Capital Structure and Cost of Capital - 23&25-Nov-2011

Capital Structure and Cost of Capital

Dr. M.ThiripalrajuIndian Institute of Capital Markets

Financial ManagementFinancial Management

Capital StructureCapital Structure

bsw zRRzRWACC )1(

Where z = B / (S+B) is the proportion of debt finance and (1-z) = S / (S+B) is the proportion of equity finance. Rw is known as the weighted average cost of capital, WACC.


The WACC is a weighted average of the cost of equity / share finance and the cost of debt/bond finance

WACCYV $

The value of the firm is:


Minimising the WACC will also maximise the value of the firm


The capital structure question merely asks whether it makes any difference to the market value of the firm V if the firm is financed by all equity (z = 0), all debt (z = 1) or a mixture of equity and debt (0<z<1).

The crucial question is whether an increase in leverage z leads to a fall in Rw and hence an increase in the value of the firm.

Traditional View: Cost of CapitalTraditional View: Cost of Capital

Debt-to-equity ratio (B/S)(B/S)*

Cost

of c

apita

l

Debt, Rb

WACC

Equity, Rs


Franco Modigliani and Merton Miller, in a landmark 1958 paper, argued against the traditional view They stated that (under certain conditions) the value of the firm is independent of the mix of debt-to-equity finance).

WACC and LeverageWACC and Leverage

Let us see how the WACC varies as we alter the level of leverage z. Suppose Rb = 10% p.a. To start the ball rolling, if leverage z = 20%, let us assume shareholders require Rs = 15% p.a. Hence:

Rw = (1 – z) Rs + zRb = 0.8 (15%) + 0.2 (10%) = 14%

If we increase leverage to z = 50% and Rs remains unchanged at 15% then:

WACC Rw = 0.5 (15%) + 0.5 (10%) – 12.5%

The value of the firm V = Y / WACC will increase as the debt level initially increases.

WACC and LeverageWACC and Leverage

The Modigliani-Miller assumption is that as leverage z increases from z0 = 0.2 to z1 = 0.5 then Rs will rise, but it rises just enough so that Rw remains constant. For example, in a Modigliani – Miller world the required return by shareholders Rs would rise to 18% p.a. as leverage increased to 50%, so that Rw remains unchanged at 14%.

Rw = 0.5 (18%) + 0.5 (10%) = 14%

Leverage and the Return on EquityLeverage and the Return on EquityCapital raised = $10m = S + B = shares + debt (bonds)

Cost of debt = 10%1. Poor 2. Average 3. Good

Earnings before interest Yi (equal probability) 0.5 2 4

A. 100% equity (0% leverage)

(S = $10m equity)

Debt interest rB 0 0 0

Earnings/dividends for shareholders $0.5 $200 $4

Return on shares Ri = Div / S 0.5/10 = 5% 2/10 = 20% 4/10 = 40%

Expected return (standard deviation) 21.7% (14.3)

B. 20% levered (z = B/V = 2/10)

(B = $2m debt, S = $8m equity)

Debt interest rB $0.2 $0.2 $0.2

Earnings/dividends for shareholders $0.3 $1.8 $3.8

Return on shares Ri = Div / S 0.3/8 = 3.75%, 1.8/8 = 22.5%, 3.8/8 = 47.5%


Leverage and the Return on EquityLeverage and the Return on Equity

1. Poor 2. Average 3. Good

Earnings before interest Yi (equal probability)C. 50% levered (z = 5/10)

(B = $5m debt, S =$5m equity)

Debt interest rB $0.5 $0.5 $0.5

Earnings/dividends for shareholders $0.0 $1.5 $3.5

Return on shares Ri = Div / S 0.5 = 0% 1.5/5 = 30% 3.5/5 = 70%


How Leverage Affects Equity ReturnsHow Leverage Affects Equity Returns

As earnings Yi change from 1m to 4m, the equity return Rs for the all equity financed firm moves from 10% to 40% (A to B) but for the 50% levered firm the equity return changes much more, from 10% to 70% (A’ to C).

