Question

This question is already answered before but all of them are wrong. here is the note from part b) that makes the difference. Please pay attention to the note.

[ Note that this cannot be equal to the price at time 5 for the stock in part a, since in part a, D₁ = 20, and will grow every year at g = 0.12. ]

This note will appear again in part b) and because of this note all the answers in other parts will be affected. If it is not the case please answer the note in detail.

here is the question:

One of the key features of the Discounted Dividend Model for valuing shares is that it can be used to predict the share price and the dividend at any time in the future.

The formula can be generalized as P_n = D_n+1 / (k–g).

So, you can predict the price at any n, as long as the following conditions are satisfied:

• you know the dividend at the end of period n, and
• both k and g are constant forever.

To illustrate this, complete the following problem:

Notation: For all n>0, let

• P_n = Price at time n  
• D_n = Dividend at time n (i.e., the end of period n)     
• Y_n = Earnings in period n
• r = retention ratio = (1 – D_n) / Y_n
• E_n = Equity at the end of year n
• k = discount rate
• g = dividend growth rate = r x ROE
• ROE = Y_n / E_n-1

We will further assume that k and ROE are constant, and that r and g are constant after the first dividend is paid.

a) Using the Discounted Dividend Model, calculate the price P₀ if D₁ = 20, k = 0.15, and g = r x ROE = 0.8 x 0.15 = 0.12.

please pay attention to b) since it has the note.

b) What will P₅ be if D₆ = 20, k = 0.15, and g = r x ROE = 0.8 x 0.15 = 0.12?

Note that this cannot be equal to the price at time 5 for the stock in part a, since in part a, D₁ = 20, and will grow every year at g = 0.12.

c) Recall that the value of any stock is equal to the present value of all future cash flows (i.e., dividends). Note that P₅ in part b above captures the value at the end of year 5 of all dividends from D₆ onward into the future (i.e., D₇, D₈, D₉, etc.).   

If P₅ = your result from part b, assuming no dividends are paid until D₆, what would the following share values be?

i. P₀, the value now, at time 0?

ii. P₁, the value 1 year from now?

iii. P₂, the value 2 years from now?

d) In part c, what is the relationship between P₂ and P₁ (i.e., P₂/P₁)? Why?

Answer 1

a)

P0 = D1 / (k–g) = 20 / (0.15 - 0.12) = 20 / 0.03 = 666.67

b)

P5 = D6 / (k–g) = 20 / (0.15 - 0.12) = 20 / 0.03 = 666.67

c)

i. Discounting the value of P5 at the discount rate, k = 0.15 and calculating present value

P0 = P5 / (1+k)^5 = 666.67 / 1.15^5 = 331.42

ii.

Discounting the value of P5 at the discount rate, k = 0.15 and calculating value one year from now

P1 = P5 / (1+k)^4 = 666.67 / 1.15^4 = 381.17

iii.

Discounting the value of P5 at the discount rate, k = 0.15 and calculating value two years from now

P2 = P5 / (1+k)^3 = 666.67 / 1.15^3 = 438.34

d) P2 / P1 = 1.15 = 1+k

The ratio of P2 / P1 is 1+k as the stock price is expected to grow at the discount rate or cost of equity (assuming no dividend is paid out).