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Common Stock Valuation

Terms:

D_o  = most recent dividend paid.

D₁   = the first dividend expected.

P_o  = actual market price of the stock today.

P^{^}_t (P hat t) = expected price of the stock at the end of year t.

g  = expected rate of growth in dividends per share.

ks = The minimum rate of return on a stock that  stockholder considers acceptable.

(minimum acceptable, or required, rate of return on the stock, considering both its riskiness and the returns available on other investments)

k^{^}s (k hat s)= expected rate of return, rate of return on a stock that a stockholder expects to receive.

k^-s (k bar s) = actual, or realized, after the fact rate of return 

(rate of return on a stock actually received by stockholders. k-s may be > or < k^s and/or ks)

D₁ / P_o = expected dividend yield, by the current price of a share of stock, during the coming year.

P^{^}₁ - P_o / P_o = expected capital gain yield on the stock during the coming year.

Expected total return = k^{^}s = expected dividend yield (D₁/P_o) plus expected capital gains yield [(P^1 - Po)/Po]

Value of Stock = P^{^}_o = PV of expected future dividends

• For any individual investor, the expected cash flows consist of expected dividends + the expected sales price of the stock. However, the sale price the current investor receives will depend on the dividends some future investor expects. Therefore, for all present and future investors, in total, expected cash flows are the expected future dividends. Hence, the value of a share of its stock must be established as the present value of that expected dividend stream.

Normal, or Constant, Growth:

Growth which is expected to continue into the foreseeable future at bout the same rate as that of the economy as a whole; g = a constant.

Dt = Do(1+g)^t 

P^o = D_o (1+g) / ks - g    =>    D₁ / ks - g        [Gordon Model] ....................[Eq. # 2 ] 

Assumptions:

1. the growth in earnings and dividends for the firm will progress at a constant rate into the future, and

2. that ks > g

Q: Hubco has just paid a dividend of Rs.2.00 and if investors expect a 7% growth rate for the company throughout the foreseeable future, then what would be the price of a stock of HubPower Company at Dec 31. 2000? when minimum acceptable rate of return by investors (ks) is 12%.

Do = 2;  D1 = 2(1.07) = Rs.2.14;  ks = 12%;  g = 7%; P^o = ??

P^o = 2(1.07) / 0.12 - 0.07 => 2.14 / 0.05 => Rs.42.80

We can obtain the same results by using equation number 1, by calculating each dividends present value at 12% discount rate up to the infinite period and add them all together, we will reach at same result Rs.42.80.

If the equation is used where ks is not greater than g, the results will be meaningless.  For example; ks = 12%; D1 = Rs.2.14; but g = 15% (> than 'ks'), then:

Do = 2;  D1 = 2(1.07) = Rs.2.14;  ks = 12%;  g = 15%; P^o = ??

P^{^}o = 2.14 / 0.12 - 0.15 => 2.14 / -0.03 => Rs.-71.33

Expected Rate of Return on a Constant Growth Stock:

Expected rate of return = Expected Div. yield + Expected growth rate or capital gain yield

                        k^{^}s =                D₁ /P_o                +        g  .....................  [Eq. #  3]    

Q:    Do = 2;  D1 = 2(1.07) = Rs.2.14;   P^o= Rs.42.80%;  g = 7%; ks = ??

k^s = 2.14 / 42.80 + 7% = 5% + 7% = 12%

Q:    Suppose the same analysis is continued for year 2001, so Po = Rs.42.80 (calculated on 31.12.2000); and D1= Rs.2.14 (dividend expected at the end of 2001) What should the stock price be at the end of 2001?

Solution:

This time the same equation 3 will be used. The only difference will be in dividend computation, as follow:

D2 = D1(1+g) or D2001 = D2000(1+g) = 2.14(1+0.07) = Rs.2.29

P^31.12.2001 = D2002 / ks - g =>     2.29/0.12-0.07 =>    Rs.45.80

Note:    P^2001= Rs.45.80 is 7% > P2000= Rs.42.80

                Po(1+g) = P^1 =>    42.80(1.07) = Rs.45.80

Thus, you would expect to make a capital gain of Rs. 3.00 (45.80 - 42.80) during 2001 and to have a capital gains yield of 7%.

Capital Gains yield ₂₀₀₁ = Capital Gain / Beginning Price => P^{^}₁ - P_o / P_o => 3/42.80 = 7%

Dividend Yield ₂₀₀₁= D₂₀₀₁ / P^{^}₂₀₀₀   ..................................................   (4)

Dividend Yield ₂₀₀₁ = 2.29 / 45.80 = 0.05 = 5%

For a constant growth stock, following conditions must hold:

1. The dividend is expected to grow forever at a constant rate, "g". This also requires that earnings grow at the rate "g".

2. The stock price is expected to grow at this same rate. [Po(1+g)]

3. The expected dividend yield (D1/Po) is a constant.

4. The expected Capital gains yield is also a constant, and it is equal to "g". [P^1 - Po / Po]

5. The expected total rate of return, k^s is equal to the expected dividend yield + the expected growth rate, "g". [ k^s = D1/Po + g ]

Supernormal, or Non-constant, Growth:

Firms typically go through life cycles. During the early part of their lives, their growth is much faster than that of the economy as a whole; then they match the economy's growth; and, if management cannot prevent it, they enter a final period when their growth is slower than that of the economy.

Supernormal growth is that phase of life cycle in which growth is much faster than that of the economy as a whole.

Following three steps are required to be computed to find the value of Supernormal (Non-constant) Growth stock:

1. Find the Dividends expected at the end of each year during the period of supernormal growth.

2. Find the expected price of the stock at the end of the supernormal growth period, at which point it has become a normal, constant growth stock.

3. Discount all the expected cash flows through the end of the supernormal growth period, and sum to find the intrinsic value of the stock, P^o.

Data: