Dividend Valuation Model If dividends are constant forever, the value of a share of stock is the
Dividend Valuation Model If dividends are constant forever, the value of a share of stock is the
present value of the dividends per share per period, in perpetuity. Let D represent the constant dividend per share of common stock expected
next period and each period thereafter, forever, P 0 represent the price of
a share of stock today, and r e the required rate of return on common stock. The required rate of return (RRR) is the return shareholders demand to compensate them for the time value of money tied up in their investment and the uncertainty of the future cash flows from these investments.
The current price of a share of common stock, P 0 , is:
D D D P 0 = --------------------- + --------------------- + … + ---------------------
( 1 + r x e ) which we can write using summation notation,
THE FUNDAMENTALS OF VALUATION
-------------------- ∑ D
The summation of a constant amount discounted from perpetuity sim- plifies to:
P 0 = D/r e
As an example, if the current dividend is $2 per share and the required rate of return is 10%, the value of a share of stock is:
P 0 = $2/0.10 = $20
Therefore, if you pay $20 per share and dividends remain constant at $2 per share, you will earn a 10% return per year on your investment every year. But dividends on common stock often change through time.
If dividends grow at a constant rate, the value of a share of stock is the present value of a growing cash flow. Let D 0 indicate this period’s (i.e., end of period 0) dividend. If dividends grow at a constant rate, g, forever, the present value of the common stock is the present value of all future dividends:
D 0 ( 1 + g 1 D 2 ) ∞ 0 ( 1 + g ) D 0 ( 1 + g ) P 0 = ---------------------------- + ---------------------------- + … + -----------------------------
Pulling today’s dividend D 0 , from each term,
P 0 = D 0 --------------------- ( ) + --------------------- ( ) +
( ) … + ----------------------
1 + r 1 1 + r 2 1 + r ( ∞ e ) ( e ) ( e ) Using summation notation:
0 ∑ --------------------
which simplifies to:
Valuation of Securities and Options
( 1 + g ) P 0 = D 0 ------------------ ( r e – g )
If we represent the next period’s dividend, D 1 , in terms of this period’s dividend, D 0 , compounded one period at the rate g (that is, D 1 =
D 0 (1+ g)) and substitute for D 0 :
0 = -------------
This equation is referred to as the Dividend Valuation Model (DVM). 1 As an example, consider a firm expected to pay a constant dividend of $2 per share, forever. If this dividend is capitalized at 10%, the value of a share is $20. If, on the other hand, the current dividend is $2 but these dividends are expected to grow at a rate of 6% per year, forever, the value of a share of stock is $53:
( + 0.06 --------------- P $2.12 = = $53
Does this make sense compared to the constant amount case where divi- dends are unchanged at $2 per year? Yes: If dividends are expected to grow in the future, the stock is worth more than if the dividends are expected to remain the same.
If today’s value of a share is $53, what are we saying about the value of the stock next year? If we move everything up one year, D 1 is no longer $2.12, but the current dividend of $2 grows at 6% to $2(1 +
0.06) 2 = $2.2472. Therefore, we expect the price of the stock at the end of one year, P 1 , to be $5:
1 = ------------------------------------ = --------------------- = $56.18
1 The Dividend Valuation Model is attributed to Myron Gordon, who popularized the constant growth model. A more formal presentation of this model can be found
in published works by Gordon entitled “Dividends, Earnings and Stock Prices,” ( Review of Economics and Statistics, May 1959, pp. 99–105) and The Investment Financing and Valuation of the Corporation (Homewood, IL: R. D. Irwin, 1962). However, the foundation of common stock valuation is laid out—for both constant and growing dividends—by John Burr Williams in The Theory of Investment Value (Amsterdam: North-Holland Publishing Company, 1938), Chapters V, VI, and VII.
THE FUNDAMENTALS OF VALUATION
EXHIBIT 9.2 The Price of a Share of Stock with a Current Dividend of $2, a 6%
Growth in Dividends, and a 10% Required Rate of Return
At the end of two years, the price will be $59.55. Since we expect dividends to grow each year, we also are expecting the price of the stock to grow through time as well. In fact, the price is expected to grow at the same rate as the dividends: 6% per period.
