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©2013 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 7 Stocks and Stock Valuation LEARNING OBJECTIVES (Slide 7-‐2) 1. Explain the basic characteristics of common stock.

2. Define the primary market and the secondary market. 3. Calculate the value of a stock given a history of dividend payments.

4. Explain the shortcomings of the dividend pricing models. 5. Calculate the price of preferred stock.

6. Understand the concept of efficient markets.

IN A NUTSHELL… In this chapter, the author covers the basic characteristics of stocks—both common and preferred—and describes how they are traded in primary and secondary markets. Considerable attention is given to the pricing of stocks and how expected dividends help determine a stock’s value. The methodology, advantages, and disadvantages associated with dividend discount models are covered next, followed by the formula for pricing preferred stock. Finally, the concept of efficient markets is discussed with particular emphasis on the role that information plays in the pricing of stocks.

LECTURE OUTLINE

7.1 Characteristics of Common Stock (Slides 7-‐3 to 7-‐11) Common stock, like bonds, represents a major financing vehicle for corporations and provides holders with an opportunity to share in the future cash flows of the issuer. Unlike bonds, however, holding common stock signifies ownership in the company, with no maturity date, and variable periodic income. This section covers the basic characteristics of common stock in comparison with bonds. A good grasp of this material is tantamount to understanding the valuation models that follow. 7.1 (A) Ownership: As part owners of the company, common shareholders are entitled to share in the residual profits of the company, and have a claim to all its assets and cash flow once the creditors, employees, suppliers, and taxes are paid off. Ownership via common stock also confers voting rights to the shareholders allowing them to participate in the management of the company by electing the board of directors, which ultimately selects the management team that runs the company’s day-to-day operations. 7.1 (B) Claim on Assets and Cash Flow (Residual Claim): right to share in the residual assets and cash flow of the issuer, once all the other stakeholders have been paid off.

184 Brooks n Financial Management: Core Concepts, 2e


7.1 (C) Vote (Voice in Management)

Standard voting rights: Typically, one vote per share provided to shareholders to vote in board elections and other key changes to the charter and bylaws. This standard can be altered by issuing several classes of stock. Non-voting stock, which is usually for a temporary period of time, or super voting rights, which provide the holders with multiple votes per share, increasing their influence and control over the company.

7.1 (D) No Maturity Date: Common stock is considered to have an infinite life since unlike bond-holders; shareholders do not have a promised future date when they will receive their investment back. 7.1 (E) Dividends and Their Tax Effect: Companies pay cash dividends periodically (usually every quarter) to their shareholders out of net income. Unlike coupon interest paid on bonds, dividends cannot be treated as a tax-deductible expense by the company. For the recipient, however, dividends are considered to be taxable income. More material on dividends and dividend policy is covered in Chapter 17. 7.1 (F) Authorized, Issued, and Outstanding Shares:

Authorized shares: is the maximum number of shares that the company may sell, as specified in the charter. Issued shares represent the number of shares that has already been sold by the company and are either currently available for public trading (outstanding shares) or held by the company for future uses such as rewarding employees (treasury stock).

7.1 (G) Treasury Stock represents non-dividend paying, non-voting shares that are being held by the issuing firm right from the time they were first issued or shares that have been later repurchased by the issuing firm in the market. 7.1 (H) Preemptive Rights which are privileges that allow current shareholders to buy a fixed percentage of all future issues before they are offered to the general public are provided to common shareholders so as to allow them to maintain their proportional ownership in the company.

7.2 Stock Markets (Slides 7-‐12 to 7-‐15) Stocks are traded in two types of markets; the primary or “first sale” market, where the firm first sells its stock, and the secondary or “after-sale” market, where previously issued shares are traded among investors themselves. 7.2 (A) Primary Markets: are markets where companies that want to “go public” are involved in selling their stock to investors, generally with the assistance and expertise of investment banking firms.

Initial public offering (IPO): is the first public equity issue of a firm. Prospectus: is a document that provides information regarding the issuing company and the impending sale of securities to potential buyers.

Chapter 7 n Stocks and Stock Valuation 185


Due diligence: step undertaken by the investment banker to ensure that all relevant information regarding the stock issue is being disclosed prior to the sale.

Firm commitment: is an underwriting arrangement under which the issuing company is assured of getting a fixed amount of funding since the investment banker essentially buys the entire stock issue from the company at an agreed upon price and then tries to sell the issue for a higher price.

Best efforts: is an arrangement whereby the investment banker pledges to give his or her best in trying to sell the shares in exchange for a small commission based on the selling price of the stock. 7.2 (B) Secondary Markets: How Stocks Trade. Secondary markets represent a forum where common stock can be traded among investors themselves, thereby providing investors with liquidity and variety. In the United States, there are three well-known secondary stock markets: The New York Stock Exchange (NYSE), and The American Stock Exchange (AMEX), are both physical trading locations with trading floors, while The National Association of Securities Dealers (NASD) and its trading system, the National Association of Securities Dealers Automated Quotation System (NASDAQ), also known as the over-the-counter market, is where registered dealers trade stocks via a telecommunication network. A Specialist is an elite member of a stock exchange assigned with the exclusive responsibility of maintaining an orderly market for an assigned group of stocks for which he or she keeps records of unfilled orders awaiting execution in the “limit order book.”

Ask price is the price that the dealer is willing to sell at. Bid price is the price that the dealer is willing to buy at.

Bid-ask spread is the dealer’s profit or difference between the bid price and the ask price. 7.2 (C) Bull Markets and Bear Markets: are terms used to describe stock market trends.

A Bull market is the label for a prolonged rising stock market, coined on the analogy that a bull attacks with his horns from the bottom up.

A Bear market is the label for a prolonged declining market, based on the analogy that a bear swipes with his paws from the top down.

7.3 Stock Valuation (Slides 7-‐16 to 7-‐21) Theoretically speaking, the value of a share of stock, like any financial asset, can be estimated as the present value of its expected future cash flow, which would include the cash dividends paid by the company (if any) and the future selling price of the stock, when sold to another buyer. The discount rate used would be the appropriate rate of return that should be earned given the riskiness of the company.

Example 1: Stock price with known dividends and sale price Agnes wants to purchase common stock of New Frontier Inc. and hold it for 3 years. The directors of the company just announced that they expect to pay an annual cash dividend



of $4.00 per share for the next 5 years. Agnes believes that she will be able to sell the stock for $40 at the end of three years. In order to earn 12% on this investment, how much should Agnes pay for this stock?

Method 1: Using an Equation

Price = ( )

( )11

11Future Price Dividend Stream1

n

n

rrr

⎛ ⎞−⎜ ⎟+⎜ ⎟× + ×⎜ ⎟+⎜ ⎟⎝ ⎠

Price = ( )

( )44

111 0.121$40.00 $4.000.121 0.12

⎛ ⎞−⎜ ⎟+⎜ ⎟× + ×⎜ ⎟+⎜ ⎟⎝ ⎠

Price = $40.00 × 0.635518 + $4.00 × 3.03734

Price = $25.42 + $12.149 = $37.57

Method 2: Using a financial calculator Mode: P/Y=1; C/Y = 1 Input: N I/Y PV PMT FV Key: 4 12 ? 4 40 Output -37.57

The problem with this approach is that, unlike bonds, which have a pre-determined coupon payment and maturity or par value, common stocks do not specify a fixed periodic dividend rate nor do they mature at a future date and pay a stated par value. Thus the timing and magnitude of cash flow from common stock ownership is uncertain and variable, making valuation difficult and more of an art than a science.

Since the main cash flow received from common stock is the periodic dividend, 4 variations of a dividend pricing model have been used to value common stock, each of which makes a different assumption about the dividend stream and the maturity of the stock; whether the dividends are constant or growing and whether we hold the stock forever or up to a point at which we sell it. The variations are:

1. The constant dividend model with an infinite horizon 2. The constant dividend model with a finite horizon 3. The constant growth dividend model with a finite horizon 4. The constant growth dividend model with an infinite horizon

Note: It is important to remind students that models can only estimate stock values based on projected cash flows. The market price that a stock trades for at any given time is a reflection of consensus estimates of future cash flow and the discount rate, and these consensus estimates change frequently, so by using these pricing models we



are trying to connect with the idea that when it comes to pricing stocks what matters is the timing and amount of cash flow. 7.3 (A) The Constant Dividend Model with an Infinite Horizon (Slides 7-‐22 to 7-‐23)

Under this model, it is assumed that the firm is paying the same dividend amount in perpetuity. That is, Div1 = Div2 = Div3 = Div4 = Div5 = Div6 = Div7 = Div8 = Div∞

Recall in Chapter 4 (equation 4.7), it was shown that for perpetuities, PV = PMT/r; where r the required rate and PMT is the cash flow.

