2003. 10

Corporate Financial Strategy and Stock Price Behavior

in a Noise Trader Model with Limited Arbitrage
Nobuyuki Isagawa

Katsuhiko Okada

Corporate Financial Strategy and Stock Price Behavior
in a Noise Trader Model with Limited Arbitrage
Nobuyuki Isagawa∗
Katsuhiko Okada **Abstract

In this paper, we attempt to explain the stock price decreases following equity offeringsand stock price increases following stock repurchases, both of which have been
identified by empirical studies. By examining corporate financial strategies in thelimited arbitrage-noise trader model presented by Shleifer and Vishny (1990, 1997), we
can explain those anomalies. When a market with an optimistic noise overvalues stock
prices, a firm can earn capital gains by selling new shares at the overvalued price. On
the other hand, when the pessimistic market undervalues stock prices, a firm can enjoy
capital gains by buying its shares at the bargain price. In the event that a noise
disappears, a firm issuing shares experiences the continual stock price decrease, and afirm repurchasing its shares experiences the continual stock price increase.

Keywords: noise trader model, equity offering, stock repurchase, stock price anomaly∗

Nobuyuki Isagawa, Graduate School of Business Administration, Kobe University, 2-1,

Rokkodai, Nada, Kobe, Hyogo, 657-8501, Japan. Phone & Fax: +81-78-803-6907,
e-mail: isagawa@rose.rokkodai.kobe-u.ac.jp**

Katsuhiko Okada, IZ company ltd., 4-2-26 Minooka-Dori, Nada, Kobe, Hyogo,657-0812, Japan. Phone: +81-78-882-7872, e-mail: katsuokada@kcc.zaq.ne.jp
1

JEL classification: G32, 35

1.

Introduction

The purpose of this paper is to examine corporate financial strategy, especially stock
issue and repurchase, and stock price behavior in the noise trader model with limited
arbitrage presented by Shleifer and Vishny (1990, 1997).1 Several phenomena andempirical findings that seem difficult to explain using the efficient markets approach
have been reported.2 Among these, we focus on two anomalies: abnormal stock pricedeclines following equity offerings, and abnormal stock price increases following open
market stock repurchases.
Loughran and Ritter (1995), Spiess and Affleck-Graves (1995), and Teoh, Welch,and Wong (1998) find that firms have experienced significant negative stock returns
after executing equity offerings. Ikenberry, Lakonishok, and Vermaelen (1995) find thatstock prices have continued to increase abnormally after firms repurchases their shareson the open market. As suggested by Loughran and Ritter (1995, pp.46-47), Spiess andAffleck-Graves (1995, p.265), Teoh, Welch, and Wong (1998, p.64), and Ikenberry,
Lakonishok, and Vermaelen (1995, p.183), these phenomena are consistent with markets
that do not revalue stock prices appropriately when equity offerings or stock

1

Shleifer and Vishny (1990) analyze corporate investment strategies; Shleifer and

Vishny (1997) go on to argue agency problems between arbitrageurs and their investors.
By contrast, we examine corporate financial strategies in a noise trader model withlimited arbitrage.
2

See Campbell and Kyle (1993), Redding (1996), Pontiff (1996), Morgan (1997),

Shleifer and Vishny (1997), Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer,
and Subrahmanyam (1998).
2

repurchases occur. In other words, neither continual stock price increases following
equity offerings nor stock price decreases following stock repurchases should beobserved in the efficient markets, which instantaneously adjust stock prices to theirfundamental values.
We analyze corporate stock issue and repurchase in a limited arbitrage-noise trader
framework based upon the model presented by Shleifer and Vishny (1990, 1997). In anoise trader model with limited arbitrage, market mispricing would not be eliminateduntil the noise disappears. In our model, an overvalued firm issues new shares, whereas
an undervalued firm buys back its outstanding shares. As the number of shares supplied
by a firm changes, the stock price changes. An equity offering that increases the numberof outstanding shares decreases the stock price. In contrast, a stock repurchasedecreasing the number of shares increases the stock price. Thus, a firm’s financial
behavior partially resolves market mispricing, and brings its stock price close to thefundamental value. In this sense, it can be interpreted that a firm provides smart money
via financial behavior.
A firm, however, never sells or buys its shares to the extent that its stock price isequal to the fundamental value, because it would lose opportunities to enjoy capitalgains by doing so. The market mispricing caused by noise remains, even after a firm
executes stock issue (repurchase), and stock price continues to decrease (increase) until
the optimistic (pessimistic) noise disappears. Therefore, a firm selling (buying) its
shares at an overvalued price (bargain price) can earn capital gains by using market
mispricing. From the viewpoint of stock price behavior, a firm experiences an abnormal
stock price decline following an equity offering, and an abnormal stock price rise
following a stock repurchase. This prediction is consistent with the empirical findings
introduced in above-referenced works.
The remainder of this paper is organized as follows. In section 2, we analyze afirm’s financial strategy in a noise trader model with limited arbitrage. In section 3, we

