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Advances in investment analysis and portfolio management


ADVANCES IN INVESTMENT
ANALYSIS AND PORTFOLIO
MANAGEMENT


ADVANCES IN INVESTMENT
ANALYSIS AND PORTFOLIO
MANAGEMENT
Series Editor: Cheng-Few Lee


ADVANCES IN INVESTMENT ANALYSIS AND PORTFOLIO
MANAGEMENT VOLUME 9

ADVANCES IN
INVESTMENT ANALYSIS
AND PORTFOLIO
MANAGEMENT
EDITED BY


CHENG-FEW LEE
Department of Finance, Rutgers University, USA

2002

JAI
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CONTENTS
LIST OF CONTRIBUTORS

vii

EDITORIAL BOARD

ix

PREFACE

xi

ENDOGENOUS GROWTH AND STOCK RETURNS
VOLATILITY IN THE LONG RUN
Christophe Faugère and Hany Shawky

1

A NOTE ON THE MARKOWITZ RISK MINIMIZATION AND
THE SHARPE ANGLE MAXIMIZATION MODELS
Chin W. Yang, Ken Hung and Felicia A. Yang

21

OPTIMAL HEDGE RATIOS AND TEMPORAL
AGGREGATION OF COINTEGRATED SYSTEMS
Donald Lien and Karyl Leggio

31

MARKET TIMING, SELECTIVITY, AND MUTUAL FUND
PERFORMANCE
Cheng-Few Lee and Li Li

41

SOURCES OF TIME-VARYING RISK PREMIA IN THE TERM
STRUCTURE
John Elder

85

STOCK SPLITS AND LIQUIDITY: EVIDENCE FROM
AMERICAN DEPOSITORY RECEIPTS
Christine X. Jiang and Jang-Chul Kim

109

PORTFOLIO SELECTION WITH ROUND-LOT HOLDINGS
Clarence C. Y. Kwan and Mahmut Parlar

133

v


vi

DEFINING A SECURITY MARKET LINE FOR DEBT
EXPLICITLY CONSIDERING THE RISK OF DEFAULT
Jean L. Heck, Michael M. Holland and David R. Shaffer

165

SHAREHOLDER HETEROGENEITY: FURTHER EVIDENCE
Yi-Tsung Lee and Gwohorng Liaw

181

THE LONG-RUN PERFORMANCE AND PRE-SELLING
INFORMATION OF INITIAL PUBLIC OFFERINGS
Anlin Chen and James F. Cotter

203

THE TERM STRUCTURE OF RETURN CORRELATIONS:
THE U.S. AND PACIFIC-BASIN STOCK MARKETS
Ming-Shiun Pan and Y. Angela Liu

233

CHARACTERISTICS VERSUS COVARIANCES: AN
EXAMINATION OF DOMESTIC ASSET ALLOCATION
STRATEGIES
Jonathan Fletcher

251


LIST OF CONTRIBUTORS
Anlin Chen

Department of Business Management,
National Sun Yat-Sen University, Taiwan

James F. Cotter

Wayne Calloway School of Business and
Accountancy, Wake Forest University, USA

John Elder

Department of Finance, College of Business
Administration, Dakota State University,
USA

Christophe Faugère

School of Business, University of Albany,
USA

Jonathan Fletcher

Department of Accounting and Finance,
University of Sthrathclyde, UK

Jean L. Heck

Department of Finance, College of
Commerce and Finance, Villanova University,
USA

Michael M. Holland

Department of Finance, College of
Commerce and Finance, Villanova University,
USA

Ken Hung

Department of Business, Management
National Dong Hwa University, Taiwan

Christine X. Jiang

Area of Finance, The Fogelman College of
Business and Economics, The Univeristy of
Memphis, USA

Jang-Chul Kim

Fogelman College of Business and
Economics, University of Memphis, USA

Clarence C. Y. Kwan

Michael G. DeGroote School of Business,
McMaster University, Canada
vii


viii

Cheng-Few Lee

Department of Finance and Economics,
Graduate School of Management, Rutgers
University, USA

