Capital Structure

\[ \def\BB#1{{\mathbb{#1}}} \def\BF#1{{\mathbf{#1}}} \def\E{{\mathbb{E}}} \def\R{{\mathbb{R}}} \]

yedlu, Winter 2024

Intro and Motivation

Consider you’re a CFO of a firm. Two fundamental questions that you must face are:

• how much should my firm finance?
• how to finance my firm out of all the possible sources/channels/products?

Capital structure theories are here to try to answer these fundamental questions. We will introduce several classical and/or salient capital structure theories that can shed some lights on the issue:

• Modigliani-Miller: irrelevance of capital structure.
• Trade-Off Theory: balancing between tax benefits and the expected cost of bankruptcy.

Modigliani-Miller

Modigliani and Miller (M&M) were pioneers when they proposed the theory of capital-structure irrelevance: total market value of the firm is unaffected by the firm’s capital structure under M&M assumptions.

This is counter-intuitive for practitioners and researchers back then, who believed that capital structure design matters a lot. Of course this is true in a world with plenty of market frictions.

What M&M has showed and contributed, nevertheless, is that we can pinpoint the friction and its mechanism of affecting firm value through capital structure adjustments.

In other words, M&M assumes a market without some market frictions and showed financing structure is irrelevant to firm value. It provided a perfect framework (or I would say playground) that we can add/adjust/relax assumptions and see how it might affect firm values. This is astonishing!

Now let’s try to understand M&M propositions in depth and get ourselves familiarized.

M&M Proposition 1

Suppose that:

• Fixing Investment Opportunity: a firm’s total cash flow to debt/equity holders are not affected by financing behavior.
• Frictionless: there are no transaction costs:
  • taxes,
  • bankruptcy costs,
  • bid-ask spreads.
• Perfect Market.
• Arbitrage-Free .

Proposition 1

Under such assumptions, the total market value of the firm, equal to the sum of market values of current debt and equity (see Happy Meal Theorem at Asset Pricing Theory), is not affected by how it is financed.

Firm value is then independent of its capital structure.

M&M Proposition 2

Proposition 2

\[\begin{aligned} r_E &= r_U + (r_U - r_D) \frac{D}{E} \end{aligned}\]

where:

Notation	Description
\(r_E\)	(Levered) cost of equity
\(\frac{D}{E}\)	(Levered) debt-equity ratio
\(r_U\)	(Unlevered) cost of capital/equity
\(r_D\)	(Levered) cost of debt

Proof

First we take a look at the unlevered firm. By definition, the market value of the unlevered firm is the value of the discounted cash flow sum:

\[\begin{aligned} V_U &= \sum_{t = 1}^{\infty} \frac{\E[x]}{(1 + r_U)^{t}} \\ &= \frac{\E[x]}{r_U} \\ \\ \implies \E[x] &= r_U V_U \end{aligned}\]

Under similar approach, the market value of the levered equity is:

\[\begin{aligned} E &= \sum_{t = 1}^{\infty} \frac{\E[x] - rD}{(1 + r_E)^{t}} \\ &= \frac{\E[x] - rD}{r_E} \\ \\ \implies r_E &= \frac{\E[x] - rD}{E} \end{aligned}\]

By M&M Proposition 1, we have:

\[\begin{aligned} V_U &= V_L = E + D \\ \\ \implies \E[x] &= r_U V_U = r_U (E + D) \end{aligned}\]

Plugging back to \(r_E\) we have:

\[\begin{aligned} r_E &= \frac{\E[x] - rD}{E} \\ &= \frac{r_U (E + D) - rD}{E} \\ &= r_U + (r_U - r) \frac{D}{E} \end{aligned}\]

\(\blacksquare\)

Intuition

Leverage makes equity return riskier.

M&M Proposition 3

Proposition 3

The investment (assets) and financing (liabilities and equities) decisions are separate and unrelated.

It is only reasonable for firm to take investments with positive NPV discounted with the firm’s aggregated cost of capital \(r_U\).

We will revisit this proposition when the M&M assumptions are relaxed in further notes. For now, however, this proposition gives a seemingly shocking result of the dividend irrelevance theorem:

Firm’s total market value is independent of its dividend policy.

But how to use M&M to show that payout policies are irrelevant of firm value? We will show by proving the following valuation model (known as Bird in the Hand theory) is wrong:

\[\begin{aligned} V_0 &= \sum_{t = 1}^{\infty} \frac{\E[d_t]}{(1+r_t)^t} \end{aligned}\]

Discussion and Proof

Let’s work on the intuition first:

• Suppose a firm decides to pay a lump-sum special dividend to its shareholders. By Bird in the Hand theory, this increases the firm’s market value.
  
• Nevertheless, to make sure that assumption Fixing Investment Opportunity still holds, the firm needs to make sure that the investment level stays fixed.
• Suppose the firm is, and intends to stay unlevered. Then, the firm will issue more equity to raise capital and maintain previous level.
• New equity issuance will dilute current shareholder value, counterbalance the positive shock of the special dividend.

Proof.