g, and the price of the stock expected in the future is illustrated in Exhibit 9.2. For a given required rate of return and dividend—in this case r e = 10% and
The relation between the growth rate of dividends,
D 0 = $2—we see that the price of a share of stock is expected to grow each period at the rate g. What if the dividends are expected to decline each year? That is, what if g is negative? We can still use the Dividend Valuation Model, but each dividend in the future is expected to be less than the one before it. For example, suppose a stock has a current dividend of $2 per share and the required rate of return is 10%. If dividends are expected to decline 6% each year, what is the value of a share of stock today? We
know that D 0 = $2, r e = 10%, and g= −6%. Therefore,
$2 1 0.06 -------------------------------- ( – ) $1.88
P 0 = = --------------- = $11.75
Two periods from now, the expected price is even lower:
Valuation of Securities and Options
P 1 = ----------------------------------- = --------------------- = $11.045
Let’s look at another situation, one in which growth is expected to change but at different growth rates as time goes on. Consider a share of common stock whose dividend is currently $3.00 per share and is expected to grow at a rate of 8% per year for five years and afterward at
a rate of 4% per year after five years. To tackle this problem, let’s break it into two manageable parts: the first five years and after five years, or:
P 0 = Present value of dividends in the first five years + Present value of dividends received after the first five years to infinity
Assuming a required rate of return of 10%,
----------------------------- 1 ----------------------------- 2 D D P D 0 = + + ----------------------------- 3 + ----------------------------- 4 + ----------------------------- 5
Dividends growing at a rate of 8% per year
D 6 D 7 D ----------------------------- -----------------------------
Dividends growing at a rate of 4% per year The present value of the dividends in the first five years is:
Present value of dividends received during the first five years $3.24 $3.4992 $3.7791 $4.0815 $4.4080
= ------------------ + --------------------- + --------------------- + --------------------- + --------------------- 1.1000 1.2100
= $2.9455 + $2.8919 + $2.8393 + $2.7877 + $2.7370 = $14.2014 The present value of dividends received after the fifth year—evaluated five
years from today—is the expected price of the stock in five years, P 5 :
D 5 ( 1 + 0.04 )
P 5 = ---------------------------------- = --------------------- = $76.4053
THE FUNDAMENTALS OF VALUATION
The price expected at the end of five years is $76.4053, which we trans- late into a value today by discounting it five years at 10%:
Present value of dividends to be received after the first five years = ----------------------------- $76.4053 = $76.4053 ------------------------- = $47.4420
Putting together the two pieces, P 0 = $14.2014 + $47.4420 = $61.6434
The value of a share of this stock is $61.6434. We can represent the Dividend Valuation Model in terms of a share’s price to earnings ratio (P/E ratio). Let’s start with the Dividend Valuation Model with constant growth in dividends:
D P 0 = ------------- 1
If we divide both sides of this equation by earnings per share, we can rep- resent the dividend valuation model in terms of the price-earnings (P/E) ratio:
D ------------ 1
------------ 0 = EPS ------------- 1
EPS 1 r e – g
Dividend payout ratio P/E = --------------------------------------------------------- r e – g
This tells us the P/E ratio is influenced by the dividend payout ratio, the required rate of return on equity, and the expected growth rate of divi- dends.
The Dividend Valuation Model makes some sense regarding the relation between the value of a share of stock, the growth in dividends, and the discount rate:
■ The greater the current dividend, the greater the value of a share of stock.
Valuation of Securities and Options
■ The greater the expected growth in dividends, the greater the value of a share of stock. ■ The more uncertainty regarding future dividends, the greater the dis- count rate and the lower the value of a share of stock.
However, the DVM has some drawbacks. How do you deal with dividends that do not grow at a constant rate? As you can see in the last example, this model does not accommodate nonconstant growth easily.
What if the firm does not pay dividends now? In that case, D 0 would be zero and the expected price would be zero. But the price of a share of stock cannot be zero. Therefore, the DVM may be appropriate to use to value the stock of companies with stable dividend policies, but it is not applicable for all firms.
