Essays on the Equilibrium Valuation of IPOs and Bonds
by
Kehong Wen
B.S.(University of Science and Technology of China) 1987
Ph.D. (The University of Texas at Austin) 1993
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Business Administration
in the
GRADUATE DIVISION
of the
UNIVERSITY of CALIFORNIA at BERKELEY
Committee in charge:
Professor Mark Rubinstein, Chair
Professor Henry Cao, Co-Chair
Professor Nils Hakansso

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n
Professor Roger Craine
May 2000
UMI Number: 9981117
Copyright 2000 by
Wen, Kehong
All rights reserved
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The dissertation of Kehong Wen is approved:
Chair Date
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University of California at Berkeley
May 2000
Essays on the Equilibrium Valuation of IPOs and Bonds
Copyright May 2000
by
Kehong Wen
1Abstract
Essays on the Equilibrium Valuation of IPOs and Bonds
by
Kehong Wen
Doctor of Philosophy in Business Administration
University of California at Berkeley
Professor Mark Rubinstein, Chair
Chapter 1 of this dissertation provides rational explanations for the IPO underperformance
puzzle. IPO underperformance is shown to arise in three equilibrium models with investor
heterogeneity and participation restrictions. The models also help explain why IPO under-
performance is concentrated in small stocks and why the average IPO return can be below
the risk-free rate. In these models, IPO residual risk acts as a source of systematic risk.
Schumpeterian creative destruction plays a key role in one of the models. This model is
extended into a dynamic setting in Chapter 2 to demonstrate that persistent after-market
underperformance is consistent with a rational expectations equilibrium. Building on the
dynamic extension, a uni…ed framework is o¤ered in Chapter 3 to address all three IPO
pricing puzzles. Many testable implications are derived and presented in detail to facilitate
future empirical work.
Chapter 4 investigates the general equilibrium implications of introducing new
2industries into the economy. It …rst establishes that the equilibrium of an N-industry pure-
exchange economy supports an N-factor Vasicek term structure of interest rates. It then
shows that industry characteristics enter as direct determinants of the yield curve, the
term premium, the forward premium, and the stock premium. Depending on the nature
of industry heterogeneity, the term structure of interest rates and the stock premium can
have qualitatively di¤erent dynamics. Depending on how industries interact, the market-
price-of-risk vector may admit di¤erent signs for its components. Consequently, risky assets
representing high impact industries can have negative return premia over bonds. This helps
explain why new industries may appear over-valued at times.
Professor Mark Rubinstein
Dissertation Committee Chair
iii
To my wife, Yunfang Lu,
and my daughter, Yanming Melinda Wen,
the stars in my life.
iv
Contents
List of Figures vi
List of Tables vii
1 A Rational Approach to IPO Underperformance 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Relation to Other Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 “Creative Destruction” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Heterogeneous Belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Preference for Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7 Long-run Underperformance: Benchmarks and Sources . . . . . . . . . . . . 33
2 A Dynamic After-market Model 36
2.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Testable Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 A Uni…ed Approach to the Three IPO Puzzles 48
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Underpricing and Hot-issue Market . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Testable Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Relation to Other Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.1 Underpricing Literature . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.2 Hot-issue Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.3 Relation between Underperformance and Underpricing . . . . . . . . 68
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Equilibrium Valuation in a Vasicek Economy with Heterogeneous Indus-
tries 70
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
v4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 The One-factor Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 The Multi-factor Vasicek Term Structure . . . . . . . . . . . . . . . . . . . 85
4.4.1 The Two-factor Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.2 Equilibrium Security Prices . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.3 Properties of the Two-factor Term Structure . . . . . . . . . . . . . 90
4.4.4 The N-factor Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5 Stock-fund Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.6 Comparative Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.6.1 Yield Curve Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.2 Term Premium and Stock Premium . . . . . . . . . . . . . . . . . . 107
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 References 112
A Proofs 118
vi
List of Figures
4.1 The solid line plots the quadratic function ± = 1+ "(1+ ")´: The dotted line
plots the quadratic function " = ¡±(± ¡ 1)=´ ¡ 1: The Relative volatility is
de…ned as ´ = ¾2M=¾
2
N : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Extending Figure 4.1 into the full parameter space R2: . . . . . . . . . . . . 107
4.3 When relative volatility ´ is increased, the solid line narrows and drop down
further into the third quadrant. The dotted line ‡atten out. . . . . . . . . . 109
4.4 When ´ is decreased, the solid line ‡atten out, while the dotted line narrows
and expands further into the …rst quadrant. . . . . . . . . . . . . . . . . . 110
vii
List of Tables
1.1 Asset payo¤s with di¤erential positive skewness . . . . . . . . . . . . . . . . 30
4.1 Yield curve dynamics in the one-factor case . . . . . . . . . . . . . . . . . . 83
viii
Acknowledgements
I thank …rst of all my committee members for their general guidance for completing the
work presented in this dissertation. I thank Jonathan Berk, Henry Cao, Sanjiv Das, Pe-
ter DeMarzo, Greg Du¢e, Dwight Ja¤ee, Matthew Spiegel, Brett Trueman, Hal Varian,
Miguel Villas-Boas, Ivo Welch, and especially Mark Rubinstein, for helpful discussions and
comments. Jay Ritter provided very detailed comments on the work presented in Chapter
1. I also thank my fellow Ph.D. students at the Haas School of Business, especially Yuan
Ma, Mark Taranto, Zane Williams, Nick Wonder, and Hong Yan for helpful discussions.
1Chapter 1
A Rational Approach to IPO
Underperformance
1.1 Introduction
For decades, researchers have been puzzled by three sets of empirical results as-
sociated with the pricing of initial public o¤erings (IPOs). Besides the well-documented
underpricing puzzle and hot-issue market puzzle1, severe long-run underperformance of
IPOs is reported recently by Ritter (1991) and Loughran and Ritter (1995), suggesting
that market ine¢ciency may be even more pervasive than previously recognized. Thus, the
IPO market, albeit small in scale, has become a leading example of anomalies against the
e¢cient market hypothesis (Fama 1998).
Current research in this area focuses on econometric issues and model speci…cation
1See Ibbotson (1975) for the …rst systematic study on underpricing, and Ibbotson and Ja¤e (1975) for
their early work on hot-issue markets. Ibbotson and Ritter (1995) provide an extensive survey of the IPO
literature.
2issues that are di¢cult to disentangle for long-window event studies (Brav and Gompers
1997, Barber and Lyon 1997, Brav 1998, Fama 1998, Loughran and Ritter 1998). Little the-
oretical work has been done, however, to o¤er insights into IPO long-run underperformance
and its relation to underpricing. Investor naivete appears to be a popular explanation for
these puzzling observations. For instance, Shiller (1991) proposes an “impresario” hypoth-
esis, which implies that investors are systematically fooled by investment bankers. Ritter
(1991) interprets some of his results as being consistent with Shiller’s hypothesis. More
recently, Teoh, Welch, and Wong (1998) argue that investors may be systematically fooled
also by earnings management of issuers. They conclude that “... it is unlikely that any
fully rational theory will be able to explain why some rational investors are willing to hold
IPOs in the after-market. Returns for what are likely to be risky and illiquid investments
are simply too low to be explained by known equilibrium models.”
The primary purpose of this chapter is to provide rational equilibrium models that
can explain low (average) after-market IPO returns2. Clearly, to develop such models, one
must go beyond the traditional Capital Asset Pricing Model (CAPM) of Sharpe (1964) and
Lintner (1965), since many assumptions underlying that model do not apply to the IPO
market. The approach taken here is to recognize and model two prominent features of the
IPO market that break the CAPM: participation restrictions and investor heterogeneity.
Two types of participation restriction are considered: share-supply restrictions and short-
sale restrictions. It is well known that almost all IPOs have lockup policies by which
insiders agree not to sell their shares until after a certain period of time (typically, 180
2Chapter 3 studies the inter-relations among all three IPO puzzles, building on a dynamic model developed
in Chapter 2.
3days or longer) has lapsed. It is also well known that shorting IPOs can be di¢cult, if not
entirely impossible. These restrictions in‡ate IPO after-market prices. Over time, however,
both restrictions are relaxed, since more shares become available when insiders sell more
of their shares or when IPO …rms issue more equities, and short sales become easier as
more shares become available for borrowing. The gradual relaxation of restrictions puts
downward pressures on price – leading to underperformance for IPOs, as compared to other
benchmark securities that have less or no such restrictions. These two e¤ects help explain
IPO underperformance, but they are not the only source of underperformance.
Investor heterogeneity contributes, in a more essential way, towards IPO underper-
formance. The fact that IPO investors are heterogeneous is evidenced by the unusually large
trading volume on the …rst public trading day of most IPOs (Miller and Reilly 1987, Hegde
and Miller 1989). The existence of large trading volumes in securities markets is one of the
important reasons for moving from the representative-agent model to explicit modeling of
investor heterogeneity (Varian 1989, Wang 1994). In developing a set of static models, I
allow, separately, for di¤erent endowments, di¤erent probability beliefs, and di¤erent pref-
erences. These models demonstrate that a certain degree of investor heterogeneity alone
can be su¢cient to generate IPO underperformance. Compounding on these heterogeneity
e¤ects are the e¤ects due to the short-sale cons
t with a rational equilibrium, some investors must have special reasons to
want to hold IPO stocks. These investors want the shares so much that they are willing to
pay high prices. Three such special reasons are considered explicitly. One, some investors
4hold IPO stocks in order to hedge their human capital risks. Two, some investors have
more favorable probability beliefs for IPO stocks than other investors do. The short-sale
constraint prevents these other investors from driving down the after-market price. Three,
some investors exhibit preference for skewness, and IPO stocks are known to have positively
skewed distributions (see e.g. Ibbotson 1975, Barber and Lyon 1997).
Another key intuition is that investors in the IPO market are looking for something
special, something that they cannot get by holding seasoned securities. More precisely, some
investors are particularly interested in the residual risk (the risk which is not spanned by
existing securities) of IPOs. IPO residual risk acts as a source of systematic risk and is priced
in equilibrium. From the point of view that innovation is the source of sustained economic
growth and economic development (Schumpeter 1934, 1942), new enterprises represent the
future of the economic system. Therefore, the risk associated with the emergence of new
industries becomes part of the economy-wide systematic risks.
This paper shows how these intuitions work by constructing and solving three
completely speci…ed models. The set of static models are mostly extensions of Lintner
(1969)3 along the aforementioned three dimensions of investor heterogeneity. Lintner (1969)
deals with the equilibrium impact of a no-short-sale constraint, in addition to a rich set of
other issues involving heterogeneous investors. I simplify his model setting to one with two
risky assets and two heterogeneous agents. Generalization to cases with more than two
risky assets and/or more than two agents is straightforward. I allow for di¤erent degrees
of short-sale constraints and derive explicitly the pricing impact by obtaining the shadow
3I thank Mark Rubinstein for singling out to me this important reference.
5price for the constraint.
In contrast to most recent IPO models, I retain the simplicity of symmetric infor-
mation in Lintner’s model. This allows me to deal with multiple risky assets fairly easily
and to extend one of the models to a multi-period setting in a straightforward manner.
This is not to deny the existence or importance of asymmetric information. However, for
the problem at hand, the results obtained here suggest that this popular assumption is not
necessary.
More importantly, I extend Lintner’s model to the case with heterogeneous endow-
ments, the case he does not consider in the paper. This case is particularly relevant in the
current study for two reasons. First, it is the most tractable case. The closed-form solution
is found to be strikingly simple and the price impacts of participation restrictions can be
clearly identi…ed with their sources. This greatly facilitates analysis and extension.
Second, an important idea related to Schumpeterian “creative destruction” (Schum-
peter 1942) is introduced in this case. Brie‡y speaking, one class of investors is endowed
with a non-tradeable asset that derives future payo¤ from existing knowledge. This estab-
lished knowledge is challenged by the introduction of new enterprises which frequently are
led by entrepreneurs equipped with innovative technologies. Initial public o¤erings of these
new enterprises provide a unique opportunity for the threatened investors to hedge against
the possible erosion of their future endowments. This hedging demand is a powerful motive
for these investors to hold IPOs in the after-market. As a result, this hedging-demand e¤ect
alone is su¢cient to explain the low expected return of IPOs.
Rational investors are willing to hold IPOs and receive low expected returns are
6also obtained in the case that investors di¤er in their probability assessments and in the
case that some investors exhibit preference for skewness. These results help strengthen the
conclusion that rational models can produce the kind of IPO underperformance observed
from data. I also …nd that in these two cases, while share-supply restriction always has an
e¤ect, a short-sale restriction may or may not have an e¤ect.