Yi-Tsung Lee

Department of Accounting, National
Chengchi University, Taiwan

Karyl Leggio

University of Missouri, USA

Li Li

Department of Finance and Economics,
Graduate School of Management, Rutgers
University, USA

Gwohorng Liaw

Department of Economics, Tunghai
University, Taiwan

Donald Lien

Department of Economics, School of
Business, University of Kansas, USA

Y. Angela Liu

Department of Business Administration,
National Chung Chen University, Taiwan

Ming-Shiun Pan

Department of Finance, Decision Sciences,
and Information Systems, USA

Mahmut Parlar

Michael G. DeGroote School of Business,
McMaster University, Canada

David R. Shaffer

Department of Finance, College of
Commerce and Finance, Villanova University,
USA

Hany Shawky

School of Business, University at Albany,
USA

Chin W. Yang

Department of Economics, Clarion University
of Pennsylvania, USA

Felicia A. Yang

Department of Economics, University of
Pennsylvania, USA


EDITORIAL BOARD
James S. Ang
The Florida State University

Chin-Wen Hsin
Yuan-Ze University

Christopher B. Barry
Texas Christian University

Dong Cheol Kim
Rutgers University

Stephen J. Brown
New York University

Stanley J. Kon
Smith-Breedan Associate, Inc.

Edwin Burmeister
Duke University

Yun Lin
National Taiwan University

Carl R. Chen
The University of Dayton

Scott C. Linn
University of Oklahoma

Ren-Raw Chen
Rutgers University

William T. Moore
University of South Carolina

Son N. Chen
National Chengchi University,
Taiwan

R. Richardson Petti
University of Houston
C. W. Sealy
University of North CarolinaCharlotte

Cheol S. Eun
Georgia Institute of Technology
Jack C. Francis
Baruch College

ix


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PREFACE
This research annual publication intends to bring together investment analysis
and portfolio theory and their implementation to portfolio management. It
seeks theoretical and empirical research manuscripts with high quality in the
area of investment and portfolio analysis. The contents will consist of original
research on:
(1) the principles of portfolio management of equities and fixed-income
securities;
(2) the evaluation of portfolios (or mutual funds) of common stocks, bonds,
international assets, and options;
(3) the dynamic process of portfolio management;
(4) strategies of international investments and portfolio management;
(5) the applications of useful and important analytical techniques such as
mathematics, econometrics, statistics, and computers in the field of
investment and portfolio management.
(6) Theoretical research related to options and futures.
In addition, it also contains articles that present and examine new and important
accounting, financial, and economic data for managing and evaluating
portfolios of risky assets. Comprehensive research articles that are too long as
journal articles are welcome. This volume of annual publication consists of
twelve papers. The abstract of each chapter is as follows:
Chapter 1.

Christophe Faugère and Hany Shawky develop an endogenous
growth model that incorporates random technological shocks to
the economy. These random technological shocks affect both
production and the depreciation of capital. We show the
existence of a long-run steady-state growth path, and characterize it. An optimal growth rate for the economy and the long-run
expected stock return are both derived. We then turn to study the
volatility of expected stock returns around this steady state. Once
tested, the model shows that deviations of de-trended capital
stock, deviations of shocks from expected values and deviations
of labor force growth from steady state together explain about
20% of the deviations of stock returns from long-term expected
xi


xii

Chapter 2.

Chapter 3.

Chapter 4.

Chapter 5.

