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Strategic Interactions and Portfolio Choice in Money Management
Theory and Evidence
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Strategic Interactions and Portfolio Choice in
Money Management: Theory and Evidence
Alvaro Pedraza Morales∗†
Abstract
I study portfolio choice of strategic fund managers in the presence of a peer-based
underperformance penalty. While the penalty generates herding behavior, correlated
trading among managers is exacerbated when a strategic setting is considered. The
equilibrium portfolios are driven by the least restricted manager, who may vary ac-
cording to the realization of returns. I compare model predictions to evidence from
the Colombian pension fund management industry, where six asset managers are in
charge of portfolio allocation for the mandatory contributions of the working popula-
tion. These managers are subject to a peer based underperformance penalty, known as
the Minimum Return Guarantee (MRG). I study trading behavior by managers before
and after a change in the strictness of the MRG in June 2007. The evidence suggests
that a tighter MRG results in more trading in the direction of peers, a behavior that
is more pronounced for underperforming managers. I show that these ﬁndings are
consistent with the qualitative and quantitative predictions of the theoretical model.
∗
The author is an Economist in the Development Research Group at the World Bank. Email address:
apedrazamorales@worldbank.org.
†
I am indebted to Pete Kyle and John Shea for their invaluable guidance. This study has beneﬁted from
the comments of Anton Korinek, Pok-sang Lam (the editor), John Rust, Giorgo Sertsios, Paula Tkac, Russ
Wermers and two anonymous referees. I also thank Shu Lin Wee, Pablo Cuba and participants at the Midwest
Macro Seminar, Inter-American Development Bank, Federal Reserve Bank of Atlanta Brownbag, University
of Maryland Macro and Finance Brownbags for comments and suggestions. The ﬁndings, interpretations,
and conclusions expressed in this paper are entirely those of the author. They do not necessarily represent
the views of the International Bank for Reconstruction and Development/World Bank and its aﬃliated
organizations, or those of the Executive Directors of the World Bank or the governments they represent.
1
JEL Classiﬁcation: G11, G23, C73, C61
Keywords: Portfolio Choice, Strategic Interactions, Relative Performance, Institutional
Investors, Pension Funds.
2
1. INTRODUCTION
In ﬁnancial markets, institutional investors manage a signiﬁcant portion of the total assets
and comprise an even greater portion of the trading volume. Given the size of the portfolio
management industry, models addressing the agency issues of delegated portfolio manage-
ment and their eﬀects on asset pricing have become popular over the last few years.
In this paper, I study portfolio choice of strategic fund managers in the presence of a peer-
based underperformance penalty. While the penalty might generate crowd eﬀects among
managers even under competitive behavior, correlated trading is potentially exacerbated
when strategic behavior is considered.
Relative performance concerns among managers may be present for several reasons. The
most common explicit compensation schemes in the asset management industry depend
linearly on the volume of assets under management and non-linearly on excess performance
relative to a benchmark (as exempliﬁed by success fees or performance bonuses). Another
implicit source for relative performance concerns is the potential increase in funds ﬂowing
towards the best performing managers. Such empirical regularities have been documented
by Chevalier and Ellison (1997) for mutual funds and Agarwal, Daniel and Naik (2004) for
the hedge fund industry. The evidence suggests that a manager will get additional money
ﬂows and thus a higher future compensation if her relative return is above a threshold.
A less studied source of relative performance concerns comes from regulation. In particu-
lar, in countries that have moved from pay-as-you-go (PAYGO) pension systems to Deﬁned
Contributions (DC) systems based on individual accounts, regulation typically includes a
Minimum Return Guarantee or an underperformance penalty levied on portfolio managers.
The rationale for such regulations is to discourage excessive risk taking by the managers of
these accounts. In most cases, the formula to determine the MRG is calculated based on
peer performance.1 Hence, managers have an explicit reason to care about the returns of
their peers.
3
Relative performance concerns generated by excess performance fees or the performance-
ﬂow relationship typically imply a convex payoﬀ based on relative performance, which gives
rise to more risk taking among managers. In contrast, an underperformance penalty repre-
sents the opposite kind of performance incentive—a serious penalty for being the loser, as
opposed to a big price for being the winner—and therefore one would expect to ﬁnd the oppo-
site sort of behavior, namely herding. More speciﬁcally, an underperformance penalty based
on peer returns introduces an explicit reason for managers to track each others’ portfolios,
possibly generating crowd eﬀects as managers minimize the risk from behaving diﬀerently
from others. Of course managers might also herd into (or out of) the same securities over
some period of time for other reasons not related to underperformance penalties. First, man-
agers may receive correlated private information, perhaps from analyzing the same indicator
(Hirshleifer, Subrahmanyam and Titman, 1994). Second, a manager might infer private in-
formation from the prior trades of better-informed managers and trade in the same direction
(Bikhchandani, Hirshleifer and Welch, 1992; Sias, 2004). Third, managers might disregard
private information and trade with the crowd due to the reputational risk of acting diﬀerently
from other managers (Scharfstein and Stein, 1990). Finally, managers might simply have
correlated speciﬁc preferences over certain types of securities. An underperformance penalty,
such as the MRG, resembles a reputational risk, in that the manager might be penalized for
having lower returns than her peers. With the MRG, the risk is explicit as the manager will
be penalized ﬁnancially if returns are below the maximum allowed shortfall relative to the
peer benchmark.
When the number of competing money managers is small, relative performance concerns
might lead to strategic behavior. In this environment, strategic interactions imply that a
manager’s optimal portfolio choice needs to take into account the impact of his trades on
other managers’ decisions. In such setting, the eﬀects from a peer performance penalty on
portfolio strategies and trading dynamics might be more pronounced.
The DC pension industries in several Latin American and Eastern European countries are
4
natural candidates to display strategic interaction, because they consist of a small number
of competing Pension Fund Administrators (PFA), who act as asset managers and make
portfolio choices on behalf of the working population.
I use detailed data on the security allocations chosen by Colombian PFAs between 2004
and 2010 to study the strategic interaction between managers under relative performance
concerns. The Colombian institutional set up satisﬁes several key conditions necessary for
testing for strategic behavior of managers. PFAs manage the savings of a captive market,
namely individual retirement accounts, so their set of competitors is restricted to other PFAs,
and excludes other asset managers. Each PFA must comply with a MRG that is calculated
based on peer performance, creating an explicit incentive to care about the portfolio choice
and performance of other managers. Since the number of PFAs is small (six), strategic
behavior might be more pronounced than with a large number of competitors. Previous
empirical work on strategic behavior has examined data on managers’ broad asset allocation
or overall portfolio returns. By using monthly detailed portfolio holdings, I am able to test
richer implications of models with strategic behavior. Finally, the Colombian government
changed the MRG formula in June 2007, increasing the maximum allowed shortfall and
thereby loosening the MRG. This policy experiment allows me to measure the change in
behavior associated with the change in the underperformance penalty, arguably holding
constant other possible explanations for correlated trading.
The evidence suggests that a tighter MRG results in more trading in the direction of
peers and a smaller cross-section dispersion of returns between pension funds. Moreover,
the ranking among managers in terms of performance seems to play a role in portfolio
balancing decisions. With a tighter MRG, underperforming managers are more likely than
their competitors to trade in the direction of their peers. This is done by buying stocks in
which the manager has smaller weights in her portfolio relative to her peers, as opposed to
selling stocks with larger weights relative to her peers.
I next present a partial equilibrium model of portfolio choice in which a small number
5
of managers behave strategically in the presence of relative performance concerns. Relative
performance concerns arise from a peer-based underperformance penalty similar to the MRG.