I then develop in Chapter 2 a dynamic extension of the static model with heteroge-
neous endowments, so that the intertemporal aspect of IPO after-market performance under
the environment of changing participation restrictions can be analyzed. I obtain closed-form
solution for the dynamic rational-expectations equilibrium. The equilibrium concept used
is the Radner (1972) equilibrium of plans, prices, and price expectations. Using the model
solution, it is readily shown that, regardless of the number of periods, a similar underper-
formance result is obtained in the dynamic context as well. The most important message
regarding long-run performance that comes out of this exercise is two-fold. On one hand,
there are legitimate reasons to expect persistent IPO underperformance over the long run,
supporting the still controversial empirical …nding of Ritter (1991). On the other hand,
attributing such …nding to things such as persistent investor overoptimism is premature.
The dynamic extension also paves the way for linking long-run performance of IPOs to their
initial pricing mechanism4.
This paper also produces some unanticipated results, the …rst of which has to do
with the degree of underperformance. It is shown that under a mild condition, the expected
rate of return from IPOs can in fact be below risk-free rate. Such a surprising result has
4This subject is taken up in Chapter 3.
7previously been observed by Loughran and Ritter (1995) in their empirical work with data
spanning more than two decades, from the 1970s to the 1990s. At a …rst glance, this severe
underperformance presents an insurmountable hurdle to a rational theory. Indeed, it is
di¢cult to imagine such a result, which implies a negative risk premium, from an equilib-
rium model with homogeneous agents. The models presented in this chapter demonstrate
that, when a su¢cient degree of heterogeneity is allowed, this counter-intuitive result can
nevertheless be consistent with a rational equilibrium. In particular, the hedging demand
driven by the economic force of creative destruction is powerful enough to support such an
equilibrium.
The second of these unanticipated results is related to the well-debated size e¤ect
in the asset pricing literature (Banz 1981, Berk 1995). Within the IPO literature, Brav
and Gompers (1997) …nd that long-run underperformance of IPOs concentrates in small
issues, after controlling for size and book-to-market as two independent pricing factors.
Interpretation of this …nding is not clear, however, mainly because the Fama-French style
three-factor model used in that study is an empirically motivated model (see Fama and
French 1992). Absent a theoretical model in which size enters in a meaningful way, “one is
merely testing whether any patterns that exist are being captured by other known patterns”
(Loughran and Ritter 2000). This paper derives CAPM-like asset pricing relationships in
which size enters in a well-speci…ed way. In view of such relationships, the result found by
Brav and Gompers can be interpreted. As will be shown in Sections 1.4-1.6, the inverse of
the size of the IPO stock appears in the coe¢cients of factors that determine IPO expected
return. Hence, size matters in cross-sectional regressions, but not as a risk factor. And since
8there is a negative sign in front of the coe¢cients, the smaller the size, the more severe the
underperformance, other things being equal.
The rest of the paper is organized as follows. Section 1.2 discusses the relation
between the explanation o¤ered here and those o¤ered in several published papers. Section
1.3 sets up the basic model structure for the three static models presented in Sections
1.4-1.6. A distinct type of investor heterogeneity is considered in each model. The most
important case is the one with heterogeneous endowments, in which the Schumpeterian
creative destruction process is formalized. This model is further developed into a multi-
period rational expectations model in Sections 2.1-2.2. Section 2.3 presents a number of
testable implications to facilitate future empirical work. Section 2.4 concludes and discusses
limitations and possible future extensions of the static and dynamic models developed in
Chapter 1 and Chapter 2. All formal proofs are collected in the Appendix.
1.2 Relation to Other Work
Miller (1977) provides an intuitive explanation for IPO underperformance based on
divergence of investor opinions, under the condition of no short sales. The partial equilib-
rium analysis is recently formalized by Morris (1996), who shows that heterogeneity of prior
beliefs and a binding short-sale constraint can support a …nitely lived price bubble. Morris
argues that this speculative bubble explains why IPOs appear over-valued immediately after
issuance, relative to their long-run values.
The explanation o¤ered by Miller or Morris relies on the hypothesis that divergence
of opinions would narrow over time after the IPO. The logic is that as the …rm grows older,
9the value of assets in place will grow relative to di¢cult-to-value growth options. Whether
or not that does happen systematically is a di¢cult empirical issue. This paper does not
rely on heterogeneous beliefs as the sole source of IPO underperformance. Other forms of
investor heterogeneity are also important, and their implications are easier to test than the
implications derived from divergence of opinions.
Both Miller and Morris assume that short sales are not permitted for any stock.
However, substantial short interests exist for seasoned equities. IPOs, on the other hand,
are much more di¢cult to short at the beginning5. This paper allows di¤erent stocks to be
di¤erentially short-sale constrained. More importantly, the constraint on IPOs is allowed
to change over time.
Allowing di¤erential short-sale constraints also distinguishes my models from most
equilibrium models with multiple securities and short-sale constraints in the literature (see
e.g. Lintner 1969, Jarrow 1980, Sharpe 1992). Explicit pricing impact of the short-sale
constraint is derived in my models by obtaining its shadow price explicitly. This permits a
clear analysis on comparative statics.
A salient feature of the IPO market emphasized here is that IPOs face more pro-
nounced restrictions when they come to the market and that these restrictions are eventually
relaxed to some “normal” levels. Presumably, these market imperfections are all public in-
formation, and hence their e¤ects should be re‡ected in prices. Contrary to the position
taken in Miller (1977), my models show that explicitly incorporating market frictions tends
5Wen (1999) contains references and discussions on related empirical facts. The key fact is that most of
the shares are closely held – the public ‡oat is typically only 10-30% of the shares outstanding immediately
after the o¤ering, and the pre-issue shareholders rarely allow thier shares to be borrowed for shorting. Also,
shares held in street name by the managing underwriter are most likely not available for shorting. I thank
Jay Ritter for pointing out this key fact.
10
to support the e¢cient market hypothesis (the semi-strong form), which merely says that
security prices re‡ect all public information. Section 4 shows, when the knowledge of market
frictions is included in the public information set, IPO prices follow a martingale process.
When the knowledge is omitted, IPO prices follow a strict supermartingale process.
Shiller (1990) presents an “impresario” hypothesis for IPOs. He argues that the
IPO market is subject to fads, and that investment bankers exploit these fads opportunis-
tically by underpricing IPOs to create excess demand, just as a manager of musical events
attempts to create an illusion of hot shows by issuing tickets at a sale price. Such temporary
fads must eventually fade away, resulting in long-run underperformance. This hypothesis
implies a signi…cant correlation between the measure of underpricing and the measure of
long-run underperformance. Empirical evidence does not appear to support this impli-
cation6. Another implication of Shiller’s hypothesis is that average cumulative abnormal
return, including the initial return, should not go below zero in the long-run. This is not
consistent with the results reported in Ritter (1991).
Similar to Shiller(1990), Ritter (1991) and Loughran and Ritter (1995) argue that
there is a “window of opportunity” in which investors are over-optimistic about new …rms’
prospects. Issuing …rms, instead of their bankers, take advantage of these swings of investor
sentiment by timing their IPOs. Firms that go public successfully within these windows
will then underperform relative to a market benchmark. While there is clear evidence of
timing in the IPO market (Lerner 1994), there is no clear reason why such a window opens
6Ritter (1991) attempts to identify a relationship between underpricing and underperformance. He in-
cludes underpricing as a right-hand-side determinant in his OLS regression for long-run returns, and …nds
that underpricing is the only regressor having insigni…cant in‡uence. A similar result is found in Hanley’s
(1993) regression.
11
from time to time.
Teoh, Welch, and Wong (1998) …nd evidence suggesting that investors may be sys-
tematically fooled by issuing …rms’ earnings management. They …nd …rms that aggressively
manage their earnings when going public tend to underperform various benchmarks in the
after-market. It is reasonable to argue that if the earnings management e¤ect exists system-
atically, short sellers should be able to take advantage of it. The reason they cannot do so
could be because of the existence of short-sales constraints. If the short sale e¤ect is taken
into consideration, I conjecture that the earnings management e¤ect will be weakened.
In contrast to the above three explanations, which in one form or another rely on
investor naivete, the explanation o¤ered in this chapter is based on investor rationality.
1.3 Model Setup
This section develops three one-period models, focusing on the equilibrium pricing
impacts of the short-sale constraint and limited share supply when investors di¤er in their
endowments, beliefs, and preferences. The basic setup for these models is outlined as follows.
Asset and Distribution. Consider a competitive, pure exchange economy in which
there are only two risky assets: asset 1 stands for the “market” (excluding IPO stocks) and
asset 2 for an IPO stock (or a portfolio of IPO stocks). Let J ´ f1; 2g. There is also a
risk-free asset, with zero net supply, available for trading. Without loss of generality, let
the exogenous interest rate be zero. The payo¤ from asset 1 at the end of the period is
M , which is normally distributed with mean ¹M and variance ¾
2
M . The payo¤ from asset
2 is aM + N , where a is a small fraction, and, in the case with heterogeneous endowments
12
N is normally distributed with mean ¹N and variance ¾
2
N . In the case with heterogeneous
preferences, N has in addition a non-zero third moment (see Section 3.3). M and N are
independent. Hence, a measures the correlation between these two assets, and N represents
the component of the IPO payo¤ which is not spanned by existing securities, i.e. N is the
residual risk of the IPO7. In the case with heterogeneous beliefs, the probability assessment
is agent-speci…c (see Section 3.2). It is natural to assume ¹M À ¹N ; ¾M À ¾N : I also
assume a2¾2M ¿ a¾2M ¿ ¾2N to re‡ect the fact that the residual component N is a much
riskier component. All parameters are positive.
Investor Preference. For simplicity, I consider only two classes of investors, A and
B. Let I ´ fA;Bg. Each investor maximizes expected utility over end-of-period wealth:
Ei[U i(W i)], i 2 I. The utility function for each agent is further specialized to the constant
absolute risk aversion (CARA) class in the …rst two cases. In the third case, preference for
skewness is introduced.
Endowment. Agents are endowed with initial …nancial wealth fW i0gi2I . They are
also endowed with non-tradeable, uncertain incomes feigi2I that are received at the end of
the period.
Participation Restriction. There is no restriction on investment in asset 1. The
total number of shares of asset 1 is normalized to one. There is a lower bound l 6 0 for
shorting asset 2. It is de…ned as the negative of the ratio of the total number of shares
available for short sale over the total number of shares outstanding. IPO shares available
7Mauer and Senbet (1992) provide an excellent discussion on why IPOs are best viewed as not being
spanned by existing securities, in a non-trivial sense. They also provide empirical evidence supporting this
view. They focus on the underpricing problem and do not develop a sequential, rational expectations model,
whereas in this paper I focus on long-run, sequential markets.
13
for trading is », which is de…ned as the ratio of the ‡oat8 over the shares outstanding. For
simplicity, share supply » is taken to be exogenous to the market.
1.4 “Creative Destruction”
Joseph Schumpeter, in one of his classics (Schumpeter 1942), characterizes capi-
talism by its incessant “Creative Destruction” process, driven by entrepreneurial creativity
and technological innovation. “The fundamental impulse that sets and keeps the capi._.talist
engine in motion comes from the new consumers’ goods, the new methods of production or
transportation, the new markets, the new forms of industrial organization that capitalist en-
terprise creates.” The competition thus created is “competition which commands a decisive
cost or quality advantage and which strikes not at the margins of the pro…ts and the outputs
of the existing …rms but at their very foundations and their very lives.”
Recently, Schumpeter’s vision of capitalism has received revived interest among
economic growth theorists (Romer 1986, 1990, Lucas 1988, Aghion and Howitt 1992, 1994).
According to the New Growth Theory, it is the endogenous production of new knowledge
that holds the key to sustained economic growth. However, as Fischer Black (1995, pp.107-
110) observes, new knowledge damages the value of old knowledge since new knowledge
“steals the market” from old knowledge and reduces its productivity. Black’s observation is
foreshadowed by Schumpeter’s view that the process of creative destruction is “the essential
fact about capitalism.”
This vision of capitalism is perhaps nowhere more apparent than in the dynamic
8The ‡oat is the number of shares that are actively tradable in the market, excluding shares subject to
lockup restrictions.
14
IPO market, where new …rms enter the public capital market for the …rst time. These
new …rms are very often led by entrepreneurs who use and develop new technologies. The
competitive threat they impose on old …rms points most directly to old technologies and
existing knowledge9. This process of creative destruction has important rami…cations in
the asset market. To construct a stock market model that re‡ects this e¤ect of creative
destruction, it is natural to consider the link between the income derived from old knowledge
and the payo¤ from the new stock. The following assumption formalizes this link:
Assumption In addition to …nancial asset WA0 , investor A is also endowed with ´ units of
a non-tradeable asset, which is interpreted as labor income derived from old knowledge.