PREFACE

values. Our estimates also implies that investors have levels of
risk aversion consistent with the literature, and that labor growth
fluctuations are not significant, due to crowding out effects.
Chin W. Yang, Ken Hung and Felicia A. Yang examine the
equivalence property between the angle-maximization portfolio
technique and the Markowitz risk minimization model is proved.
Via reciprocal and monotonic transformation, they can be made
equivalent with or without different types of short sale. Since the
Markowitz portfolio model is formulated in the standard convex
quadratic programming, the equivalence property would enable
us to apply the same well-known mathematic properties to the
angle maximization model and enjoy the same convenient
computational advantage of the quadratic program (e.g., Markowitz’s critical line algorithm).
Donald Lien and Karyl Leggio consider optimal ratios for
different lengths of hedging horizon when the highest frequency
data is generated by a cointegrated system. It is found that, after
reparameterization, a temporal aggregation of cointegrated
systems remains a cointegrated system. This result provides a
convenient method to estimate n-day hedge ratio for any integer
n. The only remaining issue concerns the possible incorrect lag
selections. Empirical results from ten futures contracts however
indicate lag selections have no effect on the estimated hedge
ratios.
Cheng-Few Lee and Li Li test various CAPM-based markettiming and selectivity models, we find that about 12% of the
funds have a statistically significant Alpha with about 4% of
the funds having a significantly positive Alpha, and 8% of the
funds having a significantly negative Alpha. About 15% of funds
show significant timing ability with about 9% funds having a
significantly positive timing coefficient and 6% of the funds
having a significantly negative timing coefficient. The Asset
Allocation funds demonstrate the most timing ability and the
Aggressive Growth funds demonstrate the least timing ability.
John Elder investigates the extent to which three observable
macroeconomic factors can explain the time-varying risk premia
in the short-end of the term structure. We employ an empirical
model that is motivated by a dynamic asset pricing model with
time-varying risk premia and time-invariant reward-to volatility
measures. We find that, in our model, two factors explain up to


Preface

Chapter 6.

Chapter 7.

xiii

65% of the temporal variation in Treasury bill returns, with the
short-end of the term structure responding significantly to
contemporaneous innovations the funds rate and shifts (or twists)
in the yield curve. Our primary new findings are that a factor
based on shifts in the yield curve may explain the time variation
in risk premia at the very short end of the term structure, and that
a factor based on innovations in the federal funds rate may be
weakly linked to the time-varying risk premia over the post-1966
sample, when the federal funds market first began to function as
a major source of bank liquidity. This latter result is somewhat
sensitive to the sample period.
Christine X. Jiang and Jang-Chul Kim use a sample of stock
splits on NYSE listed ADRs between 1994 and 1999, we study
the change in liquidity following stock splits. Our findings
suggest that cost to liquidity demanders measured by percentage
quoted and effective bid-ask spreads, split-factor adjusted quoted
depth and trading volume increases for split-up securities.
However, we observe that raw trading volume and depth both go
up after splits, suggesting that liquidity may increase because
market makers/brokers’ higher incentives in promoting the
shares for larger payments on order flows. In addition, number of
small trades and number of shareholders go up 28% and 21%,
respectively while institutional holdings pre- and post-splits are
not significantly different, also consistent with the notion that
splits provide an incentive for brokers to promote the stocks, and
their efforts seem to target small investors.
Clarence C. Y. Kwan and Mahmut Parlar consider portfolio
selection with round-lot requirements in analytical settings
where short sales are disallowed and allowed. In either case, by
exploiting some analytical properties of the objective function in
portfolio optimization, we are able to approximate the round-lot
solution without the encumbrance of any algorithmic complexities that are often associated with integer programming. The
efficient heuristic we use to solve the resulting nonlinear integer
programming problem examines only the corner points of a
‘hypercube’ surrounding the optimal fractional solution found
without the round-lot requirements. Then, by characterizing the
covariance structure of security returns with the single index
model, we establish the correspondence between the round-lot
solution and the solution without round-lot requirements for


xiv

Chapter 8.

Chapter 9.

Chapter 10.

