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Capital structure and cost of equity pdf
- 1. Capital Structure and Cost of Equity
- 2. 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
- 3. 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
- 4. Firm Value 4 Assume: NOA=0, T=0 ----------------------- OA = TA, NOCE = IS = 0 NIBCL NIBCL NIBCL NIBCL IC = EB + DB , LE = IC + NIBCL IC = OWC + NC = C V = PV(FCF) = Fair Value of IC V = IC + MVA -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Book Value LE & TA -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ CE AP ITP NIBCL NIBCL AR 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
- 5. APV Valua-on with Constant FCF Growth 0 20 40 60 80 100 120 140 160 Fair Value [$M] VU VTS D E TS FCFU 1 TSU FCF 1 V gk FCF VVV APVM gk FCF EDV FCFM + − = += − =+= 5 No assump-on yet on growth of debt, D, tax shield, TS, or present value of tax shield, VTS
- 6. Simple Firm Example 6 )τ1(EBITFCF 0OWCΔ 0TΔ 0CCDXCX OWCΔ)CCDXCX(TΔ)τ1(EBIT OWCΔNΔNOPATFCF −⋅= = = =−− −−−−+−⋅= −−= Dτ k τ)(1EBIT V k τ)(1EBIT V U TS U ⋅+ −⋅ = + −⋅ = Without Debt With Debt t 33% 33% (1-‐τ) 67% 67% kD 10% 10% D -‐$ 100,000$ IX -‐$ 10,000$ ΔT -‐$ -‐$ IDI -‐$ -‐$ ΔIC -‐$ -‐$ 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$ ΔIC -‐$ -‐$ FCF 150,000$ 150,000$ M&M assump-ons including kTS = KD
- 7. APV Valua-on with No FCF Growth $0 $20 $40 $60 $80 $100 $120 $140 $160 $180 Fair Value [$M] VU VTS D E Dτ k τ)EBIT(1 Dτ k FCF V :APVM UU ⋅+ − =⋅+= k τ)EBIT(1 k FCF V :FCFM − = = D k τ)(1Dk-‐τ)EBIT(1 D k FCFE V :FCFEM E D E + −⋅⋅− = +=
- 8. Rates of Return on Equity 8 E τ)(1Dk -‐ τ)(1EBIT E τ)(1D)k-‐(EBIT E τ)](1D)k-‐E[(EBIT E ]E[NP r D D D 0 1 E −⋅⋅−⋅ = −⋅⋅ = −⋅⋅ = = ‘Forward’ (expected) net profit on present equity fair value EB τ)IX)(1-‐(EBIT EB NP roe 1-‐ 0 − = = ‘Trailing’ net profit on present equity book value
- 9. Cost of Equity: M&M Assump-ons 9 ( ) U U U kED)τ-‐(1 kD)τE(Dτ)(1EBIT EDDτ k τ)(1EBIT V ⋅+⋅= ⋅⋅−+=−⋅ +=⋅+ −⋅ = ( ) E τ)(1DkkED)-‐(1 r E τ)(1Dk -‐ τ)(1EBIT r DU E D E −⋅⋅−⋅+⋅τ = −⋅⋅−⋅ = E D )kk()1(rr E D )kk()1(kr DUUE DUUE ⋅−⋅τ−+= ⋅−⋅τ−+= E D )kk()1(kk DUUE ⋅−⋅τ−+= But we s-ll don’t know kU M&M Assumptions FCF and Debt are perpetuities
- 10. 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] )
- 11. 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
- 12. 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
- 13. -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 −
- 14. 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 i iMi σ σ ⋅ρ=β Not enough info, investors care about ‘portfolio risk’ YesM i σ σ
- 15. 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
- 16. 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 rr rr β FMUFUU FM FU U −⋅+== − − = )rr(βrrk rr rr β FMLFEE FM FE L −⋅+== − − = 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
- 17. 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 E D )rr( )rr( )1( FM DU UL ⋅ − − ⋅τ−+β=β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅τ−+⋅β=β E D )1(1UL E D )rr()1(r)rr(r DUUFMLF ⋅−⋅τ−+=−⋅β+ )rr(βrr FMUFU −⋅+≡ )rr(βrr FMLFE −⋅+= E D )kk()1(rr DUUE ⋅−⋅τ−+= M&M assump-ons General case
