Lesson: Constant Growth

Lesson: Constant Growth, Fundamentals of Financial Management

Chapter 9

Exercise 9-17

Your broker offers to sell you some shares of Bahnsen & Co. common stock that paid a dividend of $2.00 yesterday. Bahnsen's dividend is expected to grow at 5% per year for the next 3 years. If you buy the stock, you plan to hold it for 3 years and then sell it. The appropriate discount rate is 12%.

⭐ a) Find the expected dividend for each of the next 3 years; that is, calculate D₁, D₂, and D₃, Note that D₀=$2.00.

Answer:

                D1 = 2.10,

                D2 = 2.205,

                D3 = 2.315 

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"Additional information"

To calculate the expected dividends for each of the next three years (D1, D2, and D3), we can use the dividend growth rate provided and the formula for calculating future dividends:

Dt = expected dividend a the end of the year t

D1 = D0 × (1 + growth rate)

D2 = D1 × (1 + growth rate)

D3 = D2 × (1 + growth rate)

Given:

        D0 = $2.00 (dividend paid yesterday)

        Growth rate = 5% per year

Calculating the expected dividends:

        D1 = D0 × (1 + growth rate)

        D1 = 2.00 × (1 + 5%)  =   2.00 × (1 + 0.05) =   2.10

            ⛬    D1 = $2.10

        D2 = D1 × (1 + growth rate)

        D2 = 2.10 × (1 + 0.05) =   2.205

            ⛬    D1 = $2.205

        D3 = D2 × (1 + growth rate)

        D3 = 2.205 × (1 + 0.05) =   2.31525

            ⛬    D3 = $2.31525

Therefore, the expected dividends for each of the next three years are as follows:

        D1 = $2.10

        D2 = $2.205

        D3 = $2.31525

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⭐ b) Given that the first dividend payment will occur 1 year from now, find the present value of the dividend stream; that is, calculate the PVs of D₁, D₂, and D₃ and then sum these PVs.

Answer:

                Sum of PVs = PV1 + PV2 + PV3

                                    = (D1 / (1 + 12%)^1) + (D2 / (1 + 12%)^2) + (D3 / (1 + 12%)^3)

                                   = (2.10 / (1 + 0.12)^1) + (2.205 / (1 + 0.12)^2) + (2.31525 / (1 + 0.12)^3)

                Sum of PVs = 1.875 + 1.625 + 1.535 = $5.035

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"Additional information"

To find the present value (PV) of the dividend stream, we need to discount each future dividend back to the present using the appropriate discount rate.

Given:

        D1 = $2.10 (expected dividend in year 1)

        D2 = $2.205 (expected dividend in year 2)

        D3 = $2.31525 (expected dividend in year 3)

        Discount rate = 12%

        Time horizon = 3 years

To calculate the present value of each dividend:

        PV = D / (1 + r)^t

where as:

        PV is the present value,

        D is the dividend amount,

        r is the discount rate,

        t is the number of years.

Calculating the present value of each dividend:

        PV1 = D1 / (1 + 12%)^1   =   2.10 / (1 + 0.12)^1 = 1.875

            ⛬    PV1 =  $1.875

        PV2 = D2 / (1 + 0.12)^2   =   1.625

            ⛬    PV2 =  $1.625

        PV3 = D3 / (1 + 0.12)^3   =   1.535

            ⛬    PV3 =  $1.535

To find the sum of these present values:

        Sum of PVs = PV1 + PV2 + PV3

        Sum of PVs = $1.875 + $1.625 + $1.535 = $5.035

Therefore, the present value of the dividend stream is $5.035

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⭐ c) You expect the price of the stock 3 years from now to be $34.73; that is, you expected P₃ to equal $34.73. Discounted at a 12% rate, what is the present value of this expected future stock price? In other words, calculate the PV of $34.73.

Answer:

                PV of P^3 = $34.72(0.7118) = $24.72

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"Additional information"

To calculate the present value (PV) of the expected future stock price of $34.73, we need to discount it back to the present using the appropriate discount rate.

Given:

        Expected future stock price (P3) = $34.73

        Discount rate = 12%

        Time horizon = 3 years

To calculate the present value of the expected future stock price:

        PV = P / (1 + r)^t

where PV is the present value, P is the future stock price, r is the discount rate, and t is the number of years.

