# 0.9 Growth Perpetuities and the Dividend Discount Model

A âgrowth” perpetuity is a perpetual cash flow stream (“CF”) that grows at a *constant *rate of growth, which we shall call “*g*.” It is a special case of perpetuity. If we are to receive a cash flow of $100 and its growth rate is 5%, the next cash flow would follow this formula: (Cash flow now) (1 + G) = Cash flow in next period.

CF_{1} = CF_{0} (1 + G)

$100 (1.05) = $105

The formula for a growth-perpetuity is: (Cash flow next period) Ăˇ (Discount rate – Growth rate). Symbolically, this may be expressed as:

PV = CF_{1} Ăˇ (*R – g*)

In the above example (where *I *= 0.10), if the growth rate had been 5%, the present value would be (assuming here that the *next* CF is $105, that is, [$100] [1.05] = $105:

$105 Ăˇ (.10* – *.05) = $2,100

Notice that for this formula to work, “*g*” cannot equal or exceed “*R*.”

This formula may also come in handy for cases of negative growth. Since “*g*” would be negative, in this case, the formula would require that one *add* the growth rate to the interest rate in order to determine the present value.

Suppose we say that the cash flows are $100 per year (i.e., starting with $100 next time) with a negative growth rate of 5% and a discount rate of 10%. What would the present value be? Once again, the answer would be:

$100 Ăˇ [.10 – (-.05)] = $100 Ăˇ .15 = $666.67

Of course, there is also the possibility that *g* = 0.0%.

**The Dividend Discount Model (âDDMâ)**

The DDM is attributed to Myron Gordon (1961) who asserted that a stockâs value is determined by discounting its expected future cash flows, i.e., its perpetual dividend stream. This notion is not without some controversy. In any case, the DDM is very commonly used. It only requires substituting the Dividend in the above perpetuity formula for the cash flow.

In the instance where there is no growth, the DDM conforms well to Preferred Stock. In cases where there may be some expected growth in the dividend, the formula conforms, although not necessarily perfectly, to Common Stock.

Note also that D_{1} = D_{0} (1 + G).