PREFACE

which security selection criteria in terms of risk-return trade-off
are available. This correspondence, in turn, provides useful
information regarding the sensitivity of the round-lot solution in
response to changes in return expectations. Given these nice
features, the analysis should enhance the practical relevance of
portfolio modeling for assisting investment decisions.
Jean L. Heck, Michael M. Holland, and David R. Shaffer
examine that while a major consequence of the use of debt by a
business is generally assumed to be a change to the risk of
default, theoretical work relating this risk to the lender’s required
rate of return is notably sparse. This paper defines an equilibrium
model to value debt given a non-zero probability of default by
extending previous research and then formulates the corresponding appropriate security market line. Also, a model to value debt
is synthesized that compensates a lender for both capital market
risk and default risk.
Yi-Tsung Lee and Gwohorng Liaw look at how some studies,
such as Bagwell (1992) and Bernardo and Cornell (1997),
provided evidences that the shareholders’ valuations differ
dramatically. They argued that the valuations differ substantially,
implying a significantly small supply or demand elasticity.
However, Kandel et al. (1999) indicated quite an elastic demand
for stocks of Israeli IPOs that were conducted as nondiscriminatory auctions. To resolve these controversial findings,
this paper discusses the procedure of measuring price elasticity
and provides some measures of elasticity. In addition to
indicating that Bagwell’s measure tends to underestimate the
actual elasticity, this study supplements previous work by testing
under another auction mechanism, discriminatory pricing rule,
and our results are consistent with Kandel et al.’s findings.
Anlin Chen and James F. Cotter show that private information as
well as public information is important in revising the terms of
the offer during the pre-selling period (or the waiting period) and
that when the revealed private information is positive, the
underwriter compensates the investors for this information by
underpricing the issue more than when the information is
negative. Even though the cost of compensating positive
information is quite high, the issuer still benefits from the
positive inforrnation in that the wealth transferred to the
investors is smaller under underwriter’s information acquisition


Preface

Chapter 11.

Chapter 12.

xv

activities. Furthermore, IPO long-run performance is negatively
related to the positive information revealed during the waiting
period and the underwriter prestige. Finally, IPO firms without
receiving significant information during the waiting period
survive longer after issuance.
Ming-Shiun Pan and Y. Angela Liu examines the term structure
of correlations of weekly returns for six national stock markets
namely, Australia, Hong Kong, Japan, Malaysia, Singapore, and
the U.S. We decompose stock indexes into permanent and
temporary components using a canonical correlation analysis and
then calculate short- and long-horizon return correlations from
these two price components. The empirical results for the sample
period of January 1988 to December 1994 reveal that the
relationships of return correlations among these stock markets
are not stable across return horizons. While correlations, in
general, tend to increase with return horizons, there are several
cases showing that correlations decline when investment horizons increase.
Jonathan Fletcher examines the out of sample performance of
monthly asset allocation strategies within UK industry portfolios
using linear asset pricing models and a characteristic-based
model of stock returns to forecast expected returns. We find that
strategies that use conditional versions of the asset pricing
models outperforms the strategy that uses the characteristicsbased model in terms of higher Sharpe performance and more
positive abnormal returns. In addition, these strategies provide
significant positive Jensen (1968) and Ferson and Schadt (1996)
performance measures even with binding investment constraints.
Our results support the usefulness of conditional asset pricing
models in mean-variance analysis.


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ENDOGENOUS GROWTH AND
STOCK RETURNS VOLATILITY IN THE
LONG RUN
Christophe Faugère and Hany Shawky
ABSTRACT
We develop an endogenous growth model that incorporates random
technological shocks to the economy. These random technological shocks
affect both production and the depreciation of capital. We show the
existence of a long-run steady-state growth path, and characterize it. An
optimal growth rate for the economy and the long-run expected stock
return are both derived. We then turn to study the volatility of expected
stock returns around this steady state. Once tested, the model shows that
deviations of de-trended capital stock, deviations of shocks from expected
values and deviations of labor force growth from steady state together
explain about 20% of the deviations of stock returns from long-term
expected values. Our estimates also implies that investors have levels of
risk aversion consistent with the literature, and that labor growth
fluctuations are not significant, due to crowding out effects.

1. INTRODUCTION
The efficient markets hypothesis implies that stock market prices should follow
a random walk and thus, stock returns should be unpredictable. However, many
recent studies such as Fama and French (1988a, b), Keim and Stambaugh
Advances in Investment Analysis and Portfolio Management, Volume 9, pages 1–20.
Copyright © 2002 by Elsevier Science Ltd.
All rights of reproduction in any form reserved.
ISBN: 0-7623-0887-7

