Capital Structure in Perfect Markets
Outline
This lecture is mainly based the following textbooks:
Study review and practice: I strongly recommend using Prof. Henrique Castro (FGV-EAESP) materials. Below you can find the links to the corresponding exercises related to this lecture:
- Multiple Choice Exercises - click here
\(\rightarrow\) For coding replications, whenever applicable, please follow this page or hover on the specific slides with coding chunks.
Equity Versus Debt Financing
When corporations raise funds from outside investors, they must choose which type of security to issue. The most common choices are financing through equity alone and financing through a combination of debt and equity
As a result, the Capital Structure decision of a firm is the relative proportion of debt, equity, and other securities that a firm has outstanding constitute its capital
Financial Managers are often faced with the challenge of choosing the proportions of debt and equity of the firm
When it comes to Capital Structue, a central question is: do capital structure decisions affect firm value?
Put another way, is there an optimal Capital Structure that maximizes the value of a firm?
\(\rightarrow\) As we’ll see throughout this lecture, under some conditions, the value of a firm is not affected by its capital structure decisions!
Equity Versus Debt Financing
- Let’s start our discussion by analyzing the value of a firm. Imagine an economy with two potential states: a weak and a strong economy. The firm will receive cash flows as follows:
| Date 0 (investment) | Date 1 (strong economy) | Date 1 (Weak economy) |
|---|---|---|
| -$800 | +$1,400 | +$900 |
- Assume that the cost of capital for this project is \(15\%\), and that each scenario has a \(50\%\) probability of occurrence. The value of the firm is then:
\[ \small V = \frac{\frac{1}{2} \times 1400 + \frac{1}{2} \times 900 }{(1+15\%)} = \dfrac{1,150}{(1+15\%)}=1000 \]
- Therefore, the value of the firm is \(\small \$1,000\). If you take the initial investment into consideration, the NPV for the investors is \(\small PV(CF) = -800 + 1,000=200\). Which Capital Structure would you choose to finance this project?
Case 1: all-Equity financing
Say that you decide to fund this project using \(\small100\%\) equity in your Capital Structure
The firm value today is \(\small\$1,000\). If the firm wants to finance the project 100% through equity, it could raise this amount selling equity to outside investors. In this case, the firm is called unlevered equity
From the perspective of an equity investor, we have:
| Date 0 (investment) | Date 1 (Strong economy) | Date 1 (Weak economy) |
|---|---|---|
| -$1,000 | +$1,400 | +$900 |
| Return | +40% | -10% |
- The expected return on the unlevered equity is \(\small 15\%\): \(\small \frac{1}{2} \times 40\% + \frac{1}{2} \times -10\% = 15\%\)
\(\rightarrow\) Because the risk of unlevered equity equals the risk of the project, shareholders are earning an equivalent return for the risk they are taking
Equity Versus Debt Financing
What happens to the value of a firm when we move towards a mixed Capital Structure that contains both Equity and Debt? In this situation, the firm is called levered equity
To see that in action, assume that the firm issues \(\small \$500\) of debt and \(\small\$500\) of equity, with the cost of debt being \(\small 5\%\). Will the value of the firm and the project’s NPV change?
The short answers is: no, the firm value will not change!
- As we’ll see in the upcoming slides, although the cost of debt is lower than the cost of equity (\(\small5\%\) versus \(\small15\%\)), the fact that we introduced debt in the firm’s capital structure makes equity riskier
- As a result, when we take all sources of funds into consideration, the unlevered cost of capital - i.e, the cost of capital of the firm’s assets or the underlying business - remains the same!
