To determine the price of Halliford Corporation's stock, we can use the Dividend Discount Model (DDM) with multiple stages of growth, since the retention rate and growth rate change over time.

Step 1: Calculate the growth rates for each period.

• For the first two years: The firm retains 100% of its earnings, meaning all earnings will be reinvested. The growth rate will be: $g_1 = \text{Retention Rate} \times \text{Return on Retained Earnings} = 1 \times 26\% = 26\%$

• For years 3 and 4: The firm retains 52% of its earnings. The growth rate will be: $g_2 = 0.52 \times 26\% = 13.52\%$

• From year 5 onward: The firm retains 21% of its earnings. The growth rate will be: $g_3 = 0.21 \times 26\% = 5.46\%$

Step 2: Determine the dividends for each year.

• Year 1 and Year 2: Since the firm retains 100% of its earnings, no dividends will be paid out. $D_1 = 0 \quad \text{and} \quad D_2 = 0$

• Year 3: The firm retains 52% of its earnings, meaning 48% of its earnings will be paid as dividends. The earnings in year 3 will have grown by 26% per year for two years: $\text{Earnings in Year 3} = 3.06 \times (1 + 0.26)^2 = 3.06 \times 1.5876 = 4.856$ The dividend in year 3 is: $D_3 = 4.856 \times (1 - 0.52) = 4.856 \times 0.48 = 2.331$

• Year 4: Earnings in year 4 will grow by 13.52% over year 3: $\text{Earnings in Year 4} = 4.856 \times (1 + 0.1352) = 4.856 \times 1.1352 = 5.516$ The dividend in year 4 is: $D_4 = 5.516 \times (1 - 0.52) = 5.516 \times 0.48 = 2.648$

Step 3: Calculate the terminal price in year 4 (based on dividends from year 5 onward).

From year 5 onward, the firm retains 21% of its earnings, and the growth rate of dividends will be 5.46%. The dividend in year 5 will be based on the earnings growth rate from year 4: $\text{Earnings in Year 5} = 5.516 \times (1 + 0.0546) = 5.516 \times 1.0546 = 5.818$ The dividend in year 5 is: $D_5 = 5.818 \times (1 - 0.21) = 5.818 \times 0.79 = 4.597$

The terminal price at the end of year 4 is calculated using the perpetuity formula, since the growth rate becomes constant at 5.46% from year 5 onward: $P_4 = \frac{D_5}{r - g_3} = \frac{4.597}{0.103 - 0.0546} = \frac{4.597}{0.0484} = 94.96$

Step 4: Calculate the present value of dividends and the terminal price.

Now, we discount the dividends and the terminal price back to the present using the equity cost of capital (10.3%):

• $D_3$ discounted to the present: $PV(D_3) = \frac{2.331}{(1 + 0.103)^3} = \frac{2.331}{1.342} = 1.738$

• $D_4$ discounted to the present: $PV(D_4) = \frac{2.648}{(1 + 0.103)^4} = \frac{2.648}{1.481} = 1.788$

• Terminal price $P_4$ discounted to the present: $PV(P_4) = \frac{94.96}{(1 + 0.103)^4} = \frac{94.96}{1.481} = 64.11$

Step 5: Sum the present values to find the stock price.

Finally, sum the present values of the dividends and the terminal price: $\text{Price of the stock} = 1.738 + 1.788 + 64.11 = 67.64$

Conclusion:

The estimated price of Halliford Corporation's stock is $67.64.

Would you like further details on any part of this process? Here are five related questions to expand your understanding:

1. How does the Dividend Discount Model handle different growth rates in multiple stages?
2. What factors influence a company's retention rate decisions?
3. Can you explain the importance of the equity cost of capital in stock pricing?
4. How does reinvesting retained earnings impact a company's growth?
5. What is the significance of the terminal value in stock price calculations?

Tip: Always check the consistency of growth assumptions when using models like the DDM for stock valuation.