1


2

`
CHRISTOPHE FAUGERE
AND HANY SHAWKY

(1986), French, Schwert and Stambaugh (1987), Campbell and Shiller (1988),
Chen, Roll and Ross (1986), Lo and MacKinlay (1988) and Fama (1990)
document returns predictability. These studies have shown that state variables
such as aggregate production growth and yield spreads are empirically useful
in predicting stock and bond returns.
In an attempt to explain the time-varying behavior of stock returns, two
broad categories of asset pricing models emerged. Consumption-based asset
pricing models such as in Merton (1973), Lucas (1978), Breeden (1979) and
Ceccetti, Lam and Mark (1990) relate the returns on financial assets to the
intertemporal marginal rate of substitution of consumers using a consumption
growth function. Production-based models on the other hand, relate the
marginal rate of transformation to asset returns using production functions as
in Cochrane (1991), Balvers, Cosimano and McDonald (1990) and Restoy and
Rockinger (1994).
Cochrane (1991) finds that historical stock returns are related to economic
variables such as growth rates of GNP and the investment to capital ratios.
Unfortunately however, he does not provide a formal structure for explaining
the real source and nature of economic fluctuations that might impact expected
asset returns. Shawky and Peng (1995) use a real business cycle model with
exogenous technical progress and show that technological shocks are critical
factors in explaining asset returns.
We develop an endogenous growth model with technological shocks that
affect both the final output and the depreciation rate of capital. We characterize
the properties of a particular steady state growth path, where expected growth
is constant in the long run. We then examine the volatility of stock returns
around the steady state as a function of deviations of the de-trended capital
stock, deviations from expected values of labor growth, and deviations of
shocks from their long-term mean. This leads to an empirically testable
hypothesis.
Our empirical results indicate that deviations of capital stock and
technological shocks from their long run mean are significant variables, but that
the deviations in labor growth are not, due perhaps to crowding out effects. The
theoretical model predicts both the optimal growth rate of the economy as well
as the long-term average stock market return. These are empirically testable
implications. In fact, our results imply that investors exhibit a fair degree of
consumption smoothing behavior. We also find that to be in accordance with
the sample’s expected stock return, our technology must exhibit some degree of
increasing returns to capital.
This paper is organized in five sections. In Section 2 we set up the model and
derive optimality conditions that are consistent with endogenous economic


Endogenous Growth and Stock Returns Volatility in the Long Run

3

growth. We develop the concept of steady-growth in Section 3 and proceed to
derive its equilibrium conditions. The time-series data, the methodology and
the empirical results are presented and analyzed in Section 4. The final section
provides a summary and some concluding remarks.

2. A MODEL OF ENDOGENOUS GROWTH
Consider a stochastic growth model similar to that in Brock-Mirman (1972),
where technology is affected by a random shock every period. We also assume
that growth is self-sustaining in a sense defined later on. Our goal is to
investigate how stock returns are affected by long run technological trends.
Formally our goal is to search for the optimal policy that solves

ͫ͸
ϱ

J (Y0) = max E0

ͬ

␳tU (Ct )

t=0

Ct = (1 Ϫ it ) ϫ Yt

(1)

Yt = ␪t f (vt , Kt )
Where Ct , Kt and Yt are per-unit-of-labor consumption, capital stock and
output, and vt is the rate of capacity utilization. Labor Lt is assumed to evolve
exogenously over time. The control variable is the investment rate it . The
variable ␪t is a multiplicative random shock that is i.i.d. and is defined over a
compact range [ ␪, ␪]. The value of ␪t is realized at the beginning of
period t. We assume that consumers’ preferences are represented by
U (Ct ) = C 1t Ϫ ␥/(1 Ϫ ␥), with ␥ > 0 representing the coefficient of relative risk
aversion (CRRA).
We assume that the production function has constant returns to scale in
capital and labor. In particular we use the following per-capita formulation:
f (vt , Kt ) = (A(vt Kt )␣ + B )1/␣
1

(2)

with 0 < ␣ < 1 for now. A crucial advantage of this formulation is that it allows
the economy to grow at an endogenously sustained rate. In the traditional
economic growth literature, an economy can only sustain growth by resorting
to exogenous technical progress. In a sense, the fundamental source of
economic growth is determined outside the model. The endogenous growth
literature however, has sought to incorporate the sources of economic
growth by featuring externalities (public spending, learning by doing) or
certain factors of production that can be accumulated forever (human capital).
A critical feature for achieving endogenous growth is that these externalities
counteract the natural tendency for decreasing returns to capital. In our model,


