# Stock Valuation Problem Solutions Let's address each part step by step. --- ## Given Data - Last dividend paid: \( D_ = \$6.00 \) - Capitalization (discount) rate: \( r = 11\% = .11 \) - Growth rates: as specified in each part --- ## a) Present Stock Price (No Growth) - **Formula:** \( P_ = \frac{D_1}{r} \) If no growth, \( D_1 = D_ \) - **Calculation:** \( P_ = \frac{6.00}{.11} = \$54.55 \) --- ## b) Constant Growth at 8% - **Growth rate:** \( g = 8\% = .08 \) - **Dividend next year:** \( D_1 = D_ \times (1 + g) = 6.00 \times 1.08 = \$6.48 \) - **Gordon Growth Model:** \( P_ = \frac{D_1}{r - g} = \frac{6.48}{.11 - .08} = \frac{6.48}{.03} = \$216.00 \) - **Dividend Yield:** \( \text{Dividend Yield} = \frac{D_1}{P_} = \frac{6.48}{216.00} = .03 = 3\% \) --- ## c) Supernormal Growth, Then Constant - **Growth:** 15% for 3 years, then 8% thereafter. - **Step 1: Find dividends for next 4 years.** - \( D_1 = 6.00 \times 1.15 = \$6.90 \) - \( D_2 = 6.90 \times 1.15 = \$7.935 \) - \( D_3 = 7.935 \times 1.15 = \$9.12525 \) - \( D_4 = 9.12525 \times 1.08 = \$9.85527 \) - **Step 2: Find price at year 3 (start of constant growth):** - \( P_3 = \frac{D_4}{r-g} = \frac{9.85527}{.11-.08} = \frac{9.85527}{.03} = \$328.51 \) - **Step 3: Discount all cash flows to present:** - \( P_ = \frac{6.90}{1.11} + \frac{7.935}{1.11^2} + \frac{9.12525 + 328.51}{1.11^3} \) - Calculate: - \( \frac{6.90}{1.11} = 6.216 \) - \( \frac{7.935}{1.2321} = 6.439 \) - \( \frac{9.12525 + 328.51}{1.367631} = \frac{337.63525}{1.367631} = 246.839 \) - **Total:** - \( P_ = 6.216 + 6.439 + 246.839 = \$259.49 \) --- ### I. If I bought today and sell in 2 years, what is annual rate of return and capital gain yield? - **Cash flows:** - Year 1: \( D_1 = 6.90 \) - Year 2: \( D_2 = 7.935 \) - Price in year 2: - First, find \( D_3 = 9.12525 \) - \( D_4 = 9.85527 \) - \( P_2 = \frac{D_3}{1.11 - .15} + \frac{D_4}{(1.11 - .08)} \) Actually, at year 2, the supergrowth lasts one more year, so: - At year 2, price is the PV of year 3 dividend and the price at year 3 (already calculated above). - \( P_2 = \frac{D_3 + P_3}{1.11} \) - \( D_3 = 9.12525 \), \( P_3 = 328.51 \) - \( P_2 = \frac{9.12525 + 328.51}{1.11} = \frac{337.63525}{1.11} = 304.11 \) - **Total cash flows:** - Year 1: 6.90 - Year 2: 7.935 + 304.11 = 312.045 - **Calculate IRR (annual rate of return):** - \( -259.49 + 6.90/(1+r)^1 + 312.045/(1+r)^2 = \) - Use financial calculator or iterative method. Approximate as follows: - Guess \( r = 20\% \): - \( PV = 6.90/1.2 + 312.045/1.44 = 5.75 + 216.70 = 222.45 \) (too low) - Try \( r = 15\% \): - \( 6.90/1.15 + 312.045/1.3225 = 6.00 + 236.01 = 242.01 \) (still too low) - Try \( r = 10\% \): - \( 6.90/1.10 + 312.045/1.21 = 6.27 + 257.07 = 263.34 \) (slightly high) - Try \( r = 11\% \): - \( 6.90/1.11 + 312.045/1.2321 = 6.216 + 253.25 = 259.47 \) - So, **annual rate of return ≈ 11%** (matches the discount rate). - **Capital gain yield:** - \( \text{Capital gain} = \frac{P_2 - P_}{P_} = \frac{304.11 - 259.49}{259.49} = .1719 = 17.19\% \) over 2 years, or approx 8.26% per year (using geometric average). - **Why not the same as rate of return?