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

This Page Intentionally Left Blank

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.

This Page Intentionally Left Blank

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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