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Detailed summary of managerial finance and accounting
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Corporate finance bookChapter 10: capital markets and the pricing of risk
10.8. beta and the cost of capital
Before we can estimate the risk premium of an individual stock, we need a way to assess
investors’ appetite for risk. The size of the risk premium investors will require to make a
risky investment depends upon their risk aversion. Rather than attempt to measure this
risk aversion directly, we can measure it indirectly by looking at the risk premium
investors’ demand for investing in systematic, or market, risk.
We can calibrate investors’ appetite for market risk from the market portfolio. The risk
premium investors earn by holding market risk is the difference between the market
portfolio’s expected return and the risk-free interest rate:
𝑚𝑎𝑟𝑘𝑒𝑡 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓
The market risk premium is the reward investors expect to earn for holding a portfolio
with a beta of 1 – the market portfolio itself. Consider an investment opportunity with a
beta of 2. This investment carries twice as much systematic risk as an investment in the
market portfolio. That is, for each dollar we invest in the opportunity, we could invest
twice that amount in the market portfolio and be exposed to exactly the same amount of
systemic risk. Because it has twice as much systematic risk, investors will require twice
the risk premium to invest in an opportunity with a beta of 2.
We can use the beta of the investment to determine the scale of the investment in the
market portfolio that has equivalent systematic risk. Thus, to compensate investors for
the time value of their money as well as the systematic risk they are bearing, the cost of
capital rI for an investment with beta I should satisfy the following formula:
𝑟𝐼 = 𝑟𝑖𝑠𝑘 − 𝑓𝑟𝑒𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 + 𝛽𝐼 × 𝑚𝑎𝑟𝑘𝑒𝑡 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
𝑟𝐼 = 𝑟𝑓 + 𝛽𝐼 × (𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓 )
What happens if a stock has a negative beta? A stock would have a negative risk
premium – it would have an expected return below the risk-free rate. Note that a stock
with a negative beta will tend to do well when times are bad, so owning it will provide
insurance against the systematic risk of other stocks in the portfolio. Risk-averse
investors are willing to pay for this insurance by accepting a return below the risk-free
interest rate.
For estimating the cost of capital, is often referred to as the Capital Asset Pricing
Model (CAPM), the most important method for estimating the cost of capital that is
used in practice.
10.8. beta and the cost of capital summary
• The market risk premium is the expected excess return of the market portfolio:
𝑚𝑎𝑟𝑘𝑒𝑡 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓
It reflects investors’ overall risk tolerance and represents the market price of risk
in the economy.
• The cost of capital for a risky investment equals the risk-free rate plus a risk
premium. The Capital Asset Pricing Model (CAPM) states that the risk premium
equals the investment’s beta times the market risk premium:
𝑟𝐼 = 𝑟𝑓 + 𝛽𝐼 × (𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓 )
,Chapter 14: capital structure in a perfect market
14.1. equity versus debt financing
The relative proportions of debt, equity, and other securities that a firm has outstanding
constitute its capital structure. When corporations raise funds from outside investors,
they must choose which type of security to issue.
The project’s NPV represents the value to the initial owners of the firm (in this case, the
entrepreneur) created by the project.
Equity in a firm with no debt is called unlevered equity. Because there is no debt, the
date 1 cash flows of the unlevered equity are equal to those of the project.
Equity in a firm with debt is called levered equity. Promised payments to debt holders
must be made before any payments to equity holders are distributed.
We can evaluate the relationship between risk and return more formally by computing
the sensitivity of each security’s return to the systematic risk of the economy.
In the case of perfect capital markets, if the firm is 100% equity financed, the equity
holders will require a 15% expected return. If the firm is financed 50% with debt and
50% with equity, the debt holders will receive a lower return of 5%, while the levered
equity holders will require a higher expected return of 25% because of their increased
risk.
Leverage increases the risk of equity even when there is no risk that the risk will default.
Thus, while debt may be cheaper when considered on its own, it raises the cost of capital
for equity.