Fund managers are endowed with the wealth of a group of fund investors and charge a
management fee for carrying out the investment strategy. Managers take prices/returns
as given and are strategic in the sense that they internalize that their choices aﬀect the
strategy of their peers and vice-versa due to the underperformance penalty. I calculate
the equilibrium policies and trading strategies. I calibrate the model to the data before
the change in regulation and simulate the quantitative eﬀects of a change in regulation
comparable to the one carried out by the Colombian government. The model captures the
observed change in behavior after the change in policy. In particular it can account both
qualitatively and quantitatively for the observed changes in the extent of correlated trading
and the observed increase in dispersion of returns across managers.
The model shows that the presence of relative performance concerns through a peer
based underperformance penalty aﬀects the asset allocation in two distinct ways. First, the
MRG regulation gives rise to time varying investment policies, with the managers making
procyclical trades, buying more of the asset that performed well in the previous period.
Second, given the strategic nature of the managers, the model suggests that the equilibrium
portfolios are driven by the choices of the least restricted manager (i.e. the manager that
has superior accumulated returns at any period in time), while the more restricted manager
is likely to end up selecting a portfolio that is similar to her competitor. The strategic na-
ture of the managers exacerbates the extent of procyclical trading, as both the more and
less restricted manager recognize their relative position and the action of their competitor.
For example, the overperforming manager moves heavily towards her normal portfolio (the
optimal portfolio without relative performance concerns) as she recognizes that the under-
performing manager, who is more exposed to the penalty, will rebalance her portfolio in the
same direction. The overall extent of this correlated trading is more pronounced than in a
setting with no strategic interactions among managers.
6
The rest of the document is organized as follows: In section 1.1 I review the leading
literature on strategic behavior by fund managers. The empirical evidence is presented in
section 2, where I conduct two empirical exercises that describe the overall trading behavior
of Colombian PFAs. In section 3, I introduce a model of strategic fund managers in the
presence of an underperformance penalty. I calculate the optimal portfolio choice and trading
strategies. Finally, in section 4, I present the conclusions and discuss future work.
1.1 Related Literature
This paper is related to several strands of the literature. The empirical literature on strate-
gic behavior of money managers has focused on the trading strategies of asset managers
competing for leadership to gain status, higher compensation or increased future ﬂows of
funds. The game is similar to a typical tournament, where winners get a large prize and the
losers end up with much less. In such tournaments, managers optimally increase their risk
taking to maximize the probability of reaching a top position at some target date (usually at
year end). Using U.S. data, Chevalier and Ellison (1997) document strong gambling incen-
tives among top-performing mutual funds. Examining strategic behavior in the context of
fund families, Kempf and Ruenzi (2008) document that mutual fund managers belonging to
families with a small number of funds behave diﬀerently from managers belonging to large
families. They argue that this result is driven by strategic interactions that might be more
pronounced in small fund families. For UK funds, Jans and Otten (2008) present evidence
of strategic behavior, ﬁnding that fund managers recognize the impact of their own deci-
sions on the actions of their peers, rather than treating competing managers as exogenous
benchmarks. A possible explanation for the ﬁndings of Kempf and Ruenzi (2008) and Jans
and Otten (2008) is that, with strategic managers, the interim leader expects the laggard to
increase risk, and therefore the leader also increases risk to maintain his lead (Taylor, 2003).
I complement this literature by presenting empirical evidence on the trading behavior of
Pension Fund Administrators in Colombia, where a small number of managers compete and
7
set their strategies to avoid a peer-based underperformance penalty. With only six PFAs,
it is highly likely that managers are strategic, as they recognize that the other managers
will react to their own portfolio choice as it will aﬀect each manager’s future compensation.
In contrast to the previous literature, where risk taking behavior arises as managers try to
outperform their peers, I study the eﬀects of the opposite kind of performance incentive, a
serious penalty for being the loser. In this setting one would expect to ﬁnd the opposite
outcome, meaning herding among managers.
Despite strong theoretical foundations and a common perception that professional in-
vestors herd, earlier studies found little evidence of herding behavior, and in most cases
herding was mostly associated with only particular types of assets, like small stocks (Werm-
ers (1999) for US mutual funds and Lakonishok, Shleifer and Vishny (1992) for US pension
funds). In a more recent study, however, Sias (2004) shows that changes in security positions
of institutional asset managers over a quarter are strongly correlated with the the trades of
other institutions over the previous quarter. The author also ﬁnds that changes in positions
on particular stocks are weakly but positively related to returns over the following year. The
results favor the hypothesis that herding is a result of institutions inferring information from
each other’s trades. Raddatz and Schmukler (2013) ﬁnd that Chilean pension funds, also
subject to a peer-based MRG, tend to herd, buying and selling the same assets at the same
time. The authors also ﬁnd diﬀerences in the extent of herding across assets. By comparing
the trading behavior of PFAs before and after the MRG change, I am able to identify the
eﬀects of the underperformance penalty holding other factors constant.
On the theoretical side, papers on portfolio choice with underperformance constraints
include Deelstra, Grasselli and Koehl (2003) and Tepla (2001). In a general equilibrium
framework Cuoco and Kaniel (2011) study asset price eﬀects of diﬀerent performance-based
fees for money managers, including both excess performance bonuses and underperformance
penalties. Other papers studying relative performance concerns, examine the eﬀects of del-
egation on limits to arbitrage, trading volume and price discovery. These papers include
8
Shleifer and Vishny (1997), Cuoco and Kaniel (2011), Guerrieri and Kondor (2012) and
Kaniel and Kondor (2013). However, these authors abstract from the potential role of
strategic interactions in trading behavior. In a partial equilibrium framework, Basak and
Makarov (2014) model strategic interactions between two managers competing for additional
ﬂows. My model is related to theirs in that I assume a discrete and small (two) number of
strategic managers, but I focus on the eﬀects of an underperformance penalty based on peer
returns, as opposed to the eﬀects of payoﬀs that are convex in portfolio returns due to a
ﬂow-performance relationship. In my setting herding is the optimal strategy, in contrast to
gambling which can be present when managers face a convex payoﬀ structure as in Basak
and Makarov (2014). In my model, the overperforming manager moves heavily towards her
preferred portfolio as she recognizes that the underperforming manager has to rebalance her
portfolio in the same direction. The extent of this correlated trading is more pronounced
given the strategic nature of the managers and the aggregate portfolio follows the preferences
of the least restricted manager.
2. EMPIRICAL EVIDENCE
2.1 The Colombian Private Pension Industry
In 1993 the Colombian Congress approved Law 100, which among other reforms introduced
major changes in the pension system. The country adopted a dual pension scheme, in which
a deﬁned contribution (DC) system of individual accounts was created in addition to the
already existing deﬁned beneﬁt system. Under the new system, pensions were ﬁnanced
by compulsory contributions made by both the employer and the employee. The law also
provided guiding principles for the establishment, operation and supervision of Pension Fund
Administrators (PFAs). Under the new scheme, all workers who chose the DC system were
required to select a PFA to manage their retirement accounts. The worker’s investment
decision was restricted to the choice of the PFA, while the government regulated PFAs’
9
portfolio strategies by imposing limits on speciﬁc asset classes and individual securities,2
and through other provisions such as banning short selling. Workers were allowed to switch
PFAs every six months.
