# Math Help - Price of stock (present value)

1. ## Price of stock (present value)

Ok, so the question goes like this:

A company paid a dividend per share of $8 yesterday. Because of unusual high growth, the dividend is expected to increase by 20 % annually for the next two years. Thereafter a growth of 8 % is expected in all foreseeable future. Your required annual return is 20 %. How much are you willing to pay for this stock today? Ignore any tax consequences. So here is my thinking: 1) Since it paid out yesterday the next payout will happen in one year. Since our required return is 20 %, we discount by 1.20 for each period. 2) After the first two years we have a perpetuity. The present value of the cashflow is what we're willing to pay. $P_0 = \frac{8\cdot 1.2}{1.2} + \frac{8 \cdot 1.2^2}{1.2} + \frac{8\cdot 1.2^2}{0.2-0.08} \approx 113.6$ This is not the correct answer. Where is my attempt at logic wrong? 2. You shouldn't discount both the first two payments by 1.2. They are 1 year apart from each other. You need to discount the perpetuity to the present day. The value you have written is the value of the perpetuity if the first payment is in 1 year, but it wont start until after that. 3. Oh, thank you! Didn't realize I also had to discount the perpetuity. So $PV = \frac{8\cdot 1.2}{1.2} + \frac{1\cdot 1.2^2}{1.2^2} + \frac{\frac{8\cdot 1.2^2}{0.2-0.08}}{1.2^2} = 2\cdot 8 + \frac{8}{0.12}$ This doesn't give the correct answer, why? What am I missing here? Could you please show me? 4. if you have the correct answer available please post it. 5. ## typo When i did this: Payments 9.6 in 1 year 11.52 in 2 years 11.52*1.08 = 12.44 in 3 years 13.43 in 4 years ...etc Present Values $\frac{9.6}{1.2} + \frac{9.6 \cdot 1.2}{1.2^2} + (1.2^{-2})\frac{ (9.6 \cdot 1.2 \cdot 1.08)}{0.2-0.08}$ $= 8 + 8 + (1.2^{-2}\cdot 103.68)$ $= 8 + 8 + 72$ $= 88$ The difference between my answer abnd yours is the value of the first payment in the perpetuity, i used 12.44. 6. ## Re: Price of stock (present value) Thank you! Your answer is 100 % correct, so I guess the discounted cash flow would go like this. $PV = \frac{8\cdot 1.2}{1.2} + \frac{8\cdot 1.2^2}{1.2^2} + (1.2)^{-2} \cdot \frac{8\cdot 1.2^2 \cdot 1.08}{0.2-0.08} = 8 + 8 + 72 = 88$ (I had to write it like that to see it more clearly myself) Why are we only discounting the perpetuity by 2 periods, when the perpetuity kicks in 3 periods from now? Is this because the general formula for a perpetuity assumes that the first payment will come one period from now? So if I was asked the present value of a perpetuity of$8 and the discount rate was 20 %, then it would be:

$PV = \frac{8}{0.2} = 40$

But if the first payment happened right now, it would be:

$PV = \frac{\frac{8}{0.2}}{1.2} \approx 33,33$

If this is correct, I think I understand it.

7. ## Re: Price of stock (present value)

You've almost got it

Is this because the general formula for a perpetuity assumes that the first payment will come one period from now?
yep

But if the first payment happened right now, it would be:

$PV = \frac{\frac{8}{0.2}}{1.2} \approx 33,33$
If the payment happens sooner than assumed in the general formula, the present value will increase. The PV would be $PV = 1.2 \cdot \frac{8}{0.2}$

8. ## Re: Price of stock (present value)

Originally Posted by SpringFan25
If the payment happens sooner than assumed in the general formula, the present value will increase. The PV would be $PV = 1.2 \cdot \frac{8}{0.2}$
Oh, yes, that is actually intuitive (even to my peanut brain), as getting something sooner is better than later (at least if we're not talking punishment of some kind). Thank you very much. I think I'm grasping it now with your help.