Answer to question 7(i)
Please refer to the attached photo for the timeline I used to answer this question.
This question is mostly straight forward after summarising it on a timeline. You can see how detailed my timeline is. It also has more than three cashflows so that a pattern can be easily recognised from it.
I will first find the value of the stock at time t0. I will then present value it. An easier method might be to find the value of the stock at time t0 -1 so that I will only have to work with the annuity in arrears formula. The value P is simply the present value of all future dividends in perpetuity.
P = 24 + 24(1.05)(1+3%+9%)-1 + 24(1.05)2(1+3%+9%)-2 + 24(1.05)3(1+3%+9%)-3 +…
P = 24[1 + (1.05)(1+3%+9%)-1 + (1.05)2(1+3%+9%)-2 + (1.05)3(1+3%+9%)-3 + …]
The effective interest rate used to present value the cashflows is the risk free interest rate plus the risk premium.
Now that I have P as an ordinary sum to infinity, I will use the formula for the annuity in advance to sum it up (the first receipt is on the time to which I am present valuing). Recall that the sum for the annuity in advance looks like the expression A as shown in the picture. I have to express P in the same way as A. I shall find a quantity "v" such that I have:
P = 24[1 + v + v2+ v3+ …]
Therefore, it must be that v = (1.05)(1+3%+9%)-1. I will simply take this v value and plug it into the formula A. I will then take the limit as n goes to infinite to find P. Please refer to the picture for this step. Therefore, P = R384. To find the value of the stock in the present, I simply present value P over 3 months:
The value of the stock = P(1+3%+9%)-3/12 % = R373,27
Please refer to your module 1 notes on section 2.7, more specifically section 2.7.2