** - **Total return** includes both capital gain and dividend yield. The rate of return (IRR) considers all cash flows (dividends + price appreciation), while capital gain yield considers only price appreciation. --- ### II. How much should I pay today for the stock in two years? Why is it different from \( P_ \)? - The value today to receive the stock in 2 years is the **present value of \( P_2 \)**: - \( \text{PV} = \frac{P_2}{(1.11)^2} = \frac{304.11}{1.2321} = \$246.76 \) - **Why different from \( P_ \)?** - \( P_ \) includes the value of dividends received in years 1 and 2, since if you buy today you get those dividends. If you buy in two years, you do not. --- ### III. What is \( P_9 \)? If required return is 13%, should I buy today and sell in 8 years? - **Step 1: Find \( D_{10} \) for \( P_9 \):** - Growth at 8% after year 3. - \( D_ = 6.00 \) - \( D_1 = 6.90 \) \( D_2 = 7.935 \) \( D_3 = 9.12525 \) - After that, 8% growth: - \( D_4 = 9.12525 \times 1.08 = 9.85527 \) - \( D_5 = 9.85527 \times 1.08 = 10.6437 \) - Continue up to \( D_{10} \): - \( D_4 = 9.85527 \) - \( D_5 = 9.85527 \times 1.08 = 10.6437 \) - \( D_6 = 10.6437 \times 1.08 = 11.4952 \) - \( D_7 = 11.4952 \times 1.08 = 12.4148 \) - \( D_8 = 12.4148 \times 1.08 = 13.4079 \) - \( D_9 = 13.4079 \times 1.08 = 14.4805 \) - \( D_{10} = 14.4805 \times 1.08 = 15.6581 \) - **Step 2: \( P_9 = \frac{D_{10}}{r-g} = \frac{15.6581}{.11-.08} = \frac{15.6581}{.03} = \$521.94 \)** - **Step 3: Value of stock today if selling in 8 years (discount at 13%):** - Need to discount all dividends received (years 1-9) and \( P_9 \) back to today at 13%. - \( \text{PV} = \sum_{t=1}^{9} \frac{D_t}{(1.13)^t} + \frac{P_9}{(1.13)^9} \) - You can compute this sum; it will be less than \( P_ \) at 11%, since required return is higher. - **Should you buy?** - If the **price today** is less than this calculated PV, buy. If it is higher, do not. Since required return (13%) is greater than the expected return (11%), the present value will be **lower than current price**, so you **should not buy** if you require a 13% return. --- ### V. If you buy at \( P_ \) and sell in year 3 for \$450, what is annual return (HPY)? - **Cash flows:** - Year 1: \( D_1 = 6.90 \) - Year 2: \( D_2 = 7.935 \) - Year 3: \( D_3 = 9.12525 + 450 = 459.12525 \) - **Find IRR (HPY):** - Equation: \( -259.49 + \frac{6.90}{(1+r)} + \frac{7.935}{(1+r)^2} + \frac{459.12525}{(1+r)^3} = \) - Approximate by trial: - At \( r = 20\% \): - \( 6.90 / 1.2 = 5.75 \) - \( 7.935 / 1.44 = 5.51 \) - \( 459.12525 / 1.728 = 265.58 \) - Sum = \( 5.75 + 5.51 + 265.58 = 276.84 \) - At \( r = 25\% \): - \( 6.90 / 1.25 = 5.52 \) - \( 7.935 / 1.5625 = 5.08 \) - \( 459.12525 / 1.953125 = 235.07 \) - Sum = \( 5.52 + 5.08 + 235.07 = 245.67 \) - At \( r = 23\% \): - \( 6.90/1.23 = 5.61 \) - \( 7.935/1.5129 = 5.24 \) - \( 459.12525/1.860867 = 246.68 \) - Sum: \( 5.61 + 5.24 + 246.68 = 257.53 \) - At \( r = 22\% \): - \( 6.90/1.22 = 5.66 \) - \( 7.935/1.4884 = 5.33 \) - \( 459.12525/1.81364 = 253.25 \) - Sum: \( 5.66 + 5.33 + 253.25 = 264.24 \) - So, IRR is about **22-23%** per year. - **Annual holding period yield (HPY):** **≈ 22-23%** - **Explanation**: This is the annualized return from buying at \( P_ \) and selling at $450 in year 3, including dividends. --- ## d) Payout only $3 in dividends, plowback $3 into new project growing at 11% (discount at 13%) - **Dividend paid:** $3 - **Plowed back:** $3, invested in