Athabasca University



Slide 1: Lecture 4 – Stock pricing with Dividend Growth Model

Welcome to Lecture 4: Stock pricing with the dividend growth model.

Slide 2: Five possible cases

In this lecture, we will go over five possible cases which will allow us to use the dividend growth model. These cases are:

1. No dividend growth, i.e., constant dividend,

2. constant growth,

3. non-constant growth,

4. super-normal growth, and

5. changing dollar amount of dividends (i.e., dividends are given in dollars instead of growth rates).

Slide 3: Case #1 – No growth

In the no-growth case, a company pays the same dollar amount of dividends, year after year. The timeline of cash-flows look like this:

t=0 1 2 3 4 5 (

D0 D0 D0 D0 D0 D0

Note that the valuation formula is derived using the PV(perpetuity) formula P0 = D/k, and implicitly assumes that the company is a going concern that will continue to pay dividends forever. In this formula, P0 stands for the share-price now (at time 0), D stands for the constant dividend in each year, and k is the discount rate (required rate of return on this common stock).

Note the interesting fact that, with constant dividends, the dividend growth model says that the price of the company’s shares should remain the same from one year to the next. This relationship is expressed in the form:

P0 = P1 = P2 = P3 = … = P( = D/k

As can be seen from this formula, since D and k are assumed to be constant in each year, the price of the perpetual constant dividend share will also be constant in each year.

Slide 4: Numerical Example – Case #1

Now let’s work through an example of no dividend growth.

ABC Company recently paid a dividend of $2 per share. The company has a stable dividend policy. What is ABC’s share price if the required return on its shares is 10%?

We have the following information:

D = $2

k = 0.1

t=0 1 2 3 4 5 (

$2 $2 $2 $2 $2 $2

Using the no-growth share-pricing formula, we have

P0 = D/k

Plugging in the numbers for D and k to this formula, we can get share-price:

P0 = 2/0.1 = $20

That is, ABC shares are worth $20 per share.

Slide 5: Case #2 – Constant growth

That was easy. Now, let’s move on to the slightly more complicated case of constant dividend growth.

With constant dividend growth, the cash-flow timeline looks like this:

t=0 1 2 3 4 5 (

D0 D1 D2 D3 D4 D5

g g g g g

The dividends grow at a rate of g in each year, and g is assumed to be constant. The valuation formula is derived using the PV(growing perpetuity):

P0 = D1/(k – g)

where P0 is the share price at now (at time 0), D1 is the dividend that will occur in the next period, at time 1, k is the discount rate (required rate of return), and g is the constant growth rate.

Since D1 = D0(1 + g), the valuation formula can also be written as

P0 = D0(1+g)/(k-g)

Note that the number of variables necessary to calculate the share-price has increased from 2 in the no-growth case to three in the constant-growth case.

Slide 6: Numerical Example – Case #2

Let’s work through an example of a constant dividend growth share-valuation.

DEF Corporation has just paid a dividend of $1.50. Stock market pundits anticipate that the company’s dividends will grow by 2% per year. What is DEF’s share price given that investors require a return of 11% on its stocks?

We have the following information:

D0 = $1.50

g = 0.02

k = 0.11

Using the constant growth share pricing formula, we have

P0 = D0(1+g)/(k-g)

Plugging in the numbers for D0, g, and k to this formula, we can get share-price:

P0 = 1.5(1+0.02)/(0.11 – 0.02) = 1.53/0.09 = 17

That is, DEF shares are worth $17 per share.

Slide 7: Case #3 – Non-constant growth

That was not too bad either. Now, let’s see what the non-constant growth case looks like: Well, it looks like this: The timeline of cash-flows is:

t=0 1 2 3 4 5 T-1 T (

D0 D1 D2 D3 D4 D5

g1 g2 g3 g4 g5 gT

In this case, dividends will grow at non-constant growth rates. In each period, the growth rate is different, but eventually the growth rate will taper out to a somewhat normal and constant level. D0, the most recently paid dollar dividend amount, is usually given. We represent the growth rates as g1 = growth in period 1, g2 = growth in period 2, …, and gT = constant growth starting in period T and expected to continue until forever. These should all be given (i.e., someone else has given their best to estimate these growth rates so that we could use them in our calculations).

To calculate the share-price in these non-constant growth cases, we have to calculate each of the dividends, starting from the next dividend (D1) using the first growth rate (g1), all the way to the DT, which is the first dividend to be paid at the constant growth rate. Dividends are usually calculated sequentially, that is, D1 depends on D0, D2 depends on D1, and D3 depends on D2, and so on and so forth, until DT, which depends on the dividends estimated with growth from the previous period, T-1. The calculations get confusing if there are a lot of growth rates, so the best way to get the numbers right is to write the dividend calculation out carefully, such as we have done here:

D1 = D0(1+g1)

D2 = D1(1+g2) = D0(1+g1)(1+g2)

D3 = D2(1+g3) = D0(1+g1)(1+g2)(1+g3)

DT = DT-1(1+gT) = D0(1+g1)(1+g2)(1+g3)…(1+gT)

The formula to calculate the share price of a common stock with non-constant growth is the following:

P0 = [D1/(1+k)1] + [D2/(1+k)2] + [D3/(1+k)3] + … + [DT-1/(1+k)T-1] + [DT / ((k – gT)(1 + k)T-1)]

Notice that the formula shown here can actually be divided into two groups, with the first group being the discounting of the dividends growing at non-constant rates:

[D1/(1+k)1] + [D2/(1+k)2] + [D3/(1+k)3] + … + [DT-1/(1+k)T-1]

and the second group being the discounting of the dividends growing at the constant rate:

[DT / ((k – gT)(1 + k)T-1)].

