## Computing FCFF from Net Income

FCFF is the cash flow available to the company’s suppliers of capital after all operating expenses (including taxes) have been paid and operating investments have been made. The company’s suppliers of capital include bondholders and common shareholders (plus, occasionally, holders of preferred stock, which we ignore until later). Keeping in mind that a noncash charge is a charge or expense that does not involve the outlay of cash, we can write the expression for FCFF as follows:

FCFF = Net income available to common shareholders (NI)
Plus: Net noncash charges (NCC)
Plus: Interest expense × (1 − Tax rate)
Less: Investment in fixed capital (FCInv)
Less: Investment in working capital (WCInv)

This equation can be written more compactly as: FCFF = NI + NCC + Int(1 – Tax rate) – FCInv – WCInv

Consider each component of FCFF. The starting point in the equation above is net income available to common shareholders—the bottom line in an income statement. It represents income after depreciation, amortization, interest expense, income taxes, and the payment of dividends to preferred shareholders (but not payment of dividends to common shareholders).

Net noncash charges represent an adjustment for noncash decreases and increases in net income. This adjustment is the first of several that analysts generally perform on a net basis. If noncash decreases in net income exceed the increases, as is usually the case, the adjustment is positive. If noncash increases exceed noncash decreases, the adjustment is negative. The most common noncash charge is depreciation expense. When a company purchases fixed capital, such as equipment, the balance sheet reflects a cash outflow at the time of the purchase. In subsequent periods, the company records depreciation expense as the asset is used. The depreciation expense reduces net income but is not a cash outflow. Depreciation expense is thus one (the most common) noncash charge that must be added back in computing FCFF. In the case of intangible assets, there is a similar noncash charge, amortization expense, which must be added back. Other noncash charges vary from company to company.

After-tax interest expense must be added back to net income to arrive at FCFF. This step is required because interest expense net of the related tax savings was deducted in arriving at net income and because interest is a cash flow available to one of the company’s capital providers (i.e., the company’s creditors). In the United States and many other countries, interest is tax deductible (reduces taxes) for the company (borrower) and taxable for the recipient (lender). As we explain later, when we discount FCFF, we use an after-tax cost of capital. For consistency, we thus compute FCFF by using the after-tax interest paid.5

Similar to after-tax interest expense, if a company has preferred stock, dividends on that preferred stock are deducted in arriving at net income available to common shareholders. Because preferred stock dividends are also a cash flow available to one of the company’s capital providers, this item is added back to net income available to common shareholders in deriving FCFF.

## Rules of Duration

When thinking about duration, a few general rules apply. With everything else being equal:

• The duration of any bond that pays a coupon will be less than its maturity, because some amount of coupon payments will be received before the maturity date.

• The lower a bond’s coupon, the longer its duration, because proportionately less payment is received before final maturity. The higher a bond’s coupon, the shorter its duration, because proportionately more payment is received before final maturity.

• Because zero coupon bonds make no coupon payments, a zero coupon bond’s duration will be equal to its maturity.

• The longer a bond’s maturity, the longer its duration, because it takes more time to receive full payment. The shorter a bond’s maturity, the shorter its duration, because it takes less time to receive full payment.

## Convexity

The duration measure indicates that regardless of whether interest rates increase or decrease, the approximate percentage price change is the same. However, while for small changes in yield the percentage price change will be the same for an increase or decrease in yield, for large changes in yield this is not true. This suggests that duration is only a good approximation of the percentage price change for small changes in yield.

For example, using a 5% 20-year bond selling to yield 4% with a duration of 13.09. For a 10 basis point change in yield, the estimate was accurate for both an increase or a decrease in yield. However, for a 200 basis point change in yield, the approximate percentage price change was off considerably.

The reason for this result is that duration is in fact a first (linear) approximation for a small change in yield. The approximation can be improved by using a second approximation. This approximation is referred to as “convexity.” The use of this term in the industry is unfortunate because the term convexity is also used to describe the shape or curvature of the price/yield relationship. The convexity measure of a security can be used to approximate the change in price that is not explained by duration.

## Break-even Inflation Rate

The break-even inflation rate (BEI) is a market-based measure of expected inflation. It is the difference between the yield of a nominal bond and an inflation-linked bond of the same maturity.

BEI is comprised of two elements: expected inflation ($\pi$) and risk premium for uncertainty in inflation ($\theta$).

