Multistage Dividend Growth

EXERCISE 01

Delta Services preferred shares pays no dividends before year 4. Its first dividend is €5, followed by annual growth of 0.00% forever. The required return is 8.00%.

A.Calculate today’s value.

A.

P_0=\frac{D_k}{r_E-g}\frac1{(1+r_E)^{k-1}}

P_{3}=\frac{5}{0.08-0}=62.50000000

P_0=\frac{62.50000000}{1.08^{3}}

€49.61.

Multistage Dividend Growth

EXERCISE 02

Grandi Sogni pays no dividends before year 10. Its first dividend is €8, followed by annual growth of 6.00% forever. The required return is 13.00%.

A.Calculate today’s value.

A.

P_0=\frac{D_k}{r_E-g}\frac1{(1+r_E)^{k-1}}

P_{9}=\frac{8}{0.13-0.06}=114.28571429

P_0=\frac{114.28571429}{1.13^{9}}

€38.04.

Multistage Dividend Growth

EXERCISE 03

The dividend just paid is €0.75. Dividends grow by 20.00% in each of the next two years and by 6.00% annually thereafter. The required return is 14.00%.

A.Calculate today’s share price.

A.

P_0=\frac{D_1}{1+r_E}+\frac{D_2+P_2}{(1+r_E)^2},\quad P_2=\frac{D_3}{r_E-g}

D_1=0.75(1.2)=0.9000,\quad D_2=1.0800,\quad D_3=1.1448

P_2=\frac{1.1448}{0.14-0.06}=14.3100

P_0=\frac{0.9000}{1.14}+\frac{15.3900}{1.14^2}

€12.63

Multistage Dividend Growth

EXERCISE 04

The dividend just paid is €1.00. Dividends grow by 18.00% in each of the next two years and by 6.00% annually thereafter. The required return is 12.00%.

A.Calculate today’s share price.

A.

P_0=\frac{D_1}{1+r_E}+\frac{D_2+P_2}{(1+r_E)^2},\quad P_2=\frac{D_3}{r_E-g}

D_1=1(1.18)=1.1800,\quad D_2=1.3924,\quad D_3=1.4759

P_2=\frac{1.4759}{0.12-0.06}=24.5991

P_0=\frac{1.1800}{1.12}+\frac{25.9915}{1.12^2}

€21.77

Multistage Dividend Growth

EXERCISE 05

The dividend just paid is €1.50. Dividends grow by 20.00% in each of the next two years and by 5.00% annually thereafter. The required return is 10.00%.

A.Calculate today’s share price.

A.

P_0=\frac{D_1}{1+r_E}+\frac{D_2+P_2}{(1+r_E)^2},\quad P_2=\frac{D_3}{r_E-g}

D_1=1.5(1.2)=1.8000,\quad D_2=2.1600,\quad D_3=2.2680

P_2=\frac{2.2680}{0.1-0.05}=45.3600

P_0=\frac{1.8000}{1.1}+\frac{47.5200}{1.1^2}

€40.91

Multistage Dividend Growth

EXERCISE 06

The dividend just paid is €2.00. Dividends grow by 14.00% in each of the next two years and by 6.00% annually thereafter. The required return is 10.00%.

A.Calculate today’s share price.

A.

P_0=\frac{D_1}{1+r_E}+\frac{D_2+P_2}{(1+r_E)^2},\quad P_2=\frac{D_3}{r_E-g}

D_1=2(1.14)=2.2800,\quad D_2=2.5992,\quad D_3=2.7552

P_2=\frac{2.7552}{0.1-0.06}=68.8788

P_0=\frac{2.2800}{1.1}+\frac{71.4780}{1.1^2}

€61.15

Multistage Dividend Growth

EXERCISE 07

Gillette will pay $0.65 per share one year from now. Dividends then grow by 12% per year through year 5 and by 2% thereafter. The cost of equity is 8%.

A.Forecast the dividends and terminal value.
B.Calculate the value today.

A.

Year	Dividend ($)
1	0.6500
2	0.7280
3	0.8154
4	0.9132
5	1.0228
6	1.0432

P_5=\frac{D_6}{0.08-0.02}

D_t=0.65(1.12)^{t-1}\quad(1\le t\le5)

D_6=0.65(1.12)^4(1.02)

P_5=\frac{0.65(1.12)^4(1.02)}{0.08-0.02}\approx17.387389

Terminal value at year 5 = $17.39

B.

P_0=\sum_{t=1}^{5}\frac{D_t}{1.08^t}+\frac{P_5}{1.08^5}

Component	Present value ($)
Dividends in years 1–5	3.24
Terminal value	11.83
Share price	15.07

The high-growth stage can have g > r because it is finite. The perpetual terminal stage must satisfy r > g.