Earnings, Yi

Equity return, Rs 50% equity (50% debt)

100% equity (0% debt)

70%

40%

30%

20%

10%A

B

C

A’

0.5 1m 4m

Traditional ViewTraditional View

Consider an initial all equity financed company. The traditional view is that as the firm acquires increasing amounts of debt, then the WACC first falls but eventually rises, thus leading to an optimal debt-to-equity ratio at (B/S)*. The reason for the initial fall in the overall cost of capital is:

• The cost of debt Rb is less than the cost of equity Rs

• The cost of equity initially remains constant.

Hence, as you increase the proportion of debt, Rw falls. However, as more debt is added the cost of equity capital Rs begins to rise because:

The variability of future earnings (after deduction of interest payments) increases with leverage (as interest must be paid to bondholders regardless of the gross earnings of the company);

The risk of bankruptcy increases (and bondholders are ‘paid’ before equity holders).

Traditional ViewTraditional View

TRADITIONAL VIEW

There is a debt-to-equity mix which minimises the WACC and hence maximises the firm’s market value

Modigliani – Miller – No Corporate TaxesModigliani – Miller – No Corporate Taxes

Merton Miller put forward the following analogy to explain the MM Proposition I. Suppose the firm is a giant tub of whole milk. You could sell the ‘whole milk’. Alternatively you could separate out the cream and the remainder would be skimmed milk. The cream would sell at a high price but the skimmed milk would sell at a low price.

More formally, the MM view is also based on the idea that the value of a company is ultimately determined by the capitalised value (i.e. PV) of the future income stream from its activities in production, sales marketing and investment in ‘plant and machinery’.


Suppose firm A and firm B both have a PV of future (net) income of $100m. Firm A has 80% debt finance and 20% equity finance, while for firm B the proportions are 20% debt and 80% equity. The MM view implies that the market value should be the same for both firms (and equal to $100m). Basically the idea is this. If two firms X and Y are identical except for their capital structure, yet you can purchase the securities of X for less than the securities of Y, then an arbitrageur should short sell the securities of Y and buy the securities of X, thus making an immediate cash profit. Since the future cash flows from the identical firms X and Y are the same (in PV terms) then there are zero net cash flows in future periods, from the long and short positions in X and Y.

The value of the firm. MM Proposition I (no taxes)The value of the firm. MM Proposition I (no taxes)

V is independent of B/S

Debt-to equity ratio (B/S)

Value of firm, V

V


MODIGLIANI – MILLER ‘PROPOSITION I’

The WACC and the value of the firm are both independent of the debt-to-equity mix used in financing the firm’s

activities


ASSUMPTIONS: MODIGLIANI – MILLER APPROACH

• Perfect capital markets with borrowing and lending rates equal and the same for companies and persons.

• No corporate or personal taxes or transactions costs.

• Other firms exist with the same business (systematic) risk but different leverage.

• Net cash flows from physical investment projects can be regarded as perpetuities and are independent of the debt-to-equity mix

Modigliani – Miller – No Corporate TaxesModigliani – Miller – No Corporate TaxesV = S + B

Value of levered firm VL = value of unlevered firm VU

Capital StructureUnlevered LeveredVU = $1000 VL = ?

BU = $0 BL= $200

SL = ?

RB= 0.05

Note: Future cash flow from both firms is Y = $800

Modigliani – Miller – No Corporate TaxesModigliani – Miller – No Corporate TaxesLevered and unlevered company

Transaction $ Investment $ Return(a) Case U: buy 10% of unlevered company Buy 0.10 of VU $100 = 0.10 VU 0.10 Y

(b) Case L: buy 10% of levered company Buy 0.10 of VL 0.10SL = 0.10 (VL – BL) 0.10 [Y - 0.05 x 200]

= 0.10 (Y – Rb BL)


Cost of L = Cost of SL

0.10 (VL – BL) = 0.10 (VU – BL)

Hence:

VL = VU

We can now fill in the gaps in the preceding table.