- 18. M&M Assump-on: Relate k and kU ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅⋅= V D τ-‐1kk U 18 V D τ)(1k V E k k DE ⋅−⋅+⋅= All firms with constant D/V E D )kk()1(kk DUUE ⋅−⋅τ−+= M&M restric-on of firms with constant D V D τ)(1k V D )k-‐(kτ)-‐(1 V E kk DDUU ⋅−⋅+⋅⋅+⋅=
- 19. M&M Assump-on: Hamada Equa-on 19 )rr(βrr FMLFE −⋅+= E D )1()rr()rr(rr FMUFMUFE ⋅τ−⋅−⋅β+−⋅β+= Risk free rate of return Business risk premium Risk premium due to financial (leverage) risk [ ] E D )1(1)rr(rr FMUFE ⋅τ−+⋅−⋅β+= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅τ−+⋅β=β E D )1(1UL
- 20. 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
- 21. 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(1 1 24.1 E D )τ1(1 1 β β LU = ⋅−+ ⋅= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅−+ ⋅=
- 22. 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 E D )1(1UL =⋅+⋅=⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅τ−+⋅β=β rM 12% τ 40% rF 5% Current Unlevered Prospective D/E 33% 0% 66% β 1.240 1.035 1.445 kE 13.7% 12.2% 15.1%
- 23. 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
- 24. 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/
- 25. 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
- 26. 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 U TSU k FCF D k FCF VVV = ⋅τ+=+= With Debt Without Debt t 0% 0% (1-‐τ) 100% 100% kD 10% 10% D 500,000$ -‐$ IX 50,000$ -‐$ ΔT -‐$ -‐$ IDI -‐$ -‐$ ΔIC -‐$ -‐$ 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$ ΔIC -‐$ -‐$ FCFF 450,000$ 450,000$ FCFE 400,000$ 450,000$
- 27. 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 E D kk k k 0if τ E D kk τ-‐ (1 k k DUUE DUUE ⋅)−(+= = ⋅)−(⋅)+= U U k k 0 τset V D τ-‐1k k = = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅⋅= V D k V E k k DEU ⋅+⋅=
- 28. 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% U U k FCF V k FCF V === 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
- 29. 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 $3,500,000 based on V = E + D ? 29 000,000,3$ %15 000,450$ k FCF VV U U = === With Debt Without Debt t 0% 0% (1-‐τ) 100% 100% kD 10% 10% D 500,000$ -‐$ IX 50,000$ -‐$ ΔT -‐$ -‐$ IDI -‐$ -‐$ ΔIC -‐$ -‐$ 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$ ΔIC -‐$ -‐$ FCFF 450,000$ 450,000$ FCFE 400,000$ 450,000$
- 30. 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% $450,000 D -‐ k FCF E U == =
- 31. 31 No Income Tax Example ¨ Now compute the new cost of equity, kE ¤ kE is a func-on of capital structure, D/E ¤ Equity providers expect increase return due to increased risk ¨ And compute the new cost of capital, k %0.16 000,500,2$ 000,500$ %01%51 15% k E D kk k k E DUUE =⋅)−(+= ⋅)−(+= %0.15167.0%10833.0%0.16 V D τ)(1k V E k k DE =⋅+⋅= ⋅−⋅+⋅= Increased No change
- 32. No Income Tax Example ¨ Compute the equity value, E, using FCFE 32 000,500,2$ %0.16 000,400 k FCFE 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
- 33. 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 Dτ k FCF VVV U TSU ⋅+=+=
- 34. 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 E D kk -‐ (1 k k DUUE ⋅)−(⋅)τ+= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅⋅= V D τ-‐1k k U