Despite its drawbacks, the DVM captures the valuation for many companies’ securities. We can use the DVM to take a closer look at investors’ required rate of return and the expected rate of growth in future dividends. Moreover, the DVM has been modified to allow for different types of dividend patterns. 2
Parts
» Financial Management and Analysis
» SECURITIES MARKETS The primary function of a securities market—whether or not it has a
» Stock Exchanges Stock exchanges are formal organizations, approved and regulated by
» Stock Market Indicators Stock market indicators have come to perform a variety of functions,
» Efficient Markets Investors do not like risk and they must be compensated for taking on
» THE FEDERAL RESERVE SYSTEM The United States has a central monetary authority known as the Fed-
» The Fed and the Money Supply Financial managers and investors are interested in the supply and
» Deposit Institutions Traditionally, the United States has had several types of deposit institu-
» Investment Banking The primary market involves the distribution to investors of newly
» Interest Rates and Yields Because bonds are traded in the secondary market, the price of the bond
» The Risk Premium Market participants talk of interest rates on non-Treasury securities as
» OPTIONS An option is a contract in which the writer of the option grants the
» Buying Call Options The purchase of a call option creates a position referred to as a long call
» Buying Put Options The buying of a put option creates a financial position referred to as a
» CAP AND FLOOR AGREEMENTS There are agreements available in the financial market whereby one
» I n assessing a company’s current and future cash flows, the financial
» Depreciation for Tax Purposes For accounting purposes, a firm can select a method of depreciation
» Capital Gains We tend to use the term “capital gain” loosely to mean an increase in the
» Current assets (also referred to as circulating capital and working
» Noncurrent Assets Noncurrent assets are assets that are not current assets; that is, it is not
» Deferred Taxes Along with long-term liabilities, the analyst may encounter another
» THE INCOME STATEMENT An income statement is a summary of the revenues and expenses of a
» THE STATEMENT OF CASH FLOWS The statement of cash flows is a summary over a period of time of a
» T he notion that money has a time value is one of the most basic con-
» DETERMINING THE PRESENT VALUE Now that we understand how to compute future values, let’s work the
» Shortcuts: Annuities There are valuation problems that require us to evaluate a series of level
» THE CALCULATION OF INTEREST RATES
» T here are a number of factors that affect a stock’s price and its value to
» Dividend Valuation Model If dividends are constant forever, the value of a share of stock is the
» Returns on Common Stock As we saw in the preceding section, the value of a stock is the present
» Straight Coupon Bond Suppose you are considering investing in a straight coupon bond that:
» Returns on Bonds If you invest in a bond, you realize a return from the interest it pays (if
» Coupon Bonds The present value of a bond is its current market price, which is the dis-
» Callable Bonds Some bonds have a feature, referred to as a call feature, that allows the
» RISK Whenever you make a financing or investment decision, there is some
» Financial Risk When we refer to the cash flow risk of a security, we expand our con-
» Reinvestment Rate Risk Another type of risk is the uncertainty associated with reinvesting cash
» Interest Rate Risk Interest rate risk is the sensitivity of the change in an asset’s value to
» Currency Risk In assessing the attractiveness of an investment, we estimated future cash
» 5 (Continued) Portfolio of Investment C and Investment D
» Portfolio Size and Risk What we have seen for a portfolio with two assets can be extended to
» I n Chapters 8 through 10, we discussed and practiced techniques for
» The Cost of Debt Because Congress allows you to deduct from your taxable income the
» The Cost of Common Stock The cost of common stock is the cost of raising one more dollar of com-
» INTEGRATIVE EXAMPLE: ESTIMATING THE COST OF CAPITAL FOR DUPONT
» CAPITAL BUDGETING Because a firm must continually evaluate possible investments, capital
» Investment Cash Flows When we consider the cash flows of an investment we must also consider
» Asset Disposition At the end of the useful life of an asset, the firm may be able to sell it or
» Change in Expenses When a firm takes on a new project, the costs associated with it will
» Putting It All Together Here’s what we need to put together to calculate the change in the firm’s
» The Analysis To determine the relevant cash flows to evaluate this expansion, let’s
» The Problem The new equipment costs $300,000 and is expected to have a useful life of
» T he value of a firm today is the present value of all its future cash
» Payback Period The payback period for a project is the length of time it takes to get your
» Discounted Payback Period The discounted payback period is the time needed to pay back the origi-
» Net Present Value If offered an investment that costs $5,000 today and promises to pay
» Net Present Value Decision Rule
» Profitability Index The profitability index (PI) is the ratio of the present value of change in
» Stand-Alone versus Market Risk If we have some idea of the uncertainty associated with a project’s
» Sensitivity Analysis Estimates of cash flows are based on assumptions about the economy,
» Simulation Analysis Sensitivity analysis becomes unmanageable if we change several factors
» Options on Real Assets The valuation of stock options is rather complex, but with the assis-
» OVERVIEW OF DEBT OBLIGATIONS In a debt obligation, the borrower receives money in exchange for a
» Repayment Schedule Term loans are usually repaid in installments either monthly, quarterly,
» Interest In the United States, interest is typically paid twice a year at six month
» Debt Retirement By the maturity date of the bond, the issuer must pay off the entire par
» Rating Systems In all systems the term high grade means low default risk, or conversely,
» S uppose you buy a new car that costs $20,000 and you pay cash for it.
