Problems on Stock Valuation (Soln)

5. The share of certain stock paid a dividend of Rs. 2 just now. The dividend is expected to grow
at a constant rate of 6% in the future. The required rate of return on this stock is considered to
be 12%. How much should this stock sell for now? Assuming that the expected growth rate
and required rate of return remain the same, at what price should the stock sell 2 years hence?

Soln.
Stock valuation based on single-stage dividend discount model: Growing perpetuity
0 1 2 3
k = 12%

D0 = 2 D1 D2 D3
g = 6%
1 𝐷
PV(stock) = 𝑘−𝑔

D0 = 2, g = 6%, D1 = 2 x (1+6%) = 2.12, k = 12% , PV = ?

PV(stock) today = 2.12/(0.12 -0.06) = Rs 35.33

3 𝐷
PV(stock) after 2 years = 𝑘−𝑔 = 2.382/(0.12-0.06) = Rs 39.70

6. Fizzle Limited is facing gloomy prospects. The earnings and dividends are expected to decline
at the rate of 4%. The previous dividend was Rs. 1.50. If the current market price is Rs. 8.00,
what rate of return do investors expect from the stock of Fizzle Limited?

Soln.
Stock valuation based on single-stage dividend discount model: Growing perpetuity
0 1 2 3

D0 = 1.50 D1 D2 D3
P=8 g = - 4%
D0 = 1.50, g = -4%, D1 = 1.5 x (1 – 4%) = 1.44, P = 8, k = ?

At PV = P,
𝐷1
P = 𝑘−𝑔

𝐷1
 k= + 𝑔 = 1.44/8 + (-0.04) = 0.14 = 14%
𝑃
 (cross check: 1.44/(0.14 – (-0.04)) = 8 = P)

7. The current dividend on equity share of Dizzy Limited is Rs. 2.00 Dizzy is expected to enjoy
an above normal growth rate of 18% for 6 years. Thereafter the growth rate will fall and
stabilize at 12%. Equity investors require a rate of return of 16% from Dizzy’s stock. What is
the intrinsic value of the equity share of Dizzy?

Soln.
0 1 2 3 4 5 6 7

D0 = 2 D1 D2 D3 D4 D5 D6 D7
g = 18% g = 12%
Stock valuation based on two-stage dividend discount model: 1 growing annuity + 1 growing
perpetuity

PV (stock) = PVGA on D1 (k = 16%, g = 18%, 6 years) + PV (16%,6) of [PVGP on D7 (k =

D1 = 2 x (1+0.18) = 2.36, D7 = 2 x (1+0.18)6 x (1+0.12) = 6.047

Alternatively, we could estimate PV of each year’s dividend for the first stage:
Year Growth (g) Dividend PV
0 2.000 -
1 18% 2.360 2.034
2 18% 2.785 2.070
3 18% 3.286 2.105
4 18% 3.878 2.142
5 18% 4.576 2.178
6 18% 5.399 2.216
7 12% 6.047

k 16%
PVGA (yr 1-6) 12.745
PVGP (after yr 6) 151.175
PV (PVGP) 62.049
Total Rs 74.79
For year 1-6, you can use NPV function in Excel = NPV(16%, Div1:Div6)

8. The Commonwealth Corporation’s earnings and dividends have been growing at the rate of
12% per annum. This growth rate is expected to continue for 4 years. After that the growth rate
would fall to 8% for the next four years. Beyond that the growth rate is expected to be 5%
forever. If the last dividend was Rs. 1.50 and the investors’ required rate of return on the stock
of Commonwealth is 14%, how much should be the market value per share of Commonwealth
Corporation’s equity stock?

Soln.
0 1 2 3 4 5 6 7 8 9
k = 14%

D0 = 1.50 D1 D2 D3 D4 D5 D6 D7 D8 D9
g = 12% g = 8% g = 5%
Stock valuation based on three-stage dividend discount model: 2 growing annuities + 1
growing perpetuity
D1 = 1.50 x 1.12 = 1.680, D5 = 1.50 x (1.12^4) x 1.08 = 2.549, D9 = 1.50 x (1.12^4) x (1.08^5)
x 1.05 = 3.372
1.68 1.12 4
PVGA 1 = [1 − (1.14) ] = 5.741
(0.14−0.12)
2,549 1.08 4 8.263
PVGA 2 = [1 − (1.14) ] = 8.263; PV of PVGA2 = (1.14)4 = 4.892
(0.14−0.08)
3.372 37.463
PVGP =(0,14−0.05) = 37.463, PV of PVGP = (1.14)4 = 13.133
PV(stock) = 5.741 + 4.892 + 13.133 = Rs 23.77

Though we can use growing annuity formulas as described above, it may be simpler to estimate
the PV of first 2 stages by estimating PV of each year’s dividend and summing up.
Year Growth (g) Dividend PV
0 1.50 -
1 12% 1.680 1.474
2 12% 1.882 1.448
3 12% 2.107 1.422
4 12% 2.360 1.397
5 8% 2.549 1.324
6 8% 2.753 1.254
7 8% 2.973 1.188
8 8% 3.211 1.126
9 5% 3.372

k 14%
PV (yr 1-8) 10.634
PVGP (after yr 8) 37.463
PV (PVGP) 13.133
Total Rs 23.77
For year 1-8, you can use NPV function in Excel = NPV(14%, Div1:Div8)
9. The current dividend on an equity share of International Chemicals Limited is Rs. 4.00. The
present growth rate is 20%. However, this will decline linearly over a period of 8 years and
stabilize at 10%. What is the intrinsic value per share of International Chemicals Limited if
investors require rate of return of 18%?

Soln.
Stock valuation based on DDM with different growth rates, followed by perpetuity
Year Growth (g) Dividend PV
0 20.00% 4.000 -
1 18.75% 4.750 4.025
2 17.50% 5.581 4.008
3 16.25% 6.488 3.949
4 15.00% 7.461 3.849
5 13.75% 8.487 3.710
6 12.50% 9.548 3.537
7 11.25% 10.622 3.335
8 10.00% 11.685 3.109
9 10.00% 12.853

Ke 18%
PV (yr 1-8) 29.521
PVGP (after yr 8) 160.665
PV (PVGP) 42.743
Total Rs 72.26
For year 1-8, you can use NPV function in Excel = NPV(18%, Div1:Div8)
Solution to Text Book Problems (Chapter 4)
4.

r = 5/40 = .125, or 12.5%.

5.

P0 = 10/(.08 − .05) = $333.33.

6.

First we must determine the price based on dividends per share for years 1–4. Then, we must
account for the growth in earnings per share. With next year’s EPS at $15 and EPS growing at
5% per year, the forecasted EPS at year 4 is $15 x (1.05)4 = $18.23. Therefore, the forecasted
price per share at year 4 is $18.23/.08 = $227.91. Therefore, the current price is:

18.
At a capitalization rate of 10%, Stock C is the most valuable.
For a capitalization rate of 7%, The results are:
PA = `142.86
PB = `166.67
PC = `156.48
Therefore, Stock B is the most valuable at a 7% capitalization rate.