Question

(a)       Consider a stock that pays annual dividends. It just paid $4.50 dividends per share, and the next dividend will be paid in 1 year. The dividends are expected to remain constant at $4.50 per share for the next 10 years, after which the dividends are expected to decrease at a rate of 0.5% per year. The annual cost-of-capital is 15.50%. Find the fair value of the stock today.

(b)       Consider the same stock as described in part (a), except that the stock will not pay any dividend in at the end of year 8, i.e., it will skip the year 8 dividend. All other dividends and timing of dividends are exactly the same as described in part (a). The cost-of-capital is also the same. Compute the fair value of the stock today.

(c)        Suppose that the effective 1-year risk-free rate is 6% per annum. The value of the stock in 1 year is random and not known today. The different possible values of the stock in 1 year are given in the table below. The corresponding risk-neutral probabilities are also shown in the table.

Value	$10	$12	$14	$15	$17	$22	$26
Risk Neutral Prob.	0.10	0.10	0.25	0.15	0.20	0.10	0.10

            The annual cost-of-capital for the stock is 19.45%.

           

            Find the fair value of the stock today.

Answer 1

Solution 1) After 10th year, the dividends will decline at the rate of 0.5%. This will form a perpetuity.

According to Gordon's Growth Model, present value of perpetuity at t=10 is:

= D11/(r - g)

D11 = Dividend in 11th year = 4.5*(1-0.5%) = 4.4775

r = Required rate of return = 15.5%

g = growth rate of dividends = -0.5%

Thus, present value of perpetuity at t=10 is: 4.4775/(15.5% - (-0.5%)) = 4.4775/16% = 27.98438

The cash flows are shown as below:

Solution 2) Company will omit the dividend for 8th year, hence, the cash flows are:

Solution 3) Expected Value of the stock after 1 year = 10*0.1 + 12*0.1 + 14*0.25 + 15*0.15 + 17*0.2 + 22*0.1 + 26*0.1 = 16.15

Fair value of the stock = Expected Value of the stock/(1 +Risk-free rate)^1

= 16.15/(1+ 6%)

= 15.23584906

= $15.24

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