14 terms

Why does the value of a share of stock depend on dividends?

The value of any investment depends on the present value of its cash flows; i.e., what investors will

actually receive. The cash flows from a share of stock are the dividends.

actually receive. The cash flows from a share of stock are the dividends.

The Starr Company just paid a dividend of $2.35 per share on stock. The dividends are expected to grow at a constant rate of 4 percent per year indefinitely. If investors require an 11 percent return on the stock, what is the current price? What will the price be in three years? In 15 years?

The constant dividend growth model is:

Pt = Dt × (1 + g)/(R - g)

So, the price of the stock today is:

Po = D0(1 + g)/(R - g) =

$2.35(1.04)/(.11 - .04) = $34.91

The dividend at Year 4 is the dividend today times the FVIF for the growth rate in dividends and four years, so:

P3 = D3(1 + g) / (R - g) = D0(1 + g)4 / (R - g) = $2.35(1.04)4 / (.11 - .04) = $39.27

We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so:

P15 = D15(1 + g) / (R - g) = D0(1 + g)16 / (R - g) = $2.35(1.04)16 / (.11 - .04) = $62.88

There is another feature of the constant dividend growth model: The stock price grows at the dividend growth rate. So, if we know the stock price today, we can find the future value for any time in the future we want to calculate the stock price. In this problem, we want to know the stock price in three years, and we have already calculated the stock price today. The stock price in three years will be:

P3 = Po(1 + g)^3 = $34.91(1 + .04)^3 = $39.27

And the stock price in 15 years will be:

P15 = Po(1 + g)15 = $34.91(1 + .04)15 = $62.88

Pt = Dt × (1 + g)/(R - g)

So, the price of the stock today is:

Po = D0(1 + g)/(R - g) =

$2.35(1.04)/(.11 - .04) = $34.91

The dividend at Year 4 is the dividend today times the FVIF for the growth rate in dividends and four years, so:

P3 = D3(1 + g) / (R - g) = D0(1 + g)4 / (R - g) = $2.35(1.04)4 / (.11 - .04) = $39.27

We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so:

P15 = D15(1 + g) / (R - g) = D0(1 + g)16 / (R - g) = $2.35(1.04)16 / (.11 - .04) = $62.88

There is another feature of the constant dividend growth model: The stock price grows at the dividend growth rate. So, if we know the stock price today, we can find the future value for any time in the future we want to calculate the stock price. In this problem, we want to know the stock price in three years, and we have already calculated the stock price today. The stock price in three years will be:

P3 = Po(1 + g)^3 = $34.91(1 + .04)^3 = $39.27

And the stock price in 15 years will be:

P15 = Po(1 + g)15 = $34.91(1 + .04)15 = $62.88

The next dividend payment by ABC, Inc., will be $1.99 per share. The dividends are anticipated to maintain a growth rate of 4.5 percent forever. If ABC stock currently sells for $31 per share, what is the required return?

Hint: Value of Stock (Price) = Dividend per share divided by (Discount Rate - Dividend Growth Rate)

*Must transpose the equation to:

R = (D1/P0) + g;

g is dividend growth rate and R is discount rate

What is the dividend yield and what is the expected capital gains yield?

Hint: Value of Stock (Price) = Dividend per share divided by (Discount Rate - Dividend Growth Rate)

*Must transpose the equation to:

R = (D1/P0) + g;

g is dividend growth rate and R is discount rate

What is the dividend yield and what is the expected capital gains yield?

R =

(D1/Po) + g = ($1.99 / $31) + .045 = 10.92%

Dividend yield = D1/Po = $1.99 / $31 = 6.42%

The capital gains yield, or percentage increase in the stock price, is the SAME as the dividend

growth rate:

Capital gains yield = 4.5%

The required return of a stock is made up of two parts: The dividend yield and the capital gains

yield.

(D1/Po) + g = ($1.99 / $31) + .045 = 10.92%

Dividend yield = D1/Po = $1.99 / $31 = 6.42%

The capital gains yield, or percentage increase in the stock price, is the SAME as the dividend

growth rate:

Capital gains yield = 4.5%

The required return of a stock is made up of two parts: The dividend yield and the capital gains

yield.

Mickelson Corporation will pay a $2.65 per share dividend next year. The company pledges to increase its dividend by 4.75 percent per year indefinitely. If you require a return of 11 percent on your investment, how much will you pay for the company's stock today?

