T Corp has just paid an annual dividend of $1.40 per share. Analysts expect the firm’s earnings (and therefore dividends, since the payout ratio is constant) to grow by 7.5% per year for the next five years. After that, growth is expected to slow to the industry average of 4.8% per year indefinitely. The equity cost of capital is 9.5%. What is the firm’s share price today?[Fill in the blank]单项选择题
A
a. $28.66
B
b. $35.09
C
c. $45.43
D
d. $34.72

Question

T Corp has just paid an annual dividend of $1.40 per share. Analysts expect the firm’s earnings (and therefore dividends, since the payout ratio is constant) to grow by 7.5% per year for the next five years. After that, growth is expected to slow to the industry average of 4.8% per year indefinitely. The equity cost of capital is 9.5%. What is the firm’s share price today?[Fill in the blank]单项选择题

A

a. $28.66

B

b. $35.09

C

c. $45.43

D

d. $34.72

BlackTom AI · Accepted Answer

Two-stage dividend-discount model (DDM): value equals the PV of dividends during the high-growth stage plus the PV of the terminal (constant-growth) value.

1) Knowledge points:
- If D0 is just paid, D1 = D0(1+g1).
- For first stage (years 1–5) use Dt = D0(1+g1)^t and discount at r.
- Terminal value at t = 5 is P5 = D6/(r − g2) where D6 = D5(1+g2).
- Present value of P5 is P5/(1+r)^5.
- Sum PV(dividends 1–5) + PV(P5) = P0.

2) Step-by-step calculations
Given: D0 = 1.40, g1 = 7.5% = 0.075 for 5 years, g2 = 4.8% = 0.048 thereafter, r = 9.5% = 0.095.

Compute dividends:
D1 = 1.40(1.075) = 1.50500
D2 = D1(1.075) = 1.617875
D3 = 1.739215625
D4 = 1.869656796875
D5 = 2.009881056640625
D6 = D5(1.048) = 2.106355347359375

Terminal value at t=5:
P5 = D6/(r − g2) = 2.106355347359375 / (0.095 − 0.048) = 2.106355347359375 / 0.047 = 44.81607 (approx).

3) Discount and sum
Discount factor (1+r)^5 = (1.095)^5 = 1.572925811296875.
Present value of dividends 1–5 can be computed with the geometric-sum shortcut:
q = (1+g1)/(1+r) = 1.075/1.095 = 0.98173516,
sum_{t=1..5} D0 q^t = D0 * q*(1 − q^5)/(1 − q)
= 1.40 * 0.98173516*(1 − 0.9120)/0.01826484 ≈ 6.63183.

PV of terminal value:
PV(P5) = P5/(1+r)^5 = 44.81607 / 1.572925811296875 ≈ 28.4629.

Total value:
P0 = PV(dividends 1–5) + PV(P5) ≈ 6.63183 + 28.4629 = 35.0947 ≈ $35.09.

4) Why this is correct:
The model correctly prices early high growth (7.5% for 5 years) then switches to perpetual growth (4.8%). All cash flows are discounted at the equity cost (9.5%). The algebraic geometric-sum and terminal Gordon-growth application are standard and produce P0 ≈ $35.09.

5) Why other options are incorrect:
- $28.66 and $34.72 are too low (they understate the discounted terminal value or the early dividends).
- $45.43 is too high (it likely assumes a higher perpetual growth or mis-discounts the terminal value).
Each deviates from the two-stage DDM calculation above.