Case 2: Equity and Debt Financing
- Let’s get back to our example where we added \(\small 50\%\) of debt in our Capital Structure. Because the firm needs to pay debt holders no matter the state of the economy, the Debt value (from the perspective of a debt investor) in Date 1 is:
\[ \small D_1= D_0\times(1+5\%)=500\times 1.05=\$525 \]
- We know the value of the firm (all Assets) in each state of the economy. The difference between the firm value and the debt value is the equity value: \(\small V - D = E\):
| Source | Date 0 (investment) | Date 1 (strong economy) | Date 1 (Weak economy) |
|---|---|---|---|
| Debt (D) | 500 | 525 | 525 |
| Equity (E) | ? | 875 | 375 |
| Firm (V) | 1,000 | 1,400 | 900 |
Case 2: Equity and Debt Financing (continued)
- You may have noticed something strage: if the debt cost is only \(\small5\%\), why the firm value still is \(\small\$1.000\)? Shouldn’t the value of equity increase? The answer is that the cost of equity increases for levered equity:
| Source | \(T_0\) | \(T_1\) (strong) | \(T_1\) (Weak) | \(R_{\text{Strong}}\) | \(R_{\text{Weak}}\) |
|---|---|---|---|---|---|
| Debt | 500 | 525 | 525 | 5% | 5% |
| Equity | 500 | 875 | 375 | 75% | -25% |
| Value | 1,000 | 1,400 | 900 | 40% | -10% |
- Unlevered Equity has returns of \(\small40\%\) or \(\small-10\%\) \(\rightarrow\) on average, \(\small15\%\), as before
- Debt has return of \(\small5\%\), regardless of the state of the economy
- Levered Equity has returns of \(\small75\%\) (\(\small\frac{875}{500}-1\)) or \(\small-25\%\) (\(\small\frac{375}{500}-1\)). On average, \(\small25\%\).
Case 2: Equity and Debt Financing (continued)
- The key takeaway is: Levered equity is riskier, so the cost of capital is higher:
| Source | Return sensitivity | Risk premium |
|---|---|---|
| Debt | 5% - 5% = 0% | 5% - 5% = 0% |
| Unlevered Equity | 40% - (-10%) = 50% | 15% - 5% = 10% |
| Levered Equity | 75% - (-25%) = 100% | 25% - 5% = 20% |
- Because the debt’s return bears no systematic risk, its risk premium is zero - it pays \(\small5\%\) regardless of the state of the economy!
- In this particular case, the levered equity has twice the systematic risk of the unlevered equity and, as a result, has twice the risk premium!
- Now, if you were to calculate the cost of capital of the firm, assuming \(\small 50\%\) equity and \(\small 50\%\) debt, the required return is \(\small \frac{1}{2}\times25\%+\frac{1}{2}\times 5\%=15\%\rightarrow\) the firm value remains at \(\small \$1,000\)!
Equity versus Debt Financing, a summary
- In the case of perfect capital markets, if the firm is \(\small100\%\) equity financed, the equity holders will require a \(\small15\%\) expected return
- When financingwith \(\small50\%\)/\(\small50\%\) equity and debt, debtholders will receive a return of \(\small5\%\), while the levered equity holders will require an expected return of \(\small25\%\) (because of increased risk)
- Leverage increases the risk of equity even when there is no risk that the firm will default
- Thus, while debt may be cheaper, its use raises the cost of capital for equity
\(\rightarrow\) Considering both sources of capital together, the firm’s average cost of capital with leverage is the same as for the unlevered firm!
Modigliani and Miller argued that with perfect capital markets, the total value of a firm should not depend on its capital structure
They reasoned that the firm’s total cash flows still equal the cash flows of the project and, therefore, have the same present value!
Equity versus Debt Financing, exercise
Using the same values as before, suppose the firm borrows \(\small\$700\) when financing the project. According to Modigliani and Miller, what should the value of the equity be? What is the expected return?
Because the value of the firm’s total cash flows is still \(\small\$1,000\), if the firm borrows \(\small\$700\), its equity will be worth \(\small\$300\). The firm will owe \(\small \$700 \times 1.05 = \$735\) in one year to debtholders:
- If the economy is strong, equity holders will receive \(\small 1,400 − 735 = 665\), a return of \(\small \frac{665}{300}-1 = \small 121.67\%\).
- If the economy is weak, equity holders will receive \(\small 900 − 735 = 165\), a return of \(\small \frac{165}{300}-1 = \small -45\%\).
The expected return is then:
\[ \small \frac{1}{2} \times 121.67\% + \frac{1}{2} \times -45\% = 38.33\% \]
Equity versus Debt Financing, exercise
- Note that the equity has a return sensitivity of \(\small 121.67\% − (−45.0\%) = 166.67\%\), and its risk premium is \(\small 38.33\% − 5\% = 33.33\%\):
| Source | \(R_\text{Strong}\) | \(R_\text{Weak}\) | Syst. risk | Risk premium |
|---|---|---|---|---|
| Debt | 5% | 5% | 5% - 5% = 0% | 5% - 5% = 0% |
| Equity | 122% | -45% | 122% - (-45%) = 167% | 38% - 5% = 33% |
| Firm | 40% | -10% | 75% - (-25%) = 100% | 25% - 5% = 20% |
- Again, debt increases equity risk, and the unlevered value (the value of the firm as whole) remains the same because the unlevered cost of capital is still 15%:
\[ \small \underbrace{30\%\times5\%}_{\text{Debt}}+\underbrace{70\%\times38.33\%}_{\text{Equity}}=15\% \]
Modigliani and Miller: why it makes sense?