Thus, for a stock that is expected to pay the same dividend forever, Price = Dividend/Required rate of return

Example 2: Quarterly dividends forever Let’s say that the Peak Growth Company is paying a quarterly dividend of $0.50 and has decided to pay the same amount forever. If Joe wants to earn an annual rate of return of 12% on this investment, how much should he offer to buy the stock at? Quarterly dividend = $0.50

Quarterly rate of return = Annual rate/4= 12%/4 = 3% PV = Quarterly dividend/Quarterly rate of return Price = 0.50/.03 = $16.67

7.3 (B) The Constant Dividend Model with a Finite Horizon (Slides 7-‐24 to 7-‐26)

Under this model, it is assumed that the investor holds the stock for a finite period of time, and then sells it off to another investor. If the dividends paid each period are expected to be constant over the investment horizon, the price can be estimated as the sum of the present value of an annuity (constant dividend) and that of a single sum (the selling price), similar to a typical non-zero coupon, corporate bond. Of course, one would have to estimate the future selling price, since that is not a given value, unlike the par value of a bond.

Example 3: Constant dividends with finite holding period Let’s say that the Peak Growth Company is paying an annual dividend of $2.00 and has decided to pay the same amount forever. If Joe wants to earn an annual rate of return of 12% on this investment, and plans to hold the stock for 5 years, with the expectation of selling it for $20 at the end of 5 years, how much should he offer to buy the stock at?

Method 1: Using TVM formulae Annual dividend = $2.00 = PMT ; Selling Price = $20 = FV Annual rate of return = 12%



PV = PV of dividend stream over 5 years + PV of Year 5 price

PV= ( )( )

111 1Dividend Stream Future Value

1

n

n

rr r

⎛ ⎞−⎜ ⎟+⎜ ⎟× + ×⎜ ⎟ +⎜ ⎟⎝ ⎠

Price = ( )( )

5

5

111 0.12 1$2.00 $200.12 1 0.12

⎛ ⎞−⎜ ⎟+⎜ ⎟× + ×⎜ ⎟ +⎜ ⎟⎝ ⎠

Price = $7.21 + $11.35 = $18.56

Method 2 Using a financial calculator Mode: P/Y=1; C/Y = 1 Input: N I/Y PV PMT FV Key: 5 12 ? 2 20 Output -18.56

7.3 (C) The Constant Growth Dividend Model with an Infinite Horizon (Slides 7-‐27 to 7-‐32)

This model, also known as the Gordon model (after its developer, Myron Gordon), estimates the value of stock based on the discounted value of an infinite stream of future dividends that grow at a constant rate, g. The formula is as follows:

( )

( )

( )

( )

( )

( )

( )( )

1 2 3Div 1 Div 1 Div 1 Div 10 0 0 0Price0 1 2 3 11 1 1

g g g g

rr r r

∞× + + + +

= + + + ∞++ + +

L

Where r = the required rate of return. With some algebraic derivation, this formula can be

simplified to… ( )

( )Div 10Price0

gr g× +

=−

And because, Div1 = Div0 × (1+g)

ð( )Div1Price0 r g

=−

ðAnd more generally

( )1Price n

nDivr g

+=−

,

Where, Divn+1 is the estimated next dividend of the stock with the given growth rate of the dividends, g, at time period n.



Note: For this model to be applicable, the required rate, r, must be greater than the growth rate, g. Otherwise, we will be dividing by zero, (if r = g) or by a negative (if r < g), both of which would lead to non-meaningful values.

Example 4: Constant growth rate, infinite horizon (with growth rate given) Let’s say that the Peak Growth Company just paid its shareholders an annual dividend of $2.00 and has announced that the dividends would grow at an annual rate of 8% forever. If investors expect to earn an annual rate of return of 12% on this investment how much would they offer to buy the stock for?

Div0 = $2.00; g=8%; r=12% Div1=Div0*(1+g) èDiv1=$2.00*(1.08)èDiv1=$2.16

( )1

0Price Divr g

=−

0$2.16Price

(.12 .08)=

−

Price0 = $54 Note: Remind students that r and g must be in decimals.

Example 5: Constant growth rate, infinite horizon (with growth rate estimated from past history)

Let’s say that you are considering an investment in the common stock of QuickFix Enterprises and are convinced that its last paid dividend of $1.25 will grow at its historical average growth rate from here on. Using the past 10 years of dividend history and a required rate of return of 14%, calculate the price of QuickFix’s common stock. QuickFix Enterprises’ Annual Dividends

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

$0.50 $0.55 $0.61 $0.67 $0.73 $0.81 $0.89 $0.98 $1.08 $1.25

First, estimate the historical average growth rate of dividends by using the following equation:

1

1nFVg

PV⎛ ⎞= −⎜ ⎟⎝ ⎠

Where FV = 2008 Dividend = $1.25

PV = 1999 Dividend = $0.50 n = number of years in between = 9



191.25 1

0.5g ⎛ ⎞= −⎜ ⎟

⎝ ⎠ = 0.10717

g = 10.72%

Next, use the constant growth, infinite horizon model to calculate price: Div0 = 2008 Dividend = $1.25; Div1= Div0*(1+g) è$ 1.25*(1.1072)è$1.384

r = 14%; g = 10.72% (as calculated above)

( )1

0Price Divr g

=−

0$1.384Price

(.14 .1072)=

−

Price0 = $42.19

7.3 (D) The Constant Growth Dividend Model with a Finite Horizon (Slides 7-‐33 to 7-‐37)

In cases where an investor expects to hold a stock, whose dividends are growing at a constant rate, for a limited number of years, the following adjusted formula can be used to value the stock:

( )( )0

0

1 1Price 11

nDiv g gr g r

⎛ ⎞× + +⎛ ⎞= × −⎜ ⎟⎜ ⎟⎜ ⎟− +⎝ ⎠⎝ ⎠+ ( )Price1

nnr+

where, g = the constant growth rate, r = the required rate, and n = the investor’s holding period. Pricen = Selling price in period n

Note: This formula would lead to the same price estimate as the Gordon model, if it is assumed that the growth rate of dividends and the required rate of return of the next owner, (after n years) remain the same.

Example 6: Constant growth, finite horizon The QuickFix Company just paid a dividend of $1.25 and analysts expect the dividend to grow at its compound average growth rate of 10.72% forever. If you plan on holding the stock for just 7 years, and you have an expected rate of return of 14%, how much would you pay for the stock? Assume that the next owner also expects to earn 14% on his or her investment.

Div0 = $1.25; g = 10.72%; r = 14%; n = 7 We can solve this in 2 ways.

Method 1: Use constant growth, finite horizon formula:



( )( )0

0

1 1Price 11

nDiv g gr g r

⎛ ⎞× + +⎛ ⎞= × −⎜ ⎟⎜ ⎟⎜ ⎟− +⎝ ⎠⎝ ⎠+

( )Price1

nnr+

Where based on the infinite period, constant growth model we solve for the selling price, Price7 as follows:

( )8

7Price Divr g

=−

= 1.25(1.1072)8/(.14-.1072) = $86.07

( )( )

7

01.25 1.1072 1.1072Price 1.14 .1072 1.14

⎛ ⎞× ⎛ ⎞= × −⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠+( )786.071.14

0Price = $42.195 *0.184829 + 34.40 = $42.19

Method 2: Since the growth rate is constant forever, and the required rates of return of both investors is the same, we can also use the Gordon model.

( )( )0

0

1Price

Div gr g× +

=−

è 0$1.25*(1.1072)Price(.14 .1072)

=−

= $42.19

7.3 (E) Non-‐constant Growth Dividends (Slides 3-‐38 to 7-‐41)

The above 4 models can only be used in cases where a firm is either expected to pay a constant dividend amount indefinitely, or is expected to have its dividends grow at a constant rate for long periods of time. In reality, the dividend growth patterns of most firms tend to be variable, making the valuation process complicated. However, if we can assume that at some point in the future, the dividend growth rate will become constant, we can use a combination of the Gordon Model and present value equations to calculate the price of the stock.