Using the constant growth model, the price of the stock today is:

Po = D1 / (R - g) = $2.65 / (.11 - .0475) = $42.40

Po = D1 / (R - g) = $2.65 / (.11 - .0475) = $42.40

The newspaper reported last week that Lowery Enterprises earned $45 million this year. The report also stated that the firm's return on equity is 15 percent. Lowery retains 75 percent of its earnings. What is the firm's earnings growth rate? What will next year's earnings be?

The growth rate of earnings is the return on equity times the retention ratio, so:

g = ROE × b = .15(.75)

g = .1125, or 11.25%

To find next year's earnings, we multiply the current earnings times one plus the growth rate, so:

Next year's earnings = Current earnings(1 + g)

Next year's earnings = $45,000,000(1 + .1125) Next year's earnings = $50,062,500

g = ROE × b = .15(.75)

g = .1125, or 11.25%

To find next year's earnings, we multiply the current earnings times one plus the growth rate, so:

Next year's earnings = Current earnings(1 + g)

Next year's earnings = $45,000,000(1 + .1125) Next year's earnings = $50,062,500

Metallica Bearing, Inc., is a young start-up company. No dividends will be paid on the stock over the next 12 years, because the firm needs to plow back its earning to fuel growth. The company will pay a dividend of $15 per share exactly 13 years from today and will increase the dividend by 5.5% per year thereafter. If the required return on this stock is 13%, what is the current share price?

The stock pays no dividends for 13 years, however, once the stock begins paying dividends, it will have a constant growth rate of dividends and the constant growth model will be used at that point.

The price of the stock in 12 years (NOT 13 yrs)

P12 = D13/(R - g) = $15 / (.13 - .055)

P12 = $200.00

We must find TODAY'S price of the stock, therefore, we must discount $200 by using present value:

Po= $200/(1.13)^12 (notice the exponent is NOT 13)

Po = $46.14

The price of the stock in 12 years (NOT 13 yrs)

P12 = D13/(R - g) = $15 / (.13 - .055)

P12 = $200.00

We must find TODAY'S price of the stock, therefore, we must discount $200 by using present value:

Po= $200/(1.13)^12 (notice the exponent is NOT 13)

Po = $46.14

Osbourne, Inc., has an odd dividend policy. The company has just paid a dividend of $9 per share and has announced that it will increase the dividend by $4 per share for each of the next five years, and then never pay another dividend. If you require a return of 13% on the company's stock, how much will you pay for a share today?

The price of a stock is the Present Value (PV) of the future dividends. This stock is paying five dividends, so the price of the stock is the PV of EACH dividend using the required return. The price of the stock

is:

Po = $13/1.13 + $17/(1.13)^2 + $21/(1.13)^3 + $25/(1.13)^4 + $29/(1.13)^5

P0 = $70.44

Using a financial calculator: Interest=13

CFo =0, CF1=13, CF2=17, CF3=21, CF4 =25, and CF5=29

NPV = 70.44

is:

Po = $13/1.13 + $17/(1.13)^2 + $21/(1.13)^3 + $25/(1.13)^4 + $29/(1.13)^5

P0 = $70.44

Using a financial calculator: Interest=13

CFo =0, CF1=13, CF2=17, CF3=21, CF4 =25, and CF5=29

NPV = 70.44

South Side Corporation is expected to pay the following dividends over the next four years: $11, $9, $6, and $2.50. Afterward, the company pledges to maintain a constant 5% growth rate in dividends forever. If the required return on the stock is 10%, what is the current share price?

With nonconstant dividends, we find the price of the stock when the dividends level off at a constant

growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 5, so we can find the price

of the stock in Year 4, one year before the constant dividend growth begins, as:

P4 = D4(1 + g) / (R - g) = $2.50(1.05) / (.10 - .05)

P4 = $52.50

The price of the stock today is the PV of the first four dividends, plus the PV of the Year 4 stock

price. So, the price of the stock today will be:

Po = $11/1.10 + $9/(1.10)^2 + $6/(1.10)^3 + ($2.50+52.50)/(1.10)^4

Po = $59.51

Using a financial calculator: Interest=10

CFo =0, CF1=11, CF2=9, CF3=6, and

CF4=55 (2.50+52.50)

NPV = 59.51

growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 5, so we can find the price

of the stock in Year 4, one year before the constant dividend growth begins, as:

P4 = D4(1 + g) / (R - g) = $2.50(1.05) / (.10 - .05)

P4 = $52.50

The price of the stock today is the PV of the first four dividends, plus the PV of the Year 4 stock

price. So, the price of the stock today will be:

Po = $11/1.10 + $9/(1.10)^2 + $6/(1.10)^3 + ($2.50+52.50)/(1.10)^4

Po = $59.51

Using a financial calculator: Interest=10

CFo =0, CF1=11, CF2=9, CF3=6, and

CF4=55 (2.50+52.50)

NPV = 59.51

Hughes Co. is growing quickly. Dividends are expected to grow at a 25% rate for the next three years, with the growth rate falling off to a constant 4.5% thereafter. If the required return is 10% and the company just paid a $2.40 dividend, what is the current share price?

