Discounting is a time value of money problem. The concept is simple enough but to those who haven't had to do maths for a while the actual formulae can be intimidating. If you think of a stock as simply a perpetual source of dividends and you never wish to sell it, you can make estimates in the growth of dividends and plug them into time value formula.

P0 is the value of the dividend cash flow stream at time zero (right now), D1 is the first year's dividend, D2 is the second year's dividend, D3 is the third year's dividend, right off into the future. k is the required rate of return. This might be the long term bonds rate or something higher because you want a better return to justify risk.

This equation can be shortened with summation notation. Just in case you never had to deal with this in school, the funny looking "E" thing means "take the sum of...", the little t=1 below the summation sign means "starting from 1" and the infinity up the top means "...and continuing on until you reach infinity", the t in the equation itself then stands in for all the values of t from 1 to infinity, and you add them all up.


It seems like a horrendous equation, a never-ending series that would just clog up your computer before crashing it eventually. But mathematics is full of these things, and fortunately when you constrain the series a bit it converges to form simpler, shorter sums.

For example if you assume that dividends are going to increase at a constant rate for ever, the equation becomes very simple. D0 is today's dividend yield, D1 is next year's dividend yield, k is still the required rate of return and g is the rate of growth. If growth is zero, that is this is just a steady dividend that will never increase, then the value of this stock is the dividend divided by your rate of return. If the growth rate is not zero, then the share is worth next year's dividend divided by the rate of return minus the growth rate. It isn't that difficult when you understand the notation.


This is all well and good, but no real stock is going to grow at the same rate for ever. Maybe it would be a better idea to forecast the rate of growth for just a few years and then change the growth rate to something conservative from that time on. This leads to a "two stage" model.

The summation thing works in the same way, it reads as "adding this thing up for all values of t between 1 and N", where N is the time you want to start adding it up at the other growth rate, g1 is the first growth rate, g2 is the second growth rate.

So you could use this to estimate the value of a growth stock that you think is going to have a few great years and then slow down a bit to grow at a more normal rate of growth.

Well this is all fine, but dividends are subject to the vagaries of dividend policy by management and anyway if we are going to consider ourselves as owners of the business we'd be wanting to look at earnings instead of just dividends. Putting this into the one-stage model where D=E(1-b) where D is the dividend per share, E is earnings per share, and b is the earnings retention rate (or 1-b is the dividend payout ratio) we get the following:


A rearrangement puts P0 over E0, which is the current price/earnings ratio (PER). Thus we can find a formula that calculates the correct PER for a specified growth rate.

Converting the two stage dividend model to an earnings version we get:

Where D/P0 = 1-b is the dividend payout ratio, N is the holding period in years, and (P/E)N is the price/earnings ratio expected at time period N. The other terms are as above.

I know to many people here these equations will look horrendous, but they can be programmed into a computer or programmable calculator easily enough and then it all just becomes a matter of plugging growth rates or PERs or cash flows in to figure out what is a fair price for a security. Excel has most of these functions built in among its financial functions.

You can plug growth rates into this formula and get prices, or you can plug in prices and work out growth rates (tricky, but a computer can solve it easily enough). This is why you often hear comments about a stock having a certain growth rate "built in". Expensive growth stocks already reflect a very rosy future, when you work out the k's and the g's for the stock you'll find that either the market may already have a high rate of growth factored into the equation, and the corresponding rate of growth, g, will be pretty dull.