Example 7: Non-‐constant dividend pattern The Rapid Growth Company is expected to pay a dividend of $1.00 at the end of this year. Thereafter, the dividends are expected to grow at the rate of 25% per year for 2 years, and then drop to 18% for 1 year, before settling at the industry average growth rate of 10% indefinitely. If you require a return of 16% to invest in a stock of this risk level, how much would you be justified in paying for this stock?

Step 1. Calculate the annual dividends expected in Years 1-4, using the appropriate growth rates.

D1=$1.00; D2=$1.00*(1.25)=$1.25; D3=$1.25*(1.25) = $1.56;



D4=$1.56*(1.18) = $1.84; Step 2. Calculate the price at the start of the constant growth phase using the Gordon model. P4 = D4*(1+g)/(r-g) = $1.84*(1.10)/(.16-.10) = $2.02/.06 = $33.73

Step 3. Discount the annual dividends in Years 1-4 and the Price at the end of Year 4, back to Year 0 using the required rate of return as the discount rate, and add them up to solve for the current price. P0 = $1.00/(1.16) + 1.25/(1.16)2+$1.56/(1.16)3+$1.84/(1.16)4+$33.73/(1.16)4

P0 = $$0.862+0.928+$.999+$1.016+$18.63 = $22.44 Note: This uneven cash flow stream can also be discounted by using the NPV function of the financial calculator…. CF0=0;CF1=1.00;CF2=1.25;CF3=1.56;CF4=1.84+33.73;I=16;NPV=$22.44

7.4 Dividend Model Shortcomings (Slide 7-‐42) Since dividend pricing models (constant growth or constant dividend) estimate stock prices based on future cash flow such as the dividends to be received and a required rate of return, they are difficult to apply universally. Firms with fairly erratic dividend patterns, long periods of no dividends, or declining dividend trends, in particular, are not well-suited for the application of these models. In such cases, what we need is a pricing model that is more inclusive than the dividend model, one that can estimate expected returns for stocks without the need for a stable dividend history. The capital asset pricing model (CAPM), or the security market line (SML), which will be covered in Chapter 8, is one option. It can be used to estimate expected returns for companies based on their risk, the premium for taking on risk, and the reward for waiting and not on their historical dividend patterns.

7.5 Preferred Stock (Slides 7-‐43 to 7-‐44) Preferred stock is a source of capital sold by companies whereby they offer to pay a constant dividend to the holder for as long as the stock is outstanding. Typically the stock has infinite maturity, but some issues are sold with convertibility options, in which case they are converted into shares of common stock at some pre-determined ratio. Preferred stock owners have “preferred status” over common stockholders in the case of dividend payments and liquidation payouts. They can also be issued with cumulative or non-cumulative dividend payment terms. To calculate the price of preferred stock, we use the PV of a perpetuity equation, as covered in Chapter 4, with the annual dividend (dividend rate * par value) being the PMT and the required rate of return being the discount rate. Price of Preferred Stock = Annual dividend / required rate



Example 8: Pricing preferred stock The Mid-American Utility Company’s preferred stock pays an annual dividend of 8% per year on its par value of $60. If you want to earn 10% on your investment how much should you offer for this preferred stock? Annual dividend = .08*$60 = $4.80 Price = $4.80/0.10 = $48

7.6 Efficient Markets (Slides 7-‐45 to 7-‐48) An efficient market is one in which security prices are current and fair to all traders and transactions costs are minimal. There are two forms of efficiency: operational efficiency and informational efficiency.

7.6 (A) Operational Efficiency: concerns the speed and accuracy with which trades are processed and the ease with which the investing public can access the best available prices. The NYSE, with its Super Designated Order Turnaround (SuperDOT) computer system, and the NASDAQ, with its small-order execution system (SOES), can match buyers and sellers very efficiently and at the best available price. They are therefore definitely very operationally efficient.

7.6 (B) Informational Efficiency: is the speed and accuracy with which information is reflected in the available prices for trading. In an information-efficient market, securities would always trade at their fair or equilibrium value. Since information is fairly diverse, financial economists have come up with three versions of efficient markets from an information perspective: weak form, semi-strong form, and strong form. These three forms make up what is known as the efficient market hypothesis (EMH).

Weak-form efficient markets are those in which current prices reflect the price history and trading volume of the stock. Thus, charting and other technical strategies would be useless if markets are truly weak-form efficient. Semi-strong-form efficient markets are those in which current prices already reflect not only price and volume information of the stock, but all available relevant public information as well. It is therefore futile to try to exploit publicly available news or financial statement information to routinely outperform the market. Strong-form efficient markets are those in which current prices reflect the price and volume history of the stock, all publicly available information, and even all private information. It implies that all information is already embedded in the price, and hence, there is no advantage to using insider information to routinely outperform the market. Note: These three forms of the EMH are analogous to three circles, one embedded inside the other, with the inner-most circle representing the weak-form and the outer-most circle the strong form. These circles imply that if a market is strong form efficient, it would automatically be deemed to be weak and semi-strong form efficient as well.

Despite decades of testing market efficiency, the jury is still out regarding the question of whether or not markets are truly weak or semi-strong efficient. A lot of evidence points



out to markets being fairly semi-strong efficient, but there are exceptions making the results inconclusive.

Questions 1. What are three key features of common stock?

There are many features to choose from but here is a list of three key features: (1) Residual Claim, the common stock shareholder is entitled to all assets and cash

flow of the company after the liabilities have been satisfied. (2) No Maturity Date, the stock never pays out a principal at maturity and is

considered permanent financing. (3) Vote, shares allow owners to vote on activities, charter changes, board members,

etc. 2. What are the differences between authorized, issued, and outstanding shares?

Authorized shares are the number of shares a company can sell and is set by the charter of the company. Issued shares are the authorized shares that have been sold or distributed and are available for trading. Not all issued shares are available for public trading. The shares in the “public domain” for trading are the outstanding shares. Treasury stock, for example, represents stock that although issued are being held by the company and therefore not available for trading.

3. What is the role of the investment banker in the primary sale of common stock? The investment banker is the partner to the company in the sale of common stock in the primary market. The investment banker serves as the distributor of information to potential buyers, the expert to the company on pricing and timing of the sale, and the marketer of the shares.

4. What are the potential repercussions if the investment banker does not perform the due diligence task? Failure to perform due diligence leaves the investment banker liable for potential lawsuits by those who bought the newly issued shares.

5. What is the function of a specialist in the secondary market? The specialist is assigned the job of keeping an orderly market for the trading of an assigned list of stocks in the secondary market.

6. What is a bid price, and what is an ask price? The bid price is the price offered by a willing buyer for the stock. The ask price is the price requested by a willing seller of the stock.

7. What is the difference between preferred stock and common stock? Preferred stock is a form of financing where the owner receives a pre-set dividend based on the par value of the preferred stock. Common stock’s dividends are set by the board and will change over time. Preferred stock must receive its dividends prior to dividends sent to common stock holders. Hence, preferred stock represents a preferential claim on dividends.

8. How is operational efficiency different from informational efficiency?



Operational efficiency deals with the speed and accuracy of processing a buy or sell order. Information efficiency is how quickly and accurately information is reflected in the current price of a stock.

9. What are SuperDOT and SOES, and what do they do? SuperDot and SOES are computer assisted trading programs that help process trades at the NYSE and NASDAQ respectively.

10. What does the semi-strong form of efficient markets require? Semi-strong form of efficient markets requires that an investor cannot make excess returns on a consistent basis, by using past or current publicly available information.

Prepping for Exams 1. b.

2. b. 3. d.

4. a. 5. d.

6. a. 7. a.

8. d. 9. c.

10. c.

Problems 1. Anderson Motors, Inc. has just set the company dividend policy at $0.50 per year.

The company plans on being in business forever. What is the price of this stock if

a. an investor wants a 5% return? b. an investor wants an 8% return? c. an investor wants a 10% return? d. an investor wants a 13% return? e. an investor wants a 20% return?

ANSWER Use the constant dividend infinite dividend stream model:

Price = Dividend / r a. Price = $0.50 / 0.05 = $10.00 b. Price = $0.50 / 0.08 = $6.25 c. Price = $0.50 / 0.10 = $5.00 d. Price = $0.50 / 0.13 = $3.85



e. Price = $0.50 / 0.20 = $2.50

2. Diettreich Electronics wants its shareholders to earn a 15% return on their investment in the company. At what price would the stock need to be priced today if Diettreich Electronics had a

a. $0.25 constant annual dividend forever? b. $1.00 constant annual dividend forever? c. $1.75 constant annual dividend forever? d. $2.50 constant annual dividend forever?