Hint: Dividends are expected to grow at a 25% rate for the next three years - We must find the future value for the 3 years of dividends @ 25% to determine the price at year 2 (NOT 3); then divide by (Discount Rate - Dividend Growth Rate).

To find the price at year 3, multiply the year 2 price by the constant dividend rate to find P3.

Next find the PRESENT VALUE for the differential growth - make sure EXPONENTS are correct.

Hint: Dividends are expected to grow at a 25% rate for the next three years - We must find the future value for the 3 years of dividends @ 25% to determine the price at year 2 (NOT 3); then divide by (Discount Rate - Dividend Growth Rate).

To find the price at year 3, multiply the year 2 price by the constant dividend rate to find P3.

Next find the PRESENT VALUE for the differential growth - make sure EXPONENTS are correct.

P3 = D3(1 + g)/(R - g)

We do not know D3 but we can find D3

D1=2.40(1.25)

D2=2.40(1.25)^2

D3=2.40(1.25)^3 = 4.6875

P3 = 4.6875(1.045)/(.1-.045)= $89.06

The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock

price. The price of the stock today is evaluated at the appropriate discount rate of 10%:

Po = $2.40(1.25)/1.10 + $2.40(1.25)^2/(1.10)^2 + $2.40(1.25)^3 /(1.10)^3 + $89.06/(1.10)^3

P0 = $76.26

OR

Po= $2.40(1.25)/1.10 + $3(1.25)/(1.10)^2 + $3.75(1.25)/(1.10)^3 + $89.06/(1.10)^3

Po = $76.26

ALSO another way for $89.06

With differential dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 4, so we can find the price

of the stock in Year 3, one year before the constant dividend growth begins as:

P3= $2.40(1.25)^3 (1.045)/(.10 - .045)

P3 = $89.06

We do not know D3 but we can find D3

D1=2.40(1.25)

D2=2.40(1.25)^2

D3=2.40(1.25)^3 = 4.6875

P3 = 4.6875(1.045)/(.1-.045)= $89.06

The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock

price. The price of the stock today is evaluated at the appropriate discount rate of 10%:

Po = $2.40(1.25)/1.10 + $2.40(1.25)^2/(1.10)^2 + $2.40(1.25)^3 /(1.10)^3 + $89.06/(1.10)^3

P0 = $76.26

OR

Po= $2.40(1.25)/1.10 + $3(1.25)/(1.10)^2 + $3.75(1.25)/(1.10)^3 + $89.06/(1.10)^3

Po = $76.26

ALSO another way for $89.06

With differential dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 4, so we can find the price

of the stock in Year 3, one year before the constant dividend growth begins as:

P3= $2.40(1.25)^3 (1.045)/(.10 - .045)

P3 = $89.06

Antiques R Us is a mature manufacturing firm. The company just paid a $13 dividend, but management expects to reduce the payout by 3% per year indefinitely. If you require a return of 9 % on this stock, what was the most recent dividend per share paid on the stock?

The constant growth model can be applied even if the dividends are declining by a constant percentage, just make sure to recognize the negative growth. So, the price of the stock today will

be:

Po = D0 (1+g)/(R - g) = $13(1 - .03) / [(.09 - (-.03)]

Po = $105.08

be:

Po = D0 (1+g)/(R - g) = $13(1 - .03) / [(.09 - (-.03)]

Po = $105.08

Mustaine Corporation stock currently sells for $64.85 per share. The market requires a return of 11% on the firm's stock. If the company maintains a constant 5% growth rate in dividends, what was the most recent dividend per share paid on the stock?

We are given the stock price, the dividend growth rate, and the required return, and are asked to

find the dividend. Using the constant dividend growth model, we get:

P0 = $64.85 = D0(1+g)/(R - g)

Solving this equation for the dividend gives us: D0 = $64.85(.11 - .05) / (1.05)

D0 = $3.71

find the dividend. Using the constant dividend growth model, we get:

P0 = $64.85 = D0(1+g)/(R - g)

Solving this equation for the dividend gives us: D0 = $64.85(.11 - .05) / (1.05)

D0 = $3.71

Discounted Cash Flow Valuation of Perpetuities

A perpetuity is a constant stream of identical cash flows with no end. The formula for determining the present value of a perpetuity is as follows:

PV = CF/(1+r)^1 + CF/(1+r)^2 + CF/(1+r)^3 ...