Back in our previous slides, we achieved the conclusion that the value of the firm remained at \(\small \$1,000\) regardless of the Capital Structure
That was just a direct application of the Law of One Price: the choice of firm’s leverage merely changes the allocation of value between debt and equity - i.e, you’re just slicing the pizza in different ways, but the size of the pizza remains the same!
Modigliani and Miller showed that this result holds more generally under a set of conditions referred to as perfect capital markets:
Investors and firms can trade the same set of securities at competitive market prices equal to the present value of their future cash flows
There are no taxes, transaction costs, or issuance costs associated with security trading
A firm’s financing decisions do not change the cash flows generated by its investments, nor do they reveal new information about them
Modigliani and Miller - Proposition I
Proposition I - Modigliani and Miller
In a perfect capital market, the total value of a firm’s securities is equal to the market value of the total cash flows generated by its assets and is not affected by its choice of capital structure.
Why is this important? It establishes that, under certain conditions, the irrelevance of the choice of leverage applies to more general cases
As we’ll see in the upcoming slides, this idea applies to a case where a given investor might desire a different capital structure choice than the one chosen by the firms
Regardless of the case (either a more or less leverage preference), the value of the firm for this investor is the same!
In other words, because different choices of capital structure offer no benefit to investors, they do not affect the value of the firm
Proposition I - Levering up
Let’s say the firm selects a given capital structure, but the investor likes an alternative capital structure (either more or less leveraged)
Modigliani and Miller demonstrated that if investors prefer an alternative capital structure to the one the firm has chosen, they can borrow or lend on their own personal account and achieve the same result in terms of firm value!
To illustrate that, assume the firm is an all−equity firm (zero leverage)…
- An investor who would prefer to have a levered equity firm. In perfect capital markets, he can do so by levering his own personal portfolio
- He chooses to borrow \(\small\$500\) and add leverage to his or her own portfolio (\(\small\$500\) personal + \(\small\$500\) debt)
- The investor then borrows \(\small\$500\) and buys the firm’s stock
Proposition I - Levering Up
| Source | \(T_0\) | \(T_1\) (strong) | \(T_0\) (weak) |
|---|---|---|---|
| Unlevered Equity (Firm) | 1,000 | 1,400 | 900 |
| Margin loan (Investor borrowing) | -500 | -525 | -525 |
| Levered equity (Investor’s return) | 500 | 875 | 375 |
If the cash flows of the unlevered equity serve as collateral for the margin loan (at the risk−free rate of 5%), then by using homemade leverage, the investor has replicated the payoffs to the levered equity!
As long as investors can borrow or lend at the same interest rate as the firm, homemade leverage is a perfect substitute for the use of leverage by the firm
What if investors actually want a less leveraged firm? Is it possible for them to unlever on their own?
Proposition I - Unlevering
Now, assume the firm uses debt, but investors prefer to hold unlevered equity (\(\small100\%\) equity):
- The investor can again replicate the payoffs of Unlevered Equity firm by buying both the debt and the equity of the firm!
- Combining the cash flows of the two securities produces cash flows identical to Unlevered Equity, for a total cost of \(\small\$1,000\):
| Source | \(T_0\) | \(T_1\) (strong) | \(T_0\) (weak) |
|---|---|---|---|
| Debt (Investor’s lending) | 500 | 525 | 525 |
| Levered equity (Firm) | 500 | 875 | 375 |
| Unlevered equity (Investor’s return) | 1,000 | 1,400 | 900 |
Does Proposition I hold in reality?
Modigliani and Miller showed that a firm’s financing choice does not affect its value. But how can we reconcile this conclusion with the fact that the cost of capital differs for different securities?
- When the project is financed solely through equity, the equity holders require a \(15\%\) expected return
- As an alternative, the firm could borrow at the risk-free rate of \(\small5\%\)
All in all, isn’t debt a cheaper and better source of capital than equity? Although debt does have a lower cost of capital than equity, we cannot consider this cost in isolation:
- While debt itself may be cheap, it increases the risk and therefore the cost of capital of the firm’s equity
- In the end, the savings a lower debt cost of capital are exactly offset by a higher equity cost of capital, and there are no net savings for the firm!