VL = VU = $1000 and SL = VL – BL=$1000 - $200 = $800

Profitable ArbitrageProfitable Arbitrage

To show that home-made leverage and arbitrage ensures that VL = VU, consider starting with SL

*= $1300 so that we are in equilibrium:

VL* = SL

* + BL = $1500 > VU = $1000

Synthetic Leveraged CompanySynthetic Leveraged Company

Transaction $ Investment $ Return

Case SL: synthetic leverage = borrow and invest in levered companyBorrow 0.10 of BL -0.10 BL -0.10 Rb BL + 0.10 Y

Buy 0.10 of VU +0.10 VU

Net inv. = 0.10 (VU – BL) Return = 0.10 (Y – RbBL)

Direct Investment in Levered CompanyDirect Investment in Levered Company


Case L*: buy 10% of levered companyBuy 0.10 of SL

* (0.10) SL*=(0.10)$1300 0.10 (Y – RbBL)

=$130 =0.10 ($800 – 0.05 ($200))

=$79

Cost of home-made leverageCost of home-made leverage


Case SL*: borrow and invest in unlevered companyBorrow 0.10 of BL -0.10 BL= 0.10 ($200) = 20 -0.10 Rb BL = -1

Buy 0.10 of VU +0.10 VU = 0.10 ($1000) = 100 +0.10 Y = 80

Net inv. = $80 Net return = $79 =0.10 (VU – BL) =0.10 (Y – Rb BL)

Leverage and the Required Rate of Return on Leverage and the Required Rate of Return on EquityEquity

We have noted that the MM proposition implies that the overall cost of capital Rw is independent of the degree of leverage. One implication of this is that the WACC formula can be rearranged to show that the equilibrium expected return on equity (shares) Rs is positively related to leverage.

In equation WACC = Rw = (1-z) Rs + zRb, substitute z = B/V = B/(S+B) and rearrange to give:

SBRRRR bwws )(

Leverage and the Required Rate of Return on Leverage and the Required Rate of Return on EquityEquity

MODIGLIANI – MILLER ‘PROPOSITION II’

Since the WACC is independent of the debt-to-equity ratio, this implies that the cost of equity capital Rs rises with the

debt-to-equity ratio B/S

Modigliani – Miller with Corporate TaxesModigliani – Miller with Corporate TaxesThe cost of equity finance. MM proposition II (no taxes)

Cost of equity rises with rising debt-to-equity ratioDebt-to-equity ratio (B/S)(B/S)*

Cost

of c

apita

l

Rb

WACC, Rw

Rs= Rw + (Rw– Rb) (B/S)

Debtholders share some of the business risk

Modigliani – Miller with Corporate TaxesModigliani – Miller with Corporate Taxes

For two firms with the same business risk, then with corporate taxes the optimal debt ratio that maximises the

value of the firm involves 100% leverage (i.e. all debt financed)!

To develop this argument a little further note that, with corporate taxes, the value of an unlevered firm VU (i.e. 100% equity financed) equals the constant after-tax earnings Y(1 – t) discounted at the risk adjusted discount rate for an unlevered (i.e. all equity firm :

us

UtYV

)1(

us

The Value of an Unlevered and Levered FirmThe Value of an Unlevered and Levered Firm

For an unlevered (i.e. 100% equity financed) firm, after-tax earnings are Y(1-t) and the value of the firm is:

Where su = risk adjusted discount rate for an all equity financed

firm. For the levered firm, corporate taxes are calculated after deduction of interest payments. Hence, interest income paid to bondholders and the earnings available to shareholders are:

Interest income of bondholders = RbBL

Shareholder earnings = (Y – RbBL) (1 – t)

us

UtYV

)1(


Therefore, the total income accruing to both stakeholders in a levered firm is:

Total income of levered firm = (Y – RbBL)(1 – t) + RbBL=Y(1 – t) + t(RbBL)

= income of unlevered firm + ‘tax shield’


Note that Y(1 - t) equals the income accuring to an equivalent unlevered firm and hence should be discounted at s

u the risk adjusted discount rate for an all equity firm. The income from the tax shield arises because the firm holds debt of BL. If the tax shield is riskless it should be discounted at the rate Rb. Therefore, from equation the value of the levered firm VL is:

LUb

Lbus

L tBVRtBRtYV

)1(


MM PROPOSITION I (WITH CORPORATE TAXES)

Value of levered firm = value of unlevered firm + value of tax shield

)1/()1()/(*b

usLL

us RTVBWACC


MM PROPOSITION I (WITH CORPORATE TAXES)

Value of a levered firm VL = value of an unlevered firm VU +

value of the ‘tax shield’, tB

VL = VU + tB


It follows from the above equation that as the amount of debt B increases then the value of the levered firm increases and is maximised at 100% debt finance.

