Nachman Industries just paid a dividend of D0 = $1.75. Analysts expect the company's dividend to grow by 30% this year, by 10% in Year 2, and at a constant rate of 5% in Year 3 and thereafter. The required return on this low-risk stock is 9.00%. What is the best estimate of the stockβs current market value?
49.50
59.68
To estimate the current market value of the stock, we can use the constant growth dividend valuation model, also known as the Gordon growth model. The formula for this model is as follows:
πβ = π·β / (π - π)
Where:
πβ = current market value of the stock
π·β = expected dividend in the next period
π = required return
π = growth rate
In this case, we are given that the dividend just paid (Dβ) is $1.75. To calculate the expected dividend in the next period (Dβ), we need to apply the growth rate of each year as follows:
Year 1: Dβ = Dβ Γ (1 + growth rate in Year 1)
Year 2: Dβ = Dβ Γ (1 + growth rate in Year 2)
Year 3 and thereafter: Dβ = Dβ Γ (1 + growth rate in Year 3)
Given the provided growth rates of 30% in Year 1, 10% in Year 2, and 5% in Year 3 and thereafter, we can substitute the values into the formula.
Year 1: Dβ = $1.75 Γ (1 + 0.30) = $2.275
Year 2: Dβ = $2.275 Γ (1 + 0.10) = $2.5025
Year 3 and thereafter: Dβ = $2.5025 Γ (1 + 0.05) = $2.627625
Now, we can substitute these values into the Gordon growth model formula to estimate the current market value of the stock:
πβ = $2.627625 / (0.09 - 0.05)
= $2.627625 / 0.04
= $65.690625
Therefore, the best estimate of the stockβs current market value is approximately $65.69.
To estimate the stock's current market value, we can use the dividend discount model (DDM) which calculates the present value of all future dividends.
The formula for DDM is:
V0 = D1 / (1 + r) + D2 / (1 + r)^2 + D3 / (1 + r)^3 + ... + Dn / (1 + r)^n
Where:
V0 = Current market value of the stock
D1, D2, D3, ... Dn = Expected dividends for each year (in this case, D0 = $1.75, D1 = $1.75 * (1 + 30%) = $2.275, D2 = $2.275 * (1 + 10%) = $2.5025)
r = Required return on the stock (9.00% or 0.09)
We also know that the dividend will grow at a constant rate of 5% from Year 3 and onwards.
To find the present value of the dividends after Year 3, we can use the formula for the present value of a growing perpetuity:
PV = D / (r - g)
Where:
PV = Present value
D = Expected dividend in Year 4 and onwards
g = Growth rate of the dividends (5% or 0.05)
To calculate the present value of the growing perpetuity, we need to calculate the dividend in Year 4 and onwards. Since the dividend will grow at a constant rate of 5%, we can use the formula for the future value of a growing annuity:
FV = D * (1 + g) / (r - g)
Where:
FV = Future value
D = Expected dividend in Year 3
g = Growth rate of the dividends (5% or 0.05)
r = Required return on the stock (9.00% or 0.09)
Let's calculate the present value of the dividends using the DDM formula.
V0 = ($2.275 / (1 + 0.09)) + ($2.5025 / (1 + 0.09)^2) + [(D * (1 + g) / (r - g)) / (1 + r)^3]
Using the above formula, we can calculate the present value of the dividends after Year 3 as:
PV = (D * (1 + g) / (r - g)) / (1 + r)^3
Now, substitute the values and calculate the present value of the dividends after Year 3.
PV = ($2.5025 * (1 + 0.05) / (0.09 - 0.05)) / (1 + 0.09)^3
Calculate the value:
PV = $2.5025 * 1.05 / 0.04 / (1.09)^3
PV β $21.61
Now, substitute all the values in the original DDM formula and calculate the current market value of the stock.
V0 = ($1.75 / (1 + 0.09)) + ($2.275 / (1 + 0.09)^2) + $21.61
V0 β $21.18
Therefore, the best estimate of the stock's current market value is $21.18.