# DERIVATION OF STOCK PRICE WITH NPVGO (Net Present Value of Growth Opportunities)

Please find my trying to get the well-known formula of

Stock Price equal to EPS/r + NPVGO

NPVGO, the one I see here is positive NPVGO, meaning that the company will only invest in the projects that will generate return on investment higher that its cost of capital.

I have not yet finished up with this thinking.

This is only for educational purposes.     It comes up to me that we could solve this issue of its derivation, by having two equations that will lead us to the same results.

(1) Dividend Constant Growth Model =>
Stock Price = Dividends_1/(r – g)
Here we are discounting in growth perpetuity of Dividends (and not Earnings)

(2) Growth Opportunities (GO) Model ==>

Stock Price = Earnings_1/r + NPV of first year investment/(r-g)

Here we are discounting first without growth for Earnings, and then add it the NPV of first year investments discounted at cap rate of (r-g)

Though mathematically I could get that, what is the insight that we could take from the above 2 formula?

GO Model will make more sense since we could link it to the return on investment higher than cost of capital, or the other way around, if the NPV is negative.

Next, will show you how to have both formula mathematically be proven and the example, to see whether they are the same indeed.

So we could see from the above hand-written proof, that both Models, either we are discounting the DIVIDENDS WITH GROWTH, or EARNINGS WITHOUT GROWTH and then add on it the NPV of the GROWTH OPPORTUNITIES (discounted WITH GROWTH), then the results will be the same.

Respondend 1 to Karnen:

The growth opportunities model has the right concepts in mind, but is too vague on the metric to capture the NPV. That will be the (discounted) growth in earnings over the no-growth forecast expressed by Earnings/r. See Chapter 6 : Accrual Accounting and Valuation: Pricing Earnings of the textbook : Financial Statement Analysis and Security Valuation (5th Edition, McGraw-Hill) by Stephen Penman for the full expression.

Respondent 2 to Karnen:

As for insight, it is mainly used to assess the whether the stock price is mainly from current operations or future growth.

Respondent 3 to Karnen:

I think that your problem with this issue stems from the presentation of Gordon formula. The one you are using is ok but not the full one.

If you use the following one: PV = DIV_1/(r-g) = [EPS_1 * (1-b)]/[(r – b*ROE)]

Now, the only step needed is to plug in b =0 for the no growth (and get EPS1/r) and positive b for the growth value.

Karnen to Respondent 3

Noted for that formula, but not the one I am trying to solve.

I have been able to match the Dividends Constant Growth Model against NPV Growth Opportunities Model, under (i) growth situation and(2) return on investment is higher than cost of capital. I attach my (very bad) hand-written doodle to match both models.
The interesting insight during this exercise, I noted that that the cost of capital will be the function of growth rate, as growth is a risky thing, then the higher growth rate, then this will push up the cost of capital as well. The concept that growth is a risky one, came up to me upon reading the books by Prof. Stephen Penman (Financial Statement Analysis and Security Valuation) and Douglas R Emery, John D. Finnerty and John D. Stowe (Corporate Financial Management).

Respondent 4 to Karnen:

It is just a version of decomposing an observable stock price into COV (current operations value) and FGV (future growth value), which assumes COV being measured by capitalising current no growth EPS. Then p – COV gives an estimate of FGV. Typical approach with unrealistic simplifying assumptions regarding COV.

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