14.2. Modigliani-Miller I: leverage, arbitrage, and firm value
We used the Law of One Price to argue that leverage would not affect the total value of
the firm (the amount of money the entrepreneur can raise). Modigliani and Miller showed
that this result holds more generally under a set of conditions referred to as perfect
capital markets:
1. Investors and firms can trade the same set of securities at competitive market
prices equal to the present value of their future cash flows.
2. There are no taxes, transaction costs, or issuance costs associated with security
trading.
3. A firm’s financing decisions do not change the cash flows generated by its
investments, nor do they reveal new information about them.
MM proposition I: 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.
When investors use leverage in their own portfolios to adjust the leverage choice made
by the firm, we say that they are using homemade leverage. As long as investors can
borrow or lend at the same interest rate as the firm, homemade leverage is perfect
substitute for the use of leverage by the firm.
A market value balance sheet is similar to an accounting balance sheet, with two
important distinctions. First, all assets and liabilities of the firm are included – even
intangible assets such as reputation, brand name, or human capital that are missing
from a standard accounting balance sheet. Second, all values are current market values
rather than historical costs.
Market value of equity = market value of assets – market value of debt and other
liabilities
,When a firm repurchases a significant percentage of its outstanding shares in this way,
the transaction is called a leveraged recapitalization.
14.3. Modigliani-Miller II: leverage, risk, and the cost of capital
MM proposition states that:
E+D=U=A
E = market value of equity
D = market value of debt
U = market value of equity if the firm is unlevered
A = market value of the firm’s assets
That is, 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.
By holding a portfolio of the firm’s equity and debt, we can replicate the cash flows from
holding unlevered equity. Because the return of a portfolio is equal to the weighted
average of the returns of the securities in it, this equality implies the following
relationship between the returns of levered equity (𝑅𝐸 ), debt (𝑅𝐷 ), and unlevered equity
(𝑅𝑈 ).
𝐸 𝐷
𝑅𝐸 + 𝐸+𝐷 𝑅𝐷 = 𝑅𝑈
𝐸+𝐷
This equation reveals the effect of leverage on the return of the levered equity. The
amount of additional risk depends on the amount of leverage, measured by the firm’s
market value debt-equity ratio.
MM proposition II: the cost of capital of levered equity increases with the firm’s
market value debt-equity ratio.
, 𝐷
Cost of capital of levered equity: 𝑟𝐸 = 𝑟𝑈 + 𝐸 (𝑟𝑢 − 𝑟𝐷 )
𝑟𝑤𝑎𝑐𝑐 = 𝑟𝑈 = 𝑟𝐴
That is, 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, which matches the
cost of capital of its assets.
The debt-to-value ratio, D / (E + D), which is the fraction of the firm’s total value that
corresponds to debt. With no debt, the WACC is equal to the unlevered equity cost of
capital.
A firm’s unlevered or asset beta is the weighted of its equity and debt beta:
𝐸 𝐷
𝛽𝑈 = 𝐸+𝐷 𝛽𝐸 + 𝐸+𝐷 𝛽𝐷
Recall that the unlevered beta measures the market risk of the firm’s underlying assets,
and thus can be used to assess the cost of capital for comparable investments. When a
firm changes its capital structure without changing its investments, its unlevered beta
will remain unaltered.
𝐷
𝛽𝐸 = 𝛽𝑈 + 𝐸 (𝛽𝑈 − 𝛽𝐷 )
The assets on a firm’s balance sheet include any holdings of cash or risk-free securities.
Because these holdings are risk-free, they reduce the risk – and therefore the required
risk premium – of the firm’s assets. For this reason, holding excess cash has the
opposite effect of leverage on risk and return. From this standpoint, we can view cash as
negative debt.
14.4. capital structure fallacies
MM propositions I and II state that with perfect capital markets, leverage has no effect
on firm value or the firm’s overall cost of capital.
Leverage can increase a firm’s expected earnings per share.
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