Question

Assume that a company in a year from now will have a stream of profits £100 every year for the rest of time (i.e. forever). What is the present discounted value of an income stream of £100 when the interest rate is 3%?

Answer 1

This company run by someone's grandma? Those are slim margins.

Answer 2

Must be!

Answer 3

Okay, to do this problem we have to work backwards.

Answer 4

Okay

Answer 5

First of all, do you know any equations that apply here?

Answer 6

Well, usually (when working with a single year) I'd do the following:

Answer 7

100/(1+0.03)^1

Answer 8

But in this case, that comes to £97.08, which I'm told is not the answer.. So I'm assuming I need to do this:

100/(1+0.03)^(infinity) ?

Answer 9

What would you do for two years?

Answer 10

100/(1+0.03)^1 + 100/(1+0.03)^2 ?

Answer 11

Correct, assuming you get 100 each year.

Answer 12

If so, I retract my earlier assumption of 100/(1+0.03)^(infinity) as it'd need to be a summation, right?

Answer 13

Yes.

Answer 14

So I need a way of working a summation up to the 100/(1+0.03)^(infinity) ?

Answer 15

What we have is
SUM ar^k = a SUM r^k = (a) / (1-r) when the SUM is infinite

Answer 16

http://mathworld.wolfram.com/GeometricSeries.html

Answer 17

Does this display properly for you: \[ \frac ab \]

Answer 18

I see \[ \frac ab \]

Answer 19

Okay unfortunately math on this site is broken

Answer 20

Ah, okay! Well thank you for your help so far!

Answer 21

Ah!

Answer 22

Anyway, the point is:
r + r^2 + r^3 + ... + r^infinity = r / (1 - r)

Answer 23

Do you think you can do this problem knowing that?

Answer 24

That's a big help. The only issue I'm having now is in thinking what should I treat as r. Could I work with r as 1.03 initially?

Answer 25

Hm, definitely not right

Answer 26

No, remember that r < 1

Answer 27

100/(1+0.03)^k

Answer 28

Another way to write this is
100*(1+0.03)^-k

Answer 29

Which also is
100*[(1+0.03)^-1]^k

Answer 30

So 100 + (1/3)^k ?

Answer 31

No plus signs here.

Answer 32

Identify a and r.

Answer 33

Hint: a*[r]^k

Answer 34

Really lost. Trying to get there but not I've just got 100's and 3's all over my page. I'll keep trying

Answer 35

Wait, is a = 100 and r = 100/103 ?

Answer 36

It is!! I think I'm there!! 100 * [(100/103)/(1- (100/103))]

Answer 37

Thanks so much for your help - I really appreciate it!

Answer 38

Did you get correct answer?

Answer 39

I did!