PV(D)=\sum_{t=1}^{5}\frac{0.65(1.12)^{t-1}}{1.08^t}\approx3.240577

PV(P_5)=\frac{17.387389}{1.08^5}\approx11.833565

P_0\approx3.240577+11.833565=15.074142

Multistage Dividend Growth

EXERCISE 08

The dividend just paid is 0.96. The required return is 8.500%. Annual dividend growth is 11.000%, 11.000%, 11.000%, 11.000%, 11.000% over years 1–5, then 5.200% forever.

A.Value the explicit dividends and terminal value.
B.Can no-growth value and PVGO be determined from these data?

A.

D_{t+1}=D_t(1+g_t)

Year	Dividend
1	1.06560
2	1.18282
3	1.31293
4	1.45735
5	1.61766

P_0=\sum_{t=1}^{N}\frac{D_t}{(1+r_E)^t}+\frac{D_N(1+g)}{(r_E-g)(1+r_E)^N}

P_5=\frac{1.617656(1+0.052)}{0.085-0.052}\approx 51.5689

PV(D)= \frac{1.0656}{1.085^{1}}+\frac{1.182816}{1.085^{2}}+\frac{1.312926}{1.085^{3}}+\frac{1.457348}{1.085^{4}}+\frac{1.617656}{1.085^{5}}

P_0\approx 5.1422+\frac{51.5689}{1.085^{5}}

Terminal value = 51.57; price today = 39.44.

B.

P_{no\ growth}=\frac{EPS_1}{0.085}\quad\text{(EPS is not supplied)}

PV_{dividend\ freeze}=0.96/0.085\approx11.29

Not uniquely: the payout ratio or earnings per share is missing. The fundamental no-growth value is EPS₁/rₑ. Holding the current dividend fixed at $0.96 forever gives $11.29, but that is a dividend-freeze comparison, not a fully identified no-reinvestment counterfactual.

Multistage Dividend Growth

EXERCISE 09

The dividend due in year 1 is 2.00. The required return is 12.000%. Annual dividend growth is 0.000%, 5.000%, 5.000%, 5.000%, 5.000% over years 1–5 (year 1 dividend is given), then 2.000% forever.

A.Value the explicit dividends and terminal value.
B.How is the required return obtained?

A.

D_{t+1}=D_t(1+g_t)

Year	Dividend
1	2.00000
2	2.10000
3	2.20500
4	2.31525
5	2.43101

P_0=\sum_{t=1}^{N}\frac{D_t}{(1+r_E)^t}+\frac{D_N(1+g)}{(r_E-g)(1+r_E)^N}

P_5=\frac{2.431013(1+0.02)}{0.12-0.02}\approx 24.7963

PV(D)= \frac{2}{1.12^{1}}+\frac{2.1}{1.12^{2}}+\frac{2.205}{1.12^{3}}+\frac{2.31525}{1.12^{4}}+\frac{2.431013}{1.12^{5}}

P_0\approx 7.8801+\frac{24.7963}{1.12^{5}}

Terminal value = 24.80; price today = 21.95.

B.

r_E=1.20/10=12\%

Easter pays a constant $1.20 dividend and trades at $10; the exercise assumes the same required return.

Multistage Dividend Growth

EXERCISE 10

The dividend just paid is 1.20. The required return is 12.000%. Annual dividend growth is 3.000%, 4.000%, 5.000% over years 1–3, then 6.000% forever.

A.Value the explicit dividends and terminal value.

A.

D_{t+1}=D_t(1+g_t)

Year	Dividend
1	1.23600
2	1.28544
3	1.34971

P_0=\sum_{t=1}^{N}\frac{D_t}{(1+r_E)^t}+\frac{D_N(1+g)}{(r_E-g)(1+r_E)^N}

P_3=\frac{1.349712(1+0.06)}{0.12-0.06}\approx 23.8449

PV(D)= \frac{1.236}{1.12^{1}}+\frac{1.28544}{1.12^{2}}+\frac{1.349712}{1.12^{3}}

P_0\approx 3.089+\frac{23.8449}{1.12^{3}}

Terminal value = 23.84; price today = 20.06.

Multistage Dividend Growth

EXERCISE 11

Both firms require 12% equity return, have sustainable growth 10% from the next dividend onward and payout 60%. Green Business just paid 1.20, next dividend is 1.40 and book value per share 55.50. Green Growth just paid 1.60, next dividend is 2.00 and book value 68. In an alternative scenario, Green Business keeps D1 = 1.40, grows dividends 18% in years 2–3 and 9% thereafter.

A.Find prices, ROE and market-to-book ratios.
B.Value Green Business under the alternative growth path.

A.

P_{GB}=\frac{1.40}{0.12-0.10}=70,\quad P_{GG}=\frac{2}{0.12-0.10}=100

ROE=\frac{0.10}{1-0.60}=25\%

M/B_{GB}=70/55.50=1.261261,\quad M/B_{GG}=100/68=1.470588

Use the explicitly provided next dividends; their one-time growth from the just-paid amounts need not equal the subsequent sustainable growth rate.

B.