Each unit pays o¤
Y = bM ¡ N ,
where b is assumed to be a small fraction.
Remark 1 Therefore, A’s future endowment eA = ´Y = ´(bM ¡N); where ¡N represents
the destruction e¤ect from the new enterprise (IPO …rm), in the following sense. In
the event that new knowledge prevails, old knowledge su¤ers. Moreover, the higher the
9The creative destruction e¤ect is dramatically illustrated by some internet IPOs in 1990s. Netscape, a
pioneer of the World Wide Web browser, threatened to topple the domination of Microsoft in the software
market for personal computers and network computers. Amazon.com, the pioneer of online retailing, imposed
intense competitive pressure on traditional “bricks-and-mortar” business models, with its enormous cost
advantage and superior customer service, both enabled by the proliferation of the internet. E*Trade, a
pioneer of online brokerage, has not only helped spawn a whole new industry, but also challenged the very
relevance of traditional brokers. In all these examples, it is not so much that new businesses are replacing old
businesses, since they in fact help to expand the markets, but rather that old practices are forced to change.
Hence, Microsoft started to embrace the internet; Barnes and Noble had to learn online book-selling; and
Merrill-Lynch has to walk a …ne line between going online and keeping the privilege of its army of traditional
brokers. What had been working well has become vulnerable. Knowledge and skills accumulated through
years of experience and training are becoming less relevant and less productive and hence are commanding
less rewards.
15
expected payo¤ for the new stock generated from taking on the residual risk (i.e. higher
the ¹N), the more expected damage for A’s endowment. Hence, successful creation
of a high-potential new …rm or a new industry means serious destruction of existing
human capital. It is important to note that the creative destruction process going
on between the new enterprise (aM + N) and the existing human capital (bM ¡ N)
is through the state variable N: This state variable is associated with the emergence
of the new enterprise. The process happens only in one direction: from the new
enterprise to existing knowledge. This captures the spirit of Schumpeter’s discussion
quoted above. The correlation, which can be any number between ¡1 and +1, between
payo¤s aM+N and bM ¡N , is determined by the coe¢cients a and b: The magnitude
of ´ measures the degree of erosion of old knowledge.
Remark 2 This assumption and the assumption about the IPO payo¤ (aM + N) rec-
ognize the crucial role of residual risk in a new enterprise. Here, residual risk is
non-diversi…able, since, as Schumpeter envisioned, these new competitors are forcing
their way into the mainstream. In other words, residual risk is becoming part of the
systematic risk. The “creative destruction” process makes certain investors particu-
larly interested in the IPO residual risk. Consequently, the residual risk is priced in
equilibrium.
Remark 3 The idea embodied in the above formulation is similar in spirit to Aghion and
Howitt (1994) in the sense that new technology may cause unemployment in businesses
using old technology.
It is reasonable to think that the creative destruction process must be happening
16
also in the product market, as in Aghion and Howitt (1992), and hence it may be
desirable to consider a similar relation between the payo¤ of the new stock and the
payo¤s of some existing stocks. However, this kind of relation is not considered here,
for it does not matter for the purpose of this paper. More speci…cally, the a¤ected old
stocks are tradeable assets, which will not cause IPO underperformance in the current
model.
I assume eB = 0, to emphasize that these two agents have entirely di¤erent future
endowments. B represents professional investors. For simplicity, both investors are assumed
to have the same CARA utility function with parameter ½. Each solves the following
optimization problem
max
®i; µij
E[¡e¡½W i ] (1.1)
s. t. W i = ®i + µi1M + µ
i
2(aM + N) + e
i
®i + µi1P1 + µ
i
2P2 = W
i
0; l 6 µi2, (i 2 I; j 2 J)
where, ®i is the dollar amount invested in the risk-free asset for investor i, and µij is the
share amount invested in risky asset j for investor i. P1; P2 are the market prices for risky
asset 1 and asset 2, respectively.
De…nition The equilibrium of this pure exchange …nancial market economy is a set of
portfolio holdings and security prices f®i; µij; Pjgi2I;j2J such that: (a) f®i; µijg solves
the optimization problem (1.1) for each agent, and (b) security markets clear, i.e.
®A + ®B = 0; µ
A
1 + µ
B
1 = 1; µ
A
2 + µ
B
2 = »:
In order to solve for the equilibrium, let us consider …rst the benchmark case in
17
which the short-sale constraint is absent, i.e. l = ¡1: The equilibrium for the benchmark
case is characterized by the following proposition.
Proposition 1 (Unrestricted Short Sale) If l = ¡1; the equilibrium asset prices and
the optimal portfolio holdings are:
P1 = ¹M ¡
1
2
½¾2M(1 + a» + b´);
P2 = a¹M + ¹N ¡
1
2
a½¾2M(1 + a» + b´) ¡
1
2
½¾2N(» ¡ ´);
µ
A
1 =
1
2
[1 ¡ (a + b)´]; µB1 =
1
2
[1 + (a + b)´];
µ
A
2 =
1
2
(» + ´); µ
B
2 =
1
2
(» ¡ ´):
Remark 1 The most important feature of the optimal portfolio holdings is that A holds
more than half of the IPO shares, while B holds less than half. This result is driven
by the key assumption of A’s future labor income, ´(bM ¡ N), which gives rise to a
hedging demand, ´=2; for A: In essence, this is a story of A-investors hedging their
paychecks by purchasing the IPO stock.
Do people actually behave this way when facing the competitive threat from an
emerging new industry? This is an important empirical question that lies outside the
scope of this paper. There is no question that within this model it is optimal for A to
hedge by purchasing the IPO stock. In reality, though, people may have other ways
to deal with the problem when facing possible erosion of human capital. They may
decide to invest more in themselves to pick up new skills. However, we must realize
that human capital accumulation is a slow and time-consuming process. For those
who have been successful and have already passed their best learning age, the only
18
viable choice is probably to invest in new enterprises through the capital market. It is
entirely reasonable to believe that people may be behaving this way, since those who
face the competitive threat the most are in the best position to recognize it early and
hedge accordingly. Technological leaders such as Microsoft and Intel certainly behave
this way. They are known for their strategic investments in promising start-up …rms.
Remark 2 Rearranging, P2 can be decomposed into four distinct components:
P2 = a¹M + ¹N (expected payo¤)
¡1
2
a(1 + a)½¾2M ¡
1
2
½¾2N (risk-aversion e¤ect)
+
1
2
½(a2¾2M + ¾
2
N )(1 ¡ ») (share-supply e¤ect)
+
1
2
½(¾2N ¡ ab¾2M)´ (hedging-demand e¤ect).
The …rst two terms are familiar. The last two e¤ects are especially important in the
IPO market. The thinner the ‡oat, the higher the share-supply e¤ect. When the
‡oat is equal to the number of shares outstanding, 1 ¡ » = 0, the share-supply e¤ect
vanishes. This e¤ect can be exploited to arti…cially support the IPO price. Indeed,
typically within a week after an IPO, underwriters selectively buy back shares for
issues that have fallen below their corresponding o¤er prices (Hanley et al. 1993,
Schultz and Zaman 1994, Prabhala and Puri 1998, Aggarwal 2000). By doing so,
underwriters extract shares out of the ‡oat, thereby decreasing ». A direct implication
from the result above is that the higher the total volatility of the IPO stock (a2¾2M +
¾2N ), the more e¤ective the price support.
The hedging-demand e¤ect comes from the fact that in order to hedge against
19
the possible erosion of future labor income, A is willing to pay an extra price for the
IPO. If there is no endowment di¤erence, i.e. ´ = 0; A and B simply divide up the
shares of the IPO. The di¤erence of the optimal holdings of asset 2 between A and
B is exactly ´, the di¤erence in units of endowment between the two agents. The
hedging demand term can be regarded as a measure of the “impactness” of the IPO.
The higher the impact this new business has, the higher the price it can command.
Since the a¤ected labor income can be very substantial, this hedging-demand e¤ect
can be quite signi…cant.
Using …nancial assets to hedge labor income risk is a theme that has received
increasingly sophisticated mathematical treatment in the recent …nance literature (see,
e.g. Dybvig 1990, He and Pagés 1993, Cuoco 1997, Du¢e et al. 1997). However,
this literature is mainly concerned with portfolio problems and o¤ers little help for
analyzing the speci…c pricing impact of the hedging demand. For that, one needs an
equilibrium approach in which asset prices are solved for10. The model presented here
provides an extremely simple way to characterize this impact.
Corollary 1 is immediate and will be needed to establish Theorem 1.
Corollary 1 (Short Sale Condition) Investor B will short the IPO stock if and only if
´ > ».
The condition in this corollary is not stringent. It su¢ces to have ´ close to 1, i.e.
the number of units of a¤ected endowment is greater than or slightly below one. Corollary
10For an early work, see Mayers 1972.
20
2 below establishes the condition under which the expected rate of return from the IPO
stock can be below the risk-free rate.
Corollary 2 The expected rate of return from the IPO stock is below the risk-free rate, if
and only if
´ >
»¾2N + a(1 + a»)¾
2
M
¾2N ¡ ab¾2M
:
This condition is not stringent, either. By assumption, a and b are small positive
numbers, and a2¾2M ¿ a¾2M ¿ ¾2N . Hence the condition is approximately equivalent to
´ > », the same as the short sale condition stated in Corollary 1.
Remark Corollary 2 shows that even without considering the short-sale constraint e¤ect, it
is possible to rationalize the empirical observation that average IPO return can some-
times be below the T-bill rate (see Loughran and Ritter 1995 for empirical evidence).
Corollary 3 The traditional CAPM does not hold in this economy. Instead, the following
CAPM-like relations hold:
E(R1) = ¯1E(RS) + Á1(1 ¡ ») + Á1(1 + a + b)´
E(R2) = ¯2E(RS) ¡ Á2(1 ¡ ») ¡ Á2(1 + a + b)´; (1.2)
where Rj is the rate of return of asset j, RS is the rate of return of the market portfolio
and is de…ned as RS ´ P1P1+P2R1 + P2P1+P2R2; ¯j is the traditional “beta” de…ned as
¯j ´ cov(Rj ;RS)var(RS) , and Áj is de…ned as
Áj ´
½
2Pj
¾2M¾
2
N
(1 + a)2¾2M + ¾
2
N
; j 2 J:
21
Remark 1 The relation shown in (1.2) can be thought of as a three-factor model with a
“‡oat” factor and an “endowment impact” factor, in addition to the market factor.
In contrast to the usual sense of a factor being a risk factor, here, (1 ¡ ») and ´ are
both certain. Nevertheless, this result has a clear cross-sectional implication for IPOs:
expected return is lower if the ‡oat is lower, or if the “destruction impact” is higher,
or both. These two additional factors contribute to IPO underperformance, making
it possible to have E(R2) ¡ ¯2E(RS) below zero.
The result in Corollary 3 is related to the two-beta CAPM derived in May-
ers (1972). Similar to the result obtained by Mayers, this result shows that non-
marketable assets can have a non-trivial impact on asset pricing.
Remark 2 The residual risk of IPOs is essential for the asset pricing relationship derived
here. There is a clear cross-sectional implication following from the market completion
function of IPOs: IPOs of traditional …rms like UPS, Goldman Sachs, and restaurants
should have higher expected returns than IPOs of more innovative …rms such as the
internet companies. This implication is generally consistent with the broad feature of
IPO underperformance reported in the literature (see e.g. Ritter 1991, 1998).
Remark 3 It is interesting to note that coe¢cient Áj is inversely related to market capital-
ization Pj . This implies that underperformance is especially pronounced for small-size
stocks, holding constant other parameters, especially the residual variance ¾2N . This
is consistent with the empirical …nding that IPO long-run underperformance tends to
concentrate in small IPO stocks, as reported in Ritter (1991, Table IV) and Brav and
Gompers (1997). However, it is important to note that even though size is important
22
in explaining expected IPO returns, it is not a risk factor.
Now we are ready to consider the case with short sales restrictions on the IPO.
We have the following main result of this subsection.
Theorem 1 (Restricted Short Sale) Let ¡1 6 l 6 0; the equilibrium asset prices are
given by
P1 = ¹M ¡
1
2
½(1 + a» + b´)¾2M ;
P2 = a¹M + ¹N (expected payo¤)
¡1
2
a(1 + a)½¾2M ¡
1
2
½¾2N (risk-aversion e¤ect)
+
1
2
½(a2¾2M + ¾
2
N )(1 ¡ ») (share-supply e¤ect)
+
1
2
½(¾2N ¡ ab¾2M)´ (hedging-demand e¤ect)
+½¾2N max[l ¡
1
2
(» ¡ ´); 0] (short-sale constraint e¤ect).