3examine the stock price behavior. Section 4 presents our conclusions.

2.

Financial Strategy in a Limited Arbitrage-Noise Trader Model

Consider an all-equity firm that operates for one period. Throughout the paper, t=0
represents the beginning of the period, and t=1 represents the end of the period. Prior tot=0, the total number of the firm’s outstanding shares is one. The firm generates
operating returns, V>0, at t=1. For simplicity, we assume that all participants arerisk-neutral and that the interest rate is zero.
The structure of the market follows Shleifer and Vishny (1990, 1997). The market
evaluates the firm’s stock with a noise, S, at t=0, where S>0 means an optimistic noise
and S<0 means a pessimistic noise.3 Arbitrage is limited so that the stock price of thefirm may be somewhat mispriced until t=1, at which time the noise disappears (S=0).4
The market demand schedule for the firm’s stock is given by
q t = (V + S ) / p t , t=0, 1,(1)where pt represents the stock price at t. The stock price is determined by the market
clearing condition that the total demand must be equal to the number of outstanding
shares. Thus, without additional supply of shares, the firm’s stock is undervalued if themarket has a pessimistic noise (S<0), and overvalued if the market has an optimistic
noise (S>0).

3

Barberis, Shleifer, and Vishny (1998), and Daniel, Hirshleifer, and Subrahmanyam(1998) develop the way in which investor sentiment, based on psychological theory,
leads to market mispricing.
4

Shleifer and Vishny (1990, pp.149-150) summarize fundamental risk and noise trader

risk by which arbitrage is limited. Pontiff (1996) shows that variety costs make arbitrage
limited.
4

The goal of the firm is to maximize its stock price at t=1. For example, one canconsider the situation in which the manager’s compensation scheme is based on thestock price at the end of the period (t=1). If the firm does not do anything, then its stock
price is V at t=1. Under (1), however, the firm can raise its future stock price higher than
V by selling its shares at an overvalued price or by buying shares at a bargain price. Let
the number of shares that the firm additionally issues at t=0 be denoted by n, where n>0
means an equity offering and n<0 means a stock repurchase. Since the maximum
number of shares that the firm can buy back on the market is 1, n>-1 must be held. It
follows from the market clearing condition, i.e., qt=1+n, that the stock price at t=0 is
given byp 0 (n) = (V + S ) /(1 + n) .

(2)

In order to ensure that the stock price is positive, we assume that V+S>0.
For analytical simplicity, it is assumed that the firm can lend and/or borrow at aninterest rate of zero. Since the total cash flow is V+np0, the stock price at t=1 is given by

p1 (n) =V + np0 V + n(2V + S )=.
1+ n(1 + n) 2(3)

The firm chooses the n that maximizes (3). Let the solution be denote by n*.
The first-order derivative and the second order-derivative of (3) are given by

dp1 (n) S − (2V + S )n=,dn
(1 + n) 3andd 2 p1 (n) 2[(2V + S )n − (V + 2 S )]=,dn 2
(1 + n) 4
respectively. Therefore,n * = S /(2V + S ) .

(4)

Note that n*>-1 holds under the restriction of V+S>0.
We now obtain the next proposition.

5

Proposition 1. If the market has an optimistic noise (S>0), then the firm issues its shares(n*>0). If the market has a pessimistic noise (S<0), then the firm repurchases its shares(n*<0).