Modigliani and Miller - Proposition II
- The insights from the first proposition can be used to derive an explicit relationship between leverage and the equity cost of capital, which is Modigliani Miller’s Proposition II:
Proposition II - Modigliani and Miller
The cost of capital of levered equity increases with the firm’s market value debt equity ratio:
\[ \begin{align} &r_E=r_U+\dfrac{D}{E}(r_U-r_D), \text{where:} \\ &\\ &\text{ - E = Market value of equity in a levered firm}\\ &\text{ - D = Market value of debt in a levered firm}\\ &\text{ - U = Market value of equity in an unlevered firm}\\ &\text{ - A = Market value of the firm's assets}\\ \end{align} \]
Proposition II - Modigliani and Miller (continued)
Proposition I stated that: \(\small E+D = U = A\). In words, the total market value of the firm’s securities is equal to the market value of its assets, whether the firm is unlevered or levered
Furthermore, remember that the return of a portfolio is the weighted average of the returns
So, we can write that the return on unlevered equity (\(r_U\)) is related to the returns of levered equity (\(r_E\)) and debt (\(r_D\)):
\[ \small r_U = \frac{E}{E+D} \times r_E + \frac{D}{E+D} \times r_D \]
- Rearranging terms and solving for \(r_E\)1:
\[ \small r_E = r_U + \frac{D}{E} \times (r_U - r_D) \]
Proposition II - Modigliani and Miller (continued)
- The levered equity return equals the unlevered return, plus a premium due to leverage. The amount of the premium depends on the amount of leverage, measured by the market value debt−to-equity ratio!
\[ \small r_E = \underbrace{r_U}_{\text{Risk without leverage}} + \underbrace{\frac{D}{E} \times (r_U - r_D)}_{\text{Add. risk due to leverage}} \]
- Using the previous example’s numbers (\(\small r_U=15\%\), \(\small r_D=5\%\) and a debt-to-equity ratio of 1 \(\frac{500}{500}=1\), we have:
\[ \small r_E = r_U + \frac{D}{E} \times (r_U - r_D)\rightarrow 15\%+1\times(15\%-5\%)=25\% \]
Proposition II - Modigliani and Miller (continued)
- Suppose the entrepreneur borrows \(\small\$700\) when financing the project. Recall that the expected return on unlevered equity is \(\small15\%\) and the risk−free rate is \(\small5\%\). According to Proposition II, what will be the firm’s equity cost of capital?
\[ \small r_E = r_U + \frac{D}{E} \times (r_U - r_D) = 15\% + \frac{700}{300} \times (15\%-5\%) = 38.33\% \]
- If a firm is financed with both equity and debt, then the risk of its underlying assets will match the risk of a portfolio of its equity and debt (the unlevered cost of capital or the pre-tax WACC):
\[ \small r_{WACC} = \frac{E}{E+D} \times r_E + \frac{D}{E+D} \times r_D \rightarrow 30\% \times 38.33\% + 70\% \times 5\% = 15\% \]
\(\rightarrow\) With perfect capital markets, a firm’s WACC is independent of its capital structure and is equal to its equity cost of capital if it is unlevered
Proposition II - Modigliani and Miller (example)
Honeywell (ticker: HON) has a market debt−equity ratio of \(\small0.5\). Assume its current debt cost of capital is \(\small6.5\%\), and its equity cost of capital is \(\small 14\%\). If HON issues equity and uses the proceeds to repay its debt and reduce its debt−equity ratio to \(0.4\), it will lower its debt cost of capital to \(\small 5.75\%\).
With perfect capital markets, what effect will this transaction have on HON’s equity cost of capital and WACC?
\[ \small r_{WACC} = \frac{E}{E+D} \times r_E + \frac{D}{E+D} \times r_D \rightarrow \frac{2}{2+1} \times 14\% + \frac{1}{2+1} \times 6.5\% = 11.5\% \]
- The new Cost of Equity (\(\small r_E\)) will decrease due to lower leverage and lower cost of debt:
\[ \small r_E = r_U + \frac{D}{E} (r_U - r_D) \rightarrow 11.5\% +0.4 \times (11.5\% - 5.75\%) = 13.8\% \]
Proposition II - Modigliani and Miller (example)
- The WACC, on the other hand, remains unchanged:
\[ \small r_{WACC} = \frac{1}{1+0.4} \times 13.8\% + \frac{0.4}{1+0.4} \times 5.75\% = 11.5\% \]
Levered and Unlevered Betas
- Remember that
\[ \small \beta_U = \frac{E}{D+E} \times \beta_E + \frac{D}{D+E} \times \beta_D \]
- When a firm changes its capital structure without changing its investments, its unlevered beta will remain unaltered. However, its equity beta will change to reflect the effect of the capital structure change on its risk:
\[ \small \beta_E = \beta_U + \frac{D}{E} (\beta_U - \beta_D) \]
\(\rightarrow\) Therefore, the unlevered beta is a measure of the risk of a firm as if it did not have leverage, which is equivalent to the beta of the firm’s assets!
