Capital structure and cost of equity pdf

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


¨  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 =⋅−+

⋅=

⎟⎠⎞

⎜⎝⎛ ⋅−+⋅=


¨  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


¨  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


¨  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


∑∑∑≠==

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


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

⋅+⋅=⋅+⋅

⋅+⋅=

⋅=∑=

Capital structure and cost of equity pdf

Economy & Finance

Transcript of Capital structure and cost of equity pdf