1. **Problem statement:** We are given a company, CSI, with a current stock price of $100, expected dividend $5, reinvestment rate 40%, growth rate 5%, and required return 14%. We need to find: - a-1. Expected long-run rate of return from purchasing the stock at $100. - a-2. Present value of growth opportunities (PVGO). - b. New stock price after a change in reinvestment policy for 5 years. 2. **Formulas and rules:** - Growth rate $g = b \times ROE$, where $b$ is the retention ratio (reinvestment rate). - Dividend payout ratio $= 1 - b$. - Price with growth (Gordon Growth Model): $$P = \frac{D_1}{r - g}$$ where $D_1$ is next year's dividend, $r$ is required return. - Price without growth (no reinvestment): $$P_{no\ growth} = \frac{E}{r}$$ where $E$ is earnings. - Present Value of Growth Opportunities (PVGO): $$PVGO = P - P_{no\ growth}$$ 3. **Given data:** - Current price $P = 100$ - Dividend $D_1 = 5$ - Reinvestment rate $b = 0.40$ - Growth rate $g = 0.05$ - Required return $r = 0.14$ 4. **Calculate earnings $E$:** Since dividend payout ratio is $1 - b = 0.60$, and dividend $D_1 = 5$, earnings are: $$E = \frac{D_1}{1 - b} = \frac{5}{0.60} = 8.3333...$$ 5. **a-1. Expected long-run rate of return:** The expected return $r$ is given as 14% (book return on equity). Since the stock price is $100$ and dividend and growth are consistent, the expected return is: $$r = \frac{D_1}{P} + g = \frac{5}{100} + 0.05 = 0.05 + 0.05 = 0.10 = 10\%$$ But this contradicts the given $r=14\%$. The problem states MDC pays whatever is necessary to yield 14% return, so the expected return is 14%. Alternatively, calculate expected return from price: $$r = \frac{D_1}{P} + g = \frac{5}{100} + 0.05 = 0.10 = 10\%$$ This suggests the price should be: $$P = \frac{D_1}{r - g} = \frac{5}{0.14 - 0.05} = \frac{5}{0.09} = 55.5555...$$ But price is $100$, so expected return is: $$r = \frac{D_1}{P} + g = 0.05 + 0.05 = 10\%$$ Hence, the expected long-run rate of return from purchasing at $100$ is 10%. 6. **a-2. PVGO calculation:** Price without growth: $$P_{no\ growth} = \frac{E}{r} = \frac{8.3333}{0.14} = 59.5238$$ PVGO: $$PVGO = P - P_{no\ growth} = 100 - 59.5238 = 40.4762$$ Rounded to 2 decimals: $$40.48$$ 7. **b. New stock price with changed reinvestment:** - For years 1 to 5, reinvestment rate $b = 0.80$, payout $= 0.20$. - From year 6 onward, payout $= 0.60$, reinvestment $= 0.40$. - Earnings $E = 8.3333$ (constant) Calculate dividends and growth: - Years 1-5 dividend: $$D_t = E \times (1 - b) = 8.3333 \times 0.20 = 1.6667$$ - Growth rate years 1-5: $$g_1 = b \times ROE = 0.80 \times 0.14 = 0.112 = 11.2\%$$ - From year 6 onward, payout 60%, growth rate: $$g_2 = 0.40 \times 0.14 = 0.056 = 5.6\%$$ Calculate dividends for years 1 to 5: $$D_1 = 1.6667$$ $$D_2 = D_1 \times (1 + g_1) = 1.6667 \times 1.112 = 1.8522$$ $$D_3 = D_2 \times 1.112 = 2.0597$$ $$D_4 = D_3 \times 1.112 = 2.2914$$ $$D_5 = D_4 \times 1.112 = 2.5503$$ Dividend at year 6 (start of new regime): $$D_6 = E_6 \times 0.60$$ Earnings grow at $g_1$ for 5 years: $$E_6 = E \times (1 + g_1)^5 = 8.3333 \times 1.112^5 = 8.3333 \times 1.6895 = 14.079$$ So, $$D_6 = 14.079 \times 0.60 = 8.4474$$ Price at year 5 (start of year 6) using Gordon model with $g_2$: $$P_5 = \frac{D_6}{r - g_2} = \frac{8.4474}{0.14 - 0.056} = \frac{8.4474}{0.084} = 100.56$$ Discount dividends and price back to present value at $r=0.14$: $$PV = \sum_{t=1}^5 \frac{D_t}{(1+r)^t} + \frac{P_5}{(1+r)^5}$$ Calculate each term: $$\frac{1.6667}{1.14} = 1.4623$$ $$\frac{1.8522}{1.14^2} = \frac{1.8522}{1.2996} = 1.4253$$ $$\frac{2.0597}{1.14^3} = \frac{2.0597}{1.4815} = 1.3907$$ $$\frac{2.2914}{1.14^4} = \frac{2.2914}{1.6889} = 1.3569$$ $$\frac{2.5503}{1.14^5} = \frac{2.5503}{1.9245} = 1.3247$$ $$\frac{100.56}{1.14^5} = \frac{100.56}{1.9245} = 52.27$$ Sum all: $$PV = 1.4623 + 1.4253 + 1.3907 + 1.3569 + 1.3247 + 52.27 = 59.23$$ **Final answers:** - a-1. Expected long-run rate of return: 10% - a-2. PVGO: 40.48 - b. New stock price after announcement: 59.23