PP Ltd Expects To Have Earnings This Year Of 4 Per Share

Pp Ltd Expects To Have Earnings This Year Of 4 Per Share It Plans T

Pp Ltd. expects to have earnings this year of $4 per share. It plans to retain all of its earnings for the next two years. It will retain 50% of its earnings in year 3 and year 4. It will retain 20% of earnings from year 5 onwards. Each year, retained earnings will be invested in new projects with an expected return of 25% per year. Any earnings that are not retained will be paid out as dividends. Assume that all earnings growth comes from the investment of retained earnings. The cost of capital is 12%. Calculate the price of PP today.

Paper For Above instruction

The valuation of PP Ltd. hinges on its capacity to generate earnings growth through reinvestment of retained earnings and the subsequent dividends paid to shareholders. Given the company's policy of retaining earnings at varying rates over the years, and investing those retained earnings at an expected return of 25%, we can model its future earnings, dividends, and stock price accordingly.

First, it’s essential to understand the sequence of earnings and reinvestment:

- Year 1 earnings: $4 per share

- Year 2 earnings: projected to grow from reinvested earnings

- Retention policy: 100% for years 1 and 2, 50% in years 3 and 4, and 20% from year 5 onward

- Return on retained earnings: 25%

- Cost of capital: 12%

The foundation for valuation is the Gordon growth model (or the dividend discount model), which states that the current stock price equals the dividend per share in the next year divided by the difference between the discount rate (cost of equity) and the growth rate of dividends.

To compute the stock price, we need to estimate future dividends and earnings, considering the reinvestment strategy and growth derived from retained earnings.

Step 1: Compute initial earnings and dividends

- Year 1 earnings: $4.00

- Year 1 dividend: Since all earnings are retained in Year 1, dividend = $0.

- Year 2 earnings: The company retains all earnings, which are assumed to grow from reinvested earnings.

Because no dividends are paid in Year 1 and Year 2, the company's dividends (which are paid only from earnings not retained) will start later when the retention rate drops or when earnings are explicitly paid out.

Step 2: Calculate future earnings based on reinvested earnings

The growth of earnings depends on reinvestment and return on retained earnings:

- The company reinvests all earnings in Year 1: earnings grow at a rate of 25%.

- The earnings in Year 2 are then derived from reinvestment, but since Year 2 also retains all earnings, the earnings will continue to grow similarly.

- From Year 3 onwards, retention drops to 50%, 50%, then 20%. The retained earnings are reinvested at 25% return, which yields additional earnings growth.

This creates a compounding effect on earnings, which can be modeled as follows:

Year 1:

- Earnings, E1 = $4

- Retained earnings = 100% of $4 = $4

- Reinvestment to generate future growth: Reinvested amount grows at 25% annually.

Year 2:

- Earnings, E2 = E1 × (1 + growth rate)

- Since Year 1 had 100% retention and reinvestment, Year 2 earnings are:

E2 = E1 × (1 + 25%) = $4 × 1.25 = $5

- Dividend in Year 2 = 0 (all earnings retained)

Year 3:

- Retention rate drops to 50%

- Earnings before growth: E2 = $5

- Retained amount: $5 × 50% = $2.50

- This retained amount is invested at 25%, so in Year 3, reinvested earnings generate additional growth:

E3 = E2 × (1 + 25%) = $5 × 1.25 = $6.25

- Dividend in Year 3 = (1 - 50%) × $6.25 = 50% of earnings = $3.125

Year 4:

- Retention rate: 50%

- Earnings E4 = E3 × 1.25 = $6.25 × 1.25 = $7.8125

- Dividend in Year 4 = 50% of $7.8125 = $3.90625

Year 5 and beyond:

- Retention rate: 20%

- Earnings E5 = E4 × 1.25 = $7.8125 × 1.25 = $9.765625

- Dividend in Year 5 = 80% of earnings = 0.8 × $9.765625 ≈ $7.8125

From Year 5 onwards, earnings grow at 25%, and the dividend payout ratio is 80%, meaning dividends will grow at the same or a similar proportion, depending on the retention ratio.

Step 3: Determine the growth rate of dividends from Year 5 onwards

Since the retained earnings are invested at 25%, and dividends are 80%, the dividend growth rate (g) can be approximated as:

g = retention rate × return on retained earnings = 0.20 × 25% = 5%

Alternatively, since 80% is paid out, dividend growth rate (g) remains at 25% adjusted by payout ratio; but more precisely, the ongoing growth rate in dividends aligns with reinvestment growth and payout ratio.

Given a dividend payout ratio of 80%, and reinvestment at 25%:

g = 25% × 0.20 = 5%

Thus, dividends from Year 5 onward grow at approximately 5% per year.

Step 4: Calculate the present value of dividends

- The dividend in Year 4 = $3.90625

- The dividend in Year 5 = $7.8125

- From Year 5 onwards: dividends grow at 5%

Applying the Gordon growth model to the dividend at Year 5, we get the terminal value at Year 4:

\[ P_4 = \frac{D_5}{r - g} = \frac{7.8125}{0.12 - 0.05} = \frac{7.8125}{0.07} \approx 111.607 \]

Now, discount this back to present (Year 0):

\[ P_0 = \frac{D_1}{(1 + r)^1} + \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + \frac{D_4 + P_4}{(1 + r)^4} \]

Calculating the discounted dividends:

- D1: Year 1 dividend is $0, since all earnings are retained in Year 1

- D2: Year 2 dividend is $0

- D3: $3.125, discounted back:

\[ \frac{3.125}{(1.12)^3} \approx \frac{3.125}{1.4049} \approx 2.224 \]

- D4: $3.90625, discounted back:

\[ \frac{3.90625}{(1.12)^4} \approx \frac{3.90625}{1.5735} \approx 2.482 \]

- P4: $111.607, discounted back:

\[ \frac{111.607}{(1.12)^4} \approx \frac{111.607}{1.5735} \approx 71.0 \]

Adding these components:

\[ P_0 = 0 + 0 + 2.224 + 2.482 + 71.0 \approx 75.7 \]

Therefore, the estimated current stock price of PP Ltd. is approximately $75.70.

This valuation captures the earnings growth driven by reinvestment, the policy of changing retention rates, and the expected return on retained earnings.

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