Proposition 1 states that whether the firm issues new shares or repurchases its
outstanding shares will depend on the market noise. The firm issues its new shares when
its stock price is overvalued by market optimistic noise.5 Conversely, the firm buys
back its outstanding shares when its stock price is undervalued by market pessimistic
noise.
By substituting (4) into (2), the equilibrium stock price at t=0 is given byp 0 (n * ) = (V + S ) /(1 + n * ) = V + ( S / 2) .

(5)

Recall that, at t=0, the stock price is V+S if n=0. It is clear that market mispricing ispartially eliminated by the firm’s financial behavior. In this sense, it can be interpreted
that the firm supplies smart money via an equity offering or a stock repurchase.
The firm, however, never issues or repurchases its shares to the extent that market
mispricing is fully eliminated, because the firm loses an opportunity to earn capitalgains from market mispricing if it supplies a sufficient number of shares to bring thestock price to its fundamental value at t=0.6 To demonstrate this, we calculate the stock
price at t=1. By substituting (4) into (3) and somewhat rearranging the equation, we
obtain5

Our result is consistent with the interpretation of Teoh, Welch, and Wong (1998, p.94),

that the market is too optimistic when an equity offering is executed.
6

For stock repurchases, Ikenberry, Lakonishok, and Vermaelen (1995, p.183) stressthat the most commonly cited reason of repurchases, undervaluation or good investment,
is not supported unless firms can repurchase their shares at the undervalued prices.
6V + n * p0 (2V + S ) 2
S2.
p1 (n ) ===V +4(V + S )4(V + S )1 + n**(6)

That is, the equilibrium stock price of the firm is higher than its fundamental value, V, at
t=1. The difference between p1(n*) and V can be interpreted as capital gains that the firm
can earn by implementing financial strategy.

3. Stock Price Behavior

In this section, we examine the stock price behavior during the period between t=0
and t=1. It follows from (5) and (6) that the rate of return, R, is given byR≡

p1 (n * ) − p 0 (n * )−S=.
*2(V + S )p0 (n )

Since V+S>0, the sign of R is identical to the sign of S. Therefore, we can obtain thefollowing proposition.

Proposition 2. The stock price goes down (R<0) following an equity offering (S>0), andgoes up (R>0) following a stock repurchase (S<0).

Thus, our model predicts that the stock price continues to decrease after the firm
issues new shares, and continues to increase after the firm executes a stock repurchase.
The former prediction is consistent with the empirical finding of Loughran and Ritter(1995), Spiess and Affleck-Graves (1995), and Teoh, Welch, and Wong (1998). The
latter is consistent with the finding of Ikenberry, Lakonishok, and Vermaelen (1995).
By differentiating R with V, we obtain

dR
S
=
.
dV 2(V + S ) 2
That is, R is increasing with V when S>0, and decreasing with V when S<0. The

7

operating returns of V can be interpreted as the firm size. Thus, the above relationmeans that the larger the firm size is, the larger (less) the firm experiences its stock
return following an equity offering (a stock repurchase). Spiess and Affleck-Graves(1995, p.259) report that the smaller firms tend to experience more severe
underperformance than do larger firms.

5. Conclusion

In this paper, we analyzed corporate financial strategy in the noise trader model withlimited arbitrage presented by Shleifer and Vishny (1990, 1997). When the market has
an optimistic noise, the firm can earn capital gains by issuing new equity at theovervalued price. On the other hand, when the market has a pessimistic noise, the firm
can enjoy capital gains by repurchasing its outstanding shares at the bargain price. Fromthe viewpoint of stock price behavior, our model predicts that the firm experiences acontinual stock price decrease following an equity offering, and a continual stock priceincrease following a stock repurchase. These predictions are consistent with empiricalfindings that seem difficult to explain using the efficient markets framework.
[2003.4.30 648]

References

Barberis, N., A. Shleifer, and R. Vishny, 1998, A Model of Investor Sentiment, Journalof Financial Economics 49, 307-343.

Campbell, J., and A. Kyle, 1993, Smart Money, Noise Trading, and Stock Price
Behavior, Review of Economic Studies 60, 1-34.