The optimal portfolio holdings are:
µ
A
1 =
1
2
(1 + a» ¡ b´) ¡ amin[1
2
(» + ´); » ¡ l];
µ
B
1 =
1
2
(1 ¡ a» + b´) + amin[1
2
(» + ´); » ¡ l];
µ
A
2 = min[
1
2
(» + ´); » ¡ l]; µB2 = max[
1
2
(» ¡ ´); l]:
Comparing the results in Theorem 1 to those in Proposition 1, we see that the price
of asset 1 is not a¤ected by the short-sale constraint on asset 2, although each investor’s
portfolio holdings are a¤ected by the constraint. The new e¤ect on IPO price is simply
another separate term added to the previous four terms. This new term has an “option-
like” feature: it is positive if and only if the short-sale constraint is binding for B. Moreover,
given » and ´, it is linearly increasing in l in the interval [12(» ¡ ´); 0].
23
Among the three price supporting factors, the short-sale constraint is most likely
the only one over which investment bankers can exert substantial control in the long run.
This result shows that tightening the short-sale restriction can be very e¤ective in main-
taining high after-market valuations for IPOs, especially for those IPOs of high residual
variance.
Clearly, Corollary 2 of Proposition 1 holds through when short-sale constraint
is binding, since now E(R2) is even lower. Hence the conclusion that it is possible for
rational investors to demand low returns from IPOs is reinforced by limited short sales.
The following corollary extends Corollary 3 above.
Corollary There are four factors determining expected asset returns:
E(R1) = ¯1E(RS) + Á1(1 ¡ ») + Á1(1 + a + b)´
+2Á1(1 + a)[l ¡
1
2
(» ¡ ´)]+;
E(R2) = ¯2E(RS) ¡ Á2(1 ¡ ») ¡ Á2(1 + a + b)´
¡2Á2(1 + a)[l ¡
1
2
(» ¡ ´)]+; (1.3)
where, all parameters are the same as in Corollary 3 of Proposition 1.
Remark (Cause of Long-run Underperformance) In light of this corollary, there are
at least three sources that may lead to the long-run underperformance of IPOs as
…rst documented in Ritter (1991). They are a thin ‡oat, a hedging demand due to
creative destruction, and a binding short-sale constraint. Compared to a reference
stock that has 100% ‡oat (i.e. 1 ¡ » = 0), that does not impose a competitive threat
to the established human capital (´ = 0), and that faces no short-sale constraint
24
(l = ¡1), an IPO stock commands an expected return which is lowered by an amount
Á2f(1 ¡ ») + (1 + a + b)´ + 2(1 + a)[l ¡ 12(» ¡ ´)]+g: These three factors work in
the same direction, and all of them become less signi…cant as time goes by, since
insiders will be selling (increasing »), competition will catch up (decreasing ´), and
short sales will become easier (decreasing l). When using the traditional CAPM to
measure performance of individual IPO stocks, there are the same three sources of
underperformance.
The result has a cross-sectional implication based on comparative statics. Over the
same time period, other things being equal, the larger the increase in ‡oat, the more severe
the underperformance; the higher the impact on existing human capital at the beginning, the
more severe the underperformance; the larger the decrease of a binding short-sale restriction,
the more severe the underperformance.
1.5 Heterogeneous Belief
In this subsection, the Lintner (1969) model is extended in the direction of het-
erogeneous probability beliefs, incorporating the e¤ect of a short-sale constraint. Lintner
(1969) has already investigated extensively the equilibrium impact of heterogeneous beliefs
when no short sales are allowed. The distinction here is that I focus on the di¤erential
short-sale constraint, and work out its equilibrium pricing impact explicitly.
The case of heterogeneous beliefs with short-sale constraints is important, for two
reasons. First, the IPO market is one of the most speculative segments of the capital market:
short operation history and lack of …nancial transparency may lead to a great degree of
25
divergence of beliefs or opinions among investors. Second, the di¢culty in shorting IPOs
may arti…cially in‡ate their prices, since pessimists cannot express their opinions freely and
consequently the optimists are the ones determining the equilibrium price. This is the view
expressed in Miller (1977) and in Morris (1996) in a single stock context.
However, as Jarrow (1980) shows, in a multiple securities context, a substitution
e¤ect may undo the price in‡ation caused by a no-short-sale constraint. Hence it is im-
portant to study the short-sale constraint e¤ect in a model with multiple stocks. This
subsection works out a fully speci…ed model showing that Miller’s intuition is correct when
the IPO stock is short-sale constrained but the “market” (excluding the IPO stock) is not.
For our purpose, a setting with di¤erential short-sale constraints is more reasonable than
the setting used in Jarrow (1980), since new stocks and old stocks face di¤erent constraints.
Consider again a market consisting of the two risky assets, 1 and 2. They are
jointly normally distributed. The two risk-averse investors A and B are identical except
that they have di¤erent beliefs about the mean and variance-covariance matrix of the asset
distribution. For investor i; asset 1 is Normal (¹iM ; §
i
M) and the residual component of
asset 2 is normal (¹iN ;§
i
N); where §
i
k ´ (¾2k)i; k = M;N: Furthermore, assume that these
two investors have the same CARA utility over terminal wealth. They are endowed only
with initial …nancial wealth to purchase the two risky assets. They can adjust their …nancial
position by borrowing and lending a risk-free asset with zero rate of interest.
Following a similar procedure as in the last subsection, we have a sequence of
results.
Proposition 2 If l = ¡1 (i.e. there is no short-sale constraint), then the equilibrium
26
prices and asset holdings are:
P1 = ¹M ¡ ½(1 + a»)¾2M ;
P2 = a[¹M ¡ ½(1 + a»)¾2M ] + ¹N ¡ ½»¾2N ;
µ
A
1 = ¡aµ¹;N ¡ a»µB¾;N + µ¹;M + (1 + a»)µB¾;M ;
µ
B
1 = aµ¹;N ¡ a»µA¾;N ¡ µ¹;M + (1 + a»)µA¾;M ;
µ
A
2 = µ¹;N + »µ
B
¾;N ; µ
B
2 = ¡µ¹;N + »µA¾;N ;
where, ¹j ´
¹Aj §
B
j +¹
B
j §
A
j
§Aj +§
B
j
; ¾2j ´
§Bj §
A
j
§Aj +§
B
j
; µ¹;j ´ ¹
A
j ¡¹Bj
½(§Aj +§
B
j )
, µ
i
¾;j ´
§ij
§Aj +§
B
j
, ( i 2 I;
j 2 J) and note µA¾;j + µB¾;j = 1:
Clearly, the portfolio holdings and equilibrium prices are determined by the dif-
ferences between (¹Aj ;§
A
j ) and (¹
B
j ;§
B
j ): If there are no such di¤erences, then the portfolio
holdings reduce to
µ
A
1 = 1=2; µ
B
1 = 1=2; µ
A
2 = »=2; µ
B
2 = »=2;
and the equilibrium prices reduce to the same expressions as those in Proposition 2 but
with ¹j and ¾
2
j replaced by the common mean ¹j and common variance ¾
2
j ; respectively.
The following immediate corollary shows an important consequence of heteroge-
neous belief. The result will also be needed in the derivation of Theorem 2.
Corollary (Short Sale Condition) B will choose to short the IPO if and only if ¹AN ¡
¹BN > ½»(§
A
N): Similarly, A will choose to short the IPO if and only if ¹
B
N ¡ ¹AN >
½»(§BN):
27
It is interesting to note that the short sale condition does not involve short-seller’s
belief of ¾2N ; nor anyone’s belief on asset 1’s distribution.
There are no counterparts of Corollary 2 and Corollary 3 of Proposition 1. Theo-
rem 2 in the following is the counterpart of Theorem 1.
Theorem 2 (Restricted Short Sale) Assume ¹AN ¡ ¹BN > ½»§AN and ¡1 6 l 6 0; the
equilibrium prices are
P1 = ¹M ¡ ½(1 + a»)¾2M ;
P2 = a¹M + ¹N (expected payo¤)
¡1
2
a(1 + a)½¾2M ¡
1
2
½¾2N (risk-aversion e¤ect)
+
1
2
½(a2¾2M + ¾
2
N )(1 ¡ ») (share-supply e¤ect)
+½(§AN)(l ¡ lB2 )+ (short-sale constraint e¤ect),
and the optimal portfolio holdings are
µ
A
1 = 1 ¡ µB1 ; µB1 = lB1 ¡ a(l ¡ lB2 )+;
µ
A
2 = min(» ¡ l; » ¡ lB2 ); µB2 = max(l; lB2 );
where, all parameters are the same as in Proposition 2, l
B
2 ´ ¡¹
A
N+¹
B
N+½»(§
A
N )
½(§AN+§
B
N )
= ¡µ¹;N +
»µ
A
¾;N < 0 is the optimal holding µ
B
2 when there is no short-sale constraint, and l
B
1 is the
optimal holding µ
B
1 when there is no short-sale constraint.
Remark 1 Compared to the results in Proposition 2, the price of asset 1 here is not a¤ected
by a short-sale constraint on asset 2. The net result of imposing a short-sale constraint
on the IPO stock is to increase the IPO price whenever the constraint is binding. This
28
result hence con…rms the intuition in Miller (1977), under fully speci…ed conditions.
Moreover, in this model the substitution e¤ect in Jarrow (1980) does not arise, since
the stocks are di¤erentially constrained.
Remark 2 (Comparative Statics) The amount of IPO price “in‡ation” is directly pro-
portional to the di¤erence between the allowed limit l and the desirable level l; in
the event the constraint is binding. For a given level of heterogeneity in belief, l is
…xed. The result on P2 shows that even without any change of investor opinion or
belief, just an increase or decrease of the short-sale limit l will cause P2 to increase
or decrease, as long as l > l: The more severely the constraint is binding, the more
“in‡ated” the IPO price will be. This comparative static provides a new explanation
for IPO underperformance caused by short-sale constraint: as short selling becomes
easier over time, price in‡ation decreases, and hence the IPO stock underperforms, as
compared to a reference stock which is not short-sale constrained.
On the other hand, …xing the level l, the larger the di¤erence ¹AN ¡¹BN , the larger
the price in‡ation, as long as l > l: This result supports Miller’s explanation of IPO
underperformance. The result also shows that the di¤erence between §AN and §
B
N is
not nearly as critical as the di¤erence between ¹AN and ¹
B
N .
1.6 Preference for Skewness
Investors in the IPO market are often believed to be chasing after the next “Mi-
crosoft”. Indeed, historically there are big winners that come along every now and then.
The empirical literature (e.g., Ibbotson 1975) has also shown that the IPO return distribu-
29
tion is highly skewed, with a very fat right-tail. It is hence reasonable to conjecture that at
least some investors exhibit strong preference for positive skewness. Pricing implication of
preference-for-skewness is important, for at least two reasons. One, if the skewness of IPO
returns is priced, then the right benchmark for measuring IPO returns may be di¤erent
from the usual ones used in empirical studies. Two, preference-for-skewness as a source of
investor heterogeneity may cause a short-sale-constraint premium.
The traditional CAPM is based on the mean-variance analysis, recognizing only the
…rst two moments in asset distributions. Rubinstein (1973) and Kraus and Litzenberger
(1976) extend the analysis to incorporate the e¤ects of higher-order moments, especially
the third moment. One important reason for this extension is that risky assets should be
regarded as normal goods. This requires the utility function to exhibit decreasing absolute
risk-aversion, which in turn implies a positive third derivative of the utility function with
respect to wealth. But U 000(W ) > 0 implies preference for positive skewness (see e.g. Kraus
and Litzenberger 1976). In this subsection, I extend this literature in two ways: (a) allow
di¤erential skewness in asset returns and di¤erential preferences for skewness, and (b) study
the consequence of short-sale restrictions.
Consider two classes of investors A and B: Investor A has an objective function
de…ned over the …rst three central moments of end-of-period wealth, while B’s preference
is represented only by the …rst two central moments (as in the mean-variance analysis):
E[UA(WA)] = E(WA) ¡ 1
2
½V ar(WA) +
1
3
k Skew(WA);
E[UB(WB)] = E(WB) ¡ 1
2
½V ar(WB):
The two risky assets have …nite …rst three moments, as shown in Table 1.1. All higher-order
30
Asset Payo¤ Mean Variance Skewness
1 M ¹M ¾
2
M 0
2 aM + N a¹M + ¹N a
2¾2M + ¾
2
N ±
3
N (= Skew(N))
Table 1.1: Asset payo¤s with di¤erential positive skewness
moments are assumed to be zero. As before, random payo¤s M ? N: Note that I assume
that only N; the residual component of IPO payo¤, has positive skewness ±3N to emphasize
that IPO returns are much more skewed than non-IPO returns. In addition, there is also a
risk-free asset with zero rate of return available for trading.