project growing at 11%, discount at 13% - **Value of plowed back portion:** - The PV of the growing dividend stream from the project: - \( PV = \frac{3 \times (1+

# Stock Valuation Problem Solutions ## Given Data - Last dividend paid: \( D_0 = \$6.00 \) - Capitalization (discount) rate: \( r = 11\% = 0.11 \) - Growth rates: as specified in each part --- ## a) Present Stock Price (No Growth) - **Formula:** \[ P_0 = \frac{D_0}{r} \] - **Calculation:** \[ P_0 = \frac{6.00}{0.11} = \$54.55 \] --- ## b) Present Stock Price with Constant Growth at 8% - **Growth rate:** \( g = 8\% = 0.08 \) - **Dividend next year:** \[ D_1 = D_0 \times (1 + g) = 6.00 \times 1.08 = \$6.48 \] - **Gordon Growth Model:** \[ P_0 = \frac{D_1}{r - g} = \frac{6.48}{0.11 - 0.08} = \frac{6.48}{0.03} = \$216.00 \] - **Dividend Yield:** \[ \text{Dividend Yield} = \frac{D_1}{P_0} = \frac{6.48}{216.00} \approx 0.03 = 3\% \] --- ## c) Supernormal Growth, Then Constant - **Growth:** 15% for 3 years, then 8% thereafter. ### Step 1: Calculate Dividends for the First 4 Years - \( D_1 = 6.00 \times 1.15 = 6.90 \) - \( D_2 = 6.90 \times 1.15 = 7.935 \) - \( D_3 = 7.935 \times 1.15 = 9.12525 \) - \( D_4 = 9.12525 \times 1.08 = 9.85527 \) ### Step 2: Calculate Price at Year 3 (Start of Constant Growth) - \[ P_3 = \frac{D_4}{r-g} = \frac{9.85527}{0.11-0.08} = \frac{9.85527}{0.03} = \$328.51 \] ### Step 3: Discount Cash Flows to Present Value - \[ P_0 = \frac{D_1}{(1+r)^1} + \frac{D_2}{(1+r)^2} + \frac{D_3 + P_3}{(1+r)^3} \] - Calculate: - \( \frac{6.90}{1.11} \approx 6.216 \) - \( \frac{7.935}{1.2321} \approx 6.439 \) - \( \frac{9.12525 + 328.51}{1.367631} \) - \( = \frac{337.63525}{1.367631} \approx 246.839 \) - **Total Present Value:** - \[ P_0 \approx 6.216 + 6.439 + 246.839 \approx \$259.49 \] --- ## I. Annual Rate of Return and Capital Gain Yield ### If Sold in 2 Years: - **Cash Flows:** - Year 1: \( D_1 = 6.90 \) - Year 2: \( D_2 + P_2 \) - **Calculate Price in Year 2:** - Year 3 dividend: \( D_3 = 9.12525 \) - Price in Year 2: - \( P_2 = \frac{D_3 + P_3}{1.11} = \frac{9.12525 + 328.51}{1.11} \approx \frac{337.63525}{1.11} \approx 304.11 \) - **Total Cash Flows:** - Year 1: 6.90 - Year 2: \( 7.935 + 304.11 = 312.045 \) - **Calculate Annual Rate of Return:** - Use IRR: \[ -259.49 + \frac{6.90}{(1+r)^1} + \frac{312.045}{(1+r)^2} = 0 \] - Approximate \( r \) through trial. Results show approximately \( 11\% \). ### Capital Gain Yield - \[ \text{Capital Gain} = \frac{P_2 - P_0}{P_0} = \frac{304.11 - 259.49}{259.49} \approx 0.1719 = 17.19\% \] ### Why Rate of Return Differs from Capital Gain Yield - Total return includes both dividends and capital gains. Rate of return factors in all cash flows, while capital gain yield focuses solely on price appreciation. --- ## II. Present Value to Receive Stock in Two Years - **Present Value Calculation:** - \[ PV = \frac{P_2}{(1+r)^2} = \frac{304.11}{(1.11)^2} \approx \frac{304.11}{1.2321} \approx 246.76 \] ### Explanation for Difference from \( P_0 \) - \( P_0 \) accounts for dividends received in years 1 and 2, while this PV calculation considers only the future price in two years. --- ## III. What is \( P_9 \) and Should You Buy? - **Calculate \( D_{10} \):** - Continue growth rate of 8% after Year 3. - \( D_4 = 9.85527 \) - Calculate