Question

1. (1) Suppose a company just paid a dividend of $3.6. Its manager promises to pay annual dividend growing at 6% per year. If the required return is 12% (in EAR), what would the current price be?
   (2) Now assume the manager has decided to pay quarterly dividends instead of annual dividends. It has just paid a dividend of $0.9. The next dividend will be 3 months from now, with value equal to $0.9*1.061/4, and will grow at an annual rate of 6%. That is, each dividend payment will be 1.061/4 times the previous one. What is the share price now?
   (3) Now assume the dividend will grow more slowly at a rate of 2% per year from the 40th quarterly dividend. That is, the 41st dividend will be equal to the 40th dividend multiplied by 1.021/4. What is the share price now?

Answer 1

(1) Suppose a company just paid a dividend of $3.6. Its manager promises to pay annual dividend growing at 6% per year. If the required return is 12% (in EAR), what would the current price be?

D₀ = $ 3.6; g = 6%; D₁ = D₀ x (1 + g) = 3.6 x (1 + 6%) = 3.816

Required return, K_e = 12%

Hence, Current Price = D₁ / (K_e - g) = 3.816 / (12% - 6%) = $  63.60

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(2) Now assume the manager has decided to pay quarterly dividends instead of annual dividends. It has just paid a dividend of $0.9. The next dividend will be 3 months from now, with value equal to $0.9*(1.06)^1/4, and will grow at an annual rate of 6%. That is, each dividend payment will be 1.06^(1/4) times the previous one. What is the share price now?

Frequency = Quarterly

Period = quarter

Expected divided next period = D₁ = $ 0.9 x (1.06)^1/4 = $ 0.913206462

Growth rate per period, g = growth rate per quarter = 1.06^1/4 - 1 = 1.47%
Required return per period = Required return per quarter = Effective quarterly rate = K_e = (1 + EAR)^1/4 - 1 = (1 + 12%)^1/4 - 1 = 2.87%

Hence, Current Price = D₁ / (K_e - g) = 0.913206462 / (2.87% - 1.47%) = $  64.93

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(3) Now assume the dividend will grow more slowly at a rate of 2% per year from the 40th quarterly dividend. That is, the 41st dividend will be equal to the 40th dividend multiplied by (1.02)^1/4. What is the share price now?

This has now become a two stage model,

First stage growth rage, g₁ = 1.06^1/4 - 1 = 1.47%

Second stage growth rate, g₂ = 1.02^1/4 - 1 = 0.50%

D₄₁ = D₄₀ x (1 + g₂)

D₄₀ = D₀ x (1 + g₁)⁴⁰ = 0.9 x 1.06^(40/4) x 1.02^1/4 = 1.619761996

Horizon value of future dividends at the end of 40th quarter, D_{HV, 40} = D₄₁ / (K_e - g₂) = 1.619761996 / (2.87% - 0.50%) = 68.13047228

Current price = PV of annuity D₁ growing at g₁ over (n =) 40 periods + PV of D_{HV, 40}

=$  49.43