A firm had earnings per share (eps) last year of $3 and expects that the eps will grow at a rate of 15% each year until the end of year 3. During that time all dividends are reinvested. At the end of year 4, the firm will begin paying 70% of its earnings as dividends and will have a long-run rate of 5%. The firm’s equity cost of capital is 10%. Find the firm’s current share price?
To find the firm's current share price, we need to calculate the present value of all future dividends and the future stock price at the end of year 3.
First, calculate the dividends for each year until the end of year 3, considering a 15% growth rate:
Year 1 dividend = $3 * (1 + 15%) = $3 * 1.15 = $3.45
Year 2 dividend = $3.45 * 1.15 = $3.9675
Year 3 dividend = $3.9675 * 1.15 = $4.5651
Next, calculate the future stock price at the end of year 3 using the dividend discount model:
Future stock price = Year 4 dividend / (Cost of equity capital - long-run growth rate)
Year 4 dividend = 70% * Year 3 dividend = 0.7 * $4.5651 = $3.1956
Future stock price = $3.1956 / (10% - 5%) = $3.1956 / 0.05 = $63.912
Now, calculate the present value of all future dividends:
Present value of Year 1 dividend = $3.45 / (1 + 10%) = $3.45 / 1.1 = $3.1364
Present value of Year 2 dividend = $3.9675 / (1 + 10%)^2 = $3.9675 / 1.21 = $3.2812
Present value of Year 3 dividend = $4.5651 / (1 + 10%)^3 = $4.5651 / 1.331 = $3.4321
Present value of Year 4 stock price = $63.912 / (1 + 10%)^3 = $63.912 / 1.331 = $47.9857
Finally, calculate the current share price by summing up all the present values:
Current share price = $3.1364 + $3.2812 + $3.4321 + $47.9857 = $57.8354
Therefore, the firm's current share price is approximately $57.84.
To find the firm's current share price, we need to calculate the present value of all future dividends.
Step 1: Calculate the dividends for years 1 to 4:
Year 1 dividend = EPS * (1 + growth rate) = $3 * (1 + 15%) = $3.45
Year 2 dividend = Year 1 dividend * (1 + growth rate) = $3.45 * (1 + 15%) = $3.97
Year 3 dividend = Year 2 dividend * (1 + growth rate) = $3.97 * (1 + 15%) = $4.57
Year 4 dividend = Year 3 dividend * payout ratio = $4.57 * 70% = $3.20
Step 2: Calculate the present value of dividends for years 1 to 4 using the equity cost of capital:
PV(dividend year 1) = Year 1 dividend / (1 + equity cost of capital) = $3.45 / (1 + 10%) = $3.14
PV(dividend year 2) = Year 2 dividend / (1 + equity cost of capital)^2 = $3.97 / (1 + 10%)^2 = $3.05
PV(dividend year 3) = Year 3 dividend / (1 + equity cost of capital)^3 = $4.57 / (1 + 10%)^3 = $3.72
PV(dividend year 4) = Year 4 dividend / (1 + equity cost of capital)^4 = $3.20 / (1 + 10%)^4 = $2.08
Step 3: Calculate the long-run value of the firm's dividend (at year 4) using the long-run rate and equity cost of capital:
Long-run dividend = Year 4 dividend * (1 + long-run rate) / (equity cost of capital - long-run rate) = $3.20 * (1 + 5%) / (10% - 5%) = $38
Step 4: Calculate the present value of the long-run dividend using the equity cost of capital:
PV(long-run dividend) = Long-run dividend / (equity cost of capital - long-run rate) = $38 / (10% - 5%) = $760
Step 5: Calculate the sum of the present values of dividends for each year:
Sum of PV(dividends) = PV(dividend year 1) + PV(dividend year 2) + PV(dividend year 3) + PV(dividend year 4) + PV(long-run dividend) = $3.14 + $3.05 + $3.72 + $2.08 + $760 = $771.99
Step 6: The firm's current share price is equal to the sum of the present values of dividends:
Current share price = $771.99