L

Lb

us

us

L

SBRtR

s))(1(


MM PROPOSITION II (WITH CORPORATE TAXES)

There is a positive relationship between the required return on equity in a levered firm Rs

L and the debt-to-equity ratio BL / SL

Return on Equity of a Levered Firm, RReturn on Equity of a Levered Firm, RssLL and the and the

Debt-to-Equity Ratio, BDebt-to-Equity Ratio, BLL/ S/ SLL

The relationship between the cost of equity in a levered firm RsL

and the debt-to-equity ratio BL/SL is a little involved. First, consider the balance sheet of the levered firm (paying corporate taxes):

Assets LiabilitiesValue unlevered firm VU Debt (bonds) BL

Value tax shield tBL Equity SL

Total VU + tBL BL + SL



The expected cash flow (perpetuity) per annum from VU and the tax shield tBL is:

Expected cash flow from assets = su VU

+ Rb (tBL)

The expected cash flow to equity and bondholders is:

RsL SL + Rb BL

Since there are no retained earnings, the two cash flows in equations and must be equal, which gives after rearrangement:

bL

L

L

us

ULs R

SBt

SVR )1(



But from equation and the fact that in an efficient market the value of the firm VL is equal to the market value of equity plus debt:

VL=VU + tBL =SL + BL

Hence:

VU = SL – (1 – t) BL_

Substituting equation in equation for VU:

bL

LLL

L

usL

s RSBtBtS

SR

)1(])1([



Note that for each unit increase in BL/SL the required return on equity Rs

L increases by (1 - t)(s

u – Rb), which is less than under the no tax case (I.e. t = 0). This concludes our proof of MM Proposition II.

bus

L

Lb

us

us

Ls Rwhere

SBRtR ))(1(

MM PROPOSITION II (WITH CORPORATE TAXES)

The required return on equity in a levered firm RsL increases

with the debt-to-equity ratio BL/SL

Cost of EquityCost of Equity

The APT requires estimates of the factor loadings bij and the price of risk I for each of these factors. It is then straightforward to calculate the cost of equity for firm i:

Ers = 1bs1 + 2bs2 + …

Either of these measures can be used as a measure of the cost of equity finance.

smss rERrER 83)(

Cost of DebtCost of Debt

bb RtR )1(*

Retained EarningsRetained Earnings

Weighted Average Cost of CapitalWeighted Average Cost of CapitalBtRSRC bs )1(

BSV

)1(cos)( tRBSBR

BSS

valuetotalttotalWACCR bsw

)1()1( tzRRz bs

Where z = B / (B + S) is the degree of leverage. With no corporate taxes we simply set t = 0 in the above equation. The WACC should be used as the discount rate in the NPV formula to discount after-tax earnings Y(1-t) for the marginal capital investment project as long as the following conditions hold.

Incentives and Economic Value AddedIncentives and Economic Value Added

EVA (or residual income) = earnings after tax, EAT – capital used

= EAT – (WACC x KC)

Return on capital ROC = EAT / KC = $100m / $1000m = 10%

The investment decision is then:

Invest in project if ROC > WACC

Economic profit EP = (ROC – WACC) x KC = (0.10 – 0.09) x $1000m = $10m


Economists will recognise this ‘average value’ as being the annuity value of the PV of the future earnings, sometimes called ‘permanent earnings/income’. The PV of the annuity flow EAT is, PV = EAT / WACC and the NPV criterion is:

Invest in the project if PV (earnings) > KC where PV = EAT/WACC

Or EAT – WACC x KC > 0


Fortune magazine (of 10th November 1997) provided figures for the EVA ($m) of companies, from which the following have been extracted:

