Topic 2: Valuing firms and the market for equities

• The influence of a firm’s debt-to-equity ratio on its value and required rate of return (cost of capital)

• Market efficiency and predictability

• “Anomalies” and “behavioural theories”

The firm’s stream of revenues will go partly to debt-holders; partly to equity holders. The flow to debt-holders (bond-holders) is fixed (unless leverage very high in which case firm may default) Therefore the flow to equity holders is variable – and more variable the higher the leverage (or debt/equity ratio) and usually will have nonzero beta. Do equity holders demand higher return? – of course they do. But does that mean that bond issue should be maximized: NO!

Leverage of firms and returns on equity and debt

The traditional view was vague but broadly correct. It observed that the rate of return on equity was indeed usually higher than that on bonds (riskier flow of income after interest) and that the more leverage the higher the return on equity. but the rate of return on bonds increases with high leverage (risk of default) Conclusion: issue bonds up to the point where the required return on bonds is increasing to the point where the weighted average rate of return on bonds plus equity (i.e. the “weighted average cost of capital” to the firm) starts to increase. (see picture)

Trad view of the cost of capital

Cost of capital

RE

RW

RB

z/(1 – z)

The modern theory comes to a similar conclusion, it explains precisely how fast the required return on equity increases with leverage It shows that the traditional shape of the cost of capital curves cannot easily be rationalized on the basis of equity risk and risk of bond default and it arrives at an optimal debt structure based on tax treatment of bonds and reorganization costs of default This modern understanding started with Modigliani and Miller It is based on a “no arbitrage profits” line of reasoning

Note on terms: If the proportion of a firm’s liabilities represented by debt is z

Debt to equity ratio: z

z

1

Leverage or gearing ratio: z1

1

Both of these can be measured in terms of market value or book value

Ryanair Balance Sheet March 31, 2007 Assets Current (net) €1.23 billion Non-current €3.34 billion

Aircraft=€2.9 bn; Aer Lingus stake=€0.4 bn

Liabilities Debt €2.03 billion

Fixed rate €1.1 bn; Floating rate €0.8 bn

Equity €2.54 billion Issued & share premium €0.6 bn; Retained earnings €1.9 bn

Debt/Equity ratio z

z

1= 0.8; z = 0.44 (=

54.203.2

03.2

)

Bank of Ireland Balance Sheet March 31, 2008 Assets Cash and securities € 55.5 billion

Loans €135.7 billion Other € 6.2 billion Liabilities Interbank deposits € 14.1 billion Customer deposits € 86.2 billion Securities € 60.8 billion Other debt € 29.6 billion

Equity € 6.5 billion

Debt/Equity ratio z

z

1= 29.4; z = 0.97 (=

4.197

5.64.197 )

Modigliani and Miller pointed out that an arbitrage opportunity exists unless expected rate of return / cost of capital is independent of leverage Because (in a perfect market) the investor can undo any leverage created by the firm, or generate any leverage that the firm has not Therefore, the investor must be indifferent as to the leverage chosen by the firm

All share financing Number of shares 1000 Price per share €10 Market value of shares €10,000 Market value of debt 0 State 1 State 2 Mean State 3 Operating income, € 500 1000 1500 2000 Interest, € 0 0 0 0 Equity earnings, € 500 1000 1500 2000 Earnings per share, € 0.50 1.00 1.50 2.00 Return on shares, % 5 10 15 20 Return on debt, %

After proposed debt issue and share repurchase Number of shares 500 Price per share €10 Market value of shares €5,000 Market value of debt €5,000 State 1 State 2 Mean State 3 Operating income, € 500 1000 1500 2000 Interest, € 500 500 500 500 Equity earnings, € 0 500 1000 1500 Earnings per share, € 0 1.00 2.00 3.00 Return on shares, % 0 10 20 30 Return on debt, % 10 10 10 10

(Example from Brealey and Myers)

Investors can replicate proposed leverage Buy 2 shares, borrow €10

State 1 State 2 Mean State 3 Earnings on two shares € 1 2 3 4 Less Interest on investment on €10 1 1 1 1 Net earnings on investment € 0 1 2 3 Return on €10 investment, % 0 10 20 30

Therefore the overall (“weighted average”) cost of capital, or required rate of return for the firm must be independent of the leverage: Then, writing RW as the weighted average cost of capital RE and RB as the required return on equity and debt, then

