Why is the price of this stock equal price0 + price4??? How about price 1, 2, 3 in between?

Problem DiSante and Company is a small start-up firm that will institute a dividend payment-a $0.25 dividend-for the first time at the end of this year. The company expects rapid growth over the next four years and will increase its dividend to $0.50, then to $1.50, and then to $3.00 before settling into a constant growth dividend pattern with dividends growing at 5% every year. If you believe that DiSante and Company will deliver this dividend pattern and you desire a 13% return on your investment, what price should you pay for this stock? Solution First, let's look at a time line of the dividends of DiSante and Company. T1 , T5 $0.25 $1.50 $3.00 $3.00 x 1.05 $0.50 To price this stock, we need to discount the first four dividends at 13% and then discount the constant growth portion of the dividends where the purchaser would receive the first dividend at the end of period five. Let's calculate the first four dividends: $3.00 + $0.25 $0.50 + $1.50 + PV = |(1.13)4 $0.39 + $1.04 + $1.84 = $3.49 (1.13)2 (1.13) (1.13) $0.22 We now turn to the constant growth dividend pattern, where we can use our infinite horizon constant growth model: $3.00 x (10.05) $3.15 = $39.375 price4 0.13 0.05 0.08 This figure is the price of the constant growth portion at the end of the fourth period, so we still need to discount it back to the present at the 13% required rate of return: $39.375 = $24.15 price0 (1.13) So the price of this stock with the nonconstant dividend pattern is price $3.49 + $24.15 = $27. 64