The law also determined the compensation structure of the PFAs and the Minimum
Return Guarantee (MRG). The PFAs were allowed to charge fees for collecting contributions,
managing the fund and giving beneﬁts. In particular, PFAs charge a front end load fee of
5.5% on new contributions. On average, the fee on new contributions represents close to
90% of the annual compensation of the PFAs. If a worker makes consistent contributions he
does not face any additional charges.3
The MRG is a lower threshold of returns that each individual PFA needs to guarantee
for its investors. If a PFA fails to provide at least this return, the PFA must transfer part
of its own net worth to the fund to make up the shortfall. The MRG is assessed monthly by
comparing the fund’s average annual return over the previous three years to the average of
the six PFAs.4 Between January 2004 and June 2007, the minimum return guarantee was
calculated as the average across PFAs of the average annual return over the previous three
years (Πt ), minus 30%, so that M RGt = 70%Πt . After June 2007 the government changed
the formula to M RGt = min{70%Πt , Πt − 2.6%}. For average industry returns below 8.66%,
the new formula implies a MRG equal to Πt − 2.6%, as 70%Πt > Πt − 2.6%. Eﬀectively,
for this set of returns, the new formula yielded a lower MRG (equivalently, a larger allowed
shortfall) than what would have been calculated before June 2007.5
Within this institutional setting, the MRG creates an explicit reason for each PFA to
track peer portfolios and performance. The penalty for falling too far behind the industry
average returns may lead the PFA to bankruptcy. Given the size of each PFA, and the total
value of assets under management, a typical Colombian PFA falling 50bps below the MRG
threshold would use up its entire net worth compensating its investors.6 With such a severe
penalty, one should expect that the MRG is of ﬁrst order importance when PFAs set their
strategies.
10
Data on Colombian pension funds was provided by ASOFONDOS (Colombian Associ-
ation of Pension Fund Administrators). The database includes the detailed security allo-
cations for the funds managed by each of the six PFAs, on a monthly basis for the period
2004:1 to 2010:12. Summary statistics for this data set are presented in Table 1 at two-year
intervals. As of June 2010, total assets under management were US$44.1 billion (equal to
17% of Colombian GDP). At that time, 32% of these funds were invested in Colombian
stocks, which amounted to 7.1% of the total domestic market capitalization. Throughout
the sample period, net ﬂows to these funds were positive, which reﬂects the fact that most
of the workers contributing to these funds were still young (more than 70% were younger
than 40 years old).
In addition to the pension funds, PFAs manage voluntary retirement funds in separate
accounts. These voluntary accounts supplement the compulsory retirement savings in the
pension funds. Contrary to pension funds, these accounts are subject to very few regulations.
In particular, they are not subject to the MRG and do not have limits on individual securities
or asset classes.7 Moreover, workers are typically directly involved in the asset allocation
of their voluntary portfolios. Panels D and E in Table 1 present summary statistics of the
voluntary funds.
In the following sections I present two empirical exercises suggesting that relative perfor-
mance concerns are important for the portfolio dynamics of PFAs. For this, I introduce two
measures that describe trading activity by PFAs. The ﬁrst is an aggregate fund measure
that describes how each manager rebalances her portfolio relative to the peer portfolio. The
second focuses on fund trades of individual stocks. For both empirical exercises, I focus on
the trading behavior across domestic stocks. While these represent only a fraction of the
total portfolio, correlated behavior among managers is likely to be more pronounced for these
securities, which display higher dispersion of returns than other assets in PFAs portfolios.
11
2.2 Trading Strategies and Relative Performance
In this section I introduce a measure of the direction of a PFA’s trades relative to its peers.
The objective is to summarize the trading behavior and strategies of Colombian pension
fund managers in a parsimonious way.
At the end of each month, each fund’s location is deﬁned by its portfolio weights. The
i
vector of portfolio weights for a fund i in month t is denoted wt ∈ RS +1 , where each element
i sharesist ∗pst
s = {1, 2, . . . , S } represents a domestic stock in the fund’s portfolio, wst = V Fti . Here
shares is the number of shares of stock s held by the fund, p is the stock price and V Fti is the
i
total value of the fund. The element wS +1,t in the vector of portfolio weights represents the
fund’s participation in assets other than domestic stocks (i.e. domestic corporate debt and
government debt). For each PFA i = 1, 2, ..., 6, the average peer fund portfolio has weights
i 1 i
denoted by the vector πt = 5 −i wt , where −i is the sum of all funds excluding fund i.
To measure a fund’s trading strategy, or its change in portfolio weights, I ﬁrst adjust
for passive portfolio evolution due to changes in prices. Including changes in weights due to
price changes may overstate the degree of coordination among funds. If the gross return of
stock s between period t and t+1 is deﬁned as retst , the adjusted vector of weight changes
i i
for fund i from t to t + 1 can be denoted ∆wt , where each element s is deﬁned by ∆wst =
i ×ret
wst
i st
wst +1 − i
s wst ×retst
. The last term accounts for the change in the weights due to diﬀerences in
returns among stocks in the portfolio. To measure the position of fund i’s portfolio relative to
its peers at period t, I calculate a vector of diﬀerences between the fund and its competitors,
di i i
t = πt − wt . To capture the direction of portfolio weight changes, I measure the angle
between the change of a PFA’s weights and the distance from its peers’ portfolio, as follows:
i
∆wt · di
t
directioni
t = cos(θ ) = i
(1)
||∆wt || ||dit ||
In this speciﬁcation, direction measures the correlation across securities between portfolio
weight changes for fund i and the initial distance between i and its peers.8 If fund i is moving
12
exactly towards its peers, the angle is zero and direction is equal to 1. If the manager is
rebalancing the portfolio in exactly the opposite direction of its peers, the angle is 180 degrees
and the direction measure equals -1.
Figure 1 displays two examples of the angle between the vector of weight changes for
fund manager i and the vector of initial distance between i and the other managers. This
ﬁgure assumes that there are three securities; given that the portfolio weights add up to one,
i
the third dimension is redundant. Initially, the peer portfolio πt has a larger share of stock
A than manager i. In panel (a) the manager increases her participation in stock A, moving
towards peers. In panel (b) the manager increases her participation in stock B, moving away
from the peer portfolio.
If there is a constraint on short selling, the space becomes a Simplex of portfolio weights
and the measure would be naturally biased towards higher values of direction. For example,
if fund i is currently invested only in stock B, it is located along the vertical axis, and the
only way to continue to move away from its competitors would be to move along the axis, in
which case the angle would be smaller than 180 degrees and direction would be greater than
-1. In this example, moving away from one’s peers would mean buying more of what you
already own, as opposed to short selling securities in which your peers have larger weights.
Figure 2 depicts the time series behavior of the measure of direction for both pension
funds and voluntary funds. For each month in the sample, I calculate the direction of weight
changes for each PFA over the next quarter, and take the average across PFAs. A high value
indicates that PFAs on average are moving towards their peers. Evidently, for the pension
funds, PFAs on average traded more in the direction of their peers prior to the MRG formula
change in June 2007 than after this date. For the voluntary funds, the behavior of direction
seems to be same before and after the policy change.
Table 2 presents summary statistics on direction. The statistics are split for the period
before and after the change in the MRG. For the pension funds, mean direction fell from 0.32
in the early period to 0.14 after the change in the MRG, suggesting that the policy change
13
may have aﬀected managers’ behavior.
Table 2 also reports statistics on the relative performance between pension funds before
and after the MRG change. Relative performance with respect to the peer portfolio is
i i −i
deﬁned as relt = Rt − Rt , where Rt are 36 month returns prior to t (consistent with the
i
measurement period of the MRG). The relative performance variable relt measures whether
i i
fund i is over-performing (relt > 0) or under-performing (relt < 0) at time t relative to
the other managers. After June 2007, there seems to be some increase in the cross-section
dispersion of PFA returns. If portfolios are less alike, returns are likely to vary more cross-
sectionally.
A separate question is whether managers’ strategies depend on relative performance.
i
Panel C in Table 2 presents the correlation between directionit and relt . The negative
correlation between relative performance and direction indicates that before June 2007,
PFAs with poor relative performance tended to move more strongly towards peers. After
June 2007 there is no evidence that relative performance is correlated with the direction of
trades.