`
CHRISTOPHE FAUGERE
AND HANY SHAWKY

4

given a stream of technological shocks, the economy will generate endogenous
growth due to sufficiently high marginal returns to capital in the long run.2
Capital utilization rates are included in the model because it is a way to
measure the actual flow of services provided by the capital stock in place.
These rates are exogenously determined and incorporating them in the model
leads to a better estimate of the production function.3
We assume that capital depreciates at a stochastic rate ␦t , and evolves
according to:4
Lt
(3)
Kt + 1 = [itYt + (1 Ϫ ␦t )Kt ]
Lt + 1
It is further assumed that the depreciation rate ␦t , is perfectly negatively
correlated with the shocks ␪t , and hence we write ␦t = 1 Ϫ ␮␪t , so that the above
relationship becomes:
Lt
(4)
Kt + 1 = [itYt + ␮␪t Kt ]
Lt + 1
The parameter ␮ must be such that ␮␪t < 1. The intuition for having a stochastic
depreciation rate is that the outstanding stock of capital is generally subjected
to the same type of transitory technological shocks as output.5 For example, the
productivity of labor measured in output/hour might be temporarily raised as a
result of corporate downsizing. The productivity of capital might also be
temporarily raised as a result of a credit crunch. A rise in productivity might
induce some firms to slow down the rate of depreciation of certain capital
goods.6
Next, we are solving for the social planner’s optimum, as a way to
characterize the optimal paths of consumption and investment in this
economy.
A. Optimality Conditions
The standard first order condition is:7

ͭ

␳Et

U Ј(Ct + 1)
␪t + 1 f2 (vt + 1, Kt + 1)
U Ј(Ct )

ͮ

Lt
=1
Lt + 1

(5)

Letting ␪t + 1 f2 (vt + 1, Kt + 1) = (1 + Rt + 1) measure one plus the stock market return,
we get:

ͭ

␳Et

ͮ

U Ј(Ct + 1)
(1 + Rt + 1)
U Ј(Ct )

Lt
=1
Lt + 1

(6)


Endogenous Growth and Stock Returns Volatility in the Long Run

If we define Xt + 1 =

5

Ct + 1
and let Zt + 1 = X tϪ+ ␥1(1 + Rt + 1), then the first order condition
Ct

becomes:
Lt
=1
(7)
Lt + 1
We will assume as in Hansen-Singleton (1983) that Zt + 1 is log normally
distributed ln(Zt + 1) ~ N ( ␮t , ␴ 2) conditional on the information available at t.
Following their approach we can deduce a new first order condition as:

␳Et{Zt + 1}

ͭ ͩ ͪͮ ͩ ͪ

Et {Rt + 1} = ␥Et ln

Ct + 1
Ct

+ ln

Lt + 1
Ϫ ln(␳) Ϫ ␴ 2/2
Lt

(8)

Therefore:

ͭ ͩ ͪͮ ͩ ͪ

Rt + 1 = ␥Et ln

Ct + 1
Ct

+ ln

Lt + 1
Ϫ ln(␳) Ϫ ␴ 2/2 + ␧t + 1
Lt

(9)

Where ␧t + 1 = Rt + 1 Ϫ E {Rt + 1}. Equation (9) is identical to Hansen-Singleton
(1983), except for the term involving labor growth. Hansen and Singleton state
that it was not their goal to solve for an explicit representation of equilibrium
prices in terms of the underlying shocks to technology. We take their model a
step further by looking at the determinants of consumption growth in terms of
technological progress and shocks.

3. STEADY STATE GROWTH
A characteristic of most industrialized economies is that per-capita real
variables exhibit sustained growth over long periods of time. We will use the
concept of steady state growth to describe a situation in which all state
variables grow at the same constant expected rate. This is a novel approach in
a growth model with stochastic shocks. Traditionally, the long-term stability of
the economy refers to the convergence of cumulative distributions of shocks to
a stationary distribution, as in Brock and Mirman (1972).
In order to characterize steady state growth, we need to transform the
economy by detrending real variables.8 Let g denote a particular growth rate
and define new normalized variables as:
yt = Yt /(1 + g) t ct = Ct /(1 + g) t kt = Kt /(1 + g) t
We define a Fulfilled Expectations Steady state (FESS) as a vector
(␪, g, n, i, k¯ , y¯ ), where ␪ is the expected value of the random shock, g is the


