Cascade Mining Company expects its earnings and dividends to increase by 7 percent per

year over the next six years and then to remain relatively constant thereaft er. Th e fi rm
currently (that is, as of year 0) pays a dividend of $5 per share. Determine the value of a
share of Cascade stock to an investor with a 12 percent required rate of return.

Present Value of First 6-Years' Dividends:

6 ƒÃ[Do(1 + g1)t/(1 + ke)t]; Do = $5.00; g1 = .07; ke = .12
t=1
Present Value

Year Dividend Interest Factor Present Value
t Dt = 5.00(1 + .07)t PVIF.12,t Dt x PVIF.12,t

1 5.00(1 + .07)1 = .893 $ 4.778
$5.35

2 5.00(1 + .07)2 = .797 4.563
5.725

3 5.00(1 + .07)3 = .712 4.361
6.125

4 5.00(1 + .07)4 = .636 4.168
6.554

5 5.00(1 + .07)5 = .567 3.976
7.013

6 5.00(1 + .07)6 = .507 3.805
7.504

PV (First 6-Years' Dividends) $25.651

Value of Stock at End of Year 6:

P6 = D7/(ke - g2) g2 = .00

D7 = D6(1 + g2) = 7.504(1 + .00) = $7.504

P6 = 7.504/(.12 - .00) = $62.533
Present Value of P6:

PV(P6) = P6/(1 + ke)6 = 62.533/(1 + .12)6 = 62.533 x PVIF.12,6

= 62.533 X .507 = $31.704

Value of Common Stock (Po):

Po = PV (First 6-Years' Dividends) + PV(P6)

= 25.651 + 31.704 = $57.36 (tables)

To determine the value of a share of Cascade stock, we can use the dividend discount model (DDM) formula. The DDM values a stock by calculating the present value of its future cash flows, which in this case are the dividends.

The formula for DDM is:

Value of Stock = Dividend / (Required Rate of Return - Dividend Growth Rate)

Given information:
- Dividend = $5 (current dividend per share)
- Required Rate of Return = 12%
- Dividend Growth Rate = 7% (expected annual increase in dividends)

Let's calculate the value of a share of Cascade stock step by step:

Step 1: Determine the future dividends over the next six years.
To find the dividends for each year, we can use the formula:
Dividend Year n = Dividend Year (n-1) * (1 + Dividend Growth Rate)

Year 1: Dividend = $5 * (1 + 0.07) = $5.35
Year 2: Dividend = $5.35 * (1 + 0.07) = $5.72
Year 3: Dividend = $5.72 * (1 + 0.07) = $6.11
Year 4: Dividend = $6.11 * (1 + 0.07) = $6.54
Year 5: Dividend = $6.54 * (1 + 0.07) = $7.01
Year 6: Dividend = $7.01 * (1 + 0.07) = $7.50

Step 2: Calculate the present value of dividends for the next six years.
To calculate the present value, we need to discount the future dividends by the required rate of return. The present value of each dividend can be calculated using the formula:
PV = Dividend Year n / (1 + Required Rate of Return)^n

Present Value Year 1: PV1 = $5.35 / (1 + 0.12)^1 = $4.7777 (rounded to four decimal places)
Present Value Year 2: PV2 = $5.72 / (1 + 0.12)^2 = $4.5098 (rounded to four decimal places)
Present Value Year 3: PV3 = $6.11 / (1 + 0.12)^3 = $4.3210 (rounded to four decimal places)
Present Value Year 4: PV4 = $6.54 / (1 + 0.12)^4 = $4.1665 (rounded to four decimal places)
Present Value Year 5: PV5 = $7.01 / (1 + 0.12)^5 = $4.0379 (rounded to four decimal places)
Present Value Year 6: PV6 = $7.50 / (1 + 0.12)^6 = $3.9299 (rounded to four decimal places)

Step 3: Calculate the value of the stock by summing the present values of the dividends.
Value of Stock = PV1 + PV2 + PV3 + PV4 + PV5 + PV6
Value of Stock = $4.7777 + $4.5098 + $4.3210 + $4.1665 + $4.0379 + $3.9299
Value of Stock = $25.7428 (rounded to four decimal places)

Therefore, the value of a share of Cascade stock to an investor with a 12 percent required rate of return is approximately $25.7428.

To determine the value of a share of Cascade stock, we can use the constant growth dividend discount model (DDM). The formula for DDM is:

\[ P_0 = \frac{D_0 \cdot (1+g)}{r-g} \]

Where:
- \( P_0 \) is the current price of the stock (or the intrinsic value)
- \( D_0 \) is the current dividend per share
- \( g \) is the annual growth rate of dividends
- \( r \) is the required rate of return

Given:
- \( D_0 = $5 \) (current dividend per share)
- \( g = 7\% \) (annual growth rate of dividends)
- \( r = 12\% \) (required rate of return)

We can substitute these values into the formula to find the value of a share of Cascade stock.

\[ P_0 = \frac{$5 \cdot (1+0.07)}{0.12-0.07} \]

Simplifying the equation:

\[ P_0 = \frac{$5.35}{0.05} \]

\[ P_0 = $107 \]

Therefore, the value of a share of Cascade stock to an investor with a 12 percent required rate of return is $107.