To summarize, the loosening in the MRG in June 2007 is associated with three important
changes in the data for pension funds: (F1) Less trading in the direction of peers; (F2)
Increase in cross-section dispersion of returns between funds. (F3) A disappearance of the
negative correlation between relative performance and trading in the direction of peers.
2.3 Individual Stocks and Trading Strategies
In this section I further investigate herding behavior using data on individual stock trades.
For each stock, the fund’s distance to the peer benchmark is measured as di i i
st = πst − wst ,
where the fund can be overexposed (di i
st < 0), underexposed (dst > 0) or have the same
weight (di
st = 0) as its peers. I estimate the following model of a fund’s changes in individual
stock weights:
14
M M
i
∆wst = β0 + βm xi
st + γ0 M RGt + γm M RGt · xi i
st + εst (2)
m=1 m=1
i
where ∆wst is adjusted for stock returns as in the previous section, xi
st are fund and stock
speciﬁc characteristics and M RGt is a time dummy equal to one for dates before July 2007
and zero thereafter, representing the policy change. The objective here is twofold, ﬁrst to
determine what fund based characteristics determine PFA trading on individual stocks, and
second to measure whether there was any change in the impact of these characteristics after
the MRG formula was modiﬁed.
More speciﬁcally, I set xi i i i i i i i
st = (dst , relt , dst × relt , sizet , Controlsst , M arketst ). Here sizet
is the share of assets under management of fund i relative to the industry. The vector of
Controlsst contains stock speciﬁc variables. I introduce lagged returns at one, three, six
and twelve months to account for momentum trading, deﬁned as purchasing (selling) assets
with positive (negative) past returns.9 This popular investment strategy has been widely
documented for institutional investors.10 Chan, Jegadeesh and Lakonishok (1996) suggests
that momentum trading may be caused by a delayed reaction of investors to the information
in past returns and past earnings. I also control for ﬁrm size and liquidity, as institutional
investors may share an aversion to stocks with certain characteristics, as documented by
Wermers (1999), who found evidence that US mutual funds tend to herd in small stocks.
Given the potential persistence in trading strategies, I allow the error term (εi
st ) to be
correlated within stocks and correct the standard errors as in Petersen (2009).
Finally, to verify that the results are driven by managers trading relative to their peers
and not by trading relative to a broad market benchmark, I calculate Market Distance as the
diﬀerence between the IGBC index weight on stock s and fund i’s weight in stock s for each
period, M arketi IGBC
st = Πst
i
− wst . The IGBC is a widely used value and liquidity based index
for the Colombian stock market. I also interact this measure with relative performance.
This previous speciﬁcation is motivated by Basak, Pavlova and Shapiro (2007), who ﬁnd
diﬀerent behavior in U.S. equity mutual funds depending on whether managers are ahead
15
or behind the S&P 500 index. In their speciﬁcation, the authors deﬁne risk shifting as
an increase in the absolute diﬀerence between a fund’s returns and the S&P 500 returns.
They regress this variable on an interaction between current relative returns and the market
returns. Their question is whether underperforming funds move towards or away from the
market index, thereby increasing or decreasing the size of deviations from market returns.
My speciﬁcation is analogous to theirs, in that one of my objectives is to measure whether
underperforming funds move towards or away from a reference portfolio, the peers’ portfolio
(F3), by increasing or decreasing their holdings of stocks in which they are underexposed
or overexposed. Note that while Basak, Pavlova and Shapiro (2007) only observe return
outcomes, I observe portfolio weights and thus the actual strategy of the manager. In a
setting with a small number of managers it might be hard to distinguish if changes in the
cross-section dispersion of returns are due to managers’ strategies or to the realization of
stock returns.
i 11
Table 3 documents the results of the linear regression for adjusted weight changes ∆wst .
The results suggest that regardless of relative performance, managers were more likely to
increase their holdings of stocks in which they were already overexposed after the MRG
was loosened in June 2007 than before this date. That is, there was less trading towards
peers once the MRG was loosened. This change in behavior associated with the change in
regulation is consistent with the average behavior of trading direction presented in Figure 2
and Table 2.
Figure 3 presents diﬀerences in marginal eﬀects of distance on adjustments in portfolio
∂ ∆w(M RG=1) ∂ ∆w(M RG=0)
weights before and after the policy change ∂d
− ∂d
along with corre-
sponding conﬁdence intervals. Underperforming managers (rel < 0) were more likely to
increase their holdings of stocks in which they were underexposed (d > 0) prior to June
2007 than after this date. This result for individual stocks is consistent with the decrease
after June 2007 in the correlation between direction and relative performance documented
in Table 2-Panel C. To give a sense of the quantitative importance of these estimates, the
16
results indicate that a fund lagging in returns by 200bps relative to its peers, and with 5%
underexposure in an individual stock s, would increase the weight on s by 0.42% more prior
to June 2007 than after the MRG was loosened.
For high performing managers (with relative returns above 320bps their peers), the esti-
mated marginal eﬀects on distance are negative but not statistically signiﬁcant (not shown
in Figure 3). That is, there is no evidence that the change in the strictness of the penalty
aﬀected the way top-performing managers traded stocks with overexposure, perhaps be-
cause the MRG impacts more strongly the average and worst-performing managers. Top-
performing managers, unconstrained by the MRG, might deviate from the peer portfolio
to possibly attract more funds. However, as the results indicate, the MRG policy change
doesn’t seem appropriate to identify the eﬀects of such incentives for managers in the high
end of the return spectrum.
Speciﬁcation (2) at Table 3 adds variables including distance from the market portfolio
to the benchmark speciﬁcation. These variables are insigniﬁcant both before and after the
MRG policy change. This suggests that managers’ trading strategies are sensitive to the
position relative to their PFA peers in particular, rather than to their position relative to
the market portfolio.
Columns (3) and (4) at Table 3 display the standard errors corrected for within fund
correlation. The overall results are robust to this form of clustering. I also performed
additional robustness test that are omitted for brevity but available upon request. In these
tests, I show that the main ﬁndings hold for diﬀerent measures of changes in the portfolio
weights.12
2.3.1 Buy and Sell Strategies
In a ﬁnal empirical exercise, I complement the above results by distinguishing buys and sells
of individual stocks. This is a discrete version of the previous speciﬁcation. Here, fund i’s
trading strategy for a particular stock s is measured by whether the fund buys or sells the
17
stock prior to the following period:
sharesi i
(1, 0) st+1 > sharesst
i i
buyst , sellst = (0, 1) sharesi i
st+1 < sharesst
sharesi i
(0, 0) st+1 = sharesst
corrected for stock splits at period t. In this setting, the analog to the direction measure
introduced before is as follows: When a fund buys shares in stocks in which it’s already
overexposed (underexposed), it moves away from (towards) the peer benchmark. When a
fund sells shares in stocks in which it’s already overexposed (underexposed) it moves towards
(away from) the peer benchmark.
Panel B in Table 1 shows some trading statistics for diﬀerent months within the data
set. For example, in June 2008, PFAs collectively held 44 diﬀerent stocks and each fund
on average had 26.3 stocks in its portfolio. That month, each PFA traded on average 8.33
stocks, with 6.66 of those trades as buys. In this setting a trading strategy is measured by
the probability at time t that fund i buys or sells stock s within the set of stocks owned by
all PFAs (i.e. the probability of fund i buying stock s from among the 44 total stocks is the
likelihood that s was among the 6.6 stocks that fund i bought that period).
