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Lina buys a common stock that pays annual dividends. The first dividend of $25 is at...

Lina buys a common stock that pays annual dividends. The first dividend of $25 is at the end of the third year and each subsequent annual dividend is $4 greater than the preceding one.

Calculate the price of this stock given an effective annual interest rate of 4.5 %

Solutions

Expert Solution

Let dividend for period n be Dn

Given

D3 = 25
D4 = 25 + 4
D5 = 25 + 2*4
D6 = 25 + 3*4
......

Rate of interest r = 4.5% or 0.045

Present Value of the stock = Sum of Present Value of all dividend payments = Σ Dn/(1+r)n

Hence, Value of the stock now = P0 = 25/(1+0.045)3 + (25 + 4)/(1+0.045)4 + (25 + 2*4)/(1+0.045)5 +  (25 + 3*4)/(1+0.045)6 + ....

P0 = [25/(1+0.045)3 + 25/(1+0.045)4 + 25/(1+0.045)5 + 25/(1+0.045)6 +.... ] + [4/(1+0.045)4 + 2*4/(1+0.045)5 +  3*4/(1+0.045)6 + ...]

P0 = (25/(1+0.045)3 )/ ( 1 - 1/(1+0.045)) + [4/(1+0.045)4 + 2*4/(1+0.045)5 +  3*4/(1+0.045)6 + ...]

P0 - P0/(1+0.045) = (25/(1+0.045)2 )/ 0.045 -  (25/(1+0.045)3 )/ 0.045 + [4/(1+0.045)4 + 4/(1+0.045)5 +  4/(1+0.045)6 + ...]

=> 0.045P0/(1+0.045) = (25/(1+0.045)2 )/ 0.045 -  (25/(1+0.045)3 )/ 0.045 + [4/(1+0.045)4 / (1 - 1/(1+0.045))]

=> 0.045P0/(1+0.045) = (25/(1+0.045)2 )/ 0.045 -  (25/(1+0.045)3 )/ 0.045 + (4/(1+0.045)3 )/ 0.045

=> P0 = [(1+0.045)/(0.045) ] * [(25/(1+0.045)2 )/ 0.045 -  (25/(1+0.045)3 )/ 0.045 + (4/(1+0.045)3 )/ 0.045]

=> P0 = $ 2317.59

jeff jeffy answered 1 year ago
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