8Daniel, k., D. Hirshleifer, and A. Subrahmanyam, 1998, Investor Psychology andSecurity Market Under- and Overreactions, Journal of Finance 53, 1839-1885.

Ikenberry, D., J. Lakonishok, and T. Vermaelen, 1995, Market Underreaction to Open
Market Share Repurchases, Journal of Financial Economics 39, 181-208.

Laughran, T., and J. Ritter, 1995, The New Issue Puzzle, Journal of Finance 50, 23-51.

Morgan, K., 1997, Do Noise Traders Influence Stock Prices?, Journal of Money Credit
and Banking 29, 351-363.

Pontiff, J., 1996, Costly Arbitrage: Evidence From Closed-End Funds, Quarterly
Journal of Economics 111, 1135-1151.

Redding, L., 1996, Noise Traders and Herding Behavior, IMF Working Paper.

Shleifer, A., and R. Vishny, 1990, Equilibrium Short Horizons of Investors and Firms,
American Economic Review Papers and Proceedings 80, 148-153.

Shleifer, A., and R. Vishny, 1997, The Limits of Arbitrage, Journal of Finance 52,
35-55.

Spiess, D., and J. Affleck-Graves, 1995, Underperformance in Long-run Stock Returns
Following Seasoned Equity Offerings, Journal of Financial Economics 38, 243-267.

Stephens, C., and M. Weisbach, 1998, Actual Share Reacquisitions in Open-Market
Repurchase Programs, Journal of Finance 53, 313-333.

9

Teoh, S., I. Welch, and T. Wong, 1998, Earnings Management and theUnderperformance of Seasoned Equity Offerings, Journal of Financial Economics 50,
63-99.

10

Discussion Paper
No.
2002・1

AuthorNobuyuki Isagawa

2002・2

Nobuyuki Isagawa

2002・3Mahito Okura

2002・4

Elmer Sterken
Ichiro Tokutsu
Mahito Okura
Mahito Okura
Nobuyuki Isagawa
Koji Okubayashi
Toshinori Takashina

2002・5
2002・6
2002・7
2002・8
2002・9Hajime Shimizu

2002・10 Nobuyuki Isagawa
2002･11

Nobuyuki Isagawa

2002･12Kenji Kutsuna
Richard Smith

2002・13 Kunio Miyashita
2002・14 Hajime Shimizu
2002・15 Hajime Shimizu
2002・16 Koji Okubayashi
2002・17 Fumitoshi Mizutani
Shuji Uranishi
2002・18 Kazuhisa Otogawa
2002・19 Atsuo Takenaka
2002・20 Tsuneo Nakano
2002・21 Tsuneo Nakano
2002・22 Fumitoshi Mizutani
Shuji Uranishi
2002・23 Atsushi Takao
Mahito Okura
2002・24 Fumitoshi Mizutani
2002・25 Fumitoshi Mizutani
Takuya Urakami
2002・26 Yasuyuki Miyahara
2002・27 Nobuyuki Isagawa
2002・28 Mitsutoshi Hirano

Title
Cross Holding of Shares, Unwinding Cross Holding of
Shares, and Managerial
Entrenchment
A Theory of Stock Price Behavior following Repurchase
Announcements
An Equilibrium Analysis of the Insurance Market
with Vertical Differentiation
What are the determinants of the number of bankrelations of Japanese firms?
Review Article：Adverse Selection in Insurance Market
Welfare Effect of Firm Size in Insurance Market
Does Investment Horison Affect Investment Strategy ?
Research Report of Careers and Employment Process ofWorkers over 60 years old :a Case of a Japnaese
Manufacturing Company
On the Existence of Tax Equilibrium : Incomplete Market
with Capital Income Tax
Cost of Financial Distress and Debt Restructuring :Debt Forgiveness and Equity-for-Debt Swap
Open-Market Repurchase Announcements, Actual
Repurchases, and Stock Price Behavior in Inefficient
Markets―revised version of No.2001･36―
Why Does Book Building Drive Out Auction Methodsof IPO Issuance?
Evidence and Implications from Japan
International Logistics and Modal Choice
Incomplete markets with capital income tax and public
goods
On the possibility of Pareto improving of tax-equilibrium
China-Japan Comparison of Work Organization
The Post Office vs. Parcel Delivery Companies:
Competition Effects on Costs and Productivity―revised version of No.2001･33―
Earnings Forecast and Earnings Management of JapaneseInitial Public Offerings Firms
The Classification of Foreign R&D Units
The Dutch East India Company and its Corporate
Governance―revised version of No.2000･40―
The British East India Company and its Corporate
Governance
Privatization Effects on TFP Growth and Capital
Adjustments
Some Notes on the Privatization of Postal Life Insurancein JapanPrivately Owned Railways’ Cost Function, Organization
Size and Ownership
A Private-Public Comaprison of Bus Service Operators
Principal-Multiagent Relationships with Costly
Monitoring
Unwinding of Cross Shareholding under Managerial
Entrenchment
Contingency Model for Personnel Grading System―Issues on Design and Management for Qualification
System and Job Grading System―