The total supply of asset 1 is normalized to 1; and that of asset 2 is »: 0 < » 6 1:
As before, assume that there is a lower bound ¡1 6 l 6 0 for the total fraction of shares
that can be shorted against asset 2; and that there is no such limit on asset 1.
Proposition 3 Let K ´ k±3N
½¾2N
: If K» > 3=4; an equilibrium may not exist. If 0 < K» 6
3=4; there exists a unique equilibrium, and the equilibrium prices and optimal portfolio
holdings are:
P1 = ¹M ¡
1
2
½(1 + a»)¾2M ;
P2 = a¹M + ¹N ¡
1
2
½[a(1 + a»)¾2M + »¾
2
N ] +
1
2
k±3N(1 ¡
p
1 ¡ K»)2=K2;
µ
A
1 =
1
2
+
1
2
a» ¡ a(1 ¡
p
1 ¡ K»)=K; µB1 =
1
2
¡ 1
2
a» + a(1 ¡
p
1 ¡ K»)=K;
µ
A
2 = (1 ¡
p
1 ¡ K»)=K; µB2 =
p
1 ¡ K»(1 ¡
p
1 ¡ K»)=K:
Remark The parameter K measures the relative importance of skewness (preference for
skewness) versus variance (risk aversion). An equilibrium may not exist if the former
outweighs the latter, since A then can never have enough of asset 2. When an equi-
librium does exist, it is unique. The expression for P1 is familiar. Since asset 1 is
31
not skewed, preference-for-skewness does not a¤ect its pricing. However, a skewness
premium appears in the equilibrium price of asset 2. It is proportional to k±3N , a
measure of strength of skewness. The premium is the extra price that investors are
willing to pay for a high potential stock.
The next corollary shows that a short-sale-constraint premium does not arise in
the current context.
Corollary 1 In equilibrium, no one sells short any asset. Hence the short-sale constraint
is ine¤ective.
Corollary 2 Expected rate of return from the IPO stock is below the risk-free rate, if and
only if
1 ¡ K» ¡
p
1 ¡ K» > 1
2
Ka(1 + a»)
¾2M
¾2N
: (1.4)
Remark The condition in Corollary 2 cannot be satis…ed in the current model. To see
why, divide both sides of (1.4) by
p
1 ¡ K»; the condition becomes p1 ¡ K» > 1+¢;
where ¢ ´ Ka(1+a»)¾2M
2(
p
1¡K»)¾2N
> 0: But
p
1 ¡ K» > 1 + ¢ () 1 ¡ K» > 1 + 2¢ + ¢2
() K» < ¡(2¢ + ¢2):
This is impossible since K > 0; » > 0.
This result implies that the ske._.ices ‡uctuate randomly, In-
dustrial Management Review 6, 41-49.
Schultz, Paul and Mir Zaman, 1994, Aftermarket support and underpricing of initial public
o¤erings, Journal of Financial Economics 35, 199-219.
Schumpeter, Joseph A., 1934, The Theory of Economic Development, Harvard University
Press, Cambridge, Massachussets.
——–, 1942, Capitalism, Socialism, and Democracy, Harper and Brothers, New York, esp.
Chapter VII.
Sharpe, William, 1964, Capital asset prices: A theory of market equilibrium under condi-
tions of risk, The Journal of Finance 19, 425-442.
——–, 1992, Capital asset prices with and without negative holdings, The Journal of
Finance, Vol.XLVI, No.2, 489-509.
Shiller, Robert J., 1990, Speculative prices and popular models, Journal of Economic
Perspectives 4, 55-65.
117
Teoh, Siew-Hong, Ivo Welch, and T.J. Wong, 1998, Earnings management and the long-run
market performance of initial public o¤erings, Journal of Finance, Vol.53, No.6.
Varian, Hal, 1989, Di¤erences of opinion in …nancial markets, in Courtenay Stone ed., Fi-
nancial risk: Theory, Evidence and Implications, Kluwer Academic Publishers, Nor-
well, Massachusetts.
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Financial Economics 5, 177-188.
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Economy, Vol.102, No.1, 127-168.
Wang, Jiang, 1996, The term structure of interest rates in a pure exchange economy with
heterogeneous investors, Journal of Financial Economics 41, 75-110.
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o¤erings, The Journal of Finance, Vol.44, 421-449.
Wen, Kehong, 1999, The short sale e¤ect on IPO pricing and performance, Work in
progress.
118
Appendix A
Proofs
Proposition 1.
Proof. A solves the optimization problem
max
®A;µA1 ;µ
A
2
E[¡e¡½WA ] (A.1)
s. t. WA = ®A + (µA1 + aµ
A
2 + b´)M + (µ
A
2 ¡ ´)N
®A + µA1 P1 + µ
A
2 P2 = W
A
0 : (A.2)
Because of the twin assumptions of CARA utility and normal distribution, the
problem is easily shown to be equivalent to the simpler problem of maximizing the following
objective function
E(WA) ¡ 1
2
½V ar(WA)
= ®A + (µA1 + aµ
A
2 + b´)¹M + (µ
A
2 ¡ ´)¹N
¡1
2
½[(µA1 + aµ
A
2 + b´)
2¾2M + (µ
A
2 ¡ ´)2¾2N ];
subject to the constraint (A.2). The necessary and su¢cient conditions for optimality are
given by the following …rst order conditions:
P1 = ¹M ¡ ½(µA1 + aµA2 + b´)¾2M (A.3)
P2 = a¹M + ¹N ¡ a½(µA1 + aµA2 + b´)¾2M ¡ ½(µA2 ¡ ´)¾2N (A.4)
WA0 = ®
A + µA1 P1 + µ
A
2 P2:
119
Similarly, B solves the optimization problem (without future endowment)
max
®B ;µB1 ;µ
B
2
E[¡e¡½WB ] (A.5)
s. t. WB = ®B + (µB1 + aµ
B
2 )M + (µ
B
2 )N
®B + µB1 P1 + µ
B
2 P2 = W
B
0 : (A.6)
The …rst order conditions for this problem are:
P1 = ¹M ¡ ½(µB1 + aµB2 )¾2M (A.7)
P2 = a¹M + ¹N ¡ a½(µB1 + aµB2 )¾2M ¡ ½(µB2 )¾2N (A.8)
WB0 = ®
B + µB1 P1 + µ
B
2 P2:
Now impose the market clearing conditions for asset holdings
µA1 + µ
B
1 = 1; µ
A
2 + µ
B
2 = »: (A.9)
From Walras’ Law, we know the market clearing condition for the risk-free asset is auto-
matically satis…ed.
(A.3)+(A.7), using (A.9), we get P1 as stated in Proposition 1.
(A.4)+(A.8), using (A.9), we get P2.
Then, using the results for P1 and P2, (A.3)£a¡(A.4), we get µA2 ; and consequently
µ
B
2 from the market clearing conditions. Results for µ
A
1 and µ
B
1 then follow from (A.3) and
(A.7), respectively.
Corollary 2 of Proposition 1.
Proof. The rate of return from the IPO stock is
R2 =
aM + N
P2
¡ 1: (A.10)
Its expectation is
E(R2) =
1
P2
[a¹M + ¹N ¡ P2]
=
1
P2
[
1
2
a½¾2M(1 + a» + b´) +
1
2
½¾2N(» ¡ ´)]; (A.11)
120
where, the second equality follows from Proposition 1. The necessary and su¢cient condi-
tion for E(R2) < 0 is a¾2M(1 + a» + b´) + ¾
2
N (» ¡ ´) < 0: This yields the condition stated
in the Corollary.
Corollary 3 of Proposition 1.
Proof. Similar to (A.10) and (A.11), we have
R1 =
M
P1
¡ 1; (A.12)
E(R1) =
1
P1
[
1
2
½(1 + a» + b´)¾2M ]: (A.13)
By de…nition, (A.10) and (A.12),
RS =
P1
P1 + P2
R1 +
P2
P1 + P2
R2 =
(1 + a)M + N
P1 + P2
¡ 1: (A.14)
By Proposition 1,
E(RS) =
1
P1 + P2
[(1 + a)¹M + ¹N ¡ (P1 + P2)]
=
1
P1 + P2
[
1
2
½(1 + a)(1 + a» + b´)¾2M +
1
2
½(» ¡ ´)¾2N ]: (A.15)
From(A.14), V ar(RS) = E[
(1 + a)(M ¡ ¹M) + N ¡ ¹N
P1 + P2
]2 =
(1 + a)2¾2M + ¾
2
N
(P1 + P2)2
:
From (A.12) and (A.14),
Cov(R1; RS) = Cov(
M
P1
¡ 1; (1 + a)M + N
P1 + P2
¡ 1) = (1 + a)¾
2
M
P1(P1 + P2)
:
From(A.10)and(A.14), Cov(R2; RS) =
a(1 + a)¾2M + ¾
2
N
P2(P1 + P2)
:
Hence, by de…nition,
¯1 =
Cov(R1; RS)
V ar(RS)
=
(P1 + P2)(1 + a)¾
2
M
P1[(1 + a)2¾2M + ¾
2
N ]
; (A.16)
¯2 =
Cov(R2; RS)
V ar(RS)
=
(P1 + P2)[a(1 + a)¾
2
M + ¾
2
N ]
P2[(1 + a)2¾2M + ¾
2
N ]
: (A.17)
121
Finally, from (A.13), (A.15), and (A.16), we have
E(R1) ¡ ¯1E (RS) =
½
2P1
¾2M¾
2
N
(1 + a)2¾2M + ¾
2
N
[1 ¡ » + (1 + a + b)´]:
Similarly, from (A.11), (A.15), and (A.17), we have
E(R2) ¡ ¯2E (RS) =
½
2P2
¾2M¾
2
N
(1 + a)2¾2M + ¾
2
N
[¡(1 ¡ ») ¡ (1 + a + b)´]:
Theorem 1.
Proof. If ´ 6 »; we know from Proposition 1 that the short-sale constraint
is ine¤ective. The results are identical to those in Proposition 1. If ´ > »; we know
that B-investors will short the IPO stock. If the short-sale constraint is not binding, i.e.
l < 12(» ¡ ´) < 0; the equilibrium solution is again identical to that in Proposition 1. We
need to consider only the case when the short-sale constraint on asset 2 is binding. Since
the short-sale constraint does not a¤ect A-investors, the optimal conditions for A stay the
same as in the proof of Proposition 1. For B-investors, however, the problem becomes
max
®B ;µB1 ;µ
B
2
E[¡e¡½WB ]
s. t. WB = ®B + (µB1 + aµ
B
2 )M + (µ
B
2 ¡ ´)N
®B + µB1 P1 + µ
B
2 P2 = W
B
0
µB2 = l (binding short-sale constraint).
It is easy to obtain for this problem the following …rst order conditions:
P1 = ¹M ¡ ½(µB1 + aµB2 )¾2M (A.18)
P2 = a¹M + ¹N ¡ a½(µB1 + aµB2 )¾2M ¡ ½(µB2 )¾2N + ° (A.19)
WB0 = ®
B + µB1 P1 + µ
B
2 P2; (A.20)
µB2 = l; (A.21)
where, ° is the Lagrangian multiplier for the short-sale constraint.
122
Solving the system of linear equations (A.3), (A.4), (A.9), and (A.18)-(A.21), we
have an expressions for P1 which is identical to that in Proposition 1, and
P2 = a¹M + ¹N ¡
1
2
a½¾2M(1 + a» + b´) ¡
1
2
½¾2N (» ¡ ´) +
1
2
°:
To obtain explicitly the multiplier °; (A.3)£a¡(A.4) to have
aP1 ¡ P2 = ¡¹N + ½(µA2 ¡ ´)¾2N :
Substituting in the expressions for P1 and P2; using the result µ
A
2 = » ¡ l; we have
°
2
= ½[l ¡ 1
2
(» ¡ ´)]¾2N :
To get µ
A
1 and hence µ
B
1 , plugging the result for P1 into (A.3), we have
µ
A
1 =
1
2
(1 + a» ¡ b´) ¡ a(» ¡ l):
Finally, combining the results for the case when the short-sale constraint is not
binding and the results for the case when it is binding, we have the results as stated in
Theorem 1.