Note on Short Selling: As was the case for the direction measure discussed in the
previous section, the short selling ban for these funds introduces a bias against using sales
to move away from peers. Consider the previous example. The average PFA sold 1.67 stocks
during June 2008. Given the short selling constraint, those sells must come from the set of
stocks owned in the previous month (25.8 as of May 31, 2007) not from the total set of stocks
held by the PFA industry (44 as of May 31, 2007). Hence, a measure of the probability of
selling a stock that considers the entire set of securities is naturally biased towards smaller
values, as opposed to a measure of the probability of buying a stock since a fund can buy
any stock in the peer portfolio whether owned at the beginning of the period or not. For this
18
reason I examine buying and selling strategies separately in what follows. Moreover, when
estimating the probability of selling a stock, I condition on stock ownership at the beginning
of the previous period.
I estimate a Probit speciﬁcation of the probability of buying (y = buy ) or selling (y = sell)
i
a stock as P r(yst = 1) = Φ (β0 + m βm x i
st + m γm M RGt · xi
st ), where Φ is the cumulative
distribution function of the standard normal, and the vector of independent variables x is
the same as in equation (2).
Columns one and two of Table 4 document the results of the Probit regression for the
i
probability of buying a stock (buyst ). Consistent with the results in the continuous regression,
managers were more likely to buy stocks in which they were already overexposed after the
MRG was loosened in June 2007 than before. Meanwhile, an underperforming manager
(rel < 0) was more likely to buy stocks in which she was underexposed (d > 0) prior to
June 2007 than after this date. Columns three and four of Table 4 present the results
i
from the Probit regression for the probability of selling a stock (sellst ) conditional on stock
ownership. The coeﬃcients on the interactions between MRG, Peer Distance and Relative
Performance are all indistinguishable from zero, suggesting that the policy change in June
2007 had no impact on how PFAs sold stocks in which they were overexposed, regardless
of relative performance. Given that between 2004 and 2010 the yearly net ﬂows to these
funds were about 8.5% of the value of the fund, underperforming managers had the option
of reducing their relative participation in any given stock by holding their number of shares
constant, as opposed to selling shares.
To summarize the main empirical ﬁndings, the evidence suggests that a more strict MRG
prior to June 2007 is associated with more trading in the direction of peers, and in particular
more buying of stocks in which managers were underexposed. Meanwhile, underperforming
managers traded more heavily towards the peer portfolio prior to June 2007, by buying
stocks in which they were underexposed, as opposed to selling stocks in which they were
overexposed. This asymmetric behavior between buys and sells could be explained by the
19
fact that these funds were growing within the sample period. As I will show in section 3,
these results are largely in line with the predictions of a model in which money managers
behave strategically due to relative performance concerns.
2.4 Alternative Explanations
The speciﬁcation strategy above assumes that the policy change is exogenous to the domestic
stocks’ return process. In the estimation I control for stock-speciﬁc attributes such as past
returns and trading volume. However, one cannot control for all stock characteristics that
might have changed after July 2007 and that might have induced the funds to adjust their
trading behavior. For example, PFAs might have received more good signals about the fun-
damentals of stocks in which they were underexposed prior to July 2007 than after, inducing
them to buy more of those stocks before the policy change than after. The shortcoming of
this argument is that if a PFA is underexposed in a particular stock relative to the peer
portfolio, by construction there must at least one PFA overexposed in the same stock. As
favorable new information arrives about a stock, both underexposed and overexposed PFAs
should increase their holdings. Hence, one would need some sort of argument for why PFAs
with underexposure were the only ones receiving good signals.
Another possible explanation for the results is that PFAs altered their trading strategies
due to managerial changes around the time of the policy change. For example, trading
strategies might result from changes in management within the ﬁrms or shifts in preferences
among the top investment oﬃcials. A closer look at PFAs’ CEO replacement indicates that,
while there were some changes in management over the sample period, there is no evidence of
an industry wide event before or after the MRG adjustment.13 In terms of preference shocks,
interactions between PFAs dummies and the MRG dummy should account for individual
PFA changes before and after the policy experiment. However, an industry-wide taste shock
occurring in mid 2007 would be indistinguishable from the policy experiment. While such
event is unlikely, it cannot be ruled out under the current empirical speciﬁcation.
20
3. THE GENERAL MODEL
Motivated by the above empirical ﬁndings, I consider a model in which the presence of relative
performance concerns among a small number of money managers leads to strategic behavior.
In particular, I focus on portfolio choice when there is an underperformance penalty, such as
the MRG described in the previous section. I show that the behavior of institutional asset
managers in the presence of the MRG is consistent qualitatively and quantitatively with the
observed data.
3.1 Model Assumptions
I consider a ﬁnite horizon economy t = 0, 1, . . . , T , modeled as follows:
Securities: The investment opportunities are represented by a riskless bond and a risky
stock. The bond is a claim to a riskless payoﬀ B > 0. Without loss of generality the net
interest rate is normalized to zero (B price is normalized to B = 1).
The gross stock return follows a random process with states rs = {rH , rL } and probabil-
ities {p, 1 − p}. This is a partial equilibrium model as the return process is exogenous. The
return between t and t+1 on a portfolio with fraction φt of wealth invested in the stock and
1 − φt in the risk free bond is denoted by
s s
Rt+1 (φt ) = φt rt+1 − B + B (3)
Fund Managers: I consider two fund managers i and j. Each manager chooses an
T −1
investment policy {φi
t }t=0 . For this, they are compensated at time T with a management
fee FiT , which is a function of the terminal value of their portfolio WiT and that of their
competitor WjT . Speciﬁcally I assume that
WiT WjT
FiT = F (WiT , WjT ) = βWiT + γWi0 min 0, − +x (4)
Wi0 Wj 0
21
In this speciﬁcation, the fund managers’ compensation at time T consists of two compo-
nents: a proportional fee, which depends on the ﬁnal value of the portfolio βWiT , and the
WjT
underperformance penalty γWi0 min 0, W iT
W i0
− Wj 0
+x which depends on the manager’s
performance relative to the other manager j. Here x ≥ 0 is the allowed shortfall, i.e., x = ∞
implies that the manager is unrestricted, while x = 2% means that the maximum return
shortfall allowed is 2% relative to the peer returns. Any cumulative returns below this
threshold result in a penalty that reduces the manager’s net fee, possibly to compensate the
investors for the lack of returns. The size of the penalty is modeled by γ . In the Colombian
setting, the PFAs face an underperformance penalty with γ = 1, since under the Colombian
MRG the manager pays a penalty that guarantees that the investors’ net returns are exactly
WjT
the peer benchmark Wj 0
− x.
Throughout this document I refer to a manager that pays the penalty in any given state
as the loser in that state. In this setting strategic interactions arise as each manager needs
to consider each other’s policy reaction so as to avoid the underperfomance penalty.
Fund managers are assumed not to have any private wealth. They therefore act so as to
maximize the expected utility E0 [ui (F (WiT , WjT ))] given initial wealth Wi0 , subject to the
s
period by period budget constraint Wi,t+1 = Rt+1 (φit )Wit .
The Normal Policy: In the rest of the document I refer to the normal policy φN
it
P
of
fund manager i as the optimal portfolio allocation when no relative performance concerns
are present. In this case γ = 0 and each fund manager solves a standard portfolio choice
problem. The optimal share in stocks is given by the ﬁrst order condition
H
p(rH − B )ui Rt L L
+1 (φit )Wit + (1 − p)(r − B )ui Rt+1 (φit )Wit = 0 (5)
for t = 0, 1, . . . , T − 1.
Fund managers (i, j ) are assumed to have CRRA preferences deﬁned over their ﬁnal
wealth, um (W ) = 1
1−σm
W 1−σm with m = i, j . To generate diﬀerences in the normal policy,
I assume that manager i is less risk averse than manager j , σi < σj . Using data from
22
U.S. mutual funds, Koijen (2008) documents substantial heterogeneity in estimates of fund
managers’ risk aversion. Portfolios between managers may also diﬀer because of ability,
information, or pay for performance incentives. Here, my objective is to study how the
introduction of an underperformance penalty such as the MRG causes fund managers to
deviate from their normal strategies.