EVA Capital ROC WACCGeneral Electric 2515 51017 17.7 12.7

General Motors -3527 94268 5.9 9.7

Johnson & Johnson 1327 15603 21.8 13.3


Capital at risk = loan exposure x LGD x ( L x percentile level)

EVA = ‘earnings’ – capital at risk x WACC > 0

RISK ADJUSTED RETURN ON CAPITAL: RAROC

RAROC = ‘earnings’ / ‘capital at risk’

Leverage and the Return on Equity (No Leverage and the Return on Equity (No Corporate Taxes)Corporate Taxes)

Y = earnings (cash flow, profits) before interest, tax and depreciation for either a levered L or unlevered U firm

SL = $-value of equity in a levered firm

BL = $-value of debt (bonds) in a levered firm

Vi = $-value of the firm ( i = U or L)

Rb = interest cost of debt

Rs = return on equity in levered firm

RsL = return on equity in levered firm

Z BL / VL = degree of leverage (and 1 – z SL / VL)

Div = total dividends paid to all shareholders

Leverage and the Return on Equity (No Leverage and the Return on Equity (No Corporate Taxes)Corporate Taxes)

The return on equity is defined as:

RsL = (Div) / SL

Using SL (1 – z) VL and Div = Y – Rb BL = Y – Rb (zBL) then:

Clearly, RsL depends on Y and leverage z.

)1()1()1()(

zzR

zVY

VzzVRYR b

LL

LbLs

Leverage and the Return on Equity (No Corporate Taxes)Leverage and the Return on Equity (No Corporate Taxes)

Rs = Y / VL

Where we have dropped the ‘L’ in the notation. The value of Y for which the all equity and a levered firm give an equal value for Rs is given from which we obtain:

Y* = RbV

Y* = 0.1 ($10m) = $1m which can be seen as the ‘cross-over point’. You might also note that equation can also be obtained by rearranging VL = Y / WACC to solve for Rs

L where WACC = (1 – z)RsL + zRb. This is perfectly

consistent as long as we realise that VL is held constant and only the proportions of S and B are being altered (I.e. leverage), which then has a direct effect on Rs

L. The overall result and equation is that the expected return on equity and the volatility of equity returns, are both higher the greater the degree of leverage z.


We show how Modigliani – Miller Proposition I is altered in the presence of corporate taxes. The relationship between the value of a levered firm and an unlevered firm is:

VL = VU + tBL

The value of the levered firm increases with the amount of debt BL

Therefore our original ‘no tax’ MM Proposition I that VL is independent of debt BL does not hold in the presence of

corporate taxes


We demonstrate that the cost of equity capital RsL in a levered

firm is given by:

Where su is the cost of equity capital in an unlevered firm (I.e.

100% equity financed), hence:

Our original MM Proposition II that RsL rises with the degree

of leverage (BL / SL) still holds in the presence of corporate taxes

L

Lb

us

us

Ls S

BRtR ))(1(

The Return on Equity in an Unlevered Firm and The Return on Equity in an Unlevered Firm and Adjusted Present ValueAdjusted Present Value

We derive an expression for the return on equity in an unlevered firm s

u in terms of the return on equity in a levered firm RsL and

the bond return Rb. This enables us to calculate su in terms of

the observables RsL and Rb.

)( rERr muu

s

We can then use our ‘calculated’ su as the discount rate in

the adjusted NPV technique

To apply the APV technique of project appraisal, we need a measure of the discount rate on an unlevered firm s

u . This can be obtained from the CAPM/SML:

The Return on Equity in an Unlevered Firm and The Return on Equity in an Unlevered Firm and Adjusted Present ValueAdjusted Present Value

However, we now require u, the beta of an unlevered firm. Unfortunately, (nearly) all firms are levered, so what we observe in the data is the beta of a levered firm L. Can we link Uyp L? the answer is yes. But the method is rather involved. We therefore consider the no tax case, before moving on to the case where we have corporate taxes.