Calculating the present value of the expected future stock price:

        PV = 34.73 / (1 + 12%)^3    =    34.73 / (1 + 0.12)^3 = $24.72

Therefore, the present value of the expected future stock price of $34.73, discounted at a 12% rate over 3 years, is approximately $24.72

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⭐ d) If you plan to buy the stock, hold it for 3 years, and then sell it for $34.73, what is the most you should pay for it today ?

Answer

        Maximum price = Present value of the dividend stream + Present value of the expected future stock price

        Maximum price = $5.035 + $24.72 = $29.755

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"Additional information"

To determine the maximum price you should pay for the stock today, considering your plan to hold it for 3 years and sell it for $34.73, we can calculate the present value (PV) of the expected future stock price and deduct it from the present value of the dividend stream.

Given:

        Present value of the dividend stream = $5.035 (calculated in part b)

        Present value of the expected future stock price = $24.72 (calculated in part c)

To calculate the maximum price you should pay today:

        Maximum price = Present value of the dividend stream + Present value of the expected future stock price

        Maximum price = $5.035 + $24.72 = $29.755

Therefore, the most you should pay for the stock today, considering your plan to hold it for 3 years and sell it for $34.73, is approximately $29.755.

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⭐ e) Use Equation 9-2 to calculate the present value of this stock. Assume that g=5% and that it is constant.

Answer:

        Use Equation 9-2: (P0 = D0 / r + g)

                P0 = 2.00 / (0.12 + 0.05)    =    $11.765

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"Additional information"

To calculate the present value (P0) of the stock using the formula P0 = D0 / (r + g), where D0 is the dividend paid today, r is the discount rate, and g is the constant growth rate, we can plug in the given values:

Given:

        D0 = $2.00 (dividend paid yesterday)

        r = 12% (discount rate)

        g = 5% (constant growth rate)

Calculating the present value using the formula:

        P0 = D0 / r + g

        P0 = 2.00 / (0.12 + 0.05)    =    $11.765

Therefore, the present value of the stock, using the equation P0 = D0 / (r + g), with a dividend of $2.00, a discount rate of 12%, and a constant growth rate of 5%, is approximately $11.765.

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⭐ f) Is the value of this stock dependent upon how long you plan to hold it ? In other words, if you planned holding period was 2 years or 5 years rather than 3 years, would this affect the value of the stock today, P₀ ? Explain.

Answer:

    Yes, the value of the stock today (P0) is dependent on the planned holding period. The value of a stock is influenced by the expected future cash flows it will generate, including dividends and the eventual sale price. The longer the holding period, the more dividends an investor can expect to receive, and the higher the expected future stock price.

    In the given scenario, the calculation of P0 takes into account the expected dividends over the 3-year holding period, as well as the expected future stock price at the end of the 3 years. If the holding period was 2 years instead of 3, the investor would receive one less year of dividends and would not benefit from the full growth in the stock price. This would likely result in a lower present value of the stock (P0) since there are fewer expected cash flows.

    Conversely, if the holding period was extended to 5 years, the investor would receive two additional years of dividends and would potentially benefit from further growth in the stock price. This longer-term expectation of cash flows would likely result in a higher present value of the stock (P0) compared to the 3-year holding period scenario.

    In summary, the value of a stock today is influenced by the expected future cash flows, and the holding period affects the expected dividends and future stock price, consequently impacting the present value of the stock. 

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Read more:

- Eugene F. Brigham and Joel F. Houston. Fundamentals of Financial Management.

- https://books.google.co.th/books?id=l3YcCgAAQBAJ&pg=PA337&lpg=PA337&dq=constant+growth+Your+broker+offers+to+sell+you+some+shares+of+Bahnsen+%26+Co.+common+stock+that+paid+a+dividend+of+$2.00+yesterday.+Bahnsen%27s+dividend+is+expected+to+grow+at+5%25+per+year+for+the+next+3+years.+If+you+buy+the+stock,+you+plan+to+hold+it+for+3+years+and+then+sell+it.+The+appropriate+discount+rate+is+12%25.&source=bl&ots=ZDSdCkNaVK&sig=ACfU3U3OocbaH6QE2AbBdV41vW5K5daN7A&hl=en&sa=X&ved=2ahUKEwjC77bx-PH-AhVZTGwGHX-AClUQ6AF6BAgYEAM#v=onepage&q&f=false