STOCK VALUATON
D1, D2, … DH: DIVIDENDS FOR THE FIRST H PERIODS
PHPRICE AT t = H
BUT
USING (2) IN (1)
D1 = D0(1 + g); D2 = D0(1 + g)2; …
DN = D0(1 + g)N
USING THE ABOVE IN (3)
THE TERM IN SQUARE BRACKETS IS AN INFINITE GEOMETRIC SERIES WITH COMMON RATIO
THUS,
EXAMPLE (CONSTANT GROWTH RATE)
IF g = 10%, r = 14%, D0 = $2 / SH
STOCK VALUATION WITH DIFFERENT GROWTH RATES IN DIVIDENDS
FOR MANY FIRMS, GROWTH IS HIGH INITIALLY, THEN SLOWS DOWN TO A STEADY, LOWER RATE. WE USE A VARIATION ON THE ABOVE FORMULA THEN TO PRICE THE STOCK.
PROBLEM - 1
THE FORMULA CAN NOW BE WRITTEN AS
WHERE g1 = HIGH GROWTH RATE FOR FIRST “n” YEARS
g2 = STEADY GROWTH RATE AFTERWARD, IN PERPETUITY
HERE:
g1 = 10% = 0.1 g = 6% = 0.06
D0 = $2 / SHARE r = 14% = 0.14
n = 5 YEARS
PV OF DIVIDENDS FOR FIRST 5 YEARS:
PVIF (1 YEAR, 14%)
= 2 (1.1) (0.87719)
+ 2 (1.1)2 (0.76947) + 2 (1.1)3 (0.06749)
+ 2 (1.1)4 (0.59208) + 2 (1.1)3 (0.5193)
= 1.93 + 1.86 + 1.80 + 1.73 + 1.67
= $8.99
DIV AT THE BEGINNING OF YEAR 6:
= D6 = 2 (1.1)6 = 3.41
PV OF DIVIDENDS STARTING YEAR 6:
= $22.14
TOTAL PRICE OF STOCK = 22.14 + 8.99 = $31.13
TIMELINE
t = 0 1 2 3 4 5 6 7
D0 D1 D2 D3 D4 D5 D6 D7
NOTE THAT D6 IS STOCK PRICE AT THE END OF YEAR 6; D1 IS STOCK PRICE AT THE END OF YEAR 1.
APPROXIMATING GROWTH RATES FROM RETURN ON EQUITY AND PLOW-BACK RATIO
EPS = NET INCOME
NO. OF SHARES OUTSTANDING
BOOK VALUE (BV) PER SHARE:
= BV OF EQUITY
NO. OF SHARES OUTSTANDING
ROE = EPS
BV PER SHARE
DIVIDEND PAYOUT RATIO = DIV / SH
EPS
THEN:
ASSUMPTION: RE-INVESTED EARNINGS EARN SAME RETURN AS IN THE PAST
FINDING THE IMPLIED RETURN ON EQUITY GIVEN STOCK PRICE AND GROWTH RATE IN DIVIDENDS
WE KNOW THAT:
THEREFORE:
UNDER THE ASSUMPTION OF A CONSTANT GROWTH RATE IN DIVIDENDS
STOCK VALUATION FROM THE NPV OF GROWTH OPPORTUNTIES
WE CAN DIVIDE EQUITY VALUE INTO:
VALUE OF A SHARE ASSUMING NO FUTURE INVESTMENT
+
NPV OF FUTURE GROWTH OPPORTUNITIES
(A) IF FIRM MAKES NO FUTURE INVESTMENT, VALUE = EPS1 ,
r
ASSUMING EARNINGS OF EPS1 IN PERPETUITY
(B) IF FIRM INVESTS, IT TAKES A FRACTION OF EARNINGS AND PLOWS IT BACK INTO FIRM;
NPV (OF PROJECT) = PVGO (NET PRESENT VALUE OF GROWTH OPPORTUNITIES)
NOTICE THAT ONLY THE NET PRESENT VALUE OF GROWTH OPPORTUNITIES SHOULD BE INCLUDED IN TERM TWO IN EQUATION (7); OTHERWISE IT IS DOUBLE-COUNTING. (THE INVESTMENT AMOUNT IS INCLUDED IN = EPS1
r
REARRANGING (7),
WHEN GROWTH OPPORTUNITIES ARE NON-EXISTENT,
PROBLEM 2
FIRST COMPUTE PVGO
THE FIRM HAS FIVE DIFFERENT PROJECTS, EACH REQUIRING $3 INVESTMENTS, AND EARNINGS = 3*0.20 = $0.60 PER YEAR FOREVER (IN PERPETUITY).
NPV OF EACH PROJECT, AT r = 12%
t = 0 1 2 3 4 5
$2 $2 $2 $2 $2
PVGO = $7.21
PROBLEM 3
g = 0.05 D1 = $10/SH
PLOW-BACK RATIO = 0.2
= 1 – 0.2 = 0.8
RETURN ON BOOK EQUITY
BOOK EQUITY INCREASES AS EARNINGS ARE PLOWED-BACK (AT THE SAME RATE AS DIVIDENDS, ASSUME)
0.2 (12.50) = 0.05 (BV/SH)
INCREASE IN BV
PVGO = 100 – 83.33 = $16.67 / SHARE
STOCK PRICE AS THE PRESENT VALUE OF FREE CASHFLOW PER SHARE
FREE CASH FLOW
= REVENUE – COSTS – INVESTMENT
IN THE LONG RUN, THE AMOUNT THE FIRM IS FREE TO PAY OUT (FREE CASH FLOW) IS THE SAME AS DIVIDENDS PER SHARE.
PRICE TO EARNINGS RATIO (P/E)
RECIPROCAL OF RATIO
USED FOR VALUATION BY COMPARABLES
VALUE OF EQUITY
= EARNINGS OF FIRM TO BE VALUED
X
P/E OF SIMILAR FIRMS
(a)P/E IS HIGH IF r IS LOW
(b)P/E IS HIGH IF THE FIRM HAS HIGH GROWTH OPPORTUNITIES (PVGO IS HIGH)
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