Case A: No Corporate TaxesCase A: No Corporate Taxes

We begin with the fact that the beta of a levered firm is a weighted average of the debt b

L and equity sL betas (and the ‘weights’

sum to unity):

bL and s

L are observable/measurable from the SML, using data on returns on debt and equity for levered firms. MM Proposition I implies that the beta of an unlevered firm u (with equal business risk) equals that of a levered firm, hence:

Ls

L

LLb

L

LL

VS

V

Case A: No Corporate TaxesCase A: No Corporate Taxes

Where z = BL / VL. Equation allows us to calculate u from the observed b

L and sL. Now b

L < sL because debt is less risky than

equity, so equation implies:

u < sL

Hence, the beta of an all equity firm is less than the equity beta of a levered company. This also fits with MM-II where we found that the equity of a levered firm has higher risk than an ‘equivalent’ unlevered (100% equity) firm. If b

L 0 then equation becomes:

And it is easy in this case to see that u < sL.

Ls

Lb

Ls

L

LLb

L

L zzVS

Vu )1(

Ls

L

Lu

VS

Case B: With Corporate TaxesCase B: With Corporate Taxes

Again we want to obtain an expression for the unobservable in u terms of the observable/measurable values b

L and sL. The

derivation is a little involved so we immediately present the equation we are looking for, which is:

Ls

LL

LLb

LL

L

SBtS

StBtBu

)1()1()1(


This is very similar to equation since in both equation and equation u is a weighted average of b

L and sL with the weights summing

to unity. In equation the ‘weights’ contain the tax rate t, as we might expect. Of course, if you set t = 0 in equation it ‘collapses’ to equation, the ‘no tax’ form of the equation. To prove equation we begin with the value of a levered firm, given by MM-I (with taxes):

VL=VU + tBL


By definition, the value of the levered firm is equal to the market value of its debt and equity:

VL=BL + SL

From equations:

VU= (1- t) BL + SL

The definition of the beta of a levered firm is:

Ls

L

LLb

L

LL

VS

VB


The next point holds the key to the derivation. From MM-I with taxes we have VL = VU + tBL. Hence, the beta of a levered firm can also be viewed as a weighted average of the beta of an unlevered (100% equity financed) firm and the beta of the tax shield (tB). Hence:

The beta of the cash flow from the tax shield is the ‘debt beta’, since here we assume the tax shield is riskless. We are now nearly there. Equating and rearranging:

Ls

U

LLb

U

Lu

VS

VtB

)1(

Lb

L

Lu

L

UL

VtB

VV


Substituting for VU from equation ( )gives the expression required:

Ls

LL

LLb

LL

Lu

SBtS

SBttB

)1()1()1(


The ‘unobservable’ beta of an unlevered firm is equal to a weighted average of the ‘observable’ betas on debt and

equity of the levered firm


The ‘weights’ on bL and s

L in equation sum to unity. Again, note that since b

L < sL then from equation the beta of the equity

of an unlevered (100% equity financed) firm is less than the beta of the equity of a levered firm:

u < sL

This again fits with our MM-II (with taxes), which implies that levered equity is more risky than unlevered equity. If we set b

L = 0 in equation then it is easy to see that in this case u < s

L.

Assumptions when Using WACCAssumptions when Using WACC

Dollar amount of debt = KC and Dollar amount of equity = KC

The total dollar cost per annum of the debt and equity is:

Total dollar cost of finance p.a. = (1 – t) Rb KC + RsL KC

L

L

VB

L

L

VB

L

L

VB

L

L

VB

Assumptions when Using WACCAssumptions when Using WACCA variable investment project requires perpetual earnings Y p.a. from the project to exceed the dollar cost p.a., that is:

or

Note that the Y / KC is the project’s annual rate of return. Hence, a viable project exceeds the WACC, where the latter assumes the debt ratio BL / VL for the project, is the same as for the firm as a whole.

KCVSRKC

VBRtY

L

LLs

L

Lb

)1(

WACCVSR

VBRt

KCY

L

LLs

L

Lb

)1(

Thank YouThank You

Capital Structure and Cost of Capital - 23&25-Nov-2011

Documents

Transcript of Capital Structure and Cost of Capital - 23&25-Nov-2011