subsequent dividends up to \( D_{10} \): - \[ D_{10} = D_9 \times (1.08) \] ### Calculate \( P_9 \) - \[ P_9 = \frac{D_{10}}{r-g} \] ### Present Value of Stock Today - Discount all cash flows for years 1 through 9 at 13%. - Compare this number with \( P_0 \). If lower, do not buy. --- ## IV. Annual Return on Investment (HPY) - **Selling Price:** \( P_3 = 450 \) - **Cash Flows:** - Year 1: \( D_1 = 6.90 \) - Year 2: \( D_2 = 7.935 \) - Year 3: \( D_3 + P_3 = 9.12525 + 450 = 459.12525 \) ### Calculate HPY - Use: \[ HPY = \frac{Total Cash Inflows}{Investment} - 1 \] - Total cash inflows to be calculated and compared to \( P_0 \). --- ## V. Project Recommendation - **New Project Payout:** - If CEO decides to payout only $3 in dividends and plowback $3 at growth of 11% (discount at 13%): - Calculate present stock price using new dividend model. - Analyze whether the project meets return expectations based on calculated present value of future cash flows. - Recommend based on calculated value versus current market price. If the price is undervalued, consider buying before project initiation.

# Analysis of Dividend Payout and Project Investment ## Given Data - Dividend paid: \( D = \$3.00 \) - Plowback amount: \( \$3.00 \) into a new project - Growth rate of project: \( g = 11\% = 0.11 \) - Discount rate for project: \( r = 13\% = 0.13 \) --- ## Present Stock Price Calculation ### Step 1: Calculate Future Cash Flows from Plowback The cash flows from the project are expected to grow at a constant rate indefinitely. Using the Gordon Growth Model: - **Future Cash Flows from Project:** - The value of the plowed back $3.00 can be calculated as: \[ PV = \frac{C}{r - g} \] Where: - \( C \) is the cash flow from the project, which is the amount plowed back multiplied by (1 + growth rate). ### Step 2: Calculate Value of Future Cash Flows - Cash flow from project: \[ C = 3 \text{ (initial amount)} \times (1 + g) = 3 \times (1 + 0.11) = 3 \times 1.11 = 3.33 \] - Present value of cash flows from project: \[ PV = \frac{3.33}{0.13 - 0.11} = \frac{3.33}{0.02} = 166.50 \] ### Step 3: Total Present Stock Price The total present stock price will be the sum of the present value of the dividend paid and the present value of the future cash flows from the project: \[ P_0 = \text{Dividend} + PV_{project} \] \[ P_0 = 3 + 166.50 = 169.50 \] --- ## Project Recommendation - **Recommendation:** The project has a present value of future cash flows of \$166.50, which exceeds the dividend payout. It indicates that investing in the project could yield a higher long-term return than paying out dividends. --- ## Should You Buy the Stock Before Project Initiation? - **Analysis:** - If the market price of the stock is below \$169.50, it may be a good investment opportunity, as the stock is undervalued compared to the calculated present value. - If the stock price is above \$169.50, it may not be advisable to buy, as the stock does not reflect the intrinsic value derived from future cash flows. --- ## Conclusion - The analysis shows that investing in the new project is beneficial, given the expected growth rate and the present value of future cash flows. The decision to buy the stock should depend on its current market price relative to the calculated present stock price of \$169.50.