`
CHRISTOPHE FAUGERE
AND HANY SHAWKY

6

long-run expected growth rate of consumption, n is the long run growth rate of
¯ y¯ ) is defined as follows:
the labor force, and the vector (i, k,
¯
lim ln(it ) = ln(i ); lim ln(kt ) = ln(k)
t→ϱ

t→ϱ

lim ln( yt ) = lim Et {ln( yt + 1)} = ln( y¯ )
t→ϱ

t→ϱ

with lim Et {ln(Ct + 1 /Ct )} = g and lim Et {ln(Lt + 1 /Lt )} = n
t→ϱ

t→ϱ

and lim ln(␪t ) = Et {ln(␪t + 1)} = ln(␪)
t→ϱ

A Fulfilled Expectations Steady state is an equilibrium where the sequence of
shock realizations converge to the expected value of the shock, and the long run
expected growth rate is actually realized.9 Our next proposition proves the
existence of such a steady state.
Proposition 1: Assume there is a sequence of ex-post shocks which
converges to ␪, such that capacity utilization rates converge to a constant and
the stock of capital grows at a constant rate in the long run, then a FESS
(␪, g, n, i, k¯ , y¯ ) exists and:

ͫͩ ͪ ͬ

g = lim Et {ln(Ct + 1/Ct )} = (1/␥) ln
t→ϱ

␳A1/␣v¯ ␪
+ ␴ 2/2
1+n

R¯ = lim Et {Rt + 1} = ln(A1/␣v¯ ␪) = ␥ ln (1 + g) + ln(1 + n) Ϫ ln(␳) Ϫ ␴ 2/2
t→ϱ

Proof: Assume that there exists a growth rate ␬ > 0 such that:
lim ln(␪t) = ln(␪) and lim ln(k˜ t ) = ln(k˜ )
t→ϱ

t→ϱ

With k˜ t = Kt /(1 + ␬) t. Therefore actual sequences of capital stocks grow to
infinity. As lim ln(vt ) = ln(¯v) this implies that:
t→ϱ

lim Et {Rt + 1} = lim Et {ln(␪t + 1 f2 (vt + 1, Kt + 1))} = ln(A1/␣v¯ ␪)
t→ϱ

t→ϱ

(10)

Where ln(␪) = Et {ln(␪t + 1)}. We conclude from the Euler equation that:

ͫͩ ͪ ͬ

lim Et {ln(Ct + 1 /Ct )} = (1/␥) ln
t→ϱ

␳A1/␣v¯ ␪
+ ␴ 2/2 = g
1+n

(11)

So that consumption grows at a constant expected growth rate.10 From Eqs (10)
and (11) we can easily derive the second equality expressing the long-term


Endogenous Growth and Stock Returns Volatility in the Long Run

7

expected rate R¯ as a function of the growth rate and other parameters. From the
capital accumulation equation we know that:
lim ln(k˜ t + 1) = lim ln(it(Av␣t + BK tϪ ␣)1/␣ + ␮) + lim ln(k˜ t )
t→ϱ

t→ϱ

t→ϱ

+ lim ln(␪t ) Ϫ ␬ Ϫ lim ln(Lt + 1 /Lt )
t→ϱ

t→ϱ

(12)

As lim ln(␪t ) = ln(␪) this implies
t→ϱ

lim ln(A1/␣it vt + ␮) = n + ␬ Ϫ ln(␪)

(13)

t→ϱ

so that lim ln(it vt ) = constant. Since ln(vt ) converges to a constant ln(¯v) thus the
t→ϱ

log of the rate of investment ln(it ) converges to a constant ln(i ). If output is to
grow at the same rate as consumption we have to set ␬ = g. In order to fulfill
the last condition of existence of an FESS:
lim ln( yt ) = lim Et{ln( yt + 1)} = lim ln( y¯ )
t→ϱ

t→ϱ

t→ϱ

(14)

We need the following to hold true:
lim ln(␪t ) + ␣ Ϫ 1 lim ln[A + B (vt Kt ) Ϫ ␣ ] + lim ln(vt ) + lim ln(kt )
t→ϱ

t→ϱ

= lim Et ln(␪t + 1) + ␣

t→ϱ

Ϫ1

t→ϱ

t→ϱ

Ϫ␣

lim ln[A + B (vt + 1 Kt + 1) ] + lim ln(vt + 1)
t→ϱ

t→ϱ

+ lim ln(kt + 1) = ln( y¯ )

(15)

t→ϱ

For some y¯. Again, this is true when lim ln(␪t ) = ln(␪).
t→ϱ

Q.E.D.