» Limited Liability The corporate form of doing business is attractive to owners of a busi-
» Stock Ownership We can classify a corporation according to whether its shares of stock
» Voting Rights Common shareholders are generally granted rights to
» Corporate Democracy Corporate democracy gives owners of the corporation a say in how to
» Methods of Repurchasing Stock
» Dividends Although a firm’s board of directors declares a dividend on its preferred
» Sinking Funds Because there is no legal obligation to pay the preferred dividend and
» DEBT VERSUS EQUITY The combination of debt and equity used to finance a firm’s projects is
» CAPITAL STRUCTURE AND TAXES We’ve seen how the use of debt financing increases the risk to owners;
» Interest Tax Shield An interesting element introduced into the capital structure decision is
» Unused Tax Shields The value of a tax shield depends on whether the firm can use an interest
» PUTTING IT ALL TOGETHER As a firm increases the relative use of debt in the capital structure, its
» A s we saw in Part Three, managers base decisions about investing in
» CASH MANAGEMENT Cash flows out of a firm as it pays for the goods and services it pur-
» The Baumol Model The Baumol Model is based on the Economic Order Quantity (EOQ)
» The Miller-Orr Model The Baumol Model assumes that cash is used uniformly throughout the
» The Check Clearing Process The process of receiving cash from customers involves several time-
» RECEIVABLES MANAGEMENT When a firm allows customers to pay for goods and services at a later
» Captive Finance Subsidiaries Some firms choose to form a wholly-owned subsidiary—a corporation
» The Economic Order Quantity Model The Economic Order Quantity (EOQ) model helps us determine what
» Just-in-Time Inventory The goal of the just-in-time (JIT) inventory model is to cut down on the
» Monitoring Inventory Management We can monitor inventory by looking at financial ratios in much the
» Add-on-interest Another way of stating interest is with add-on interest, where the total
» Trade Credit Trade credit is granted by a supplier to a customer purchasing goods or
» Commercial Paper Commercial paper is an unsecured promissory note with a fixed matu-
» Types of Inventory Financing There are several different types of loan arrangements that involve
» SPECIALIZED COLLATERALIZED BORROWING ARRANGEMENT FOR FINANCIAL INSTITUTIONS
» RATIOS AND THEIR CLASSIFICATION
» RETURN-ON-INVESTMENT RATIOS Return-on-investment ratios compare measures of benefits, such as earn-
» The Du Pont System The returns on investment ratios give us a “bottom line” on the perfor-
» LIQUIDITY Liquidity reflects the ability of a firm to meet its short-term obligations
» PROFITABILITY RATIOS We have seen that liquidity ratios tell us about a firm’s ability to meet its
» Using a Benchmark To interpret a firm’s financial ratios we need to compare them with the
» INTEGRATIVE EXAMPLE: FINANCIAL ANALYSIS OF WAL-MART STORES 6
» Dilutive Securities For a company having securities that are dilutive—meaning they could
» ANALYSTS’ FORECASTS There are many financial services firms offering projections on different
» PRICE-EARNINGS RATIO Many investors are interested in how the earnings are valued by the mar-
» FREE CASH FLOW Cash flows without any adjustment may be misleading because they do
» NET FREE CASH FLOW There are many variations in the calculation of cash flows that are used
» Using Cash Flow Information The analysis of cash flows provides information that can be used along
» THE GLOBAL ECONOMY Many countries export a substantial portion of the goods and services
» FOREIGN CURRENCY Doing business outside of one’s own country requires dealing with the cur-
» The Euro The European Union consists of 15 European member countries that
» Global Equity Market In 1985, Euromoney surveyed several firms that either listed stock on a
» Currency Swaps When issuing bonds in another country where the bonds are not denom-
» Currency Option Contracts In contrast to a forward or futures contract, an option gives the option
» A s an alternative to the issuance of a corporate bond, a corporation
» WHAT RATING AGENCIES LOOK AT IN RATING ASSET-BACKED SECURITIES
» Third-Party Guarantees Perhaps the easiest form of credit enhancement to understand is insur-
» EXAMPLE OF AN ACTUAL STRUCTURED FINANCE TRANSACTION
» Accounting for Capital Leases
» FEDERAL INCOME TAX REQUIREMENTS FOR TRUE LEASE TRANSACTIONS
» Direct Cash Flow from Leasing When a firm elects to lease an asset rather than borrow money to pur-
» S tructured financing is a debt obligation that is backed by the value of
» CREDIT IMPACT OBJECTIVE While the sponsor or sponsors of a project financing ideally would pre-
» A business that maximizes its owners’ wealth allocates its resources
» Budgeting In budgeting, we bring together analyses of cash flows, projected income
» Taxes and Transaction Costs The Black-Scholes option pricing model ignores taxes and transaction
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