, …)
4. Modern Por}olio Theory • Harry Markowitz 1990
5. Op-ons • Myron Scholes and Robert Merton 1997
The M&M Proposi-ons
¨ Provide fundamental insights into corporate finance
¨ Franco Modigliani ¤ formerly professor at MIT ¤ 1985 Nobel Prize winner
¨ Merton Miller ¤ formerly professor at the University of Chicago
¤ 1990 Nobel Prize co-‐winner
24
http://nobelprize.org/nobel_prizes/economics/laureates/
Irrelevance or indifference ?
These proposi-ons are also referred to as “Irrelevance Theorems” or “Indifference Theorems”
“showing what doesn’t ma~er can also show, by implica-on, what does” Merton Miller
25
M&M Proposi-on 1
Assume no income tax: τ = 0 thus no tax shield ¤ The firm may have debt ¤ Capital structure and leverage are
irrelevant to firm value
¤ The firm’s value is due to its asset’s expected free cash flow and risk, not how the assets are financed
¤ The alloca-on of FCF between debt and equity providers is irrelevant to firm value
26
U
UTSU
kFCF
DkFCFVVV
=
⋅τ+=+=
With Debt Without Debt t 0% 0%(1-‐τ) 100% 100%kD 10% 10%D 500,000$ -‐$ IX 50,000$ -‐$ ΔT -‐$ -‐$ IDI -‐$ -‐$ Δ IC -‐$ -‐$
EBIT 450,000$ 450,000$ τ·∙EBIT -‐$ -‐$
EBIT·∙(1-‐τ) 450,000$ 450,000$ IX·∙(1-‐τ) 50,000$ -‐$
NP 400,000$ 450,000$
IX·∙(1-‐τ) 50,000$ -‐$ IDI·∙(1-‐τ) -‐$ -‐$
ΔT -‐$ -‐$ NOPAT 450,000$ 450,000$
Δ IC -‐$ -‐$ FCFF 450,000$ 450,000$ FCFE 400,000$ 450,000$
M&M Proposi-on 2
• No income tax: τ = 0 thus no tax shield • Leverage does increase the expected return on equity, rE, due to
increased risk to the shareholders, and thus increases the cost of equity, kE
¤ But leverage does not change the cost of capital, k, from the unleveraged cost of capital, kU. Therefore leverage does not increase the value of the firm.
27
EDkk k k
0if τEDkk τ-‐ (1 k k
DUUE
DUUE
⋅) − (+=
=
⋅) − (⋅)+=
U
U
k k0 τset
VDτ-‐1k k
=
=
⎟⎠⎞
⎜⎝⎛ ⋅⋅=
VDk
VEk k DEU ⋅+⋅=
6%
8%
10%
12%
14%
16%
18%
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
D / E
k
kE
kU
kD
28
Proposi-on 2: No income tax
τ=0%
kU=15%
kD=10%
UU k
FCFVkFCFV ===
kD is assumed not a func-on of D/E
The rate cost advantage of using more debt capital is exactly offset by the increased rate cost of the equity due to increased risk
Example: No Income Tax ¨ M&M assump-ons ¨ τ=0%, kU=15%, kD=10%, D=$0 ¨ FCF = $450,000
¨ Now the firm borrows $500,000 ¤ D=DB=$500,000
¨ Is the firm’s value s-ll $3,000,000 or has it increased to $3,500,000 based on V = E + D ?