Levered and unlevered Betas (example)
In August 2018, Reenor had a market capitalization of 140 billion. It had debt of 25.4 billion as well as cash and short−term investments of 60.4 billion. Its equity beta was 1.09 and its debt beta was approximately zero. What was Reenor’s enterprise value at time? Given a risk−free rate of 2% and a market risk premium of 5%, estimate the unlevered cost of capital of Reenor’s business.
- Net Debt is \(\small 25.4 − 60.4=-35\) billion. As a result, the Enterprise Value is \(\small 140−35=105\) billion
\[ \small \beta_U = \frac{E}{E+D} \times \beta_E + \frac{D}{E+D} \times \beta_D = \frac{140}{105} \times 1.09 + \frac{-35}{105} \times 0 = 1.45 \]
\[ \small r_U = 2\% + 1.45 \times 5\% = 9.25\% \]
Capital Structure Fallacies
We will discuss now two fallacies concerning capital structure:
- Leverage increases earnings per share (EPS), thus increase firm value
- Issuing new equity will dilute existing shareholders, so debt should be issued
\(\rightarrow\) As we’ll see, using Modigliani and Miller’s proposition under perfect capital markets, we can conclude that both claims are incorrect - in other words, these actions do not change the value of a firm
Capital Structure Fallacies - EPS
Assume that LVI’s EBIT is not expected to grow in the future and that all earnings are paid as dividends. Is the increase in expected EPS lead to an increase in the share price?
Without leverage, expected earnings per share and therefore dividends are \(\small\$1\) each year, and the share price is \(\small\$7.50\)
Let \(r_U\) be LVI’s cost of capital without leverage. The value LVI as a perpetuity is simply:
\[ \small P = 7.50 = \frac{Div}{r_U} = \frac{EPS}{r_U} = \frac{1}{r_U} \]
\(\rightarrow\) Therefore, current stock price implies that \(\small r_U = \frac{1}{7.50} = 13.33\%\)
Capital Structure Fallacies - EPS (continued)
- The market value of LVI without leverage is \(\small \$7.50 \times 10\) million shares = \(\small \$75\) million. Assume that LVI uses debt to repurchase \(\small 15\) million worth of the firm’s equity. Then, the remaining equity will be worth \(\small 75\) − \(\small \$15\) = \(\small \$60\) million. After the transaction, \(\frac{D}{E} = \frac{1}{4}\), thus, we can write:
\[ \small r_E= r_U +\frac{D}{E} \times (r_U - r_D) = 13.33\% + 0.25 \times (13.33\% - 8\%) = 14.66 \]
- Also, note that the EPS will actually increase to \(\small 1.1\): because total earnings (earnings times number of shares) are \(\small \$10,000,000\) and now the firm has only \(\small 8,000,000\) (since \(\small \$15,000,000/\$7.5=2,000,000\)) has been bought back with debt, we have:
\[ \small EPS=\dfrac{Earnings}{Share}=\dfrac{\overbrace{10,000,000}^{\text{EBIT}}-\overbrace{8\%\times15,000,000}^{\text{Interest}}}{8,000,000}=\dfrac{8,800,000}{8,000,000}=1.1 \]
Capital Structure Fallacies - EPS (continued)
- Given that expected EPS is now \(\small \$1.10\) per share, the new value of the shares equals:
\[ \small P=\frac{1.10}{14.66\%} = 7.50 \]
\(\rightarrow\) Thus, even though EPS is higher, due to the additional risk, shareholders will demand a higher return. These effects cancel out, so the price per share is unchanged
As Earnings-per-Share (EPS), Price-Earnings (P/E) and even ROE ratios are affected by leverage, we cannot reliably compare these measures across firms with different capital structures!
