Hi. Here's the question we're working with, for clarity:
4.1 Use an interest rate of 13% p.a. and a dividend growth rate of 7% to calculate the present value of:
b. A share which just paid a dividend yesterday, which was R15.50.
The answer given is correct, \(15.5 \times \frac1j \)
The answer you've suggested would be the present value if you were receiving your next dividend tomorrow, i.e. you were going to be paid 15.5 x 1.07 tomorrow and then annually in perpetuity, increasing each year. If you multiply the answer you've suggested by \(v_i\), thereby discounting back a year, then you will get the correct answer.
To make this clearer, take a look at the cashflows or draw a timeline (always a good idea). Let's look at the cashflows here. Our last payment was 15.5 yesterday and our next payment is in a year's time. This payment would have grown by 7%. We have:
$$ PV = 15.5 \times 1.07 \times (v + 1.07v^2 + 1.07^2v^3 + ...) $$
$$ = 15.5 \times (1.07v + 1.07^2v^2 + 1.07^3v^3 + ...) $$
$$ = 15.5 \times a _\infty , [in \ arrears, \ at \ j] $$
$$ =15.5 \times \frac1j$$
$$ j = 1.13/1.07 -1 $$