In this paper, I appeal to the Nash equilibrium notion to characterize managers’ strategic
interactions in the presence of relative performance concerns. Below, I deﬁne the structure
of the game between managers at time 0, where each manager draws up a plan of how she
is going to invest throughout the whole time period {0, 1, . . . , T }. This deﬁnition of the
game eases exposition, and is without loss of generality, since neither manager would want
to deviate from a policy chosen initially at any subsequent date t, so that the equilibrium
policies are time-consistent.
Information sets: I consider a complete information game at time 0, in which each
manager knows all the primitives and parameters of the model described above, namely the
stock return process, own initial wealth and risk aversion and those of the other manager.
Finally, each manager knows the functional form of the underperformance penalty (γ and
x).
Strategy sets: A strategy of manager i is a function φi (t, Wit , Wjt ) deﬁned over the space
{0, 1, . . . , T } × (0, +∞) × (0, +∞), where φi (t, Wit , Wjt ) is manager i’s investment policy at
time t for given values of wealth under management, Wit , and that of her opponent, Wjt .
For convenience, I will use φit as a shorthand notation for manager i’s time t investment
strategy and drop its arguments.
−1
Manager’s payoﬀs: The manager’s payoﬀs for policy vectors {φit , φjt }T
t=0 are given
as follows. First, period T wealth is obtained by substituting φit and φjt into the dynamic
wealth process of each manager. Given terminal wealth WiT and WjT , fees are computed
according to (4), yielding the ﬁnal payoﬀ.
23
3.2 Two Period Model (T = 1): Portfolio Choice
I start by solving the model in a two period version of the above economy. The objective here
is to show how best response and equilibrium policies are calculated and to address in the
simplest environment the eﬀects of the underperformance penalty on managers’ equilibrium
portfolios.
In a two period and two state economy (s = {H, L}), for a given initial level of wealth
under management Wi0 , a portfolio allocation φi0 by manager i at period 0 determines two
possible values of ﬁnal wealth WiH L
1 and Wi1 according to equation (3). For each value of ﬁnal
wealth there is an associated management fee, calculated using (4). With no underperfor-
mance penalty γ = 0, the manager’s income is a proportional fee on the ﬁnal wealth under
management (βWis
1 ). With the underperformance penalty, the net fee depends on whether
S
Wj 1
manager i’s returns are above or below the peer benchmark Wj 0
− x for each of the two states.
Hence a portfolio choice φi0 , for a given choice φj 0 , is eﬀectively a choice of a possible pair
of management fees in both high and low states of the economy.
Figure 4 depicts the optimization problem in the space of returns. In the left panel, the
normal policy is such that in both the high and low states, manager i’s returns are above
her peer or below her peer by less than x. In this case the underperformance penalty is not
binding and the manager’s optimal portfolio is her normal policy. In the right panel, if the
normal policy was played, the manager’s returns would be below the peer benchmark in the
low state (loser in low). The manager optimally chooses a portfolio with less exposure to the
ˆi0 < φN P ), generating smaller returns in the high state but greater returns in
risky asset (φ i
the low state than in her normal policy. Basically the manager is giving up a higher income
in the high state to increase the income in the low state in order to reduce the penalty. In
this example, manager i still pays the penalty in the low state, but less than what she would
have paid had she played her normal policy.
ˆ to denote best response functions and φ∗ to
In the rest of the document I will use φ
24
denote equilibrium policies.
PROPOSITION 1. For a given manager j portfolio choice φj 0 , manager i’s best response
ˆi0 is given by
φ
x
ai + bi φj − ci x φN P
≤ φj 0 − (loser in high) (6)
i rH −B
ˆi0 = φN
φ i
P
φj 0 − x
< φN
i
P
< φj 0 + x
(7)
rH −B B −r L
x
ai + bi φj + ci x φN P
≥ φj 0 + (loser in low) (8)
i B −r L
with bi , bi , ci , ci ≥ 0. Switching subscripts i and j above yields manager j’s best response. The
proof of the proposition and deﬁnitions for ai , ai , bi , bi , ci and ci in terms of the underlying
parameters of the model are presented in Appendix C.
DEFINITION 1. The Shifting Region is the region in the parameter space such that a
ˆ = φN P .
manager’s best response policy is to play a strategy diﬀerent than her normal policy φ
In this two period economy, this region is deﬁned by the conditions in (6) and (8). In
x x
other words, we are in the shifting region if φN
i
P
∈ −∞, φj 0 − r H −B
φj 0 + B −rL
,∞
L
Wj H
Wj
1 1
which is the region to the left of Wj 0
− x and below Wj 0
− x in Figure 4.
In the Shifting Region, bi , bi ≥ 0 indicates that the manager optimally chooses a portfolio
that is shifted towards her peer, buying more or less of the risky asset depending on the
other manager’s portfolio. If x increases (e.g. the allowed shortfall is larger, implying a
looser MRG) the shifting region becomes smaller, which means that the manager will play
her normal policy for a larger set of parameters.
Moreover, in the shifting region, a larger (smaller) x implies a smaller (larger) shift.
As an example, consider the best response when manager i plays to lose in the low state
(8). Here the manager selects an allocation with a greater share in the risky asset than her
ˆi0 > φj 0 ). More speciﬁcally, the allocation is such that the manager pays the
competitor (φ
underperformance penalty in the low state (loser in Low). A larger x implies a greater share
in the risky asset, or less shift from her normal policy. On the contrary, a smaller x implies
25
a smaller share in the risky asset or a larger absolute shift. A similar analysis can be made
for the best response in (6). Note that when the manager is playing to pay the performance
penalty in the high state (loser in high) she has a lower share of the risky asset than her
competitor. As x decreases, her participation in the risky asset increases, moving farther
away from her normal policy and closer to her peer.
To summarize, the maximum allowed shortfall x determines both the size of the shifting
region and the size of the shift in the best response functions.
COROLLARY 2. In the shifting region, if the manager is playing to lose in the low state,
ˆi0 ∈ φj 0 +
the best response satisﬁes φ x
, φN P
. If the manager is playing to lose in the
B −rL i
ˆi0 ∈ φN P , φj 0 −
high state, the best response satisﬁes φ x
,
i rL −B
Corollary 2 states that within the shifting region, the maximum shift is up to the point
where the maximum allowed shortfall is met.
ˆi0 =
COROLLARY 3. If γ >> β , manager i’s best response in the shifting region is φ
x
φj 0 + B − ˆi0 = φj 0 − Hx when playing to lose
when playing to lose in the low state, and φ
rL r −B
in the high state.
Corollary 3 refers to the case when the size of the underperformance penalty γ is signiﬁ-
cantly larger than the proportional fee β . When facing a large enough penalty the manager
shifts her strategy to the point where she is never below the underperfomance benchmark,
and the optimal portfolio implies that the manager hits the maximum allowed shortfall ex-
s s
WjT
WiT
actly in either the high or the low state, such that Wi0
= Wj 0
− x, in which case the manager
s
does not pay the penalty and receives the proportional fee βWiT . In ﬁgure 4, this would
β
look like a horizontal line in the left shifting region as β +γ
→ 0. The manager is giving up a
higher income in the high state, to guarantee that in the low state she doesn’t have to pay
the underperfomance penalty.
Given the deﬁnition of the MRG in the Colombian case, where the PFA is required to
pay in full (γ = 1) any shortfall in returns below the benchmark, the penalty is signiﬁcantly
26
greater than the average proportional fee β = 0.008 on the assets under management (see
Appendix B for more details). In this case, corollary 3 suggests that the managers will chose
a strategy to avoid the penalty in every state. As it turns out, as of December 2013, no PFA
has ever fell below the MRG threshold. Even in the turmoil of October 2008, the PFA with
the lowest returns managed to have returns 118bps above the MRG (this is the closest any
PFA was to the MRG in the sample period).