Corollary of Theorem 1.
Proof. The proof is similar to that for Corollary 3 of Proposition 1, and is omitted.
Proposition 2.
Proof. A-investors solve the following optimization problem:
max
®A;µA1 ;µ
A
2
EA[¡e¡½WA ]
s. t. WA = ®A + µA1 M + µ
A
2 (aM + N)
®A + µA1 P1 + µ
A
2 P2 = W
A
0 :
We know
EA(WA) = ®A + (µA1 + aµ
A
2 )¹
A
M + µ
A
2 ¹
A
N
V arA(WA) = (µA1 + aµ
A
2 )
2§AM + (µ
A
2 )
2§AN :
123
The …rst order conditions for optimality are then:
P1 = ¹
A
M ¡ ½(µA1 + aµA2 )§AM (A.22)
P2 = a¹
A
M + ¹
A
N ¡ a½(µA1 + aµA2 )§AM ¡ ½µA2 §AN (A.23)
WA0 = ®
A + µA1 P1 + µ
A
2 P2:
Similarly, the optimality conditions for B-investors’ problem are:
P1 = ¹
B
M ¡ ½(µB1 + aµB2 )§BM (A.24)
P2 = a¹
B
M + ¹
B
N ¡ a½(µB1 + aµB2 )§BM ¡ ½µB2 §BN (A.25)
WB0 = ®
B + µB1 P1 + µ
B
2 P2
The market clearing conditions for the risky assets are:
µA1 + µ
B
1 = 1; µ
A
2 + µ
B
2 = »: (A.26)
We can now solve the system of six linear equations (A.22) - (A.26) for six un-
knowns.
(A.22)£§BM+(A.24)£§AM =) P1 = ¹M ¡ ½(1 + a»)¾2M ; where the parameters ¹M
and ¾2M are as de…ned in Proposition 2. The expression for P2 is not so easy to obtain. We
need to solve for portfolio holdings …rst.
(A.22) - (A.24) =)
0 = ¹AM ¡ ¹BM ¡ ½(1 + a»)§AM
+½(§AM + §
B
M)µ
B
1 + ½a(§
A
M + §
B
M)µ
B
2 : (A.27)
(A.23) - (A.25) =)
0 = a(¹AM ¡ ¹BM) + ¹AN ¡ ¹BN ¡ a½(1 + a»)§AM ¡ ½»§AN
+a½(§AM + §
B
M)µ
B
1 + ½[a
2(§AM + §
B
M) + §
A
N + §
B
N ]µ
B
2 : (A.28)
From equations (A.27) and (A.28), we obtain µ
B
1 and µ
B
2 easily. They are:
µ
B
1 = a
¹AN ¡ ¹BN ¡ ½»§AN
½(§AN + §
B
N)
¡ ¹
A
M ¡ ¹BM ¡ ½(1 + a»)§AM
½(§AM + §
B
M)
;
µ
B
2 =
¡¹AN + ¹BN + ½»§AN
½(§AN + §
B
N )
= ¡µ¹;N + »µA¾;N :
124
Now from (A.25) we get the expression for P2.
Clearly, the proof can be generalized to the case with more than two agents and
more than two risky assets, in a straightforward manner.
Theorem 2.
Proof. By assumption, ¹AN ¡¹BN > ½»§AN ; so according to the Corollary of Propo-
sition 2, B will short sell asset 2. If the short-sale constraint is not binding, then the
equilibrium solutions are the same as in Proposition 2.
If the short-sale constraint is binding, it is binding only for B. So in this case,
the optimality conditions for A are the same as (A.22) and (A.23). B; however, solves the
following maximization problem with one more constraint:
max
®B ;µB1 ;µ
B
2
EB[¡e¡½WB ]
s. t. WB = ®B + µB1 M + µ
B
2 (aM + N)
®B + µB1 P1 + µ
B
2 P2 = W
B
0
µB2 = l:
The …rst order conditions for B’s optimality are:
P1 = ¹
B
M ¡ ½(µB1 + aµB2 )§BM (A.29)
P2 = a¹
B
M + ¹
B
N ¡ a½(µB1 + aµB2 )§BM ¡ ½µB2 §BN + ° (A.30)
µB2 = l; (A.31)
WB0 = ®
B + µB1 P1 + µ
B
2 P2:
Now solve the system of seven linear equations in (A.22), (A.23), (A.26), and
(A.29-A.31) for seven unknowns, including the Lagrangian multiplier °. We …nd
° = ½(§AN + §
B
N)[l ¡
¹BN ¡ ¹AN + ½»§AN
½(§AN + §
B
N )
] = ½(§AN + §
B
N)(l ¡ lB2 );
µ
B
1 =
¹BN ¡ ¹AN + ½(1 + a»)§AN
½(§AN + §
B
N)
¡ al = lB1 + alB2 ¡ al:
125
Consequently,
P2 = a
¹BM§
A
M + ¹
A
M§
B
M ¡ ½(1 + a»)§AM§BM
§AM + §
B
M
+
¹BN§
A
N + ¹
A
N§
B
N ¡ ½»§AN§BN
§AN + §
B
N
+ ½§AN (l ¡ lB2 ):
Obviously, P1 is the same as before, and µ
B
2 is just l:
Combining the solutions for the cases when the short-sale constraint is binding
and when it is not, we have the results as stated in the Theorem.
Proposition 3.
Proof. A solves the following optimization problem:
max
®A;µA1 ;µ
A
2
E(WA) ¡ 1
2
½V ar(WA) +
1
3
k Skew(WA)
s. t. WA = ®A + µA1 M + µ
A
2 (aM + N) (A.32)
WA0 = ®
A + µA1 P1 + µ
A
2 P2:
Use (A.32) to …nd expressions for E(WA); V ar(WA); and Skew(WA); then sub-
stitute those back into the objective. The Lagrangian is
LA = ®A + (µA1 + aµA2 )¹M + µA2 ¹N
¡1
2
½[(µA1 + aµ
A
2 )
2¾2M + (µ
A
2 )
2¾2N ] +
1
3
k(µA2 )
3±3N
¡¸A(®A + µA1 P1 + µA2 P2 ¡ WA0 ): (A.33)
Note that the Lagrangian has a cubic term which may turn the function into a
convex one over some region. Hence the …rst order conditions are no longer su¢cient to
guarantee a local maximum. We must verify this property after …nding the solutions.
As necessary conditions, the …rst order conditions give
P1 = ¹M ¡ ½(µA1 + aµA2 )¾2M (A.34)
P2 = a¹M + ¹N ¡ ½[a¾2MµA1 + (a2¾2M + ¾2N)µA2 ] + k±3N(µA2 )2: (A.35)
126
B solves
max
®B ;µB1 ;µ
B
2
E(WB) ¡ 1
2
½V ar(WB)
s. t. WB = ®B + µB1 M + µ
B
2 (aM + N)
WB0 = ®
B + µB1 P1 + µ
B
2 P2:
The …rst order conditions for optimality are:
P1 = ¹M ¡ ½(µB1 + aµB2 )¾2M (A.36)
P2 = a¹M + ¹N ¡ ½[a¾2MµB1 + (a2¾2M + ¾2N)µB2 ]: (A.37)
Market clearing conditions are µA1 + µ
B
1 = 1 and µ
A
2 + µ
B
2 = »:
(A.34) and (A.36) =)
P1 = ¹M ¡
1
2
½(1 + a»)¾2M : (A.38)
(A.35) and (A.37) =)
P2 = a¹M + ¹N ¡
½
2
[a(1 + a»)¾2M + »¾
2
N ] +
1
2
k±3N(µ
A
2 )
2: (A.39)
The last term 12k±
3
N (µ
A
2 )
2 is clearly the skewness premium for the IPO stock. It needs to
be further determined.
(A.36) and (A.38) =) µB1 =
1
2
(1 + a») ¡ aµB2 : (A.40)
Plugging this into (A.37), we have
P2 = a¹M + ¹N ¡
½
2
a(1 + a»)¾2M ¡ ½(¾2N)µB2 : (A.41)
Comparing (A.41) to (A.39), and using the market clearing conditions, we have a
quadratic equation determining the portfolio holding µA2 :
K
2
(µA2 )
2 ¡ µA2 +
1
2
» = 0;
where K ´ k±3N
½¾2N
: Solving this equation, we have µ
A
2 =
1§p1¡K»
K :
127
We now eliminate one root: At the optimum, LA must be concave with respect
to µA2 : But
@2LA
@(µA2 )
2 = ¡½(a2¾2M + ¾2N) + 2k±3N (µA2 ): This is a linear, increasing function with
the unique zero at ½(a
2¾2M+¾
2
N )
2k±3N
: By model assumption, a¾M is much smaller than ¾N ; so
the zero is approximately at ½¾
2
N
2k±3N
= 12K : Hence, to guarantee
@2LA
@(µA2 )
2 6 0; we must have the
smaller root µ
A
2 =
1¡p1¡K»
K ; and require that
µ
A
2 =
1 ¡ p1 ¡ K»
K
6 ½(a
2¾2M + ¾
2
N)
2k±3N
: (A.42)
At the optimum, LA must in general satisfy a second order condition. When the
constraint is linear in the control variables, the second order condition implies that the
Hessian matrix of LA is negative semide…nite, evaluated at the optimum. The elements of
the Hessian matrix are as follows.
@2LA
@(µA1 )
2
= ¡½¾2M ;
@2LA
@µA2 @µ
A
1
= ¡½a¾2M ;
@2LA
@(µA2 )
2
= ¡½(a2¾2M + ¾2N ) + 2k±3N(µA2 ):
We require, in addition to @
2LA
@(µA2 )
2 6 0; that ( @
2LA
@(µA1 )
2 )(
@2LA
@(µA2 )
2 ) ¡ ( @
2LA
@µA2 @µ
A
1
)2 > 0 =)
½2¾2M¾
2
N ¡ 2½¾2Mk±3NµA2 > 0:
Or, the optimal holding µ
A
2 6 12K =) 1¡
p
1¡K»
K 6 12K =) 0 < K» 6 3=4:
Note that when this condition is satis…ed, condition (A.42) is automatically sat-
is…ed. So, if 0 < K» 6 3=4; we have the expressions for P1 from (A.38), for P2 from
(A.39) after plugging in µ
A
2 =
1¡p1¡K»
K ; and optimal portfolio holding µ
A
2 ; and µ
B
2 =
» ¡ µA2 =
p
1¡K»¡(1¡K»)
K : From (A.40), we have µ
B
1 =
1
2(1 + a») ¡ a(» ¡ 1¡
p
1¡K»
K ) =
1
2(1 ¡ a») + a1¡
p
1¡K»
K ; and …nally µ
A
1 =
1
2(1 + a») ¡ a1¡
p
1¡K»
K :
Corollary 1 of Proposition 3.
Proof. It is easy to see from Proposition 3 that 0 < µA2 < »; which implies that
0 < µ
B
2 < »: Consequently,
1
2¡ 12a» < µ
A
1 ; µ
B
1 <
1
2+
1
2a»: By assumption, a is a small fraction
and 0 < » 6 1; hence, 0 < µA1 ; µ
B
1 < 1:
Corollary 2 of Proposition 3.
128
Proof.
E(R2) =
1
P2
(a¹M + ¹N ¡ P2)
=
1
P2
f½
2
[a(1 + a»)¾2M + »¾
2
N ] ¡
1
2
k±3N(
1 ¡ p1 ¡ K»
K
)2g:
It follows that E(R2) < 0 () a(1 + a»)¾
2
M
¾2N
+ » <
k±3N
½¾2N
(1¡p1¡K»)2
K2 =
1
K (2 ¡
2
p
1 ¡ K») ¡ » () 1 ¡ K» ¡ p1 ¡ K» > 12Ka(1 + a»)
¾2M
¾2N
:
Corollary 3 of Proposition 3.
Proof. The proof is similar to the proof of Corollary 3 of Proposition 1, and is
omitted.
Theorem 3.
Proof. The proof is by backward induction.
First, we show all statements in Theorem 3 are true for time period T ¡ 1: The
problem is the same as the one solved in proving Theorem 1 with some simple substitutions.
The only thing new here is the indirect utility functions. We …rst derive the results based
on Theorem 1, and then do the appropriate substitutions.
By de…nition,
JA(WA0 ; 0) = max
®A;µA1 ;µ
A
2
E[¡e¡½WA ]
= ¡e¡½[E(WA)¡ 12½V ar(WA)]
= ¡ expf¡½[®A + (µA1 + aµA2 + b´)¹M + (µA2 ¡ ´)¹N
¡1
2
½[(µ
A
1 + aµ
A
2 + b´)
2¾2M + (µ
A
2 ¡ ´)2¾2N ]g:
The second equality follows from the joint-normality property.