ANSWER Use the constant dividend infinite dividend stream model:

Price = Dividend / r

a. Price = $0.25 / 0.15 = $1.67 b. Price = $1.00 / 0.15 = $6.67 c. Price = $1.75 / 0.15 = $11.67 d. Price = $2.50 / 0.15 = $16.67

3. Singing Fish Fine Foods has a current annual cash dividend policy of $2.25. The price of the stock is set to yield a 12% return. What is the price of this stock if the dividend will be paid

a. for 10 years? b. for 15 years? c. for 40 years? d. for 60 years? e. for 100 years? f. forever?

ANSWER Use the finite constant dividend model except with f (use infinite constant dividend model) Price = Dividend × (1 – 1/(1+r)n) / r

a. Price = $2.25 × (1 – 1/(1.12)10 / 0.12 = $2.25 × 5.6502 = $12.71 b. Price = $2.25 × (1 – 1/(1.12)15 / 0.12 = $2.25 × 6.8109 = $15.32 c. Price = $2.25 × (1 – 1/(1.12)40 / 0.12 = $2.25 × 8.2438 = $18.54 d. Price = $2.25 × (1 – 1/(1.12)60 / 0.12 = $2.25 × 8.3240 = $18.73 e. Price = $2.25 × (1 – 1/(1.12)100 / 0.12 = $2.25 × 8.3332 = $18.75 f. Price = $2.25 / 0.12 = $18.75



4. Pfender Guitars has a current annual cash dividend policy of $4.00. The price of the stock is set to yield an 8% return. What is the price of this stock if the dividend will be paid

a. for 10 years and then a liquidating or final dividend of $25.00? b. for 15 years and then a liquidating or final dividend of $25.00? c. for 40 years and then a liquidating or final dividend of $25.00? d. for 60 years and then a liquidating or final dividend of $25.00? e. for 100 years and then a liquidating or final dividend of $25.00? f. forever with no liquidating dividend?

ANSWER Use the finite constant dividend model liquidating dividend except with f (use infinite constant dividend model) Price = Dividend × (1 – 1/(1 + r)n) / r + Liquidating Dividend × (1/(1 + r)n)

a. Price = $4.00 × (1 – 1/(1.08)10 / 0.08 + $25.00 × 1/1.0810

= $4.00 × 6.7101 + $25 × 0.4632 = $26.84 + $11.58 = $38.42 b. Price = $4.00 × (1 – 1/(1.08)15 / 0.08 + $25.00 × 1/1.0815

= $4.00 × 8.5595 + $25 × 0.3152 = $34.24 + $7.88 = $42.12 c. Price = $4.00 × (1 – 1/(1.08)40 / 0.08 + $25.00 × 1/1.0840

= $4.00 × 11.9246 + $25 × 0.0460 = $47.70 + $1.15 = $48.85 d. Price = $4.00 × (1 – 1/(1.08)60 / 0.08 + $25.00 × 1/1.0860

= $4.00 × 12.3766 + $25 × 0.0099 = $49.51 + $0.24 = $49.75 e. Price = $4.00 × (1 – 1/(1.08)100 / 0.08 + $25.00 × 1/1.08100 = $4.00 × 12.4943 + $25 × 0.0005 = $49.98 + $0.01 = $49.99 f. Price = $4.00 / 0.08 = $50.00

5. King Waterbeds has an annual cash dividend policy that raises the dividend each year by 4%. Last year’s dividend was $0.40 per share. What is the price of this stock if


ANSWER Use the constant growth dividend model with an infinite dividend stream:

Price = Last Dividend × (1 + g) / (r – g)

a. Price = $0.40 × (1.04) / (0.05 – 0.04) = $0.4160 / 0.01 = $41.60 b. Price = $0.40 × (1.04) / (0.08 – 0.04) = $0.4160 / 0.04 = $10.40 c. Price = $0.40 × (1.04) / (0.10 – 0.04) = $0.4160 / 0.06 = $6.93 d. Price = $0.40 × (1.04) / (0.13 – 0.04) = $0.4160 / 0.09 = $4.62 e. Price = $0.40 × (1.04) / (0.20 – 0.04) = $0.4160 / 0.16 = $2.60



6. Seitz Glassware is trying to determine its growth rate for its an annual cash dividend. Last year’s dividend was $0.25 per share. The stock’s target return rate is 10%. What is the stock’s price if

a. the annual growth rate is 1%? b. the annual growth rate is 3%? c. the annual growth rate is 5%? d. the annual growth rate is 7%? e. the annual growth rate is 9%?

ANSWER Use the constant growth dividend model with an infinite dividend stream:

Price = Last Dividend × (1 + g) / (r – g) a. Price = $0.25 × (1.01) / (0.10 – 0.01) = $0.2525 / 0.09 = $2.81 b. Price = $0.25 × (1.03) / (0.10 – 0.03) = $0.2575 / 0.07 = $3.68 c. Price = $0.25 × (1.05) / (0.10 – 0.05) = $0.2625 / 0.05 = $5.25 d. Price = $0.25 × (1.07) / (0.10 – 0.07) = $0.2675 / 0.03 = $8.92 e. Price = $0.25 × (1.09) / (0.10 – 0.09) = $0.2725 / 0.01 = $27.25

7. Miles Hardware has an annual cash dividend policy that raises the dividend each year by 3%. Last year’s dividend was $1.00 per share. Investors want a 15% return on this stock. What is the stock’s price if

a. the company will be in business for 5 years and not have a liquidating dividend? b. the company will be in business for 15 years and not have a liquidating dividend? c. the company will be in business for 25 years and not have a liquidating dividend? d. the company will be in business for 35 years and not have a liquidating dividend? e. the company will be in business for 75 years and not have a liquidating dividend? f. forever?

ANSWER Use the constant growth dividend model with a finite dividend stream:

Price = Last Dividend × (1 + g) / (r – g) × [1 – ((1+g) / (1+r))n] a. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))5] = $1.03 / 0.12 × [1 - 0.5764] = $8.58 × [0.4236] = $3.64 b. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))15] = $1.03 / 0.12 × [1 - 0.1915] = $8.58 × [0.8085] = $6.94 c. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))25] = $1.03 / 0.12 × [1 - 0.0636] = $8.58 × [0.9364] = $8.03 d. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))35] = $1.03 / 0.12 × [1 - 0.0211] = $8.58 × [0.9789] = $8.40



e. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))75] = $1.03 / 0.12 × [1 - 0.0003] = $8.58 × [0.9997] = $8.58 f. Price = $1.00 × (1.03) / (0.15 – 0.03) = $1.03 / 0.12 = $8.58

8. Sia Dance Studios has an annual cash dividend policy that raises the dividend each year by 2%. Last year’s dividend was $3.00 per share. The company will be in business for forty years with no liquidating dividend. What is the price of this stock if


ANSWER Use the constant growth dividend model with a finite dividend stream:

Price = Last Dividend × (1 + g) / (r – g) × [1 – ((1+g) / (1+r))n]

a. Price = $3.00 × (1.02) / (0.09 – 0.02) × [1 – ((1.02) / (1.09))40] = $3.06 / 0.07 × [1 - 0.0703] = $43.71 × [0.9297] = $40.64 b. Price = $3.00 × (1.02) / (0.11 – 0.02) × [1 – ((1.02) / (1.11))40] = $3.06 / 0.09 × [1 - 0.0340] = $34.00 × [0.9660] = $32.85 c. Price = $3.00 × (1.02) / (0.13 – 0.02) × [1 – ((1.02) / (1.13))40] = $3.06 / 0.11 × [1 - 0.0166] = $27.82 × [0.9834] = $27.35 d. Price = $3.00 × (1.02) / (0.15 – 0.02) × [1 – ((1.02) / (1.15))40] = $3.06 / 0.13 × [1 - 0.0082] = $23.54 × [0.9918] = $23.35 e. Price = $3.00 × (1.02) / (0.17 – 0.02) × [1 – ((1.02) / (1.17))40] = $3.06 / 0.15 × [1 - 0.0041] = $20.40 × [0.9959] = $20.32

9. Fey Fashions expects the following dividend pattern over the next seven years:

Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7

$1.00 $1.10 $1.21 $1.33 $1.46 $1.61 $1.77

Then the company will have a constant dividend of $2.00 forever. What is the price of this stock today if an investor wants to earn

a. 15%? b. 20%?

