To estimate the value of Procter & Gamble (PG) shares at the end of 2021 using the dividend-discount model (DDM), we follow these steps:

Step 1: Calculate the dividends for the first 5 years (2022-2026)

Given that the dividend increases by 8% per year for the first 5 years, we calculate the dividends as follows:

• $D_1$ (2022) = $3.40 \times (1 + 0.08) = 3.40 \times 1.08 = 3.672$
• $D_2$ (2023) = $D_1 \times (1 + 0.08) = 3.672 \times 1.08 = 3.966$
• $D_3$ (2024) = $D_2 \times (1 + 0.08) = 3.966 \times 1.08 = 4.283$
• $D_4$ (2025) = $D_3 \times (1 + 0.08) = 4.283 \times 1.08 = 4.625$
• $D_5$ (2026) = $D_4 \times (1 + 0.08) = 4.625 \times 1.08 = 4.995$

Step 2: Calculate the terminal value

Starting from 2027, the dividends grow at 3% indefinitely. The terminal value at the end of 2026 is based on the dividend in 2027 (which is $D_6$) and the growth rate of 3%. We use the formula for the terminal value:

$\text{Terminal Value at 2026} = \frac{D_6}{r - g}$

Where:

• $D_6$ is the dividend in 2027,
• $r = 0.06$ is the cost of equity,
• $g = 0.03$ is the long-term growth rate.

First, calculate $D_6$: $D_6 = D_5 \times (1 + 0.03) = 4.995 \times 1.03 = 5.145$

Now calculate the terminal value: $\text{Terminal Value at 2026} = \frac{5.145}{0.06 - 0.03} = \frac{5.145}{0.03} = 171.50$

Step 3: Calculate the present value of the dividends and the terminal value

We now discount the dividends and the terminal value back to the end of 2021 using the formula for the present value:

$PV = \frac{D_t}{(1 + r)^t}$

We calculate the present value for each dividend and the terminal value:

• $PV(D_1) = \frac{3.672}{(1.06)^1} = 3.463$
• $PV(D_2) = \frac{3.966}{(1.06)^2} = 3.526$
• $PV(D_3) = \frac{4.283}{(1.06)^3} = 3.591$
• $PV(D_4) = \frac{4.625}{(1.06)^4} = 3.657$
• $PV(D_5) = \frac{4.995}{(1.06)^5} = 3.725$

Now, discount the terminal value: $PV(\text{Terminal Value}) = \frac{171.50}{(1.06)^5} = 128.185$

Step 4: Sum the present values

Finally, we sum up the present values of the dividends and the terminal value:

$\text{Value of PG Stock} = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(D_5) + PV(\text{Terminal Value})$ $\text{Value of PG Stock} = 3.463 + 3.526 + 3.591 + 3.657 + 3.725 + 128.185 = 146.147$

Conclusion:

The estimated value of Procter & Gamble’s stock at the end of 2021 is approximately $146.15 per share.

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Would you like more details on any step or have any questions?

Here are some related questions:

1. How does the dividend growth rate impact the valuation of a stock?
2. What happens to stock value if the cost of equity increases?
3. How is the terminal value calculated in different valuation models?
4. Why is it important to discount future dividends?
5. Can dividend-discount models be applied to non-dividend-paying companies?

Tip: Always compare different growth scenarios (optimistic, realistic, pessimistic) when estimating stock prices using models like DDM.