Growth Opportunities and PVGO

EXERCISE 01

Eta has growth opportunities worth €10 per share. An otherwise identical all-payout company, Psi, has earnings of €2 per share and a 10% cost of equity.

A.Calculate Eta’s price.

A.

P_0=EPS_1/r_E+PVGO=2/0.10+10=30

Price = €30.

Growth Opportunities and PVGO

EXERCISE 02

Summer expects EPS₁ = €7.50 and D₁ = €4. Dividends grow at 10% forever; required equity return is 15%.

A.Calculate PVGO per share.

A.

PVGO=P_0-\frac{EPS_1}{r_E}

P_0=\frac4{0.15-0.10}=80

PVGO=80-\frac{7.50}{0.15}

€30.00 per share.

Growth Opportunities and PVGO

EXERCISE 03

ABM expects earnings per share of $12 next year. It plans a $6 dividend and reinvests the remaining earnings at a constant 24% ROE. The required return is 18%. Assume payout and ROE remain constant.

A.Calculate growth and the share price.
B.Calculate the forward P/E ratio and PVGO.

A.

b=1-\frac{6}{12}=0.50,\qquad g=b\,ROE=12\%

P_0=\frac{6}{0.18-0.12}=100

B.

P/E_1=\frac{100}{12}=8.33

PVGO=P_0-\frac{EPS_1}{r_E}=100-\frac{12}{0.18}=33.33

\frac{PVGO}{P_0}=\frac{100-12/0.18}{100}\approx33.33\%

Growth opportunities account for 33.33% of the share price.

PVGO is the present value of future positive-NPV reinvestment opportunities. The no-growth benchmark distributes all earnings, not merely the current dividend.

Growth Opportunities and PVGO

EXERCISE 04

Next-year earnings per share are 20.00; the dividend is 8.00. Retained earnings earn 20.000% and the required equity return is 18.000%. Payout and ROE remain constant.

A.Calculate growth, price, forward P/E and PVGO.

A.

g=(1-D_1/EPS_1)ROE,\quad P_0=\frac{D_1}{r_E-g}

g=(1-8/20)(0.2)=0.12

P_0=\frac{8}{0.18-0.12}\approx 133.3333

P/E_1=\frac{133.3333}{20}\approx 6.6667

PVGO=P_0-\frac{EPS_1}{r_E}\approx 133.3333-20/0.18=22.2222

PVGO/P_0\approx 22.2222/133.3333=0.166667

Measure	Value
Growth	12.000%
Price	133.33
Forward P/E	6.67
PVGO	22.22
PVGO / price	16.667%

Growth Opportunities and PVGO

EXERCISE 05

Next-year earnings per share are 2.50; the dividend is 1.50. Retained earnings earn 15.000% and the required equity return is 10.000%. Payout and ROE remain constant.

A.Calculate growth, price, forward P/E and PVGO.

A.

g=(1-D_1/EPS_1)ROE,\quad P_0=\frac{D_1}{r_E-g}

g=(1-1.5/2.5)(0.15)=0.06

P_0=\frac{1.5}{0.1-0.06}\approx 37.5

P/E_1=\frac{37.5}{2.5}\approx 15

PVGO=P_0-\frac{EPS_1}{r_E}\approx 37.5-2.5/0.1=12.5

PVGO/P_0\approx 12.5/37.5=0.333333

Measure	Value
Growth	6.000%
Price	37.50
Forward P/E	15.00
PVGO	12.50
PVGO / price	33.333%

Growth Opportunities and PVGO

EXERCISE 06

A.How is share value decomposed into a no-growth value and PVGO? Does delaying an otherwise identical positive-NPV opportunity increase its present value?

A.

P_0=\frac{EPS_1}{r_E}+PVGO

Share value equals the value of the sustainable earnings stream without reinvestment plus the NPV of future investments. With positive discount rates and otherwise identical cash flows, delaying the same opportunity lowers its present value. A higher accounting growth rate alone does not guarantee value creation: returns on new investment must exceed the appropriate opportunity cost.

Growth Opportunities and PVGO

EXERCISE 07

RWE expects EPS of €6, pays all earnings as dividends and trades at €60. It considers a 75% payout ratio, investing retained earnings at 12% ROE.

