Question

Formally derive and discuss the divident discount model used for a common stock valuation

Answer 1

Price or value of a common stock should be present value of all the future cash flows (dividends) it’s expected to generate.

Basic Dividend Discount Model: Let’s assume the stock pays a dividend D_t in the time period t for an infinite period of time (going concern basis). Then,

where V_s = Intrinsic value of a share of common stock; D_t = Expected dividend per share on the common stock in period t; K_s = Required rate of return on the common stock (cost of equity calculated by CAPM)

From here it can take any of the following two simplifications:

1. Zero Dividend Growth Model
2. Constant Dividend Growth Model
3. Variable Dividend Growth Model

Zero Dividend Growth Model: In case the dividend remains constant over a period of time, i.e. dividend growth is zero then D_t = D and above equation can be simplified as:

If we multiply both side by 1 / (1 + K_s) then,

If we subtract second equation from first one, we will get (on the RHS, only first term in first equation will remain, second term in the first equation will be cancelled by first term in second equation and so on)

Constant Growth Dividend Model: In this case the dividend grows by a constant growth “g” which implies D_t+1 = D_t x (1 + g) and if D₁ = first dividend due at the end of first year = D₀ x (1 + g) then the above formula changes to

If we multiply both side by (1 + g) / (1 + K_s) and subtract the resultant equation from the equation above and simplify the way we did in Zero Dividend Growth Model above, we will get

The equation above is also called “Gordon Constant Dividend Growth Model”.

Points to note:

• Note that all the dividends, growth rates and cost of equity (discount rate) are annual rates. Quarterly dividend may be annualized before applying these equations for valuations.
• Further all these equations give price a year prior to the first dividend is due. If dividend payments start many years in future, use these formulae to calculate the price just a year prior to when dividend payments start. The discount the price backward to build in the impact of non-dividend payment period in the prices.
• Prices, dividend and earnings grow at the same rate. Hence, growth rate in one can be used as a substitute for other in these equations. Share price will grow at the same rate as earnings and dividends in future.

Relationship between cost of equity and constant growth rate

Sl. No.	Difference = Ks - g	Price of the stock
1.	Small changes in difference will lead to	Relatively larger changes in stock price. Stock price is very sensitive to this difference
2.	If difference widens	Leads to drastic fall in prices
3.	If difference narrows down	Leads to a large increase in prices

Variable dividend growth model:

• When dividends grow at a different rate in initial few years and then stabilize over the years, variable dividend growth model is used.
• Variable dividend growth model projects dividends separately for every year till the time it becomes stable. Terminal value is calculated using Gordon constant growth model from the year dividend stability is achieved.   Present value of all the dividends and terminal value is calculated and added together to arrive at the value of the stock.