For the FESS to exist we impose a transversality condition that
lim (␳(1 + g)␥)t = 0. In other words, we need ␳(1 + g)␥ < 1.11 Proposition 1 gives
t→ϱ

exact closed form solutions for the optimal expected growth rate and the long
run expected stock return. The long run expected stock return equals the long
run expected productivity of capital. From a comparative statics perspective,
we see that the long run rate of growth would rise with larger expected
productivity, discount factor, and variance ␴ 2. The expected growth rate would
drop with faster population growth, and a larger degree of risk aversion.12
A. Deviations from the Steady State
We follow the Real Business Cycle literature (Kydland-Prescott, 1982), and
linearize the economy around the steady state (FESS). Even though an


`
CHRISTOPHE FAUGERE
AND HANY SHAWKY

8

economy subjected to arbitrary shocks does not necessarily converge to the
FESS, this steady state offers an interesting benchmark to look at macroeconomic fluctuations. It reproduces the stylized facts of actual economies,
while still accounting for the random nature of shocks. One additional
advantage is that by linearizing, we can construct a simple testable hypothesis
about these fluctuations, without having to know the actual shape of optimal
solutions.
The first step is to rewrite the first order conditions using normalized
variables. Let us recall that ct = Ct /(1 + g)t, and then we have:

ͭ ͩ ͪͮ ͩ ͪ ͩ ͪ

Rt + 1 = ␥Et ln

ct + 1
ct

+ ln

Lt + 1
(1 + g)␥
Ϫ ␴ 2/2 + ␧t + 1
Ϫ ln
Lt


(16)

Let et = Ct /Yt be the consumption rate, then we have:

ͭ ͩ ͪͮ ͭ ͩ ͪͮ ͩ ͪ
ͩ ͪ

Rt + 1 = ␥Et ln

Ϫ ln

et + 1
et

+ ␥Et ln

yt + 1
yt

+ ln

(1 + g)␥
Ϫ ␴ 2/2 + ␧t + 1


Lt + 1
Lt

(17)

We denote with a ‘hat’ variables that represent deviations from the FESS. Thus
¯ is the deviation of the stock market return, from its long run
Rˆ t + 1 = (Rt + 1 Ϫ R)
trend R¯ = ln(A1/␣v¯ ␪). The variable ˆlt + 1 = (ln(Lt + 1 /Lt ) Ϫ n) is the deviation of labor
force growth rate from its long-term value. We also define yˆ t = ln( yt /¯y),
kˆ t = ln(kt /k¯ ), vˆ t = ln(vt /¯v), and ␪ˆ t = ln(␪t/␪¯ ).
Because the representative agent’s problem can be solved after we normalize
the variables, analogous first order conditions imply that the optimal
consumption rate decision can be rewritten as et = 1 Ϫ it( yt ) = e( yt ).13 If we
define a monotonic transformation ln(et ) = Q(ln( yt )), then the function ln(et )
can be linearized around the FESS so that in effect we have:
eˆ t = ln(et /¯e) ≈ a ϫ ln( yt /¯y) = a ϫ yˆ t and Et ln(et + 1/¯e) ≈ a ϫ Et ln( yt + 1/¯y) (18)
Where a = QЈ(ln( y¯ )). The variable a represents the elasticity of the rate of
consumption with respect to income, along the FESS.14 In the long run, we
obtain the following expression for the return on the market:

ͭ ͩ ͪͮ

Rˆ t + 1 = ␥(1 + a)Et ln

yt + 1
yt

+ ˆlt + 1 + ␧t + 1

(19)


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