A.Find the required return, growth and new price.

A.

r_E=6/60=10\%,\quad g=0.25(12\%)=3\%

P_0=\frac{6(0.75)}{0.10-0.03}=64.29

The price increases to €64.29 because reinvestment earns more than the cost of equity.

Growth Opportunities and PVGO

EXERCISE 08

ABC expects earnings per share of $5 next year and initially plans to pay a $3 dividend. Retained earnings earn 15%. The cost of equity is 12%. Consider a permanent increase in the dividend to $4, with earnings unchanged in year 1.

A.Value the shares under the two payout policies.
B.Should ABC increase the dividend?

A.

g_{D=3}=(1-3/5)(0.15)=0.06

P_{D=3}=3/(0.12-0.06)=50

g_{D=4}=(1-4/5)(0.15)=0.03

P_{D=4}=4/(0.12-0.03)\approx44.44

Policy	Retention	Growth	Share price ($)
Dividend $3	40%	6%	50.00
Dividend $4	20%	3%	44.44

g=b\,ROE,\qquad P_0=\frac{D_1}{r_E-g}

B.

No, under these assumptions. Retained earnings fund projects earning 15%, above the 12% opportunity cost. Distributing more leaves fewer positive-NPV investments and reduces the share value from $50.00 to $44.44.

Growth Opportunities and PVGO

EXERCISE 09

At year 1, a firm can retain €60 to fund an investment that adds €50 to distributable earnings in years 2 and 3. Subsequent earnings return to their original level. The discount rate is 10%.

A.Calculate the opportunity’s value today.

A.

PVGO_0=\sum_t\frac{\Delta CF_t}{(1+r_E)^t}

PVGO_0=-\frac{60}{1.10}+\frac{50}{1.10^2}+\frac{50}{1.10^3}

€24.34.

Growth Opportunities and PVGO

EXERCISE 10

Gamma’s next-year dividend is €10 and next-year EPS €15. Dividends and earnings grow at 5% through year 4. From year 5 it distributes all earnings, with no further growth. The required return is 8%.

A.Value the share today.
B.Calculate PVGO relative to immediately paying out €15 forever.

A.

P_0=\sum_{t=1}^{4}\frac{D_t}{(1+r_E)^t}+\frac{EPS_5/r_E}{(1+r_E)^4}

Year	1	2	3	4	5
Dividend	10.000000	10.500000	11.025000	11.576250	18.232594

EPS_5=15(1.05)^4=18.23259375,\quad P_4=\frac{18.23259375}{0.08}=227.90742188

P_0=35.52220666+\frac{227.90742188}{1.08^4}

€203.04.

B.

PVGO=P_0-\frac{EPS_1}{r_E}

PVGO=203.04096542-\frac{15}{0.08}

€15.54.

Growth Opportunities and PVGO

EXERCISE 11

FriedKitchen has 5,000 shares, book equity of 50,000, market-to-book of 1.25 and EPS of 1.50. It is initially a debt-free cash cow paying all earnings. A proposal retains 40% of earnings in years 1–3. Total earnings are 7,500, 7,650 and 7,803 in those years, then 7,959 annually from year 4, with full payout thereafter.

A.Find the initial share price and cost of equity.
B.Value the expansion policy and its growth opportunities.
C.Should the proposal be adopted?

A.

P_0=\frac{50000(1.25)}{5000}=12.50

R_E=\frac{EPS_1}{P_0}=\frac{1.50}{12.50}

12%.

B.

E_0=\sum_{t=1}^{N}\frac{Div_t}{(1+R_E)^t}+\frac{TV_N}{(1+R_E)^N}

E_0=\frac{4500}{1.12}+\frac{4590}{1.12^2}+\frac{4681.8+7959/0.12}{1.12^3}

Equity = 58,218.21; price = 11.6436; PVGO = -4,281.79.

C.

No. Reinvestment earns approximately 5%, below the 12% required return. Growth destroys value. The calculation uses the stated 7,959 terminal earnings; retaining the unrounded growth calculation would give 7,959.06 and a slightly different value.