= CF/r

The concept of a perpetuity is often used to explain the concepts of the Dividend Discount Model (DDM).

To value a company using the DDM, calculate the value of dividend payments that are estimated that the stock is anticipated to generate for future years.

Zero Growth: Div/r = Po

(P is the price at time 0, and r is the discount rate).

Po = Div1/(1+r)^1 + Div2/(1+r)^2 + .... = Div/r

A perpetuity is a constant stream of identical cash flows with no end. The formula for determining the present value of a perpetuity is as follows:

PV = CF/(1+r)^1 + CF/(1+r)^2 + CF/(1+r)^3 ...

= CF/r

The concept of a perpetuity is often used to explain the concepts of the Dividend Discount Model (DDM).

To value a company using the DDM, calculate the value of dividend payments that are estimated that the stock is anticipated to generate for future years.

Zero Growth: Div/r = Po

(P is the price at time 0, and r is the discount rate).

Po = Div1/(1+r)^1 + Div2/(1+r)^2 + .... = Div/r

Discounted Cash Flow Valuation of Perpetuities (cont)

If the company is expected to GROW the equation would have to reflect the relationship between required rate of return and expected growth rate.

Constant Growth DDM or the Gordon Model.

Po = Div/(R-g)

Po = Div/(1+r) + Div(1+g)/(1+r)^2 + Div(1+g)^2/(1+r)3 + .... = Div/(r-g)

If the company is expected to GROW the equation would have to reflect the relationship between required rate of return and expected growth rate.

Constant Growth DDM or the Gordon Model.

Po = Div/(R-g)

Po = Div/(1+r) + Div(1+g)/(1+r)^2 + Div(1+g)^2/(1+r)3 + .... = Div/(r-g)

Gillette Corporation will pay an annual dividend of $0.65 one year from now. Analysts expect this dividend to grow at 12% per year thereafter until the fifth year. After then, growth will level off a 2% per year. According to the dividend-discount model, what is the value of a share of Gillette stock if the firm's equity cost of capital is 8%?

$.65/1.08 + $.65(1.12)/1.082 + $.65(1.12)2 /1.083 + $.65(1.12)3 /1.084 + $.65(1.12)4 / 1.085 + (.65(1.02)(1.12)4 /(.08 -.02))/ 1.085 = $15.07

First 5 dividends = $.65/1.08 + $.65(1.12)/1.082 + $.65(1.12)2 /1.083 + $.65(1.12)3 /1.084 + $.65(1.12)4 / 1.085 = $3.24

Value on date 5 of the rest of the dividend payments

(.65(1.02)(1.12)4 /(.08 -.02))/ 1.085 = $17.39

PV = $17.39/1.085 = $11.83

$3.24 + $11.83 = $15.07

First 5 dividends = $.65/1.08 + $.65(1.12)/1.082 + $.65(1.12)2 /1.083 + $.65(1.12)3 /1.084 + $.65(1.12)4 / 1.085 = $3.24

Value on date 5 of the rest of the dividend payments

(.65(1.02)(1.12)4 /(.08 -.02))/ 1.085 = $17.39

PV = $17.39/1.085 = $11.83

$3.24 + $11.83 = $15.07

Colgate-Palmolive Company has just paid an annual dividend of $0.96. Analysts are predicting an 11% per year growth rate in earnings over the next five years. After then, Colgate's earnings are expected to grow at the current industry average of 5.2% per years. If Colgate's equity cost of capital is 8.5% per years and its dividend payout ratio remains constant, what price does the dividend-discount model predict Colgate stock should sell for?

First 5 dividends = (.96)(1.11)/1.085 + (.96)(1.11)2/(1.085)2 + (.96)(1.11)3/(1.085)3 + (.96)(1.11)4/(1.085)4 + (.96)(1.11)5/(1.085)5 = 5.14217

Value on date 5 of the rest of the dividend payments

(.96)(1.11)5 (1.052)/((1.085 - .052) = 51.5689

PV = 51.5689/(1.085)5 = $34.2957

Price = $34.2957 + $5.1421 = $39.4378

Value on date 5 of the rest of the dividend payments

(.96)(1.11)5 (1.052)/((1.085 - .052) = 51.5689

PV = 51.5689/(1.085)5 = $34.2957

Price = $34.2957 + $5.1421 = $39.4378