Question

4. Consider a bank stock that you want to value using a 3-stage DDM:

a) Calculate the DDM value of the bank if last year’s dividend (i.e. Div0) was $3.50 and you

project the dividend will grow at 10%, 5%, and 2% for year 1-5, 6-10, and 11- infinity,

respectively. The bank’s cost of equity is 8.5%.

b) Instead of paying a dividend for years 1 – 5, the bank could spend the cash on acquiring

a competitor, which would raise the bank’s dividend to $5.00 in year 6, growing at 10%

until year 20, after which (i.e. year 21 to infinity) the growth rate would drop to 2%.

Should the bank engage in the acquisition, why or why not?

Answer 1

Solution a) The bank has paid $3.50 as last years' dividend, therefore D_{0 =} 3.50

Step 1: To calculate the Present Value of expected dividends for First Year to Tenth Year

Therefore, D_{1 =} D₀ (1+g)

Where, D₁ is the expected dividend to be paid at the end of year 1
g = growth rate
D_{0 =} Last years' dividend

Year	Dividend $	Discounting Factor @ 8.5%	Present Value of Dividends $
1	3.8500	0.9217	3.548
2	4.2350	0.8495	3.597
3	4.6585	0.7829	3.647
4	5.1244	0.7216	3.698
5	5.6368	0.6650	3.749
6	5.9186	0.6129	3.628
7	6.2146	0.5649	3.511
8	6.5253	0.5207	3.398
9	6.8515	0.4799	3.288
10	7.1941	0.4423	3.182
		Total Present Value of Dividends $	35.245

Step 2: Calculation of the value of the share at the end of year 10 when the dividend is growing at a constant rate.

D_{11 =} D10( 1+g) = 7.1941 ( 1 + 0.02 ) = $7.3380

Using constant growth Model,

P_{10 =} D₁₁ / (K_e - g)

Where, P₁₀ is the price of the share at the end of Tenth Year
  D₁₁ is the expected dividend at the end of year 11
g is the growth rate at which share is expected to grow constantly i.e 2%
K_e is the rate of return expected by the investors i.e. 8.50%

Therefore, P_{10 =} 7.3380/ (0.085- 0.02) = $112.8924

Step 3: Calculation of the Present value of the share at the end of year 10 when the dividend is growing at a constant rate

  P₁₀ * discounting factor @ 8.50% for the Tenth Year

= $112.8924 * 0.4423 = $49.93

Step 4: Calculation of the price of the share today

P_{0 =} (Present Value of the dividends from first year to tenth year) + (Calculation of the Present value of the share at the end of year 10)

= $35.245 + $49.93

= $85.176

Hence, the price of the stock today is $85.176.

Solution b) The bank has paid $5 at the end of Fifth Year, therefore D_{5 = $}5

Step 1: To calculate the Present Value of expected dividends for First Year to Twentieth Year:

Year	Dividend $	Discounting Factor @ 8.5%	Present Value of Dividends $
1	0.0000	0.9217	0.000
2	0.0000	0.8495	0.000
3	0.0000	0.7829	0.000
4	0.0000	0.7216	0.000
5	0.0000	0.6650	0.000
6	5.0000	0.6129	3.065
7	5.5000	0.5649	3.107
8	6.0500	0.5207	3.150
9	6.6550	0.4799	3.194
10	7.3205	0.4423	3.238
11	8.0526	0.4076	3.283
12	8.8578	0.3757	3.328
13	9.7436	0.3463	3.374
14	10.7179	0.3191	3.421
15	11.7897	0.2941	3.468
16	12.9687	0.2711	3.516
17	14.2656	0.2499	3.564
18	15.6921	0.2303	3.614
19	17.2614	0.2122	3.664
20	18.9875	0.1956	3.714
		Total Present Value of Dividends $	$50.698

Step 2: Calculation of the value of the share at the end of year 20 when the dividend is growing at a constant rate.

D_{21 =} D20( 1+g) = 18.9875 ( 1 + 0.02 ) = $19.3672

Using constant growth Model,

P_{20 =} D₂₁ / (K_e - g)

Where, P₂₀ is the price of the share at the end of Twentieth Year
  D₂₁ is the expected dividend at the end of year 21
g is the growth rate at which share is expected to grow constantly i.e 2%
K_e is the rate of return expected by the investors i.e. 8.50%

Therefore, P_{20 =} 19.3672/ (0.085- 0.02) = $297.958

Step 3: Calculation of the Present value of the share at the end of year 20 when the dividend is growing at a constant rate

  P₂₀ * discounting factor @ 8.50% for the Twentieth Year

= $297.958 * 0.1956 = $58.285

Step 4: Calculation of the price of the share today

P_{0 =} (Present Value of the dividends from first year to Twentieth year) + (Calculation of the Present value of the share at the end of year 20)

= $50.698 + $58.285

= $108.983

Hence, the price of the stock today is $108.983.

As the Value of the Share is higher for the second option, the bank is recommended to engage in the acquisition.