By dividend discount model (DDM), firm’s equity price at time \(t\) equals the discounted future dividend \(d\) and equity price \(p\):

\[\begin{aligned} p_t &= \frac{d_{t+1} + p_{t+1}}{1 + r_U} \end{aligned}\]

Let \(q\) be the time-dependent number of shares of the firm with the following dynamics:

\[\begin{aligned} q_{t+1} &= q_t + m_{t+1} \end{aligned}\]

where \(m_{t+1}\) is the number of new share issued at time \(t+1\). We also specify that:

• \(D_{t+1} = q_t d_{t+1}\) be the total amount of dividend paid at time \(t+1\);
• \(V_t = q_t p_t\) be the total market value of the firm;
  • then \(V_{t+1} = q_{t+1} p_{t+1}\)
• \(\Delta E_{t+1} = m_{t+1} p_{t+1}\) be the total amount of incremental cash flow from financing

We can first denote the firm’s total market value with:

\[\begin{aligned} V_t &= q_t p_t \\ &= q_t \frac{d_{t+1} + p_{t+1}}{1 + r_U} \\ &= \frac{D_{t+1} + q_t p_{t+1}}{1 + r_U} \end{aligned}\]

Further dissecting and deriving:

\[\begin{aligned} q_t p_{t+1} &= (q_{t+1} - m_{t+1}) p_{t+1} \\ &= q_{t+1} p_{t+1} - m_{t+1} p_{t+1} \\ &= V_{t+1} - \Delta E_{t+1} \\ \\ \implies V_t &= \frac{D_{t+1} + V_{t+1} - \Delta E_{t+1}}{1 + r_U} \end{aligned}\]

Under M&M assumption on the fixed investment opportunity, we have this relationship with firm’s cash flow \(X\) and investment scale \(I\): decrease of cash from special dividends should be counterbalanced by increase of cash from equity issuance

\[\begin{aligned} \Delta E_{t+1} &= I_{t+1} - (X_{t+1} - D_{t+1}) \end{aligned}\]

Thus:

\[\begin{aligned} V_t &= \frac{D_{t+1} + V_{t+1} - I_{t+1} + X_{t+1} - D_{t+1}}{1 + r_U} \\ &= \frac{V_{t+1} - I_{t+1} + X_{t+1}}{1 + r_U} \end{aligned}\]

Through iteration we have:

\[\begin{aligned} V_t &= \sum_{\tau = 0}^{\infty} \frac{X_{t + \tau} - I_{t + \tau}}{(1 + r_U)^{\tau}} \end{aligned}\]

which shows that firm value is irrelevant of payout policies.

\(\blacksquare\)

Relaxing M&M

M&M is not only elegant at constructing the theoretical framework of capital structure irrelevance, it is also the backbone where corporate finance research rely on.

Though the assumptions assumed are considered to be unrealistic, the M&M Proposition can be a perfect framework point of studying market frictions. Here are some examples.

Assumptions Relaxed	Research Agenda
Taxes	Tax Benefits
Imperfect Markets	Cost of Financial Distress (Bankruptcy Cost)
Fixing Investment Opportunity	Agency Costs/Asymmetric Information/…
Arbitrage-Free	Limited Arbitrage Models

Optimal Debt Level - Tradeoff Theory

Under classic M&M framework, leverage ratio (\(D/V\)) should have no effect on a company’s market value. This implies that firms under these sets of assumption should be indifferent to assume any given debt level.

Nonetheless, we will now relax two assumptions under the M&M framework and introduce the static tradeoff theory.

Tax Benefits

It is not hard to notice that debt and equity are taxed differently for a company. Cash flows to

• debts are tax-exempt at company level (and thus reduce tax basis);
• equity are taxed under corporate rate at company level.

Modigliani and Miller introduced corporate taxes in their subsequent works (1963). Under the new settings, the capital structure is no longer irrelevant to a firm’s value. We will show that corporate taxes bring benefits to the firm.

Modigliani, Franco, and Merton H. Miller. 1963. “Corporate Income Taxes and the Cost of Capital: A Correction.” The American Economic Review 53 (3): 433–43. https://www.jstor.org/stable/1809167.

Example

Let:

• \(\tau_c\) be the non-zero corporate tax rate;
• \(\tilde{x}\) be the perpetual static cash flow (annual);
• \(r\) be the debt interest rate.
• debt level is non-zero and deterministic (hence discountable at \(r\));
• \(rD \tau_c \leq \E[\tilde{x}](\tau_c)\), i.e. tax benefits should not exceed unlevered firm payable taxes.

We would like to show that the levered firm has a higher firm value than the unlevered firm, all else being equal.

Proof

The unlevered firm value would then be:
\[\begin{aligned} V_U &= \sum_{t} \frac{\E[\tilde{x}](1-\tau_c)}{[1 + r_u]^t} \\ &= \frac{\E[\tilde{x}](1-\tau_c)}{r_u} \end{aligned}\]

As for the levered firm, we would first dissect cash flows. For equity and debt holders,
\[\begin{aligned} \tilde{x}_E &= \E[\tilde{x} - rD](1-\tau_c) \\ \tilde{x}_D &= rD \\ \end{aligned}\]

Total cash flow to firm is then:
\[\begin{aligned} \tilde{x} &= \E[\tilde{x}](1-\tau_c) + rD \tau_c \end{aligned}\]

Firm’s current market value then would be:
\[\begin{aligned} V_L &= V_U + \BB{PV}(rD \tau_c) \\ &= V_U + \sum_t \frac{rD \tau_c}{r} \\ &= V_U + D \tau_c \\ &> V_U \end{aligned}\]

\(\blacksquare\)

We call \(D \tau_c\) the interest tax shield. Under such setting, 100% would be the optimal debt level. We, however, never observe this in real life: there are also costs of debt that we must account for and strike a balance.

Cost of Financial Distress

Having way too much debt is intuitively and theoretically dangerous for a firm – the risk of bankruptcy and relative costs (direct/indirect) are scaled-up.

Direct costs of financial distress are the costs directly related to the event of bankruptcy, such as attorney fees, filing fees. Indirect costs, however, arise from different sources and are often hard to quantify.