Question

How The Constant Growth Dividend Model with a Finite Horizon is derived? Thanks!

Price Price - Diego : 0) (1-C5") Price = Prenota (r-g (1 + r

Answer 1

Current price is the sum of the present values of all future cash flows. The constant growth dividend model with a finite horizon is used when an investor holds a stock for a given number of years, whose dividends are growing at a constant rate g.

The first part of the formula is the present value of a growing annuity. Second part is the present value of the price at time = n, discounted to time = 0, at the rate r.

Present value of a growing annuity forms a geometric series like

D1/(1+r) + D1*(1+g)/(1+r)^2 + ....... + D1*(1+g)^n-1/(1+r)^n

This can be rewritten as

D1/(1+r) + D1/(1+r)*(1+g)/(1+r) + D1/(1+r)*((1+g)/(1+r))^2 + D1/(1+r)*((1+g)/(1+r))^3 + ....... + D1/(1+r)*((1+g)/(1+r))^n-1

This is a geometric series with a ratio of (1+g)/(1+r). Sum of geometric series is given by a*(1 - c^n)/(1-c) where a = first term and c = common ratio. Using this formula, in the series above, we get

D1/(1+r)*[1 - ((1+g)/(1+r))^n]/[1 - (1+g)/(1+r)]

= D1/(1+r)*[1 - ((1+g)/(1+r))^n]/[(1+r-1-g)/(1+r)]

= = D1/(1+r)*[1 - ((1+g)/(1+r))^n]/[(r-g)/(1+r)]

Multiplying by (1+r)/(1+r), we get

D1*[1 - ((1+g)/(1+r))^n]/[r-g]

D0*(1+g)/(r-g)*[1 - ((1+g)/(1+r))^n]

Second part of the formula is easily derived by discounting the price Price_n at time = n to time = 0, at the rate r, to get

Price_n/(1+r)^n

Putting both parts together, we get

Current price = D0*(1+g)/(r-g)*[1 - ((1+g)/(1+r))^n] + Price_n/(1+r)^n