ANSWER There are two dividend patterns here; the first is a constant growth pattern for the next seven years and then a constant dividend forever. Solve each part separately and then add the two parts.



Part one: Constant growth for seven years, first find growth rate and note that there are six changes in the dividend stream over the seven years. g = ($1.77 / $1.00)1/6 – 1 = 1.771/6 – 1 = 10% Next, use the finite dividend growth model Price = Dividend × (1 + g) / (r – g) × [1 – ((1+g) / (1+r))n] Part two: Use the constant dividend (infinite period) model and then discount the price at period 7 back to the present Price7 = Dividend8 / r Price0 = Price7 / (1 + r)7

a. Part One Price = $1.00 (1.10) / (0.15 – 0.10) × [ 1 – (1.10/1.15)7] Part One Price = $22.00 × 0.2674 = $5.88 Part Two Price = ($2.00 / 0.15) / (1.15)7 = $13.33 / 2.66 = $5.01 Price = $5.88 + $5.01 = $10.89 b. Part One Price = $1.00 (1.10) / (0.20 – 0.10) × [ 1 – (1.10/1.20)7] Part One Price = $11.00 × 0.4561 = $5.02 Part Two Price = ($2.00 / 0.20) / (1.20)7 = $10.00 / 3.5832 = $2.79

Price = $5.02 + $2.79 = $7.81

10. Staton-Smith Software is a new up-start company and will not pay dividends for the first five years of operation. It will then institute an annual cash dividend policy of $2.50 with a constant growth rate of 5% with the first dividend at the end of year six. The company will be in business for 25 years total. What is the price of this stock if an investor wants a. a 10% return? b. a 15% return? c. a 20% return? d. a 40% return?

ANSWER Calculate the price at the beginning of the sixth year (end of the fifth year) with the finite constant growth dividend model and then discount the price by five years to the present value.

a. Price5 = $2.50 / (0.10 – 0.05) × [1 – (1.05/1.10)20] Price5 = $50.00 × 0.6056 = $30.28 Price0 = $30.28 / 1.105 = $30.28 / 1.6105 = $18.80 b. Price5 = $2.50 / (0.15 – 0.05) × [1 – (1.05/1.15)20] Price5 = $25.00 × 0.8379 = $20.95 Price0 = $20.95 / 1.155 = $20.95 / 2.0114 = $10.41 c. Price5 = $2.50 / (0.20 – 0.05) × [1 – (1.05/1.20)20] Price5 = $16.67 × 0.9308 = $15.51 Price0 = $15.51 / 1.205 = $15.51 / 2.4883 = $6.23 d. Price5 = $2.50 / (0.40 – 0.05) × [1 – (1.05/1.40)20]



Price5 = $7.14 × 0.9968 = $7.12 Price0 = $7.12 / 1.405 = $7.12 / 5.3782 = $1.32

11. Fenway Athletic Club plans to offer its members preferred stock with a par value of $100 and a 6% annual dividend rate. If a member wants the following returns, what price should he or she be willing to pay?

a. Theo wants a 10% return. b. Jonathan wants a 12% return c. Josh wants a 15% return d. Terry wants an 18% return

ANSWER Use the constant dividend model with infinite horizon èPrice = Dividend / r

a. Theo’s Price = $100 × 0.06 / 0.10 = $60.00 b. Jonathan’s Price = $100 × 0.06 / 0.12 = $50.00 c. Josh’s Price = $100 × 0.06 / 0.15 = $40.00

d. Terry’s Price = $100 × 0.06 / 0.18 = $33.33

12. Yankee Athletic Club has preferred stock with a par value of $50 and an annual 6% cumulative dividend. Given the following market prices for the preferred stock, what is each investor seeking for his or her return?

a. Alex is willing to pay $40.00 b. Derek is willing to pay $30.00 c. Mark is willing to pay $20.00 d. Johnny is willing to pay $15.00

ANSWER Use the constant dividend formula rearranged for the return, r = Dividend/ Price where the dividend is $50.00 × 0.06 = $3.00

a. Alex’s return = $3.00 / $40.00 = 0.075 or 7.5% b. Derek’s return = $3.00 / $30.00 = 0.10 or 10% c. Mark’s return = $3.00 / $20.00 = 0.15 or 15% d. Johnny’s return = $3.00 / $15.00 = 0.20 or 20%

13. Villalpondo Winery wants to raise $10 million from the sale of preferred stock. If the winery wants to sell 1 million shares of preferred stock, what annual dividend will they have to promise if investors demand

a. a 12% return? b. an 18% return?



c. an 8% return? d. a 6% return? e. a 9% return? f. a 7% return?

ANSWER Use the constant dividend formula rearranged for the dividend,

Dividend = r × Price

And the price of the shares is $10,000,000 / 1,000,000 = $10.00 per share a. A 12% return means: Dividend = 12% × $10.00 = $1.20 per year b. A 18% return means: Dividend = 18% × $10.00 = $1.80 per year c. A 8% return means: Dividend = 8% × $10.00 = $0.80 per year d. A 6% return means: Dividend = 6% × $10.00 = $0.60 per year e. A 9% return means: Dividend = 9% × $10.00 = $0.90 per year f. A 7% return means: Dividend = 7% × $10.00 = $0.70 per year

14. Find the annual growth rate of the dividends for each of the firms listed in the table below.

Dividend Payment per Year

Firm 1999 2000 2001 2002 2003 2004 Loewen $1.00 $1.05 $1.10 $1.16 $1.22 $1.28

Morse $1.00 $0.90 $0.81 $0.73 $0.66 $0.59 Huddleston $1.00 $1.00 $1.00 $2.00 $2.00 $2.00

Meyer $0.00 $0.00 $0.25 $0.50 $0.75 $1.00

ANSWER Either find the change each year and then average the change or g = (Dividend 2004 / Dividend 1999)1/5 – 1

Loewen g = ($1.28 / $1.00)0.20 – 1 = 0.0506 = 5.06% Morse g = ($0.59 / $1.00)0.20 – 1 = -0.10 = negative 10%

Huddleston g = ($2.00 / $1.00)0.20 – 1 = 0.1487 = 14.87% Meyer g = can only measure for the years 2001 to 2004, ($1.00 / $0.25)1/3 – 1 = 0.5874 = 58.74%

15. Using Yahoo! Finance (http://finance.yahoo.com/) and ticker symbol PEP, find PepsiCo’s historical dividend payment and current price. Historical dividends are



available in the historical price section. Use these payments to find the annual dividend growth rate. (If you have a quarterly pattern be sure to annualize this quarterly growth rate.) Now, find the required rate of return for this stock, assuming that the future dividend growth rate will remain the same and the company has an infinite horizon. Does this return seem reasonable for PepsiCo?

ANSWER: (Note that we used 1998-‐2010 annual dividends for PepsiCo) For this problem a thirteen-year dividend history is used and the following are the prior thirteen year dividends adjusted for stock splits with a price on January 3, 2011 of $65.75: 1998 $0.515 1999 $0.535 2000 $0.555 2001 $0.575 2002 $0.595 2003 $0.63 2004 $0.85 2005 $1.01 2006 $1.16 2007 $1.425 2008 $1.65 2009 $1.775 2010 $1.89 To find the average growth rate for the thirteen years with twelve changes we have:

($1.89 / $0.515)1/12 -1 = 0.1144 or 11.44% Now by using the constant growth formula and the current price of $65.75 we solve for r:

Price = Last Dividend × (1 + g) / (r – g) $65.75 = $1.89 × (1 + 0.1144) / (r – 0.1144) $65.75 = $2.11 / (r – 0.1144)

r – 0.1144 = $2.11 / $65.75 r = 0.03209 + 0.1144 = 0.1464 or 14.64% This looks like a reasonable required return for a stock.

16. Using Yahoo! Finance (http://finance.yahoo.com/) and ticker symbol HPQ, find Hewlett-Packard’s recent dividend payments and current price. Historical dividends are available in the historical price section. Use these payments to find the annual dividend growth rate. (If you have a quarterly pattern be sure to annualize this quarterly growth rate.) Now, find the required rate of return for this stock assuming that the future dividend growth rate will remain the same and the company has an infinite horizon. Does this return seem reasonable for Hewlett-Packard?



