Halliford Corporation expects to have earnings this coming year of $ 3.245 per share. Halliford plans to retain all of its earnings for the next two years.​ Then, for the subsequent two​ years, the firm will retain 48% of its earnings. It will retain 23% of its earnings from that point onward. Each​ year, retained earnings will be invested in new projects with an expected return of 25.7% per year. Any earnings that are not retained will be paid out as dividends. Assume​ Halliford's share count remains constant and all earnings growth comes from the investment of retained earnings. If​ Halliford's equity cost of capital is 10.6%​, what price would you estimate for Halliford​ shares?

To estimate the price of Halliford Corporation's shares, we can use a **multi-stage dividend discount model (DDM)**, since the firm plans to retain different percentages of earnings over various stages. The stock price is based on the future dividends discounted to the present value using the cost of equity capital.

### Key Information:
- **Earnings per share (EPS) next year (Year 1):** $3.245
- **Retention ratios:**
  - 100% for the first two years (no dividends in years 1 and 2)
  - 48% retention in years 3 and 4 (dividends in these years are paid out from the remaining 52%)
  - 23% retention from year 5 onward (dividends from 77% of earnings from this point forward)
- **Return on retained earnings (ROE):** 25.7%
- **Cost of equity (discount rate):** 10.6%

The process for calculating the stock price involves several steps:

### 1. **Calculate future earnings:**
Since Halliford plans to reinvest retained earnings at 25.7%, the earnings will grow based on this rate during the retention period.

For the first two years, all earnings are reinvested:
- **Year 1 Earnings (E1):** $3.245
- **Year 2 Earnings (E2):** $ E1 \times (1 + 0.257) $
- **Year 3 Earnings (E3):** $ E2 \times (1 + 0.257) $
- **Year 4 Earnings (E4):** $ E3 \times (1 + 0.257) $

### 2. **Calculate dividends for years 3 and 4:**
From year 3 onward, the firm retains only 48% of its earnings, paying out 52% as dividends. Similarly, for year 5 onward, the retention rate drops to 23%, with 77% paid as dividends.

Dividends are calculated as:
- **Year 3 Dividend (D3):** $ E3 \times (1 - 0.48) $
- **Year 4 Dividend (D4):** $ E4 \times (1 - 0.48) $

### 3. **Calculate dividends from year 5 onward:**
From year 5 onward, the firm retains 23% and pays out 77% as dividends. We assume that earnings will continue to grow indefinitely at the sustainable growth rate.

- **Growth rate from year 5 onward (g):** $ 23\% \times 25.7\% = 5.911\% $
- **Year 5 Dividend (D5):** $ E5 \times (1 - 0.23) $
- The stock price at the end of year 4 will be the present value of all dividends from year 5 onward, using the Gordon growth model:
  \[
  P_4 = \frac{D5}{(r - g)}
  \]
  where $ r $ is the cost of equity and $ g $ is the long-term growth rate.

### 4. **Discount all dividends and future stock price to the present value:**
Finally, discount all the dividends (from years 3 and 4) and the future stock price at year 4 to present value using the equity cost of capital (10.6%).

Let me calculate each of these steps.The estimated price for Halliford Corporation's shares is approximately **$93.12**.

Would you like more details on the calculations, or do you have any further questions?

Here are 5 related questions to expand on this concept:
1. How does a change in the equity cost of capital affect the stock price?
2. What happens to the stock price if the retention rate in later years increases?
3. How does the dividend discount model compare with other valuation methods?
4. What role does the return on retained earnings play in this stock valuation?
5. How would inflation influence the assumptions used in this model?

**Tip:** The retention rate directly affects a company's growth; higher retention rates lead to more reinvestment and potentially higher earnings growth.

Calculate the estimated stock price of Halliford Corporation using a multi-stage dividend discount model (DDM). This problem involves key finance concepts like future earnings growth, retention rates, return on equity (ROE), and dividend payouts. Suitable for undergraduate finance students.

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