29
000,000,3$ %15000,450$
kFCFVV
UU
=
===
With Debt Without Debt t 0% 0%(1-‐τ) 100% 100%kD 10% 10%D 500,000$ -‐$ IX 50,000$ -‐$ ΔT -‐$ -‐$ IDI -‐$ -‐$ Δ IC -‐$ -‐$
EBIT 450,000$ 450,000$ τ·∙EBIT -‐$ -‐$
EBIT·∙(1-‐τ) 450,000$ 450,000$ IX·∙(1-‐τ) 50,000$ -‐$
NP 400,000$ 450,000$
IX·∙(1-‐τ) 50,000$ -‐$ IDI·∙(1-‐τ) -‐$ -‐$
ΔT -‐$ -‐$ NOPAT 450,000$ 450,000$
Δ IC -‐$ -‐$ FCFF 450,000$ 450,000$ FCFE 400,000$ 450,000$
30
No Income Tax Example
¨ The value remains $3,000,000 since ¤ the firm’s FCF remains at $450,000 and ¤ kU and rU remain at 15%
n kU is not a func-on of capital structure
¨ However the equity value is reduced to $2,500,000 (debt is senior to equity)
¨ Actually a firm raising debt in this scenario intends to use it to buy back equity so that capital structure changes, but not total capital
$2,500,000 $500,000 -‐ 15%
$450,000
D -‐ kFCF
EU
==
=
31
No Income Tax Example
¨ Now compute the new cost of equity, kE ¤ kE is a func-on of capital structure, D/E
¤ Equity providers expect increase return due to increased risk
¨ And compute the new cost of capital, k
%0.16000,500,2$000,500$%01%51 15% k
EDkk k k
E
DUUE
=⋅) − (+=
⋅) − (+=
%0.15167.0%10833.0%0.16 VD
τ)(1kVE
k k DE
=⋅+⋅=
⋅−⋅+⋅=
Increased
No change
No Income Tax Example
¨ Compute the equity value, E, using FCFE
32
000,500,2$%0.16000,400
kFCFE E
E
===
$-‐
$500,000
$1,000,000
$1,500,000
$2,000,000
$2,500,000
$3,000,000
$3,500,000
Value
VU EU E*
D
D = $0 D = $500,000
M&M Proposi-on 1
¨ Income tax included: τ > 0 ¤ If the firm has debt, D>0, then the firm does have a tax shield
¤ Capital structure and leverage are relevant to firm value n The present value of the tax shield increases its unlevered value by τ·∙D
n The firm’s value is due to its asset’s expected free cash flow and risk, as well as how the assets are financed
n The alloca-on of FCF between debt and equity providers is relevant to firm value
33
DτkFCFVVV
UTSU ⋅+=+=
M&M Proposi-on 2
• Income tax included: τ > 0 ¤ If the firm has debt, D>0, then the firm does have a tax shield ¤ Leverage increases the risk to shareholders and thus increases the
expected (demanded) return on equity, rE , and the cost of equity, kE ¤ However the tax shield decreases the risk to shareholders rela-ve to
the no tax scenario
¤ Leverage, D/V, decreases the cost of capital, k, from the unleveraged cost of capital, kU
34
EDkk -‐ (1 k k DUUE ⋅) − (⋅)τ+=
⎟⎠⎞
⎜⎝⎛ ⋅⋅=
VDτ-‐1k k U
Example with Income Tax 35
tax % 33.0%kU 15.00%kD 10.0%
6%
8%
10%
12%
14%
16%
18%
0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5
D / E
k
kE
kU
kD
36
Determine costs of capital and value under four levels of debt
Capital Structure Example A: Tax / DB=$0
B: Tax / DB=$250,000
C: Tax / DB=$500,000
D: Tax / DB=$750,000