Up to this point, I have characterized the best response functions. In this setting with the
competition restricted to a small number of managers, one should expect that the managers
anticipate each others’ reactions to their strategies. In order to describe the behavior of
these strategic managers I appeal to Nash Equilibrium, in which strategies are mutual best
responses.
DEFINITION 2. Nash Equilibrium A pure-strategy Nash equilibrium is a pair of port-
folio choices (φi∗ , φj ∗ ) that solve the ﬁxed point equations φ∗ ˆ ˆ ∗
and φ∗
it = φit φjt (φit ) jt =
ˆit φ∗
ˆit φ
φ .
jt
Proposition 4 describes these equilibrium policies for each manager and the conditions
that determine when each one of these equilibria is played.
PROPOSITION 4. The Nash equilibrium policies are given by:
NP NP
φi , φj Φi0 ≥ φN
i
P
and Φj 0 ≤ φN
j
P
(9)
φN P NP
i , aj + bj φi − cj x Φi0 ≥ φN P
and Φj 0 > φN P
(10)
i j
φ∗ ∗
i0 , φj 0 =
ai + bi φN j
P
+ ci x, φN
j
P
Φi0 < φN
i
P
and Φj 0 ≤ φN
j
P
(11)
(Φi0 , Φj 0 ) Φi0 < φN P
and Φj 0 > φN P
(12)
i j
(ai +bi aj )+(ci −bi cj )x (aj +bj ai )−(cj −bj ci )x
where Φi0 = 1−bi bj
, Φj 0 = 1−bi bj
. Moreover, when x > (rH −B ) φN
i
P
− φN
j
P
and x > (B − rL ) φN
i
P
− φN
j
P
the conditions in (9) are always satisﬁed and managers play
their normal policy. rH − B > B − rL is a necessary condition for (10) to be an equilibrium
and rH − B < B − rL is a necessary condition for (11) to be an equilibrium.
27
The equilibrium portfolios take a simple form: (i) Both managers play their normal policy
if (9) holds. (ii) One manager plays her normal policy and the other shifts her portfolio if
(10) or (11) hold. (iii) Both shift their portfolio towards their competitor if (12) holds.
As expected, for a large enough x, both managers play their normal policies, as they are
guaranteed under their normal policies of not paying the penalty in either state. Within
the shifting region, when condition (12) is satisﬁed, both managers shift their portfolios
towards each other, but they do so taking into account that the other manager will also
shift, resulting in a shift that is less than in a non-strategic case. In this equilibrium it is
as if both managers agree to move from their normal portfolio towards the other manager.
As a result, neither individual needs to shift as much. This case is illustrated in panel (a)
of Figure 5. The equilibrium portfolios in (10) and (11) are particularly interesting as they
clearly illustrate the impact of strategic behavior on equilibria. In each case the equilibrium
portfolios are highly shifted toward one of the two managers. In (10), the less risk averse
manager (i throughout this document) plays her normal policy and the more risk averse
manager (j) shifts her portfolio, buying a higher share in the risky asset than her normal
policy. This case is illustrated in panel (b) of Figure 5. Suppose that manager i initially
conjectures that manager j will play her normal policy. In the shifting region, i responds by
ˆA , manager j is in the
ˆA in Figure 5). For φ
playing a portfolio that is shifted towards j (φi i
shifting region and responds by increasing her share in the risky asset, thus moving towards
ˆB in Figure 5, for which j
i. In response, manager i plays a portfolio with less shift, as in φi0
responds with still more shifting towards i. In equilibrium, i plays her normal policy and j
does all the shifting even though relative to their normal policies both managers are in the
shifting region. Note that rH − B > B − rL is a necessary condition for this equilibrium to
exist. With returns skewed to the right, the more risk averse manager j is more exposed to
the underperformance penalty, as manager i can select a portfolio to avoid the penalty in
the low state, at the same time that manager j has to pay the penalty in the high state.
To summarize the previous discussion, the shifting region and size of the shift depend
28
among other things on the tightness of the MRG constraint x. A prominent characteristic
of the equilibria with strategic managers and an underperformance penalty based on peer
performance is that managers might end up playing portfolios that are heavily shifted towards
the normal policy of one of the two managers, more so than in a non-strategic environment.
3.3 Three Period Model (T=2): Trading Strategies
To study how underperformance penalties aﬀect manager’s trading strategies, and how this
relationship depends on the manager’s current level of underperformance, I solve a three
period version of the model with periods t=0,1 and 2 (details are presented in Appendix A).
The structure of the equilibrium policies in the intermediate period t = 1 is the same as in
the initial period of the two period case. However the conditions under which each manager
shifts from the normal policy and the size of the shift varies according to the accumulated
returns. More speciﬁcally, Proposition 5 states that in the shifting region, the outperforming
manager is likely to play her normal policy, while the underperfoming manager does all the
shifting. Here the outperforming manager realizes not only that she is far away from the
penalty threshold, but that her competitor is lagging in returns and is facing the risk of
paying a higher penalty.14
The main results of the three period model are summarized in Proposition 6. In period 0,
either both managers select their normal policies, one shifts and the other plays her normal
policy or both shift. After the ﬁrst period, if high returns are realized in period 1, the
manager with the greater share in the risky asset (manager i throughout this document)
will either increase the share of the risky asset in her portfolio or continue to play her
normal policy if that was her equilibrium strategy in the ﬁrst period. Manager j, who has
a smaller share in the risky asset is now more vulnerable to the underperformance penalty,
will then buy more shares of the risky asset, shifting more from her normal policy. Here both
managers end up (weakly) increasing their shares in the risky asset, thus trading in the same
direction. If instead low returns are realized in period 1, both managers (weakly) decrease
29
their participation in the risky asset. Using the terminology from the previous section, the
model states that the overperforming manager will move toward her normal policy while the
underperforming manager moves toward her peer. The game is to follow the leader, where
the interim winner moves away from the peer portfolio and the interim loser tries to catch
up to minimize the risk of paying the underperformance penalty in the last period.
The size of these portfolio changes, depends among other things on the strictness of the
MRG (x). This correlated trading may look as if both managers are chasing returns, but
in fact they are simply chasing each other. When a manager is overperforming she is less
exposed to the penalty, so she can now move toward her normal policy, which happens to be
more of the asset that performed well. The underperforming manager is more constrained
by the penalty and realizes that the leading manager will move toward her normal policy, so
she will have to shift her portfolio more in this direction.
The model shows that the introduction of relative performance concerns through a peer
based underperformance penalty aﬀects the asset allocation in important ways. First, the
MRG regulation gives rise to time varying investment policies, with the managers making
procyclical trades, buying more of the asset that performed well in the previous period. Sec-
ond, given the strategic interactions of the managers, the model suggests that the equilibrium
portfolios and individual shifts are driven by the least restricted manager, and depending
on the parameters, the more restricted manager may end up doing all the shifting. Hence
the combined portfolio might be highly tilted towards the preferences of the best-performing
manager.
As an additional exercise, I calibrate the three period model to the estimated data prior
to June 2007 and calculate the change in the trading behavior predicted by the model
in response to a change in regulation equivalent to the change in the Colombian MRG
formula introduced in June 2007 (details are presented in Appendix B). The model does a
reasonably good job explaining the observed magnitudes of the decline in correlated trading,
the observed increase in dispersion of returns between managers and the observed reduction
30
in the correlation between trading direction and relative performance.
