Consider the stock of Davidson Company that will pay an annual
dividend of $2 in the coming year. The dividend is expected to grow at
a constant rate of 5 percent permanently.  The market requires a
12-percent return on the company.

a. What is the current price of a share of the stock?



b. What will the stock price be 10 years from today?

Hi redd1986!
The fair value of a stock should be the present discounted value of
its stream of dividends, using the required rate of return as the
discount rate.

The stream of dividends will be something like:

Year 1 = 2
Year 2 = 2*(1.05) = 2.1
Year 3 = 2*(1.05)^2 = 2.205
...

Therefore, the price of the stock today should be (assuming the first
dividend payment happens exactly one year from now):

P = 2/(1.12) + 2*(1.05)/(1.12)^2 + 2*(1.05)^2/(1.12)^3 + 2*(1.05)^3/(1.12)^4 + ...

Now, recall the following formula for infinite sums of this form:

1 + x^1 + x^2 + x^3 + ... = 1/(1-x)
when |x|<1

So, we multiply the above equation by 1.05

(1.05)*P = 2*(1.05/1.12) + 2*(1.05/1.12)^2 + 2*(1.05/1.12)^3 + ...

and rearrange to get

(1.05/2)*P = (1.05/1.12) + (1.05/1.12)^2 + (1.05/1.12)^3 + ...

1 + (1.05/2)*P = 1 + (1.05/1.12) + (1.05/1.12)^2 + (1.05/1.12)^3 + ...

So, calling 'x' to (1.05/1.12) (which gives 0.9375):

1 + (1.05/2)*P = 1 + x + x^2 + x^3 + ...

1 + (1.05/2)*P = 1/(1-x)

1 + (1.05/2)*P = 1/(1-0.9375)

1 + (1.05/2)*P = 16

(1.05/2)*P = 15

P = 28.57

So the current fair value of the stock should be $28.57.


What will the stock price be 10 years from today?

The procedure is exactly the same as before. However, the dividends
are different. When "standing" in year 10, the stock has just yielded
a 2*(1.05)^10 dividend. So the dividend in the coming year is
2*(1.05)^11 = 3.42. So, in order to find the value of the stock we
simply repeat the previous procedure using 3.42 instead of 2. This
will give you that the value of the stock 10 years from now will be
$48.85.


I hope this helps! If you have any questions regarding my answer,
please don't hesitate to request a clarification. Otherwise I await
your rating and final comments.

Best wishes!
elmarto

Very well put

If this is a theoretical company (there are many real Davidson
companies) then you need to know that, although there is a method for
calculating fair value of stock options, stock prices are what someone
is willing to pay for the stock - there is no way to accurately
predict that or future stock prices - even billion dollar IPOs (such
as a well known search engine) can be priced at, say $100 or so and
quickly double - if the underwriters knew it was worth $200/share they
would have priced it closer to $190 and made a lot more money for the
owners and themselves.

I'm glad the client was satisfied, but fair value is only vaguely
related to price. If it were that simple then prices wouldn't go up
and down so much.

Thank you for your comments and generous tip!

Hi siliconsamurai,
I disagree with the statement that fair value is vaguely related to
price. The change in stock prices, as I see it, reflects the
ever-changing expectations regarding how well the company will do in
the future. The "unrealistic", if you wish, part in this question is
assuming that investors know that dividends will grow exactly 5% a
year, when it is clear that it's extremely unlikely for a company to
grow at a steady rate while the economy and expectations around it
change all the time.

Regards,
elmarto

Hi, elmarto. =)

Could you please clarify why you used $3.42 for part (b)?  I don't
understand your reference to "standing" in year 10.

The problem asks about the stock price 10 years from today.  I'm
confused.  Wouldn't the dividend stream be something like this:

We're in Year 0, the dividend for the next year: $2 (given)
We're in Year 1, the dividend for the next year: $2 * 1.05^1
We're in Year 2, the dividend for the next year: $2 * 1.05^2
We're in Year 3, the dividend for the next year: $2 * 1.05^3
We're in Year 4, the dividend for the next year: $2 * 1.05^4
We're in Year 5, the dividend for the next year: $2 * 1.05^5
We're in Year 6, the dividend for the next year: $2 * 1.05^6
We're in Year 7, the dividend for the next year: $2 * 1.05^7
We're in Year 8, the dividend for the next year: $2 * 1.05^8
We're in Year 9, the dividend for the next year: $2 * 1.05^9
We're in Year 10, the dividend for the next year: $2 * 1.05^10

TIA for helping me understand your solution.