CP

Stock Repurchases 531

valueofoperationsplusthevalueoftheextracash.Wecanfindthepricepershare,P 0 ,

bydividingthetotalvaluebythenumberofsharesoutstanding,n 0 :

. (14-2)

Wecaneasilysolvethisforthevalueofoperations:Vop
P 0 (n 0 )Extracash
$40

$5
$35million

Now consider the repurchase. P is the repurchase price and n is the number

of shares that will be outstanding after the repurchase. We can multiply the unknown

repurchase price by the number of shares that are repurchased, and this must equal the

extra cash that is being used in the repurchase:

. (14-3)

Since the company will have no extra cash after the repurchase, the stock price will be

the value of operations divided by the remaining shares of stock:

. (14-4)

We know the current price (P 0 ), the current number of shares (n 0 ), and the amount

of extra cash. This leaves three remaining unknown variables (P, n, and Vop) and three

equations, so we can solve for the unknown variables.^14 The solution shows that P 


P 0 
$20. In other words, the repurchase itself does not change the stock price.

However, the repurchase does change the number of outstanding shares. Rewriting

Equation 14-4,

(14-4a)

Asacheck,wecanseethatthetotalmarketcapitalizationbeforetherepurchasewas

$40million,$5millionwasusedtorepurchaseshares,andthetotalmarketcapitaliza-

tionaftertherepurchase is $35 million
P(n)
$20(1.75million).Thisshouldmake

sense,sincetherepurchaseitselftransferred$5millionofcorporateassetstotheindi-

vidual shareholders. Notice that the aggregate wealth of the shareholders didn’t

change. It was $40 million prior to the repurchase, and it is $40 million afterward

($35millioninstockand$5millionincash).Noticealsothatarepurchaseof250,000

shares of stock at a price of $20 equals the $5 million in cash used to repurchase the

shares.

To summarize, the events leading up to a repurchase (the sale of a division, a re-

capitalization, or the generation of higher than normal free cash flows) can certainly

change the stock price, but the repurchase itself doesn’t change the stock price.

n

Vop

P

$35 million

$20

1.75 million.

P

Vop

n

P(n 0 n)
Extra cash

P 0 

VopExtra cash

n 0

(^14) We can rewrite Equation 14-3 as Extra cash
Pn 0 Pn and Equation (14-4) as Vop
Pn. We then sub-
stitute these expressions for extra cash and Vopinto Equation 14-2 and solve for P, which results in P
P 0.

Distributions to Shareholders: Dividends and Repurchases 527