ANSWER For this problem Hewlett-Packard has had a constant dividend for the past ten years of $0.32 per year and with a current price of $42.74 as of January 3, 2011 we can estimate the required return from the constant dividend model which is same as the constant growth model with g equal to zero:

Price = Last Dividend / (r) $42.74 = $0.32 / r r = $0.32 / $42.74

r = 0.007487or 0.7487% This looks like an unreasonably low required return for a stock.

17. Using Yahoo! Finance, update the dividends for Coca-Cola for the last ten years. Find both the arithmetic growth rate and the geometric growth rate of the dividends.

ANSWER

Year Dividend Change (percent) 2010 $1.76 $1.76-$1.64 = $0.12 (.12/1.64 = 7.32% 2009 $1.64 $1.64 – $1.52= $0.12 ($0.12/1.52 = 7.89%) 2008 $1.52 $1.52 – $1.36 = $0.16 ($0.16/$1.36 = 11.76%) 2007 $1.36 $1.36 – $1.24 = $0.12 ($0.12 / $1.24 = 9.68%) 2006 $1.24 $1.24 – $1.12 = $0.12 ($0.12 / $1.12 = 10.71%) 2005 $1.12 $1.12 – $1.00 = $0.12 ($0.12 / $1.00 = 12.00%) 2004 $1.00 $1.00 – $0.92 = $0.08 ($0.12 / $0.92 = 8.70%) 2003 $0.92 $0.92 – $0.80 = $0.12 ($0.12 / $0.80 = 15.00%) 2002 $0.80 $0.80 – $0.72 = $0.08 ($0.12 / $0.72 = 11.11%) 2001 $0.72 Average Change 10.46%

Geometric growth rate = ($1.76 / $0.72)1/9 – 1 = 10.44%

P/Y = 1, C/Y = 1 INPUT 9 ? -0.72 0 1.76 KEYS N I/Y PV PMT FV COMPUTE 10.44

18. Using Yahoo! Finance, update the dividends for Johnson & Johnson for the last ten years. Find both the arithmetic growth rate and the geometric growth rate of the dividends.

ANSWER

Year Dividend Change (percent)



2010 $2.11 $2.11 – $1.90 = $0.21 ($0.21/$1.90 = 11.05%) 2009 $1.90 $1.90 – $1.795 = $0.105 ($0.105/$1.795 = 5.85%) 2008 $1.795 $1.795 – $1.62 = $0.175 ($0.175/$1.62 = 10.80%) 2007 $1.62 $1.62 – $1.455 = $0.165 ($0.165 / $1.455 =

11.34%) 2006 $1.455 $1.455 – $1.275 = $0.18 ($0.18 / $1.275 = 14.12%) 2005 $1.275 $1.275 – $1.095 = $0.18 ($0.18 / $1.095 = 16.44%) 2004 $1.095 $1.095 – $0.925 = $0.17 ($0.17 / $0.925 = 18.38%) 2003 $0.925 $0.925 – $0.80 = $0.125 ($0.125 / $0.80 = 15.63%) 2002 $0.80 $0.80 – $0.70 = $0.10 ($0.10 / $0.70 = 14.29%) 2001 $0.70 Average Change 13.1% (Arithmetic Growth Rate)

Geometric growth rate = ($2.11 / $0.70)1/9 – 1 = 13.04%

P/Y = 1, C/Y = 1 INPUT 9 ? -0.55 0 1.795 KEYS N I/Y PV PMT FV COMPUTE 13.04

19. Using Yahoo! Finance, update the dividends of Wal-Mart for the last ten years. Find the arithmetic growth rate and the geometric growth rate of the dividends.

ANSWER

Year Dividend Change (percent) 2010 $1.212 $1.212 – $1.092 = $0.12 ($0.12/$1.092=10.99%) 2009 $1.092 $1.092 – $0.952=$0.14 ($0.14/$0.952= 14.71%) 2008 $0.952 $0.952 – $0.88 = $0.072 ($0.072/$0.88 = 8.18%) 2007 $0.88 $0.88 – $0.672 = $0.208 ($0.208 / $0.672 = 30.95%) 2006 $0.672 $0.672 – $0.60 = $0.072 ($0.072 / $0.60 = 12.00%) 2005 $0.60 $0.60 – $0.52 = $0.08 ($0.08 / $0.52 = 15.38%) 2004 $0.52 $0.52 – $0.36 = $0.16 ($0.16 / $0.36 = 44.44%) 2003 $0.36 $0.36 – $0.30 = $0.06 ($0.06 / $0.30 = 20.00%) 2002 $0.30 $0.30 – $0.28 = $0.02 ($0.02 / $0.28 = 7.14%) 2001 $0.28 Average Change 19%

Geometric growth rate = ($1.212 / $0.28)1/9 – 1 = 17.68% P/Y = 1, C/Y = 1

INPUT 9 ? -0.28 0 1.212



KEYS N I/Y PV PMT FV COMPUTE 17.68

20. Using Yahoo! Finance, update the dividends of Intel for only the last six years. Find the arithmetic growth rate and the geometric growth rate of the dividends.

ANSWER

Year Dividend Change (percent) 2010 $0.632 $0.632 – $0.56 = $0.072 ($0.072/$0.56 = 12.86%) 2009 $0.56 $0.56 – $0.546 = $0.014 ($0.014/$0.546 = 2.56%) 2008 $0.546 $0.546 – $0.452 = $0.094 ($0.094/$0.452 = 20.80%) 2007 $0.452 $0.452 – $0.40 = $0.052 ($0.052 / $0.40 = 13.00%) 2006 $0.40 $0.40 – $0.32 = $0.08 ($0.08 / $0.32 = 25.00%) 2005 $0.32

Arithmetic growth rate = (12.86%+2.56%+20.8%+13%+25%)/5 = 14.84%

Geometric growth rate = ($0.632 / $0.32)1/5 – 1 =14.58% P/Y = 1, C/Y = 1

INPUT 5 ? -0.32 0 0.632 KEYS N I/Y PV PMT FV COMPUTE 14.58

21. Using the answer to Problem 17 on the Coca-Cola growth rates and the current trading price, determine the current required rate of return for the company.

ANSWER Coca-Cola price was at $65.22 as of January 3, 2011

Coca-Cola’s ( )$1.76 1 0.10440.1044

$65.22r

× +⎛ ⎞= +⎜ ⎟⎝ ⎠

= 0.1342 or 13.42%

22. Using the answer to Problem 18 on the Johnson & Johnson growth rates and the current trading price, determine the current required rate of return for the company.

ANSWER Johnson and Johnson’s price was at $62.82

Johnson and Johnson’s ( )$2.11 1 0.13.040.1304

$62.82r

× +⎛ ⎞= +⎜ ⎟⎝ ⎠

=.1684 or 16.84%



23. Using the answer to Problem 19 on the Wal-Mart growth rates and the current trading price, determine the current required rate of return for the company.

ANSWER Wal-Mart’s price was at $54.56

Wal-Mart’s ( )$01.212 1 0.17680.1768

$54.56r

× +⎛ ⎞= +⎜ ⎟⎝ ⎠

= .2029 or 20.29%

24. Using the answer to Problem 20 on Intel growth rates and the current trading price, determine the current required rate of return for the company.

ANSWER Intel’s price was at $20.85 on January 3, 2011

Intel’s ( )$0.632 1 0.14580.1458

$20.85r

× +⎛ ⎞= +⎜ ⎟⎝ ⎠

= 0.1805 or 18.05%

25. Given the growth rates for Coca-Cola, Johnson & Johnson, Wal-Mart, and Intel from the dividend history in Problems 21 through 24, what price would you predict for each stock if they all had a required return of 18%? Why are Wal-Mart and Intel prices troublesome?

ANSWER

Coca-Cola, ( )$1.76 1 0.1044$25.71

0.18 0.1044P

× +⎛ ⎞= =⎜ ⎟−⎝ ⎠

Johnson and Johnson, ( )$2.11 1 0.1304$48.09

0.18 0.1304P

× +⎛ ⎞= =⎜ ⎟−⎝ ⎠

Walmart, ( )$1.212 1 0.1768$445.71

0.18 0.1768P

× +⎛ ⎞= =⎜ ⎟−⎝ ⎠

Intel, ( )$0.632 1 0.1458$21.18

0.18 0.1458P

× +⎛ ⎞= =⎜ ⎟−⎝ ⎠

Wal-Mart’s predicted price is unrealistically high since the required rate is so close to the growth rate. Intel’s price on the other hand seems okay.