The fundamental difference between the pricing formula as applied to, for example, a three-month T-bill and its application to, for example, a default-free zero-coupon bond relates to their investment horizons. The relative certainty about the real payoff from a three month T-Bill and thus the relative certainty about the amount of consumption that the investor will be able to undertake with the payoff means that the investment in the T-Bill will be a good hedge against possible bad consumption outcomes. In other words, the payoff, in real terms, from a three month T-Bill is highly unlikely to fall if the investor loses his or her job during the T-Bill’s three month investment horizon. The low, probably zero, correlation between the T-Bill’s payoff with bad consumption outcomes will mean that the risk premium needed to tempt an investor to invest in the T-Bill will be close to zero.

However, it is unlikely that the same level of certainty would apply, for example, to a 20-year default-free conventional government bond. For such a bond, it would seem reasonable to assume that the risk premium would be higher than that related to a one- or three-month T-Bill. Note that the cash flow in Equation 10 is still certain, but only in nominal terms. Because investors will naturally have less confidence in their ability to form views about future inflation over 20 years relative to their abilities to form those views over three months, the greater uncertainty about the real value of the bond’s payoff will cause investors to demand a premium in compensation for this uncertainty, represented by πt,s.

The difference between the yield on, for example, a zero-coupon default-free nominal bond and on a zero-coupon default-free real bond of the same maturity is known as the break-even inflation (BEI) rate. It should be clear from the discussion earlier that this break-even inflation rate will incorporate the inflation expectations of investors over the investment horizon of the two bonds, θt,s, plus a risk premium that will be required by investors to compensate them predominantly for uncertainty about future inflation, πt,s. Although the evolution of real zero-coupon default-free yields over time should be driven mainly by the inter-temporal rate of substitution, the evolution of their nominal equivalents will, in addition, be driven by changing expectations about inflation and changing perceptions about the uncertainty of the future inflation environment. We can see this evolution by plotting the constant maturity zero-coupon break-even inflation rates over time.

## Modigliani–Miller Theorem

The Modigliani–Miller theorem is a set of two propositions on corporate capital structure. It was first proposed by Franco Modigliani and Merton Miller in 1958.

## Two Propositions (No Taxes)

Modigliani and Miller made some very serious assumptions. The most important two are that there are no taxes and no costs of financial distress. Additional assumptions will be discussed in the next section. The two propositions are still true when the no taxes assumption is relaxed.

Proposition I : the market value of any firm is independent of its capital structure.

This means a firm cannot change its total value just by splitting its cash flows into different streams: The firm’s value is determined by its real assets, not by how it is financed. Thus capital structure is irrelevant as long as the firm’s investment decisions are taken as given.

Firms can not create value simply by changing the company’s capital structure.

Proposition II: the cost of equity is a linear function of the company’s debt/equity ratio.

More specifically, expected return on equity = expected return on assets + (expected return on assets – expected return on debt) * debt-equity ratio.

The mathematical representation (which can be derived from the WACC formula) is:

$r_E = r_0 + (r_0 - r_D)(D/E)$

According to this proposition, as the company increases its use of debt financing, the cost of equity rises. We know from MM Proposition I that the value of the company is unchanged and the weighted average cost of capital remains constant if the company changes its capital structure. What Proposition II then means is that the cost of equity increases in such a manner as to exactly offset the increased use of cheaper debt in order to maintain a constant WACC.

The risk of the equity depends on two factors: the risk of the company’s operations (business risk) and the degree of financial leverage (financial risk). Business risk determines the cost of capital, whereas the capital structure determines financial risk.

The expected rate of return on the common stock of a levered firm increases in proportion to the debt–equity ratio (D/E), expressed in market values; the rate of increase depends on the spread between $r_A$, the expected rate of return on a portfolio of all the firm’s securities, and $r_D$, the expected return on the debt.

Note that $r_E = r_A$ if the firm has no debt.

## Assumptions

• Expectations are homogeneous. This means investors agree on the expected cash flows from a given investment. This means that all investors have the same expectations with respect to the cash flows from an investment in bonds or stocks.

• Bonds and shares of stock are traded in perfect capital markets. This means that there are no transactions costs, no taxes, no bankruptcy costs, and everyone has the same information. In a perfect capital market, any two investments with identical cash flow streams and risk must trade for the same price.

• Investors can borrow and lend at the risk-free rate.

• There are no agency costs. This means that managers always act to maximize shareholder wealth.

• The financing decision and the investment decision are independent of each other. This means that operating income is unaffected by changes in the capital structure.

• No costs of asymmetric information

• debtholders have prior claim to assets and income relative to equityholders, the cost of debt is less than the cost of equity