residual theory

To discount a perpetuity, you multiple by 1/r (where r is the rate of interest).

So, the first part of the third item in your equation is 10 / 0.1 which is discounting the perpetuity.

However, that would assume that the first dividend was in 1 years time and was thereafter every year.

I would guess that the reason they have done more is because the actual pattern of dividends is different, but without seeing the actual question I cannot tell.

Hi Vipin,

I hope you have understood this theory.

Several days ago, I was stuck at this same question too. You must have read the FTC book, the same as I did. Anyway, as I was saying, I got stuck at this question. And I was hoping that someone else in other part of the world had the same problem, and as always, I turned here, hoping there would be an answer ready for me.
Alas, no such luck.
today, I reread this theory and this question, and I was enlightened. Just as Opentuition Tutor Mr. Moffat said, the theory is about discounting cash flows.
So, here I am, writing my following understandings, hoping it might shed some light on someone who, by chance, do not understand this question and theory, and wind up here by googling. 🙂

The Residual Theory is all about the present value of future cash flows. It argues that as long as present value is the same, it doesn’t matter when the cash flows in. And it argues that company shall invest its retentions in projects to generate future cash flows. Only the residua profits that do not meet any projects would be distributed to the shareholders. The cash flow here obviously is dividends.

So here the question is about this situation:

On one scenario, the company do not invest its profits to any projects, and the dividends are distributed annually at 10%. This dividend is a perpetuity, with annual cash flow of 10c at 10% discount rate, thus the PV is calculated at 10c/0.1 = 100c

The other situation is the company rather than distributes its 10c per share dividends to its shareholders at the end of first year, it sees a one-year investment opportunity. And the investment rate of return is 10% annually. So it invests the money to the project at the end of year 1, ie. T1. The return is recorded at the end of year 2, ie T2. So apart from the usual annual dividend of 10c (T2), there is a further cash flow that is generated from the investment opportunity, with return to be 10c * 1.1 = 11c (Note, this cash flow is at the T2). From T3 onwards, the dividends payment is unchanged at 10c each year. So for this part of cash flows, it is perpetuity again and the calculation of “PV” is 10c/0.1 (But we should take note that this perpetuity is not recorded from T1 onward, rather it is from T3 onward, hence the PV is only discounted back at T2.

Now we see all the money is at T2 (normal dividend of T2, investment opportunity cashflow earned at T2, and the perpetuity arrived at T2), and we want to discount them back to T0, to arrive at the true PV.

So the calculation is listed as the following:
PV = (10 + 11)/1.1^2 + 10/0.1 * 1/1.1^2 = 100c

The present value of these two scenarios are the same, at 100c.

One thing should blame the FTC text book, there is an error in the formula. It lacks the parenthesis of the first part of the calculation ( there is no parenthesis between 10 + 11, creating confusion for us students)

Hope you all understand now.