26. Assume that Exxon-Mobil’s price dropped to $30 overnight. Given the dividend growth rate of Exxon-Mobil of 5.07% and the last annual dividend of $1.28, what is



the implied required rate of return necessary to justify the new lower market price of $30.00?

ANSWER

Exxon-Mobil’s ( )$1.28 1 0.05070.0507

$30.00r

× +⎛ ⎞= +⎜ ⎟⎝ ⎠

= 0.0955 or 9.55%

27. Peterson Packaging Incorporated does not currently pay dividends. The company will start with a $0.50 dividend at the end of year three and grow it by 10% for each of the next six years until it nearly reaches $1.00. After six years of growth, it will fix its dividend at $1.00 forever. If you want a 15% return on this stock, what should you pay today, given this future dividend stream?

ANSWER The first step is to look at the timing and amount of the cash flow that you will receive as a shareholder.

Expected Dividend Stream of Peterson Packaging T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 … T ∞

--- $0.00 $0.00 $0.50 $0.55 $0.61 $0.67 $0.73 $0.81 $0.89 $1.00 …$1.00 We notice that there are three distinct dividend patterns -- a period of no dividends, T1 to T2; a period of constant growth dividends, T3 to T9; and a period of constant dividends, T10 to T∞. To price this stock we need to determine the present value of each pattern.

The first pattern, no dividends, is rather easy to value. It is zero. The second pattern is a constant growth dividend stream with a finite horizon.

But we need to realize that this pattern does not start until the end of year three. Therefore, when we apply the dividend growth model we will be getting a price for the end of year two and we will receive seven dividends:

( )3

21Price 11

ngDivr g r

⎛ ⎞+⎛ ⎞= × −⎜ ⎟⎜ ⎟⎜ ⎟− +⎝ ⎠⎝ ⎠

( )

7

2$0.50 1 0.10Price 1

0.15 0.10 1 0.15⎛ ⎞+⎛ ⎞= × −⎜ ⎟⎜ ⎟⎜ ⎟− +⎝ ⎠⎝ ⎠

Price2 = $10.00 × (0.2674) = $2.674 The price at the end of period 2 is a future value. Now we must discount this future value at 15% for its present value:



( )1 nFVPVr

=+

Price0 = ( )2$2.674 $2.674 $2.02

1.32251 0.15PV = = =

+

The final dividend pattern is a perpetuity and we can use equation 7.1 here:

109Price Dividend

r=

9$1.00Price $6.670.15

= =

And again, we must discount this future value at 15% for its present value:

( )0 9$6.67Price $1.901 0.15

= =+

Finally, adding the three pieces we get:

Price of Stock = $0.00 + $2.02 + $1.90 = $3.92 Although we now have a way of pricing stocks through discounting dividends, we must realize that dividends are not a promised future cash flow. In fact, firms can increase, reduce, or even suspend cash dividends. And even though past dividends may be a good predictor of future dividends, the timing and amount of dividends can and do vary across time for a company. The dividend models are really expected dividend models, and if we were to write these models correctly we would use an expectations operator with the dividends, E(Div0) and E(Div1) instead of Div0 and Div1.



Solutions to Advanced Problems for Spreadsheet Applications 1. Dividend history and dividend growth rates

Date Dividend Quarterly Change Annual Dividend Annual Change2/7/1990 $0.17500 5/1/1990 $0.17500 0.00%

7/31/1990 $0.19375 10.71%11/6/1990 $0.19375 0.00% $0.737502/5/1991 $0.19375 0.00%

4/30/1991 $0.20625 6.45%8/6/1991 $0.20625 0.00%

11/5/1991 $0.20625 0.00% $0.81250 10.17%2/4/1992 $0.20625 0.00%5/5/1992 $0.20625 0.00%8/4/1992 $0.20625 0.00%

11/3/1992 $0.20625 0.00% $0.82500 1.54%5/15/2008 $0.65000 12.07%8/15/2008 $0.65000 0.00%

11/14/2008 $0.65000 0.00% $2.53000 11.95%2/12/2009 $0.65000 0.00%5/15/2009 $0.65000 0.00%8/17/2009 $0.68000 4.62%

11/16/2009 $0.68000 0.00% $2.66000 5.14%

Quarterly AVERAGE CHANGE 1.80% 7.07%

Quarterly Geometric Change 1.73% 6.98%

Simple Average Qtr to Yr 7.19%

Simple Geo QTR to YR 6.93%

Compounding Average QTR to YR 7.39%

Compounding Geo QTR to YR 7.11%



2. Changing stock price year by year. Year Required Return Most recent Dividend Dividend Growth Rate Stock Price

1 11.20% 2.80$ 4.00% 40.44$ 2 12.15% 2.80$ 4.00% 37.16$ 3 10.98% 2.80$ 4.00% 45.12$ 4 11.45% 2.80$ 4.00% 43.97$ 5 12.06% 2.80$ 4.00% 42.27$ 6 12.98% 2.80$ 4.00% 39.45$ 7 11.55% 2.80$ 4.00% 48.80$ 8 10.83% 2.80$ 4.00% 56.11$ 9 10.22% 2.80$ 4.00% 64.07$ 10 11.73% 2.80$ 4.00% 53.62$ 11 11.98% 2.80$ 4.00% 54.02$ 12 11.42% 2.80$ 4.00% 60.42$ 13 10.96% 2.80$ 4.00% 66.99$ 14 12.15% 2.80$ 4.00% 59.49$ 15 11.55% 2.80$ 4.00% 66.79$

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Solutions to Mini-‐Case Lawrence’s Legacy A This case requires students to think in a critical and applied way about the nature of common stock investments, primary and secondary markets, the implications of EMH, stock valuation models, and the limitations of stock valuation models.

1. Why do you think Lawrence specified to invest money in stocks rather than bonds or certificates of deposit? This question can be the stimulus for a broad ranging discussion of the nature of common stock.



Lawrence wants the memorial trust fund to provide meaningful grants in perpetuity. Common stock represents ownership of a percentage of a corporation. The nominal value of corporate assets, sales, earnings and dividends should all be able to increase with inflation. If the corporation is successful, the original value of an investment can increase many times over. Because stock represents a percentage of the company’s value, over the long run, stock prices and dividends can also be expected at least to keep pace with inflation and sometimes much more. Bonds and CD’s on the other hand, have fixed maturity values, fixed interest payments, and do not offer the growth opportunities available with stock. Trust funds that are invested too conservatively are safer in the short-term, but often provide income that becomes insignificant over time. Perhaps some students will have received “scholarships” of $50 or $100 at their high school graduations. When these scholarships were first created, those would have been meaningful amounts, but have become somewhat insignificant.

2. How will the trust obtain the cash to make the grants if the dividends do not amount to 5% of the portfolio’s value? Unless stocks are specifically purchased for high dividend yields rather than growth, it is unlikely that dividend income will be sufficient for the grants. Some stock will need to be sold from time to time in order to generate cash. You could say that we will be making our own dividends.

3. What is the difference between common stock and preferred stock? Preferred stock usually pays a fixed dividend. As a result, the price moves more closely with interest rates than with the growth and profitability of the company. In other words, preferred stock acts more like a bond than like common stock, and is probably not the kind of thing Mr. Lawrence had in mind. Common stock represents ownership of a percentage of the company, so its value goes up when the company prospers, and diminishes when the company earnings or expected future earnings go down. Unlike preferred stock, common stock has voting rights. Because of the voting rights, even an inefficient, poorly managed company can be valuable because another company or group of investors may find it cheaper to buy a controlling interest in the company and change management than to duplicate the company’s assets.

4. How do we know if we are paying a fair price for the stock that we purchase? There is no way to know in advance if a stock will turn out to be a good or bad investment, but it is reassuring to know that for larger companies, millions of shares are bought and sold every day. The price we pay will be very close to the price paid by the buyer just before or after us. Many stock purchases and sales take place between large institutional investors managed by people with advanced degrees in finance or economics, and whose only job is to know everything there is to know about the companies they invest in. If these investors agree to pay a particular price, and equally sophisticated investors agree to sell at the same price, then the price must reflect all the information it is legally possible to have. These conditions describe what is known as the semi-strong form of the efficient market hypothesis, which implies that no investor has an unfair advantage over any other investor.

