Development of the Constant Growth Model

# Development of the Constant Growth Model - to the equation...

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(1) P 0 = D 0 (1+g) 1 /(1+r) 1 + D 0 (1+g) 2 /(1+r) 2 + … + D 0 (1+g) -1 /(1+r) -1 + D 0 (1+g) /(1+r) Now multiply both sides of Equation 1 by (1+r)/(1+g) to get Equation 2: (2) P 0 (1+r)/(1+g) = D 0 + D 0 (1+g) 1 /(1+r) 1 + … + D 0 (1+g) -2 /(1+r) -2 + D 0 (1+g) -1 /(1+r) -1 Now subtract Equation 1 from Equation 2 P 0 (1+r)/(1+g) - P 0 = D 0 - D 0 (1+g) /(1+r) If we assume that the discount rate is greater than the growth rate, then the last term approaches 0 in the limit and can be discarded, which leaves us with: P 0 (1+r)/(1+g) - P 0 = D 0 Multiplying through we can rearrange terms as follows: P 0 (1+r) - P 0 (1+g) = D 0 (1+g) P 0 - P 0 + P 0 r -P 0 g = D 0 (1+g) = P 0 (r-g) P 0 = D 0 (1+g)/(r-g) = D 1 /(r-g) __________ Note that if growth is zero, then D1 = D for all time periods and this reduces down
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Unformatted text preview: to the equation for a perpetuity: P = D/r __________ When using the perpetuity equation, we always find the value of the perpetuity one period before the first cash flow. Thus, if our first cash flow is at Year 13, we will find the value at Year 12: V 12 = CF 13 / r On the other hand, if our first cash flow takes place at Year 54, we will find the value at Year 53, etc.: V 53 = CF 54 / r Where “r” is the appropriate discount rate to apply to the cash flows....
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