We know that ®A = WA0 ¡ (µA1 P1 + µA2 P2): Using the expressions for P1 and P2 in
Theorem 1, we have
JA(WA0 ; 0) = ¡ expf¡½[WA0 + (b¹M ¡ ¹N )´ + fA(¢)]g
= ¡ expf¡½[WA0 + ´E(Y jF0) + fA(¢)]g;
where fA(¢) is a complicated function of »; ´; l and …xed parameters including ¾2M and ¾2N :
129
In the dynamic context, appropriate substitutions lead to
JA(WAT¡1; T ¡ 1) = ¡e¡½[W
A
T¡1+´E(Y jFT¡1)+fAT¡1(¢)]:
The value function for B is obtained in a similar fashion. Since there is no future
endowment for B; the second term in the exponent does not appear. This completes the
proof for t = T ¡ 1.
Now suppose all statements are true for t + 1; t + 2; :::; T ¡ 1: We show in the
following that the results are true for time period t: The following Lemma will be needed.
Lemma 1 Suppose the expressions for equilibrium prices are true for all future periods
t + 1; :::; T ¡ 1; then, at time t, the random component of P1;t+1 is aT¡t¡1M Mt+1; that
of P2;t+1 is aaT¡t¡1M Mt+1 + a
T¡t¡1
N Nt+1; and that of E(Y jFt+1) is baT¡t¡1M Mt+1 ¡
aT¡t¡1N Nt+1:
Proof of Lemma 1. By hypothesis, repeated iterations, and by the law of
iterated expectations, we have
P1;t+1 = E(MT jFt+1) + deterministic terms = aT¡t¡1M Mt+1 + d.t.s.
P2;t+1 = E(aMT + NT jFt+1) + d.t.s = aaT¡t¡1M Mt+1 + aT¡t¡1N Nt+1 + d.t.s.
E(Y jFt+1) = E(bMT ¡ NT jFt+1) = baT¡t¡1M Mt+1 ¡ aT¡t¡1N Nt+1:
(continuing proof of Theorem 3)
At period t; A solves the optimization problem
max
®At ;µ
A
1;t;µ
A
2;t
E[JA(WAt+1; t + 1)jFt] = E[¡e¡½(W
A
t+1+´E(Y jFt+1)jFt] (A.43)
s. t. WAt+1 = ®
A
t + µ
A
1;tP1;t+1 + µ
A
2;tP2;t+1
®At + µ
A
1;tP1;t+1 + µ
A
2;tP2;t+1 6 WAt :
130
Substituting WAt+1 into the objective, and by Lemma 1 all stochastic terms in the exponent
are jointly normally distributed. The problem is equivalent to the problem
max
®At ;µ
A
1;t;µ
A
2;t
E[WAt+1 + ´E(Y jFt+1)jFt] ¡
1
2
½V ar[WAt+1 + ´E(Y jFt+1)jFt] (A.44)
s. t. ®At + µ
A
1;tP1;t+1 + µ
A
2;tP2;t+1 6 WAt :
We have
E[WAt+1 + ´E(Y jFt+1)jFt] = ®At + µA1;tE[P1;t+1jFt] + µA2;tE[P2;t+1jFt] + ´E(Y jFt):
And by the Lemma, we have
V ar[WAt+1 + ´E(Y jFt+1)jFt] = (µA1;t + aµA2;t + b´)2(aT¡t¡1M )2¾2M
+(µA2;t ¡ ´)2(aT¡t¡1N )2¾2N :
The …rst order conditions, which are necessary and su¢cient, of the optimization problem
give
P1;t = E[P1;t+1jFt] ¡ ½¾2M(µA1;t + aµA2;t + b´)(aT¡t¡1M )2; (A.45)
P2;t = E[P2;t+1jFt] ¡ a½¾2M(µA1;t + aµA2;t + b´)(aT¡t¡1M )2 (A.46)
¡½¾2N(µA2;t ¡ ´)(aT¡t¡1N )2:
B solves the optimization problem
max
®Bt ;µ
B
1;t;µ
B
2;t
E[JB(WBt+1; t + 1)jFt] = E[¡e¡½W
B
t+1 jFt] (A.47)
s. t. WBt+1 = ®
B
t + µ
B
1;tP1;t+1 + µ
B
2;tP2;t+1
®Bt + µ
B
1;tP1;t+1 + µ
B
2;tP2;t+1 6 WBt :
Note that as before, we consider …rst the problem without the short-sale constraint. We have
E[WBt+1jFt] = ®Bt + µB1;tE[P1;t+1jFt] + µB2;tE[P2;t+1jFt];
and by the Lemma, we have
V ar[WBt+1jFt] = (µB1;t + aµB2;t)2(aT¡t¡1M )2¾2M + (µB2;t)2(aT¡t¡1N )2¾2N :
131
Solving B’s optimization problem, we have
P1;t = E[P1;t+1jFt] ¡ ½¾2M(µB1;t + aµB2;t)(aT¡t¡1M )2; (A.48)
P2;t = E[P2;t+1jFt] ¡ a½¾2M(µB1;t + aµB2;t)(aT¡t¡1M )2 ¡ ½¾2NµB2;t(aT¡t¡1N )2: (A.49)
The market clearing conditions are:
®At + ®
B
t = 0; µ
A
1;t + µ
B
1;t = 1; µ
A
2;t + µ
B
2;t = »t: (A.50)
(A.45)+(A.48) and (A.50) yield
P1;t = E[P1;t+1jFt] ¡ 1
2
½¾2M(1 + a»t + b´)(a
T¡t¡1
M )
2: (A.51)
(A.46)+(A.49) and (A.50) give
P2;t = E[P2;t+1jFt] ¡ 1
2
½¾2Ma(1 + a»t + b´)(a
T¡t¡1
M )
2
¡1
2
½¾2N(»t ¡ ´)(aT¡t¡1N )2: (A.52)
It follows from (A.48), (A.49), (A.51), and (A.52) that optimal holdings of asset 2 are
µ
A
2;t =
1
2
(»t + ´); µ
B
2;t =
1
2
(»t ¡ ´): (A.53)
We obtain the same short sale condition as before: B will short asset 2 if and only
if ´ > »t: Using (A.51), (A.45) gives optimal holdings for asset 1:
µ
A
1;t =
1
2
(1 + a»t ¡ b´) ¡ a
1
2
(»t + ´); (A.54)
µ
B
1;t =
1
2
(1 ¡ a»t + b´) + a
1
2
(»t + ´):
Now, assume ´ > »t so that B will short asset 2. The short-sale constraint maybe
binding for B. If it is not binding, we have the same solution as above. If it is binding, only
B’s problem is changed. B solves the following equivalent problem:
max
®Bt ;µ
B
1;t;µ
B
2;t
f®Bt + µB1;tE[P1;t+1jFt] + µB2;tE[P2;t+1jFt]
¡1
2
[(µB1;t + aµ
B
2;t)
2(aT¡t¡1M )
2¾2M + (µ
B
2;t)
2(aT¡t¡1N )
2¾2N ]g
s. t. ®Bt + µ
B
1;tP1;t+1 + µ
B
2;tP2;t+1 = W
B
t
lt = µ
B
2;t:
132
The necessary and su¢cient conditions for optimality are now
P1;t = E[P1;t+1jFt] ¡ ½¾2M(µB1;t + aµB2;t)(aT¡t¡1M )2; (A.55)
P2;t = E[P2;t+1jFt] ¡ a½¾2M(µB1;t + aµB2;t)(aT¡t¡1M )2
¡½¾2NµB2;t(aT¡t¡1N )2 + °; (A.56)
where ° is the Lagrangian multiplier for the binding short-sale constraint.
(A.45)+(A.55) and (A.50) yield
P1;t = E[P1;t+1jFt] ¡ 1
2
½¾2M(1 + a»t + b´)(a
T¡t¡1
M )
2; (A.57)
the same as (A.51).
(A.46)+(A.56) and (A.50) give
P2;t = E[P2;t+1jFt] ¡ 1
2
½¾2Ma(1 + a»t + b´)(a
T¡t¡1
M )
2
¡1
2
½¾2N(»t ¡ ´)(aT¡t¡1N )2 +
1
2
°: (A.58)
To obtain the explicit expression for the shadow price °; note that the binding short-sale
constraint gives µ
B
2 = lt; and µ
A
2 = »t ¡ lt: Now a£(A.45)¡(A.46):
aP1;t ¡ P2;t = aE[P1;t+1jFt] ¡ E[P2;t+1jFt] (A.59)
+½¾2N (»t ¡ lt ¡ ´)(aT¡t¡1N )2:
And a£(A.56)¡(A.57):
aP1;t ¡ P2;t = aE[P1;t+1jFt] ¡ E[P2;t+1jFt] (A.60)
+
1
2
½¾2N(»t ¡ ´)(aT¡t¡1N )2 ¡
1
2
°:
(A.59) and (A.60) imply 12° = ½¾
2
N (lt ¡ 12(»t ¡ ´))(aT¡t¡1N )2: Plugging back into (A.57), we
have the equilibrium price of asset 2 when the short-sale constraint is binding:
P2;t = E[P2;t+1jFt] ¡ 1
2
½¾2Ma(1 + a»t + b´)(a
T¡t¡1
M )
2
¡1
2
½¾2N(»t ¡ ´)(aT¡t¡1N )2 + ½¾2N(lt ¡
1
2
(»t ¡ ´))(aT¡t¡1N )2: (A.61)
133
From (A.45), (A.57), and µ
B
2 = lt; we have
µ
A
1;t =
1
2
(1 + a»t ¡ b´) ¡ alt and µB1;t =
1
2
(1 ¡ a»t + b´) + alt:
Combining these results with those when the short-sale constraint is not binding,
we have the equilibrium prices and optimal portfolio holdings of period t as stated in
Theorem 3 (re-arrange terms in (A.61) as done before). In particular, the min and max
functions in the solution are all results of the short-sale constraint.
Finally, the value functions for A and B are
JA(WAt ; t) = ¡e¡½E[W
A
t+1+´E(Y jFt+1)jFt]+ 12½V ar[W
A
t+1+´E(Y jFt+1)jFt]
JB(WBt ; t) = ¡e¡½E[W
B
t+1jFt]+12½V ar[W
B
t+1jFt];
where, E[W
A
t+1jFt]
= ®A;t + µ
A
1;tE[P1;t+1jFt] + µA2;tE[P2;t+1jFt]
= WAt ¡ µA1;tfP1;t ¡ E[P1;t+1jFt]g ¡ µA2;tfP2;t ¡ E[P2;t+1jFt]g
= WAt + µ
A
1;t
1
2
½¾2M(1 + a»t + b´)(a
T¡t¡1
M )
2 + µ
A
2;tf
1
2
½¾2Ma(1 + a»t + b´)(a
T¡t¡1
M )
2
+
1
2
½¾2N (»t ¡ ´)(aT¡t¡1N )2 ¡ ½¾2N(a2N)T¡t¡1max[lt ¡
1
2
(»t ¡ ´); 0]g;
E[´E(Y jFt+1)jFt] = ´E(Y jFt);
V ar[WAt+1 + ´E(Y jFt+1)jFt]
= (µ
A
1;t + aµ
A
2;t + b´)
2(aT¡t¡1M )
2¾2M + (µ
A
2;t ¡ ´)2(aT¡t¡1N )2¾2N ;
E[W
B
t+1jFt]
= ®B;t + µ
B
1;tE[P1;t+1jFt] + µB;tE[P2;t+1jFt]
= WBt ¡ µB1;tfP1;t ¡ E[P1;t+1jFt]g ¡ µB2;tfP2;t ¡ E[P2;t+1jFt]g
= WBt + µ
B
1;t
1
2
½¾2M(1 + a»t + b´)(a
T¡t¡1
M )
2 + µ
B
2;tf
1
2
½¾2Ma(1 + a»t + b´)(a
T¡t¡1
M )
2
+
1
2
½¾2N(»t ¡ ´)(aT¡t¡1N )2 ¡ ½¾2N(a2N )T¡t¡1max[lt ¡
1
2
(»t ¡ ´); 0]g;
134
V ar[W
B
t+1jFt] = (µB1;t + aµB2;t)2(aT¡t¡1M )2¾2M + (µ
B
2;t)
2(aT¡t¡1N )
2¾2N :
So,
JA(WAt ; t) = ¡e¡½[W
A
t +´E(Y jFt)+fAt (¢)]
JB(WBt ; t) = ¡e¡½[W
B
t +f
B
t (¢)];
where fAt (¢) and fBt (¢) are complicated but deterministic functions of »t; ´; lt and other …xed
parameters.