Notice that the section: DT/(k-gT) is simply the formula for pricing shares with constant-growth which we learned previously, in Case #2. So, the logic of this formula is fairly straight-forward. Simply calculate the present values of all the non-constant growth dividends, and then calculate the future share-price at the future time when the growth rate turns constant (PT-1 = DT / ((k – gT)). Then, calculate the present value of that share-price (PT-1 / (1+k)T-1), and add the present value of this future share-price to the present values of all the non-constant growth dividends. This, in turn, gives us the share-price of this common stock with non-constant growth at time 0.

Slide 8: Numerical Example – Case #3

The best way to get a handle on how to value shares with non-constant growth is, of course, through practice, practice, practice. So, let’s work through an example.

GHI Inc., has just paid a $1.20 per-share dividend. Due to a new project taken on by GHI, the company’s dividends are expected to grow by 5% next year, 7% in two years, and 9% in three years, then 3% thereafter. What is GHI’s current share-price, given that the required return on its stocks is 12%?

We have information as follows:

D0 = $1.20

g1 = 0.05

g2 = 0.07

g3 = 0.09

g4 = 0.03

k = 0.12

We first calculate the expected dividends in the next four years:

D1 = D0(1+g1) = 1.2(1+0.05) = 1.26

D2 = D1(1+g2) = 1.26(1+0.07) = 1.3482

D3 = D2(1+g3) = 1.3482(1+0.09) = 1.469538

D4 = D3(1+g4) = 1.469538(1+0.03) = 1.5136 2414

Slide 9: Numerical Example – Case #3 (cont.)

The fourth year’s dividend allows us to use the constant-growth dividend formula to calculate the share-price at Year 3:

P3 = D4/(k-g) = 1.51362414/((0.12-0.03) = 1.51362414/0.09 = 16.818046

The share-price at time 0 can then be calculated as the sum of the present values of all future cash-flows:

P0 = PV(D1) + PV(D2) + PV(D3) + PV(P3)

= [1.26/1.12] + [1.3482/1.122] + [1.469538/1.123] + [16.818046/(1.123)]

1.125 + 1.07477678 + 1.04598812 + 11.97075295

= 15.21651786

≈ $15.22

That is, GHI’s shares are worth about $15.22 per share.

Slide 10: Case #4 – Super-normal growth

Well, that was a bit scary, but we are brave people studying finance, so, let’s push on and leave more practice time for later.

The next case occurs when a company is expected to undergo a period of supernormal growth. This is, actually, just a special case of the non-constant growth case. We are just adding an extra category here to trip you up and check that you are still awake.

So…we have a cash-flow stream that looks like this:

t=0 1 2 3 4 5 (

D0 D1 D2 D3 D4 D5

g1 g2 g3 g4 g5

This timeline looks like the one for Case #3, with the difference that the first few growth rates are super-normal growth rates, and the rest of the growth rates after that, to infinity, are smaller and involve normal growth.

Here are the steps for calculating current share-price in a case of super-normal growth.

- Calculate individual dividends for each super-normal growth year as well as normal growth dividend for one year beyond the super-normal growth year (like we did in the non-constant growth case).

- Then, calculate the share-price at time T-1, where T is the first year of normal constant growth using the formula (just like we did in the non-constant growth case):

PT-1 = DT/(k – normal growth rate)

- Then, calculate the share-price as the present value of super-normal dividends plus the present value of the share-price at T-1.

See, this is all very normal and familiar from the non-constant growth rate case.

Slide 11: Numerical Example – Case #4

As usual, let’s work on an example to make sure we understand how things work.

The earnings and dividends of JKL Fencelink Inc., are expected to grow by 30% per year for the next two years, then by 20% per year for another two years, before settling down to a normal growth rate of 8% per year forever. If JKL has just recently paid a dividend of $0.50 per share, what is its current share-price according to the dividend growth model? Assume a required rate of return of 10%.

We see that JKL has

D0=$0.50

g1=0.3

g2=0.3

g3=0.2

g4=0.2

g5=0.08

k=0.1

We calculate the dividends as follows:

D1 = D0(1+g1) = 0.5(1.3)=0.65

D2 = D1(1+g2) = 0.65(1.3) = 0.845

D3 = D2(1+g3) = 0.845(1.2) = 1.014

D4 = D3(1+g4) = 1.014(1.2)=1.2168

D5 = D4(1+g5) = 1.2168(1.08)=1.314144

Slide 12: Numerical Example – Case #4

We then use the fifth dividend, which is the first dividend with constant normal growth rate of 8%, to calculate our future share-price at Year 4:

P4 = D5/(k-g) = 1.314144/(0.1-0.08)=65.7072

Now we have all the information we need to calculate present values at time 0:

PV(D1) = 0.65/1.1 = 0.59090909

PV(D2)=0.845/1.1 ^2=0.69834711

PV(D3)=1.014/1.1^3=0.76183321

PV(D4) = 1.2168/1.1^4 = 0.83109077

PV(P4) = 65.7072/1.1^4=44.87890171

Slide 13: Numerical Example – Case #4 (cont.)

