Question (Discounted Dividend Model):

Pn = Price at time n On = Dividend at time n Yn = Earnings in period n r = retention ratio = (Yn- Dn) / Ym= 1 - D Ym= 1 - dividend payout ratio En = Equity at the end of year n k = discount rate g = dividend growth rate = r x ROE ROE = Yn / En-i 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 Po if D1 = 20, k = .15, g = rx ROE = .8 x .15 = .12, and Y1 = 100 per share B. What, then, will Ps be if: Do = 20, k = .15, and g = r x ROE = .8 x .15 = .12? C. D . If Ps = your result from part B, and assuming no dividends are paid until Ds, what would be Po? P1? P2? E . Again, assuming the facts from part B, what is the relationship between P2 and Pi (i.e., P2/Pi)? Explain why this is the result. If k = ROE, we can show that the price Po doesn't depend on r. To see this, let g = r x ROE, and ROE = Ym / En-1, and since r = (Ym - Dn) / Ym , then Di= (1 - r) x Y1 and Po = D1 / (k - g) Po = [(1 - r) x Y1] / (k - g) Po = [(1 - r) x Yi] / (k - g), but, since k = ROE = Y1 / Eo Po = [(1 - r) x Y1] / (ROE - r x ROB) Po = [(1 - r) x Y1] / (Y1 / Eo- r x Y1 / Eo) Po Po [(1 - r) x Yi] / (1 - r) x Y1 / Eo), and cancelling (1 - r) Y1 / (Y1/Eo) = Y1 x (Eo / Y1) = Eo So, you see that r is not in the final expression for Po, indicating that r (i.e., retention ration or, equivalently, dividend policy) doesn't 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