Therefore, most analysts prefer to use performance measures and valuation multiples that are based on the firm’s earnings before interest has been deducted
For example, the ratio of enterprise value to EBIT (or EBITDA) is more useful when analyzing firms with very different capital structures than is comparing their P/E ratios
Capital Structure Fallacies - Equity Issuances and Dilution
It is sometimes (incorrectly) argued that issuing equity will dilute existing shareholders’ ownership value, so debt financing should be used instead
To see that, suppose Jet Sky Airlines (JSA) currently has no debt and \(\small 500\) million shares of stock outstanding, which is currently trading at a price of \(\small \$16\). Last month, the firm announced that it would expand and the expansion will require the purchase of \(\small \$1\) billion of new planes, which will be financed by issuing new equity:
- The current (prior to the issue) value of the equity and the assets of the firm is \(\small \$8\) billion
- Therefore, \(\small 500\) million shares \(\times\) \(\small \$16\) per share leads to a firm value of \(\small \$8\) billion
Capital Structure Fallacies - Equity Issuances and Dilution (continued)
- Suppose now JSA sells \(\small 62.5\) million new shares at the current price of \(\small \$16\) per share to raise the additional \(\small \$1\) billion needed to purchase the planes:
| Assets | Before | After |
|---|---|---|
| Cash | - | $1,000 |
| Existing Assets | $8,000 | $8,000 |
| Total Value | $8,000 | $9,000 |
| # Shares (out) | 500 | 562.5 |
| Value per share | $16 | $16 |
\(\rightarrow\) As a consequence, share prices don’t change. Any gain or loss associated with the transaction will result from the NPV of the investments the firm makes with the funds raised
Why should I bother about Modigliani and Miller?
Modigliani and Miller is truly about the conservation of the value principle in perfect financial markets: with perfect capital markets, financial transactions neither add nor destroy value, but instead represent just a repackaging of risk and therefore return
As a result, what really matters for the firm value is to find good investment opportunities that increase the future cash-flows!
Practitioners often question why Modigliani and Miller’s results are important if, after all, capital markets are not perfect in the real world
- While it is true that capital markets are not perfect, all scientific theories begin with a set of idealized assumptions from which conclusions can be drawn
- When we apply the theory, we must then evaluate how closely the assumptions hold, and consider the consequences of any important deviations
Why should I bother about Modigliani and Miller?
You may well think…but we are not in a perfect capital market, so why should I bother? That goes hand in hand with the Modigliani and Miller findings:
- If capital markets were perfect (as in the Modigliani and Miller world), then the value of a firm should not be affected by any financial transaction…
- … this implies that if there is a financial transaction that appears to be a good, it must be that it is exploiting some type of market imperfection!
Knowing how these imperfections affect the firm value is important for business and policy considerations
- For example, what happens if you remove the tax-shield from debt interest payments? What if the government decides to tax dividends differently?
- These (and other) actions may create/remove market imperfections that will tweak the real-practice results away/closer to the Modigliani-Miller findings!
Practice
Important
Practice using the following links:
Appendix
Levered and Unlevered \(\beta\)
- Recall that the unlevered beta, \(\beta_U\), is simply:
\[ \small \beta_U = \frac{E}{E+D}\times \beta_E + \frac{D}{E+D} \times \beta_D \]
- Putting it into \(\beta_E\) terms, we have:
\[ \small \begin{align} &\beta_U = \frac{(E\times \beta_E +D\times \beta_D)}{E+D}\\ &\rightarrow E\times \beta_E =(E+D)\times\beta_U - D\times \beta_D\\ &\rightarrow\beta_E =\dfrac{(E+D)\times\beta_U - D\times \beta_D}{E}=\beta_U+\dfrac{D}{E}\times \beta_U-\dfrac{D}{E}\times \beta_D\\ &\rightarrow \beta_E= \beta_U+\dfrac{D}{E}\times(\beta_U-\beta_D) \end{align} \]
Levered and Unlevered Cost of Capital
- Note that, as we can lever (or unlever) \(\beta\), we can do the same with the required returns from equity, \(r_E\):
\[ \small r_U = \frac{E}{E+D}\times r_E + \frac{D}{E+D} \times r_D \]
- Putting it into \(r_E\) terms and using the same rationale, we have:
\[ r_E= r_U+\dfrac{D}{E}\times(r_U-r_D) \]
- In words, if leverage increases (through higher debt-to-equity ratios), the equity sensitivity should increase as well, and the required returns for equity should increase!