That is all we need to calculate the current share-price,

P0 = Sum of PV(all cash flows)

= 0.59090909 + 0.69834711 + 0.76183321 + 0.83109077 + 44.87890171

=$47.76

That is, JKL’s shares are worth $47.76 per share.

Slide 14: Case #5 – Changing dollar dividends

In this fifth type of dividend growth, dividends are stated in dollar amounts and the share-price in a future period is given (i.e., estimated by some poor soul in a back room and given to the finance department for further use). A timeline for this case looks something like this:

t=0 1 2 3 4 5 (

D0 D1 D2 D3 P4

The steps for calculating current share-price with changing dollar dividends are as follows:

Step 1 - Calculate the present value of each of the given dividends using the PV(lump sum) formula:

PV = CFt / (1 + k)t

Step 2 - Calculate the present value of the future share-price estimated at time T.

Step 3 - Calculate the current share-price as the sum of the present values of the individual dividends plus the present value of the ending share-price.

Slide 15: Numerical Example – Case #5

Let’s work on an example.

MNO Omnipresence Ltd has just paid a dividend of $2.50 per share. The company has undertaken a new project which will require certain levels of cash investment over the next 5 years, and therefore the company has announced that it will reduce dividends to $1.00, $1.20, $1.40, $1.60, and $1.80 per share for each of the next 5 years, respectively. At the end of the 5 years, MNO estimates that its share-price will rise to $75 per share. What is the current share-price of MNO, given a required return of 10%?

So, for MNO, the dividends are given as:

D1=1

D2=1.2

D3=1.4

D4=1.6

D5=1.8

The ending share-price is given as:

P5=75

The required return is

k=0.1

We first calculate the present values of the dividends:

PV(D1)=1/1.1=0.90909091

PV(D2) =1.2/1.1^2 = 0.99173554

PV(D3) = 1.4/1.1^3 = 1.05184072

PV(D4) = 1.6/1.1^4 = 1.09282153

PV(D5) = 1.8/1.1^5 = 1.11765838.

Slide 16: Numerical Example – Case #5 (cont.)

Last, but not least, we calculate the present value of the share-price at Year 5. Watch out for the time period in which the share-price occurs. In this case, the share-price is at the end of the 5th year, which means that we need to discount this share-price by 5 years in order to get the present value at time 0.

PV(P5)=75/1.1^5=46.56909923

The current share-price is then calculated as the sum of the PV of the dividends and the PV of the ending share-price:

P0

= Sum(all future cash-flows)

= 0.90909091 + 0.99173554 + 1.05184072 + 1.09282153 + 1.11765838 + 46.56909923

= 51.73

That is, MNO’s shares are worth $51.73 per share.

Slide 17: General process for calculating stock price

And that, as they say, is that! So, the general process for calculating stock prices is as follows:

Step 1. Write down all the available information.

Step 2. Figure out if it is a case of zero-growth, constant growth, non-constant growth, super-normal growth, or different dollar amounts of predicted dividends.

Step 3. Do the calculations according to the case type. The basic principle to follow is this one:

P0 = PV(All future dividends or cash flows)

Slide 18: Practice makes easy-peasy

Try this question on. It is a bit more difficult than the ones that we have discussed in this lecture, but you can do it!

PQ Recon Corporation has just paid a dividend of $1 per share. To undertake a project, the company plans to reduce dividends to $0 for the next 3 years, after which dividends will grow (from their current level) for 3 years at 30% per year, before settling down to normal growth of 10% per year. If the company does not take on this project, dividends will grow at the normal rate of 10%. Is it worthwhile for this company to take on this project, given a required rate of return of 15%?

Answer:

There are two alternatives for the company: Do the project, or don’t do the project. Therefore, we compare the share-price without the project, versus the share-price with the project.

Without project:

D1=1(1.1)=1.1

P0 = 1.1/(0.15-0.1) = 22

With project:

D4=1(1.3)=1.3

D5=1.3(1.3)=1.69

D6=1.69(1.3)=2.197

D7=2.197(1.1)=2.4167

P6=2.4167/(0.15-0.1)=48.334

PV(D4)=1.3/1.15^4 =0.74327922

PV(D5)=1.69/1.15^5 = 0.84022868

PV(D6)=2.197/1.15^6 = 0.94982373

PV(P6) = 48.334/1.15^6 = 20.896122

P0 = PV(all future cash flows) = 0.74327922 + 0.84022868 + 0.94982373 + 20.896122 = 23.43.

Since the share-price will actually increase with the project, it is worthwhile for the company to take on this project.

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