Let's break down the problem step by step: --- **Problem Restatement:** - Ninja Co. will pay a dividend of **$5.00**, which will **increase by 5% each year for the following three years**, and then **grow at an annual rate of 6% forever**. - You expect a **6% required return** on your invested capital. - **What price would you pay for a share in this company?** --- ### **Step 1: Find the Dividends for the Next Years** #### **Given:** - \( D_0 = \$5.00 \) (This is the next dividend) - Growth for next 3 years: 5% per year - After that: 6% forever #### **Calculate the next 3 dividends:** \[ D_1 = D_0 \times (1 + 0.05) = 5.00 \times 1.05 = \$5.25 \] \[ D_2 = D_1 \times 1.05 = 5.25 \times 1.05 = \$5.5125 \] \[ D_3 = D_2 \times 1.05 = 5.5125 \times 1.05 = \$5.7881 \] --- ### **Step 2: Calculate the Stock Price at the End of Year 3** After year 3, dividends grow at **6% forever**. The price at the end of year 3 (\(P_3\)) is the present value of all future dividends from year 4 onwards. \[ P_3 = \frac{D_4}{r - g} \] Where: - \( D_4 = D_3 \times 1.06 = 5.7881 \times 1.06 = \$6.1354 \) - \( r = 0.06 \) (required return) - \( g = 0.06 \) (perpetual growth rate) \[ P_3 = \frac{6.1354}{0.06 - 0.06} \] But, **if \( r = g \), the present value formula doesn’t apply (it would be undefined).** Typically, the required return should be higher than the perpetual growth rate. Double-check if the required return and perpetual growth are both 6%. If so, the price theoretically approaches infinity, which is not practical. Let's proceed under the assumption that there's a typo, and the required return should be higher than the perpetual growth rate, for example, **required return = 8%**. Let's recalculate \( P_3 \) **assuming required return \( r = 0.08 \) (8%)**. \[ P_3 = \frac{6.1354}{0.08 - 0.06} = \frac{6.1354}{0.02} = \$306.77 \] --- ### **Step 3: Discount All Cash Flows to Present Value** \[ P_0 = \frac{D_1}{(1+r)^1} + \frac{D_2}{(1+r)^2} + \frac{D_3}{(1+r)^3} + \frac{P_3}{(1+r)^3} \] Let’s use \( r = 8\% = 0.08 \): \[ P_0 = \frac{5.25}{1.08^1} + \frac{5.5125}{1.08^2} + \frac{5.7881}{1.08^3} + \frac{306.77}{1.08^3} \] Calculate each term: - \( 1.08^1 = 1.08 \) - \( 1.08^2 = 1.1664 \) - \( 1.08^3 = 1.2597 \) \[ P_0 = \frac{5.25}{1.08} + \frac{5.5125}{1.1664} + \frac{5.7881}{1.2597} + \frac{306.77}{1.2597} \] \[ P_0 = 4.86 + 4.73 + 4.60 + 243.61 = \$257.80 \] --- ### **Step 4: Final Answer** **If the required return is 8% (and perpetual growth is 6%),** \[ \boxed{P_0 = \$257.80} \] --- #### **Note:** If the required return is actually **6%**, the formula for a perpetuity with growth doesn't work (division by zero). Please double-check the required return value as this is likely a typo in the problem. --- **If you have the correct required return, substitute it into the formula above. The process will remain the same.**