P_0=\frac{1.40}{1.12}+\frac{1.40(1.18)}{1.12^2}+\frac{1.40(1.18)^2+1.40(1.18)^2(1.09)/(0.12-0.09)}{1.12^3}

54.37

Value is lower than 70 because the lower perpetual growth rate outweighs the faster initial growth.

Multistage Dividend Growth

EXERCISE 12

SmartCap is debt-free. Its next dividend is 6, growing 5% in years 2–3, then 3% forever from year 4. Comparable equity beta is 1.25, D/E 0.20, risk-free debt 2%, market premium 5% and corporate tax 30%. Long-run payout is 40%. InfoSys has 20 million shares and perpetual earnings of 80 million, fully distributed, with the same required return. It can invest 30 million immediately to earn incremental cash profits of 35, 40 and 45 million over years 1–3; assume no further working-capital or capital-spending requirements.

A.Estimate SmartCap’s required return and share value.
B.Find SmartCap’s implied long-run ROE.
C.Value InfoSys before and after its investment.

A.

\beta_U=\frac{1.25}{1+0.70(0.20)}

R_E=0.02+0.05\beta_U=0.07482456

P_0=\frac{6}{1+R_E}+\frac{6(1.05)}{(1+R_E)^2}+\frac{6(1.05)^2+6(1.05)^2(1.03)/(R_E-0.03)}{(1+R_E)^3}

Share value = 138.78.

B.

ROE=\frac{g}{b}=\frac{0.03}{0.60}

5%, below the required return. Long-run reinvestment destroys value.

C.

P_{cash\ cow}=\frac{80/20}{R_E}=53.46

PVGO=-30+\frac{35}{1+R_E}+\frac{40}{(1+R_E)^2}+\frac{45}{(1+R_E)^3}=73.43

P_{new}=\frac{4}{R_E}+\frac{PVGO}{20}

57.13

InfoSys creates value because incremental NPV is positive. Earnings can be used here only under the explicit cash-flow assumption.

Multistage Dividend Growth

EXERCISE 13

Alfa and Beta are all-equity firms, each with book equity 27,000, 100 shares, next EPS 18, payout 50% and required return 8%. Reinvestment ROE is 10% for Alfa and 6% for Beta. Separately, Gamma just paid dividend 3, growing 10%, 12%, and 6% in years 1–3, then 3% forever. Gamma’s required return is 10%.

A.Compute dividends, growth, price, PVGO and valuation multiples.
B.Value Gamma’s shares.

A.

D_1=pEPS_1,\quad g=(1-p)ROE,\quad P_0=\frac{D_1}{R-g}

D_1=0.5(18)=9,\quad BVPS=27000/100=270

P_{cash\ cow}=18/0.08=225

Metric	Alfa	Beta
Growth	0.5×10% = 5%	0.5×6% = 3%
Price	9/(8%−5%) = 300	9/(8%−3%) = 180
PVGO	300−225 = 75	180−225 = −45
Forward P/E	300/18 = 16.6667	180/18 = 10
Market/book	300/270 = 1.1111	180/270 = 0.6667

Alfa reinvests above its required return and creates value. Beta reinvests below it and destroys value; the difference is investment profitability, not just earnings growth.

B.

D_1=3(1.10)=3.30,\quad D_2=3.30(1.12)=3.696

D_3=3.696(1.06)=3.91776,\quad D_4=3.91776(1.03)

P_0=\frac{3.30}{1.10}+\frac{3.696}{1.10^2}+\frac{3.91776+3.91776(1.03)/(0.10-0.03)}{1.10^3}

52.31

Multistage Dividend Growth

EXERCISE 14

Colgate-Palmolive just earned EPS 1.20 and paid dividend 0.96. Earnings and dividends grow 11% for three years and 5.20% thereafter. Equity cost is 8.50%. Separately, a risk-free five-year zero-coupon with face 100 trades at 82.50. Assume a flat yield curve for the coupon-bond comparisons.

A.Price the equity and calculate growth value.
B.Find the zero-coupon yield and price a three-year 3% coupon bond.
C.Would a fifteen-year 6% coupon bond trade above or below par?

A.

P_0=\sum_{t=1}^3\frac{0.96(1.11)^t}{1.085^t}+\frac{0.96(1.11)^3(1.052)}{(0.085-0.052)1.085^3}

Price = 35.78.

P_{no\ growth}=\frac{1.20}{0.085}=14.12

PVGO relative to the stated current-earnings no-growth scenario = 21.67.

This counterfactual freezes current EPS at 1.20; using next-year EPS as a different no-growth benchmark changes the PVGO convention.

B.

y=\left(\frac{100}{82.50}\right)^{1/5}-1=0.03922410

P=3a_{3|y}+\frac{100}{(1+y)^3}=97.44

C.

Above par under the stated flat curve, because 6% exceeds 3.9224%. Without an assumption about the fifteen-year yield, the five-year zero alone would not establish the answer.