- 35. 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. 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$
- 37. Debt used to buy back equity so that IC remains constant A B C D D=DB -‐$ 250,000$ 500,000$ 750,000$ Input EB 1,005,000$ 755,000$ 505,000$ 255,000$ IC 1,005,000$ 1,005,000$ 1,005,000$ 1,005,000$ =EB+DB VTS -‐$ 82,500$ 165,000$ 247,500$ τ·∙D V 2,010,000$ 2,092,500$ 2,175,000$ 2,257,500$ =VU + τ·D E 2,010,000$ 1,842,500$ 1,675,000$ 1,507,500$ =VL -‐ D E/EB 2.00 2.44 3.32 5.91 D/E 0.000 0.136 0.299 0.498 D/V 0.000 0.119 0.230 0.332 kE 15.00% 15.45% 16.00% 16.67% =kU+(1-‐τ)(kU-‐kD)·D/E k 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 / k IX -‐$ 25,000$ 50,000$ 75,000$ =kDD FCFE 301,500$ 284,750$ 268,000$ 251,250$ =FCF-‐(1-‐τ)·kD·D E 2,010,000$ 1,842,500$ 1,675,000$ 1,507,500$ = FCFE / kE roic 30.00% 30.00% 30.00% 30.00% =NOPLAT/IC EP 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/k V 2,010,000$ 2,092,500$ 2,175,000$ 2,257,500$ =IC+MVA rE 15.00% 15.45% 16.00% 16.67% =(EBIT-‐IX)(1-‐τ)/E roe 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$
- 38. 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 VTS VTS D=$0 D=$250,000 D=$500,000 D=$750,000
- 39. 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 -‐Dτ k τ)EBIT(1 V U ⋅+ − =
- 40. 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
- 41. 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
- 42. Deriva-on of the Beta Risk Factor ¨ Calculate por}olio variance ¤ Split into market propor-onal variance and firm specific variance ij M 1j ji M 1i 2 P σwwσ ⋅⋅= ∑∑ == )σσβ(βwwσ ijε M 1j 2 Mjiji M 1i 2 P ∑∑ == +⋅⋅⋅⋅= 2 Mjiijε ε 2 Mjiij σββσσ σσββσ ij ij ⋅⋅−≡ +⋅⋅≡ ij M 1j ji M 1i M 1j 2 Mjiji M 1i 2 P wwww ε ==== σ⋅⋅+σ⋅β⋅β⋅⋅=σ ∑∑∑∑ 42
- 43. Deriva-on of the Beta Factor ¨ Split ¨ Firm specific covariance is assumed zero. Split the variances and covariances ij M 1j ji M 1i M 1j 2 Mjiji M 1i 2 P wwww ε ==== σ⋅⋅+σ⋅β⋅β⋅⋅=σ ∑∑∑∑ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ σ⋅⋅+σ⋅+ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ σ⋅β⋅β⋅⋅+σβ=σ ε ≠ == ε = ≠ === ∑∑∑∑∑∑ iji M ij 1j ji M 1i 2 M 1i 2 i M ij 1j 2 Mjiji M 1i M 1i 2 M 2 i 2 i 2 P wwwwww Market propor-onal Firm specific variance covariance variance covariance ∑∑∑ ≠ == ε = σ⋅β⋅β⋅⋅+σ+σ⋅β⋅=σ M ij 1j 2 Mjiji M 1i 2 M 1i 2 M 2 i 2 i 2 P ww)(w i 43
- 44. Deriva-on of the Beta Factor ∑∑∑ ≠ == ε = σ⋅β⋅β⋅⋅+σ+σ⋅β⋅=σ M ij 1j 2 Mjiji M 1i 2 M 1i 2 M 2 i 2 i 2 P ww)(w i 22 M 2 i 2 i iεσ+σ⋅β=σ 2 MMiiM σ⋅β⋅β=σ 2 MiiM σ⋅β=σ 2 M iM i σ σ =β 2 Mjiij σββσ ⋅⋅= 44 Systemic and non-‐systemic (firm specific) risk Systemic risk only
- 45. Deriva-on of the Beta Factor 2 M iM i σ σ =β )rr(rr FM2 M iM Fi −⋅ σ σ += Sub into CAPM formula 2 M FM iM Fi rrrr σ − = σ − Price of risk MiiMiM σ⋅σ⋅ρ=σ M i iMi σ σ ⋅ρ=β 45
- 46. Reference: More About Beta 46 EDVVV TSU +=+= E D r E V rr V D rr V E r V E r V D rr 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 V E V D assume V E V D V V V V βwβw βwβ EDU UTS ED TS TS U U 2211 M 1i iiP ⋅β+⋅β=β β=β ⋅β+⋅β=⋅β+⋅β ⋅+⋅= ⋅= ∑= V E r V D r V V r V V r rwrw rwr ED TS TS U U 2211 M 1i iiP ⋅+⋅=⋅+⋅ ⋅+⋅= ⋅= ∑=