Hence all statements of Theorem 3 are true for period t; if they are true for
t + 1; :::; T ¡ 1: By induction, the Theorem is true for all periods.
Corollary 1 of Theorem 3.
Proof. From Theorem 3,
P2;t = E[P2;t+1jFt] ¡ 1
2
½¾2Ma(1 + a»t + b´)(a
T¡t¡1
M )
2
¡1
2
½¾2N (»t ¡ ´)(aT¡t¡1N )2 + ½¾2N(lt ¡
1
2
(»t ¡ ´))+(aT¡t¡1N )2:
This implies, through repeated iterations, and by the law of iterated expectations, that
P2;t = E[P2;t+sjFt] ¡
t+s¡1X
u=t
[
1
2
½¾2Ma(1 + a»u + b´)(a
T¡u¡1
M )
2 (A.62)
+
1
2
½¾2N (»u ¡ ´)(aT¡u¡1N )2 ¡ ½¾2N(lu ¡
1
2
(»u ¡ ´))+(aT¡u¡1N )2]:
To have E[R2;t;sjFt] < 0; or E[P2;sjFt] < P2;t; it is su¢cient that
1
2
½¾2Ma(1 + a»u + b´)(a
T¡u¡1
M )
2 +
1
2
½¾2N(»u ¡ ´)(aT¡u¡1N )2
< ½¾2N(lu ¡
1
2
(»u ¡ ´))+(aT¡u¡1N )2; for all t 6 u 6 t + s ¡ 1:
Certainly, it is also su¢cient that
¾2Ma(1 + a»u + b´)(a
T¡u¡1
M )
2 + ¾2N(»u ¡ ´)(aT¡u¡1N )2 < 0; for all t 6 u 6 t + s ¡ 1;
which gives the condition in Corollary 1.
Corollary 2 of Theorem 3.
135
Proof. By de…nition, Rj;t;s =
Pj;t+s
Pj;t
¡ 1: From Theorem 3, we have
P1;t = E[P1;t+sjFt] ¡ 1
2
½¾2M
t+s¡1X
u=t
(1 + a»u + b´)(a
T¡u¡1
M )
2; (A.63)
and P2;t as expressed in (A.62). Hence,
Et(R1;t;s) =
1
P1;t
1
2
½¾2M
t+s¡1X
u=t
(1 + a»u + b´)(a
T¡u¡1
M )
2; (A.64)
Et(R2;t;s) =
1
P2;t
1
2
½
t+s¡1X
u=t
[¾2Ma(1 + a»u + b´)(a
T¡u¡1
M )
2 + ¾2N(»u ¡ ´)(aT¡u¡1N )2
¡2¾2N(lu ¡
1
2
(»u ¡ ´))+(aT¡u¡1N )2]: (A.65)
The rate of return for the market portfolio from t to t + s is
RS;t;s =
P1;t
P1;t + P2;t
R1;t;s +
P2;t
P1;t + P2;t
R2;t;s =
P1;t+s + P2;t+s
P1;t + P2;t
¡ 1: (A.66)
By (A.63) and (A.62),
Et(RS;t;s) =
1
P1;t + P2;t
1
2
½
t+s¡1X
u=t
[¾2M(1 + a)(1 + a»u + b´)(a
T¡u¡1
M )
2
+¾2N(»u ¡ ´)(aT¡u¡1N )2 ¡ 2¾2N(lu ¡
1
2
(»u ¡ ´))+(aT¡u¡1N )2]: (A.67)
In order to evaluate conditional variances, we need the following Lemma.
Lemma 2 The random component of P1;t+s is aT¡t¡sM Mt+s, and that of P2;t+s is aa
T¡t¡s
M Mt+s+
aT¡t¡sN Nt+s. Conditional variance V ar[Mt+sjFt] = ¾2M
sP
u=1
(as¡uM )
2: Similarly, V ar[Nt+sjFt] =
¾2N
sP
u=1
(as¡uN )
2:
Proof of Lemma 2. The …rst statement follows from the proof of Lemma 1.
For the second statement, note that
Mt+s = aMMt+s¡1 + "M;t+s = :::
= asMMt +
sX
u=1
as¡uM "M;t+u:
The result then follows from the i.i.d. assumption. Similarly for the last statement.
136
From (A.66),
V art(RS;t;s) =
V art(P1;t+s + P2;t+s)
(P1;t + P2;t)2
=
(1 + a)2(aT¡t¡sM )
2V art(Mt+s) + (a
T¡t¡s
N )
2V art(Nt+s)
(P1;t + P2;t)2
=
(1 + a)2¾2M
sP
u=1
(aT¡t¡uM )
2 + ¾2N
sP
u=1
(aT¡t¡uN )
2
(P1;t + P2;t)2
=
(1 + a)2¾2MaM;t;s + ¾
2
NaN;t;s
(P1;t + P2;t)2
: (A.68)
We have de…ned:
aM;t;s ´
sX
u=1
(aT¡t¡uM )
2; aN;t;s ´
sX
u=1
(aT¡t¡uN )
2: (A.69)
The second and the third equalities follow from Lemma 2. Using the same Lemma, we have
the following covariance:
Covt(R1;t;s; RS;t;s)
= Et[
P1;t+s ¡ Et(P1;t+s)
P1;t
P1;t+s + P2;t+s ¡ Et(P1;t+s + P2;t+s)
P1;t + P2;t
]
=
Et[a
T¡t¡s
M (Mt+s ¡ EtMt+s)((1 + a)aT¡t¡sM (Mt+s ¡ EtMt+s) + aT¡t¡sN (Nt+s ¡ EtNt+s)]
P1;t(P1;t + P2;t)
=
(1 + a)(aT¡t¡sM )
2V art(Mt+s)
P1;t(P1;t + P2;t)
=
(1 + a)¾2M
sP
u=1
(aT¡t¡uM )
2
P1;t(P1;t + P2;t)
=
(1 + a)¾2MaM;t;s
P1;t(P1;t + P2;t)
: (A.70)
Similarly,
Covt(R2;t;s; RS;t;s) =
a(1 + a)¾2MaM;t;s + ¾
2
NaN;t;s
P2;t(P1;t + P2;t)
: (A.71)
Consequently, (A.68), (A.70), and (A.71) give
¯1;t;s =
Covt(R1;t;s; RS;t;s)
V art(RS;t;s)
=
(P1;t + P2;t)(1 + a)¾
2
MaM;t;s
P1;t[(1 + a)2¾2MaM;t;s + ¾
2
NaN;t;s]
; (A.72)
¯2;t;s =
Covt(R2;t;s; RS;t;s)
V art(RS;t;s)
=
(P1;t + P2;t)[a(1 + a)¾2MaM;t;s + ¾
2
NaN;t;s]
P2;t[(1 + a)2¾2MaM;t;s + ¾
2
NaN;t;s]
: (A.73)
137
Finally, direct calculation using (A.64), (A.67), and (A.72) gives
Et(R1;t;s) ¡ ¯1;t;sEt(RS;t;s)
= Á1;t;s
sX
u=1
[(1 + a)(aT¡t¡uN )
2aM;t;s ¡ a(aT¡t¡uM )2aN;t;s](1 ¡ »t+u¡1)
+Á1;t;s(1 + a + b)aM;t;s aN;t;s ´
+2Á1;t;s(1 + a)aM;t;s
sX
u=1
(¾T¡t¡uN )
2(lt+u¡1 ¡ 1
2
(»t+u¡1 ¡ ´))+:
Similarly, (A.65), (A.67), and (A.73) give
Et(R2;t;s) ¡ ¯2;t;sEt(RS;t;s)
= ¡Á2;t;s
sX
u=1
[(1 + a)(aT¡t¡uN )
2aM;t;s ¡ a(aT¡t¡uM )2aN;t;s](1 ¡ »t+u¡1)
¡Á2;t;s(1 + a + b)aM;t;s aN;t;s ´
¡2Á2;t;s(1 + a)aM;t;s
sX
u=1
(¾T¡t¡uN )
2(lt+u¡1 ¡ 1
2
(»t+u¡1 ¡ ´))+:
Corollary 3 of Theorem 3.
Proof. By de…nition of Q0, and Theorem 3, we have
EQ0 [P2;t+1jFt] = E[P2;t+1jFt] ¡ 1
2
½¾2Ma(1 + a)(a
2
M)
T¡t¡1 ¡ 1
2
½¾2N(a
2
N)
T¡t¡1:
By Theorem 3 again,
P2;t = E
Q0 [P2;t+1jFt]
+
1
2
½[¾2Ma
2(a2M)
T¡t¡1 + ¾2N(a
2
N)
T¡t¡1](1 ¡ »t)
+
1
2
½[¾2N(a
2
N)
T¡t¡1 ¡ ab¾2M(a2M)T¡t¡1]´
+½¾2N (a
2
N)
T¡t¡1max[lt ¡ 1
2
(»t ¡ ´); 0]
> EQ0 [P2;t+1jFt]:
Theorem 4.
Proof. Note that ³0 = »0; and ³1 = »1 ¡ »0:
138
From the rational expectations assumption, we have the asset prices:
P2;2 = aM2 + N2
= a(aMM1 + "M;2) + aNN1 + "N;2
= aa2MM0 + aaM"M;1 + a"M;2 + a
2
NN0 + aN"N;1 + "N;2;
P2;1 = aaMM1 + aNN1 ¡ 1
2
½¾2Ma[1 + a»1 + b´]
¡1
2
½¾2N(»1 ¡ ´) + ½¾2N [l1 ¡
1
2
(»1 ¡ ´)];
P o2;0 = P2(0) ¡
C(l1)
»0
= aa2MM0 + a
2
NN0 ¡
1
2
½¾2Ma[1 + a»1 + b´]
¡1
2
½¾2N(»1 ¡ ´) + ½¾2N [l1 ¡
1
2
(»1 ¡ ´)]
¡1
2
½¾2Maa
2
M(1 + a»0 + b´) ¡
1
2
½¾2Na
2
N(»0 ¡ ´)
+½¾2Na
2
N [0 ¡
1
2
(»0 ¡ ´)] ¡
C(l1)
»0
:
It is clear that W2 is normally distributed. Its mean is
E(W2) = ³0P
o
2;0 + ³1E[P2;1] + (1 ¡ ³0 ¡ ³1)(aa2MM0 + a2NN0)
= aa2MM0 + a
2
NN0 + ½¾
2
N (³0 + ³1)l1 ¡ C(l1)
+½[¡1
2
¾2M(1 + a
2
M)a(1 + b´) + ¾
2
N(1 + a
2
N )´] ³0
+½[¡1
2
¾2Ma(1 + b´) + ¾
2
N´] ³1 + ½[¡
1
2
¾2M(1 + a
2
M)a
2 + ¾2N(1 + a
2
N)] ³
2
0
+2½(¡1
2
¾2Ma
2 ¡ ¾2N) ³0³1 + ½(¡
1
2
¾2Ma
2 ¡ ¾2N) ³21;
and its variance is
V ar(W2) = V ar[³1(aaMM1 + aNN1)
+(1 ¡ ³0 ¡ ³1)(aaMM1 + a"M;2 + aNN1 + "N;2)]
= V ar[(1 ¡ ³0)(aaMM1 + aNN1) + (1 ¡ ³0 ¡ ³1)(a"M;2 + "N;2)]
= (a2a2M¾
2
M + a
2
N¾
2
N)(1 ¡ ³0)2 + (a2¾2M + ¾2N)(1 ¡ ³0 ¡ ³1)2:
139
Forming Lagrangian for the equivalent objective E (W2)¡ 12½V ar(W2); and setting
the …rst order condition, we have
½¾2N(³0 + ³1) = C
0(l1)
or
½¾2N»1 = C
0(l1) (A.74)
This equation has an unique solution l1 in the interval (l¤1; 0): To see this, note that
C0(l1) is a monotone increasing function of l1: By assumption, C0(l¤1) = 0; and lim
l1 " 0
C0(l1) =
+1: Hence, for any given positive constants ½; ¾2N ; ³0; and ³1; C 0(l¤1) < ½¾2N(³0 + ³1) <
lim
l1 " 0
C0(l1): Monotonicity of C0(l1) implies that equation (A.74) has a unique solution l1 2
(l¤1; 0):
Since the left hand side of (A.74) is increasing in ¾2N ; it is clear that l1 must be
increasing (i.e. less negative) in ¾2N : This implies that the underpriced amount (or, “money
left on the table”) is more when the residual risk is higher.
._.

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