t 33% 33% 33% 33%(1-‐τ) 67% 67% 67% 67%kD 10% 10% 10% 10%DB=D -‐$ 250,000$ 500,000$ 750,000$ IX -‐$ 25,000$ 50,000$ 75,000$ ΔT -‐$ -‐$ -‐$ -‐$ IDI -‐$ -‐$ -‐$ -‐$
EBIT 450,000$ 450,000$ 450,000$ 450,000$ τ·∙EBIT 148,500$ 148,500$ 148,500$ 148,500$
EBIT·∙(1-‐τ) 301,500$ 301,500$ 301,500$ 301,500$ IX·∙(1-‐τ) -‐$ 16,750$ 33,500$ 50,250$
NP 301,500$ 284,750$ 268,000$ 251,250$
IX·∙(1-‐τ) -‐$ 16,750$ 33,500$ 50,250$ IDI·∙(1-‐τ) -‐$ -‐$ -‐$ -‐$
ΔT -‐$ -‐$ -‐$ -‐$ NOPAT 301,500$ 301,500$ 301,500$ 301,500$
ΔIC -‐$ -‐$ -‐$ -‐$ FCF 301,500$ 301,500$ 301,500$ 301,500$ FCFE 301,500$ 284,750$ 268,000$ 251,250$
Debt used to buy back equity so that IC remains constant
A B C DD=DB -‐$ 250,000$ 500,000$ 750,000$ InputEB 1,005,000$ 755,000$ 505,000$ 255,000$ IC 1,005,000$ 1,005,000$ 1,005,000$ 1,005,000$ =EB+DBVTS -‐$ 82,500$ 165,000$ 247,500$ τ·∙D
V 2,010,000$ 2,092,500$ 2,175,000$ 2,257,500$ =VU + τ·DE 2,010,000$ 1,842,500$ 1,675,000$ 1,507,500$ =VL -‐ DE/EB 2.00 2.44 3.32 5.91 D/E 0.000 0.136 0.299 0.498D/V 0.000 0.119 0.230 0.332kE 15.00% 15.45% 16.00% 16.67% =kU+(1-‐τ)(kU-‐kD)·D/Ek 15.00% 14.41% 13.86% 13.36% =kU·(1-‐τ·D/V)V 2,010,000$ 2,092,500$ 2,175,000$ 2,257,500$ = FCF / kIX -‐$ 25,000$ 50,000$ 75,000$ =kDDFCFE 301,500$ 284,750$ 268,000$ 251,250$ =FCF-‐(1-‐τ)·kD·DE 2,010,000$ 1,842,500$ 1,675,000$ 1,507,500$ = FCFE / kEroic 30.00% 30.00% 30.00% 30.00% =NOPLAT/ICEP 150,750$ 156,694$ 162,186$ 167,277$ =IC·(roic-‐k)MVA 1,005,000$ 1,087,500$ 1,170,000$ 1,252,500$ = EP/kV 2,010,000$ 2,092,500$ 2,175,000$ 2,257,500$ =IC+MVArE 15.00% 15.45% 16.00% 16.67% =(EBIT-‐IX)(1-‐τ)/Eroe 30.00% 37.72% 53.07% 98.53% =(EBIT-‐IX)(1-‐τ)/EB
Capital Structure Example
kD 10.0%kU 15.0%τ 33.0%DB -‐$ VU 2,010,000$ E 2,010,000$ EB 1,005,000$
EBIT 450,000$ NOPAT 301,500$ FCF 301,500$
Capital Structure Example
$1,000,000
$1,200,000
$1,400,000
$1,600,000
$1,800,000
$2,000,000
$2,200,000
$2,400,000
VU VU VU VU
E
VTS
E E E
D D D
VTSVTS
D=$0 D=$250,000 D=$500,000 D=$750,000
39
Op-mal Capital Structure
V
Value according to simple firm assump-ons PV(financial distress) Actual firm value Value of unleveraged firm Op-mal D/E ra-os under each assump-on D/E
distress) alPV(Financi -‐Dτk
τ)EBIT(1 VU
⋅+−
=
40
Essen-al Points
¨ Proposi-on 1 ¤ Firm value is due only to the expected return and risk on firm
opera-ons, FCF, unless there is a tax shield due to debt and income tax. In that case the addi-onal value is due to the present value of the tax shield.
¨ Proposi-on 2 ¤ Debt (leverage) increases risk to shareholders and thus
increases the cost of equity, kE, and the expected return on equity, rE
¤ The tax shield reduces the risk to the shareholder and thus the cost of equity. The tax shield increases the value of the firm.