Date(in Japanese)

1／ 2002(in Japanese)

1／ 20022／20023／2002(in Japanese)(in Japanese)(in Japnaese)

3／20023／20023／20024／2002(in Japnaese)

4／2002(in Japnaese)

4／20025／20025／2002(in Japnaese)(in Japnaese)

6／20026／20026／20027／20027／20028／2002(in Japanese)(in Japanese)

8／20028／2002(in Japanese)

8／20028／2002(in Japanese)

9／20029／20029／2002
10／2002
10／2002(in Japanese)

11／2002

Discussion Paper
No.
Author
2002・29 Atsushi Takao
2002・30 Yasuhiro Shimizu
2002・31 Hideki Murakami

2002・32 Atsushi Takao
2002・33 Nobuyuki Isagawa
2002・34 Takuji Hara
2003・1 Katsuhiko Kokubu
2003・2 Kazuhiro Tanaka
Fumitoshi Mizutani
2003・3
Shuji Uranishi
Yang Jia Yin
2003・4
Koji Okubayashi
2003・5

Mitsutoshi Hirano

2003・6

Kazuhisa Otogawa

2003・7

Nobuyuki Isagawa
Tadayasu Yamashita

2003・8Kenji Kutsuna
Marc Cowling

2003・9

Noriko Masumura

2003・10

Nobuyuki Isagawa
Katsuhiko Okada

Title
On the Emergency Effect of the late FUKUZAWA,
Yukichi in Introducing of Modern Insurance into Japan
Amortization of Intangible Assets before Newark
Morning Ledger
An Economic Analysis of Duopolistic Competition
between Gulliver and Dwarf airlines:
The case of Japanese Domestic Air Markets
Simulation Model to Consider the Penetration of Modern
Insurance in Japan −Confirmation of Emergence andPhase Transition of New Japanese Risk-Bearing System
by Evolutional Economics
Mutual Shareholding and Unwinding of Mutual
Shareholding as Stockpile for Business Recovery
TLOs in Japan in the life science area : present and future
Environmental Accounting for Corporate Management
Embeddedness and Entrenchment of Managers
The Effects of Privatization on TFP Growth and Capital
Adjustments
Labour Turnover of Japan-Affiliated Companies inShanghai
Generation and Resolution of Asymmetric Information in
Human Resource Management –In the Cases of Job
Rotation in Two Retailers in JapanMarket Liquidity around Quarterly Earnings
Announcements
Lender’s Risk Incentive and Borrower’s Risk Incentive
Determinants of Small Business Loan Approval:
Evidence from Japanese Survey after 1997 Financial
CrisisVoluntary Corporate Disclosure and the Cost of Debt:
The Case of Quarterly Reporting
Corporate Financial Strategy and Stock Price Behavior
in a Noise Trader Model with Limited Arbitrage(in Japanese)Date
11／2002(in Japanese)

11／2002
11／2002(in Japanese)

12／2002

12／2002(in Japanese)(in Japanese)(in Japanese)

12／2002
1／20032／20032／2003(in Japanese)

3／2003(in Japanese)

3／20033／2003(in Japanese)

3／20033／2003(in Japanese)

4／20034／2003