BEW zRRzR )1( is constant

And therefore

z

zRRRR BWWE

1

)(

As long as RB is constant, required rate of return on equity

increases linearly with z

z

1

(and therefore increases at an accelerating rate with z)

MM Prop II

MM Prop I

Therefore the overall (“weighted average”) cost of capital, or required rate of return for the firm must be independent of the leverage: Then, writing RW as the weighted average cost of capital RE and RB as the required return on equity and debt, then

BEW zRRzR )1( is constant

And therefore

z

zRRRR BWWE

1

)(

As long as RB is constant, required rate of return on equity

increases linearly with z

z

1

(and therefore increases at an accelerating rate with z)

Cost of

capital RE

RW

RB

MM view of the cost of capital

z

z

1

Actually, the risk of default does start to impact RB noticeably after a while. This means that RB starts to slope up. But there is a corresponding slowdown in RE, in accordance with

z

zRRRR BWWE

1

)( .

RW is still flat!

MM view of the cost of capital – with risk of default

z/(1 – z)

Cost of capital

RE

RW

RB

But there’s more: in most countries, bond interest is deductible before payment of company/corporation tax. Therefore firms can reduce corporation tax burden (and leave more to be distributed to shareholders + bondholders) by increasing leverage! On the other hand, the more leverage, the more risk of default on bonds And default triggers sizable reorganization and renegotiation costs This risk can be reduced by limiting leverage Optimum leverage balances tax and expected bankruptcy costs

Let’s look at the tax dimension: We can work through some of this in more detail if we take a firm whose earnings are CONSTANT at Y per annum. If there is no tax, then the value of the firm is:

B

B

E

B

R

BR

zR

BRY

)(

The value of an unlevered firm (B = 0) is )0(ER

Y

The value of an unlevered firm (B = 0) is )0(ER

Y

Suppose also that the firm has to pay corporate profits tax on earnings at rate τ after payment of interest An unlevered firm’s after tax earnings will be Y(1 – τ) And the market value of an unlevered firm will be

)0(

)1(

ER

Y

The tax paid will depend on the amount of bonds The stream of earnings attributable to shareholders is Y – T – RBB = Y(1 – τ) – RBB(1 - τ) And that to bondholders is RBB The market value V of such a firm will be the sum of these three each discounted at the appropriate rate:

BR

Y

R

BR

R

BR

R

Y

EB

B

B

B

E

)0(

)1()1(

)0(

)1(

So: the value of the leveraged firm is equal to the value of the unleveraged firm plus the tax shelter B

BVV UL

A similar line of reasoning gives a tax-adjusted cost of equity capital:

z

zRRRzR BEEE

1

))0()(1()0()(

it still rises with the debt/equity ratio (despite the tax saving)

Subtract in reorganization costs…growing with B

BVV UL – C(B)

and we arrive at an intermediate optimal value of leverage and there may be other reasons for adjusting leverage: e.g. agency costs (debt acts as a discipline on managers)…

Usefulness of the analysis of leverage. We may be able to infer from a pricing model, such as the APT, what the appropriate rate of return on the earnings of the firm, should be. The appropriate rate of return on its equity will, however, depend on its leverage. For example, suppose we believe a multi-factor model of market rates of return such as that used by Burmeister, Roll and Ross (1994).

They fitted the model:

iMMiSSiIIFi bbbRR

where

iR is expected return on asset i

RF is risk-free rate λI (= – 4.32) is the estimated “risk-premium” for unexpected inflation λS (= 1.49) is the estimated “risk premium” for unexpected aggregate

sales data λM (= 3.96) is the estimated “risk premium” for that part of the change

in the S&P index unpredicted by the other two factors and the b’s are the sensitivities of asset i to (or weightings on) the three factors

If asset 1 is the equity of an unleveraged listed firm “Kellyair” and we estimate b1I = – 0.50; b1S = 2.75; b1M = 1.30. Then we can calculate that expected excess return on Kellyair is 11.41 If the risk-free rate is 6%, then both RW and RE for the unleveraged Kellyair are 17.41. (-0.5*-4.32 + 2.75*1.49 + 3.96*1.30 = 11.41)

A new airline (Murphyair) with 150% debt-to-equity ratio (z = 0.6) is offering its shares. We think its business model is very similar to Kellyair. But should we pay the same price for its shares. NO. We can use the MMII formula as a first approximation:

z

zRRRR BWWE

1

)( .

comes out at RE = 34.53 (=17.41+11.41*1.5)