Suppose that in 2018 the expected dividends of the stocks in a broad market index equaled $240 million when the discount rate was 8% and the expected growth rate of the dividends equaled 6%. Using the constant-growth formula for valuation, if interest rates increase to 8.5%, the value of the market index will change by __________.[Fill in the blank]单项选择题
A
a. −10%
B
b. −20%
C
c. −25%
D
d. −33%

Question

Suppose that in 2018 the expected dividends of the stocks in a broad market index equaled $240 million when the discount rate was 8% and the expected growth rate of the dividends equaled 6%. Using the constant-growth formula for valuation, if interest rates increase to 8.5%, the value of the market index will change by __________.[Fill in the blank]单项选择题

A

a. −10%

B

b. −20%

C

c. −25%

D

d. −33%

BlackTom AI · Accepted Answer

The constant-growth (Gordon) valuation formula is P = D / (r − g), so the market index value equals expected dividends divided by the difference between the discount rate and the growth rate.

1) Relevant knowledge points and concepts
- Gordon (constant-growth) model: P = D / (r − g), where D is next-period expected dividends, r is the required return (discount rate), and g is the constant growth rate.  
- A change in r (with D and g held constant) changes P inversely and proportionally to 1/(r − g).

2) Step-by-step reasoning
- Given: D = $240 million, initial r0 = 8% = 0.08, growth g = 6% = 0.06.  
- Compute initial index value P0 using P0 = D / (r0 − g).

3) Calculations with intermediate steps
- r0 − g = 0.08 − 0.06 = 0.02.  
- P0 = 240 / 0.02 = 12,000 (million dollars).  
- New discount rate r1 = 8.5% = 0.085, so r1 − g = 0.085 − 0.06 = 0.025.  
- P1 = 240 / 0.025 = 9,600 (million dollars).  
- Percentage change = (P1 − P0) / P0 = (9,600 − 12,000) / 12,000 = −2,400 / 12,000 = −0.20 = −20%.

4) Why the correct answer is correct
- The formula shows price ∝ 1/(r − g). Increasing r from 8% to 8.5% raises the denominator from 0.02 to 0.025, reducing the price to 0.02/0.025 = 0.8 of its original level, a 20% decline. These numeric steps produce −20%.

5) Why the other choices are incorrect
- a. −10%, c. −25%, d. −33% do not match the computed change. Only −20% follows from the exact application of P = D/(r − g) with the given numbers.

6) Conclusion
- The value of the market index falls by −20%, which matches the provided correct answer (b. −20%).

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