9.2 Notation: Let

P_n = Price at time n

D_n = Dividend at time n

Y_n = Earnings in period n r = retention ratio = (Y_n D_n) / Y_n = 1 D_n/ Y_n = 1 - dividend payout ratio E_n = Equity at the end of year n k = discount rate

g = dividend growth rate = r x ROE

ROE = Y_n / E_n-1 for all n>0. We will further assume that k and ROE are constant, and that r and g are constant after the first dividend is paid.

A. Using the Discounted Dividend Model, calculate the price P₀ if

D₁ = 20, k = .15, g = r x ROE = .8 x .15 = .12, and Y₁ = 100 per share

B. What, then, will P₅ be if: D₆ = 20, k = .15, and g = r x ROE = .8 x .15 = .12?

C. If P₅ = your result from part B, and assuming no dividends are paid until D₆, what would be P₀? P₁? P₂?

D. Again, assuming the facts from part B, what is the relationship between P₂ and P₁ (i.e., P₂/P₁)? Explain why this is the result.

E. If k = ROE, we can show that the price P₀ doesnt depend on r. To see this, let

g = r x ROE, and ROE = Y_n / E_n-1, and since r = (Y_n D_n) / Y_n , then D₁ = (1 r) x Y₁ and

	P0	=	D1 / (k g)
	P0	=	[(1 r) x Y1] / (k g)
	P0	=	[(1 r) x Y1] / (k g), but, since k = ROE = Y1 / E0
	P0	=	[(1 r) x Y1] / (ROE r x ROE)
	P0	=	[(1 r) x Y1] / (Y1 / E0 r x Y1 / E0)
	P0	=	[(1 r) x Y1] / (1 r) x Y1 / E0), and cancelling (1 r)
	P0	=	Y1 / (Y1/E0) = Y1 x (E0 / Y1) = E0

So, you see that r is not in the final expression for P₀, indicating that r (i.e., retention ration or, equivalently, dividend policy) doesnt matter if k = ROE.

Check that changing r from .8 to .6 does not change your answer in part A of this question by re-calculating your result using r = .6