¤ Leverage does not lower the cost of capital except in the case of tax advantaged debt
Essen-al Points
• Calculate cost of equity capital, kE • For any firm with a historical record of market equity price
• Introduc-on to the CAPM model • Understand β risk and • Cost of equity capital and equivalence with expected return rate
• Calculated unleveraged cost of capital, kU, from kE in the case of a simple firm • Explored the rela-onships between k, kU, kD, and kE for a simple firm • Differen-ated between risk free return, expected return on business opera-ons, and addi-onal expected return due to financial leverage
41
Deriva-on of the Beta Risk Factor
¨ Calculate por}olio variance ¤ Split into market propor-onal variance and firm specific variance
ij
M
1jji
M
1i
2P σwwσ ⋅⋅= ∑∑
==
)σσβ(βwwσijε
M
1j
2Mjiji
M
1i
2P ∑∑
==
+⋅⋅⋅⋅=
2Mjiijε
ε2Mjiij
σββσσ
σσββσ
ij
ij
⋅⋅−≡
+⋅⋅≡
ij
M
1jji
M
1i
M
1j
2Mjiji
M
1i
2P wwww ε
====
σ⋅⋅+σ⋅β⋅β⋅⋅=σ ∑∑∑∑
42
Deriva-on of the Beta Factor
¨ Split
¨ Firm specific covariance is assumed zero. Split the variances and covariances
ij
M
1jji
M
1i
M
1j
2Mjiji
M
1i
2P wwww ε
====
σ⋅⋅+σ⋅β⋅β⋅⋅=σ ∑∑∑∑
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛σ⋅⋅+σ⋅+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛σ⋅β⋅β⋅⋅+σβ=σ ε
≠==
ε=
≠===
∑∑∑∑∑∑ iji
M
ij1j
ji
M
1i
2M
1i
2i
M
ij1j
2Mjiji
M
1i
M
1i
2M
2i
2i
2P wwwwww
Market propor-onal Firm specific
variance covariance variance covariance
∑∑∑≠==
ε=
σ⋅β⋅β⋅⋅+σ+σ⋅β⋅=σM
ij1j
2Mjiji
M
1i
2M
1i
2M
2i
2i
2P ww)(w
i 43
Deriva-on of the Beta Factor
∑∑∑≠==
ε=
σ⋅β⋅β⋅⋅+σ+σ⋅β⋅=σM
ij1j
2Mjiji
M
1i
2M
1i
2M
2i
2i
2P ww)(w
i
22M
2i
2i iε
σ+σ⋅β=σ
2MMiiM σ⋅β⋅β=σ
2MiiM σ⋅β=σ
2M
iMi σ
σ=β
2Mjiij σββσ ⋅⋅=
44
Systemic and
non-‐systemic
(firm specific)
risk
Systemic
risk only
Deriva-on of the Beta Factor
2M
iMi σ
σ=β
)rr(rr FM2M
iMFi −⋅
σ
σ+=
Sub into CAPM formula
2M
FM
iM
Fi rrrrσ
−=
σ
−
Price of risk
MiiMiM σ⋅σ⋅ρ=σ
M
iiMi σ
σ⋅ρ=β
45
Reference: More About Beta 46
EDVVV TSU +=+=
EDr
EVrr
VDrr
VEr
VEr
VDrr
DUE
DUE
EDU
⋅−⋅=
⋅−=⋅
⋅+⋅=
Use the following weighted averages when leverage (D/V and E/V) is constant Note that the sums below are for por}olios, not through -me Cannot use with M&M, but can use for M&E and H&P
VE
VD assume
VE
VD
VV
VV
βwβw
βwβ
EDU
UTS
EDTS
TSU
U
2211
M
1iiiP
⋅β+⋅β=β
β=β
⋅β+⋅β=⋅β+⋅β
⋅+⋅=
⋅=∑=
VEr
VDr
VV
rVV
r
rwrw
rwr
EDTS
TSU
U
2211
M
1iiiP
⋅+⋅=⋅+⋅
⋅+⋅=
⋅=∑=