5. For what are we actually paying when we buy a share of stock?



Kraska intends to use the following examples to answer this question.

a. ABC Inc. preferred stock pays a constant dividend of $5.00 per year. Assume that investors require a 9% rate of return. This situation describes the simplest pattern of future cash flows. Assume we require a 9% rate of return on this investment. Mathematically $5.00 = Price × .09, so $5.00/.09 = $55.56. From the formula, it is obvious that if the required rate of return goes up, the price goes down, and vice versa.

b. DEF Inc. common stock that recently paid s a dividend of $1.50. The estimated growth rate of dividends is 6% per year and the required rate of return is 11%. If we assume a constant growth rate, we can use a variation on the simple formula used in question A, which is Price = Next Dividend/ (Required Rate of Return – Growth Rate of Dividends), that is Price = 1.50(1.06)/(.11-.06) = $31.80

c. GBH Inc pays no dividend and reinvests all of its earnings into rapid growth. However, GBH is expected to begin paying dividends in 5 years. The first dividend will be $5.00; dividends will grow at 5% per year; the required rate of return throughout the period is 15%.

To the extent that our assumptions are correct, in 4 years the next dividend will be $5.00, so at that time the stock should be worth $5.00/(.15-.05) = $50.00. the next question is what would we pay today for something that will be worth $50.00 in 4 years.

$50.00/ (1.15)4 = $28.59 is our estimated value. Or we could use a financial calculator:

N i/y PV PMT FV 4 15 -28.59 0 50

This solution tells us that if a stock does not pay dividends, its price is based on what we expect it to be worth to someone else at some point in the future.

6. Why do stock prices change so quickly and by so much? Notice that the formulas we used depend on forecasts of variables that in themselves depend on many other forecasts, most of which are very difficult to predict. Future dividends depend on future earnings which depend on future costs, future sales volume and future prices. Future sales depend on competition, changes in technology, geo-political events and so on. The discount rate or required rate of return depends on future inflation, interest rates, and perceptions of how risky the company is. These estimates all change with every piece of news, and frankly sometimes with the mood of investors. Realistically, the market is always looking for the correct price of a stock and never quite finding it, but the large volume of sales, rapid flow of information, insider trading laws, and the self-interest of investors assure that we are all trading on the same information. Paradoxically, this is true whether or not we even have the information since the price is presumably determined by those who do have it. This is another illustration of the efficient market hypothesis.



Additional Problems with Solutions 1. Pricing constant growth stock, with finite horizon. The Crescent Corporation just

paid a dividend of $2.00 per share and is expected to continue paying the same amount each year for the next 4 years. If you have a required rate of return of 13%, plan to hold the stock for 4 years, and are confident that it will sell for $30 at the end of 4 years, how much should you offer to buy it at today?

ANSWER (Slides 7-‐49 to 7-‐50) In this case, we have an annuity of $2 for 4 periods, followed by a lump sum of $30, to be discounted at 13% for the respective number of years.

Using a financial calculator Mode: P/Y=1; C/Y = 1

Input: N I/Y PV PMT FV Key: 4 13 ? 2 30 Output -24.35

2. Constant growth rate, infinite horizon (with growth rate estimated from past history. Using the historical dividend information provided below to calculate the constant growth rate, and a required rate of return of 18%, estimate the price of Nigel Enterprises’ common stock. Nigel Enterprises’ Annual Dividends

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

$0.35 $0.45 $0.51 $0.65 $0.75 $0.88 $0.99 $1.10 $1.13 $1.30

ANSWER (Slides 7-‐51 to 7-‐53) First, estimate the historical average growth rate of dividends by using the following equation:

1

1nFVg

PV⎛ ⎞= −⎜ ⎟⎝ ⎠

Where FV = 2010 Dividend = $1.30

PV = 2001 Dividend = $0.35 n = number of years in between = 9

191.30 1

0.35g ⎛ ⎞= −⎜ ⎟

⎝ ⎠ = 0.157

g = 15.7%



Next, use the constant growth, infinite horizon model to calculate price: Div0 = 2010 Dividend = $1.30; Div1= Div0*(1+g) è$1.30*(1.157)è$1.504

r = 18%; g = 15.7% (as calculated above)

( )1

0Price Divr g

=−

0$1.504Price

(.18 .157)=

−

Price0 = $65.40

3. Pricing common stock with multiple dividend patterns. The Wonder Products Company is expanding fast and therefore will not pay any dividends for the next 3 years. After that, starting at the end of year 4, it will pay a dividend of $0.75 per share to its common shareholders and increase it by 12% each year until it pays $1.50 at the end of year 10. After that it will pay $1.50 per year forever. If an investor wants to earn 15% per year on this investment, how much should he pay for the stock?

ANSWER (Slides 7-‐54 to 7-‐56) First lay out the dividends on a time line.

Expected Dividend Stream of The Wonder Products Co.

T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 … T ∞

--- $0.00 $0.00 $0.00 $0.75 $0.84 $0.94 $1.05 $1.18 $1.32 $1.50 …$1.50

Note: There are 3 distinct dividend payment patterns Years 1-3, no dividends; Years 4-10, dividends grow at 12%; Years 11 onwards, zero-growth in dividends.

Next, Calculate the price at the end of Year 10, i.e. when the dividend growth rate is zero. Price10 = Div11/r = 1.50/.15 = $10;

Discount each dividend and the Year 10 price back to time 0 at 15% ð0.75/(1.15)4+0.84/(1.15)5+0.94/(1.15)6+1.05/(1.15)7+1.18/(1.15)8+1.32(1.15)9+

ð1.50/(1.15)10+10/(1.15)10 ðPrice = $5.25



4. Pricing non-constant growth common stock. The WedLink Corporation just paid a dividend of $1.25 to its common shareholders, and announced that it expects the dividends to grow by 25% per year for the next 3 years, then drop to a growth rate of 16% for an additional 2 years, after which it expects the dividends to converge to the industry median growth rate of 8% per year. If investors are expecting 12% per year on WedLink’s stock, calculate the current stock price.

ANSWER (Slides 7-‐57 to 7-‐59) Determine the dividend per share in Years 1-5 using the stated annual growth rates:

D1=$1.25*(1.25)=$1.56; D2=$1.56*(1.25)=$1.95; D3=1.95*(1.25)=$2.44

D4=$2.44*(1.16)=$2.83; D5=$2.83*(1.16)=3.28 Next, Calculate the price at the end of Year 5; using the Gordon Model,

Using r = 12% and g = 8% (constant growth phase) i.e. P5 = D5(1+g)/(r – g) èP5 = $3.28*(1.08)/(.12-.08)è3.54/.04=$88.56

Finally calculate the present value of all the dividends in Years 1-5 and the price in Year 5, by discounting them at 12% for the respective number of years è1.56/(1.12) + 1.95/(1.12)2+ 2.44/(1.12)3+ 2.83/(1.12)4+ 3.28/(1.12)5+88.56/(1.12)5} = $58.60

5. Pricing common stock with constant growth and finite life versus infinite life.

(a) The ANZAC Corporation plans to be in business for 30 years. They announce that they will pay a dividend of $3.00 per share at the end of one year, and continue increasing the annual dividend by 4% per year until they liquidate the company at the end of 30 years. If you want to earn a rate of return of 12% by investing in their stock, how much should you pay for the stock?

(b) If the company was to announce that it would continue increasing the dividend at 4% per year forever, how much more would you be willing to pay for its stock, assuming your required rate of return is still 12%?

ANSWER (A) (Slides 7-‐60 to 7-‐63) Div1 = $3.00; r = 12%; g = 4%; n = 30

Using the formula for a growing annuity we can solve for the current price.

( )

30Div 11Price 10 1g

r g r

⎛ ⎞+⎛ ⎞⎜ ⎟= × −⎜ ⎟⎜ ⎟− +⎝ ⎠⎝ ⎠

( )

30

0$3.00 1.04Price 1.12 .04 1.12

⎛ ⎞⎛ ⎞= × −⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠

0Price =$37.5*0.89174 = $33.44



ANSWER (B) If the growth rate is 4% forever, the price of the stock can be figured out by using the Gordon Model;

( )1

0Price Divr g

=−

è $3.00/(.12 - .04) è$37.50

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