4. CONCLUSIONS
In this paper I study portfolio choices of strategic fund managers in the presence of a peer-
based underperformance penalty. The penalty generates herding behavior, and the extent
of correlated trading is exacerbated when a strategic setting is considered.
I document empirical evidence suggesting strategic behavior of asset managers when
facing a peer-based underperformance penalty. The evidence is taken from the Colombian
pension industry, where six Pension Fund Administrators compete to manage the saving
accounts of the working population, and are subject to a Minimum Return Guarantee based
on peer performance. The evidence suggests that a tighter MRG is associated with more
trading in the direction of peers and a smaller cross-section dispersion of returns between
pension funds. Moreover, the ranking among managers in terms of performance seems to
aﬀect how PFAs rebalance their portfolio. When the MRG is tight, underperforming man-
agers are more likely than their competitors to trade in the direction of their peers; this is
not true when the MRG is slack. Underperforming managers rebalance their portfolio by
buying more heavily stocks in which the manager is underexposed relative to her peers, as
opposed to selling stocks in which she is overexposed. Since these pension funds were in an
accumulation stage during the sample period, with new ﬂows accounting for an average of
8.5% the value of the fund each year, managers were presumably able to reduce their par-
ticipation in stocks to which they were overexposed simply by maintaining a ﬁxed number
of shares.
There is an interesting time dimension that is not studied in this paper: managers’
behavior close to the MRG evaluation date. Unfortunately, since the data provided was for
monthly portfolio holdings, same as the MRG evaluation period, I cannot study whether
PFAs altered their trading strategies by the end of the month. This is an important analysis
but it requires access to portfolio data at higher frequencies.
31
Finally, I present a model of portfolio choice and strategic interactions among managers
facing a peer-based underperformance penalty similar to the MRG. In the model, two fund
managers select the trading strategy on behalf of their investors and are compensated based
on their assets under management and their relative performance. The model shows that the
introduction of a peer based underperformance penalty induces managers to make procyclical
trades, buying more of the asset that performed well in the previous period. Such behavior
is exacerbated when strategic managers are considered and the aggregate portfolio of the
managers is highly tilted towards the preferences of the least restricted manager.
In the model analyzed in this paper asset prices and returns are exogenous. Given the
size of the assets under management by the PFAs, their procyclical trading might have
price eﬀects; for example, procyclical trading might increase both volatility and correlation
between stock prices. It would be interesting to study asset prices in a general equilibrium
version of the above model where fund managers interact with other market participants. The
challenge for such an extension is that since there is only a small number of managers, they
might be strategic about the eﬀect of their trading on asset prices, recognizing their market
power. Previous work starting with Lindenberg (1979) suggests that these considerations
might reduce the size of the trades but the direction of trading would still be the same.
Finally, since I have data on individual security allocations for PFAs, any model prediction
could be further tested with individual stocks. These extensions are left for future work.
32
APPENDIX A: THREE PERIOD MODEL SOLUTION
The timeline of events is as follows: In period t = 0, each manager chooses some φi0 and
s
φj 0 .15 Returns for the risky asset r1 are realized and managers enter period t = 1 with a new
s s s s
level of wealth R1 W0 (to simplify notation R1 ≡ R1 (φ0 ) where R1 (φ0 ) is deﬁned according
(3)). In this period each manager chooses a portfolio allocation φi1 and φj 1 . The last period
returns (t=2) for the risky asset r2 are realized, managers observe their relative wealth, and
fees are calculated depending on their relative performance. Here I deﬁne a trading strategy
as the change in portfolio allocation between period 0 and 1 (φi1 − φi0 ). I solve this problem
by backward induction.
Period t=1 equilibrium policies: Starting at period 1, each manager observes the
s s
realized returns Ri 1 and Rj 1 given portfolio allocations at time 0, φi0 and φi0 , and the
realization of the uncertainty s at the high or low state. At this point, the best response
functions and equilibrium policies are calculated as in the two period economy in the previous
section. The only diﬀerence is that the underperformance penalty for manager i now satisﬁes
S S
WjT
WiT s S s S
W i0
− Wj 0
+ x = Ri 1 R2 (φi1 ) − Rj 1 R2 (φj 1 ) + x using the dynamic wealth process.
The shifting region, which was given by conditions in (6) and (8) in Proposition 1 for the
two period case, is now
Rj 1 x − B (Rj 1 − Ri1 ) Rj 1 x − B (Rj 1 − Ri1 )
φN P
i1 ∈ −∞, φj 1 − φj 1 + ,∞ (A1)
R i1 Ri1 (rH − B ) Ri1 Ri1 (B − rL )
The best response functions in period 1 for this case are similar to Proposition 1 and are
omitted here. Instead in Proposition 5 I present the equilibrium policies for both managers
s s
at period t = 1 for a given pair of accumulated returns Ri 1 and Rj 1 .
PROPOSITION 5. The Nash equilibrium policies for managers i and j at period t = 1 for
s s
a given pair of accumulated returns Ri 1 and Rj 1 by manager i and j respectively are given
33
by:
NP NP
φi , φj Φi1 ≥ φN
i
P
and Φj 1 ≤ φN
j
P
(A2)
φN P NP
i , aj + bj φi − cj x Φi1 ≥ φN P
and Φj 1 > φN P
(A3)
i j
φ∗ ∗
i1 , φj 1 =
ai + bi φN j
P
+ ci x, φN
j
P
Φi1 < φN
i
P
and Φj 1 ≤ φN
j
P
(A4)
(Φi1 , Φj 1 ) Φi1 < φN P
and Φj 1 > φN P
(A5)
i j
(ai +bi aj )+(ci −bi cj )x (aj +bj ai )−(cj −bj ci )x
where Φi0 = 1−bi bj
, Φj 0 = 1−bi bj
and the coeﬃcients are deﬁned as:
s
Rj s
Rj s s
1 1 ci Ri Ri
ai = ai − γ 1 − s
Ri
/Ai , bi = s bi ,
Ri
ci = s ,
Ri
aj = aj − γ 1 − s
Rj
1
/Aj , bj = s bj ,
Rj
1
1 1 1 1 1
cj
cj = s
Rj
and Aj , Ai , ai , aj , bi , bj , ci , cj were deﬁned in the proof of Proposition 1 in Appendix
1
C. Moreover,
s NP
i. If x > (rH −B ) Ri 1 φi
s NP
− Rj 1 φj
s
+ Ri s L s NP
1 − Rj 1 B and x > (B −r ) Ri1 φi
s NP
− Rj 1 φj −
s s
Ri 1 − Rj 1 B , the conditions in (A2) are always satisﬁed and managers play their nor-
mal policy at t = 1.
s H s L
ii. Ri 1 r − B > Rj 1 B − r is a necessary condition for (A3) to be an equilibrium
s H s L
iii. Ri 1 r − B < Rj 1 B − r is a necessary condition for (A4) to be an equilibrium
Intuitively, a manager that enters period 1 with smaller returns than her competitor
s
Ri
s
Rj
1
<1 is more likely to pay the underperformance penalty at t=2. From (A1), the
1
shifting region for i becomes larger as her accumulated relative returns with respect to her
peer are smaller.
Nash equilibrium strategies at period 1 are functions of the portfolio choices by both
managers at period 0 and the state of returns s at period 1. Formally φ∗ ∗
i1 = φi1 (φi0 , φj 0 , s).
Period t=0 equilibrium policies: With this set up one can write the maximization
problem for manager i as a portfolio choice φi0 at period 0 that maximizes expected terminal
utility for a given policy by her peer φj , with the equilibrium portfolios in period 1 for
s
realized returns r1 given by Proposition 5.
34
s
Formally, the wealth process can be written as WiT = R1 S
(φi0 )R2 (φ∗
i1 (φi0 , φj 0 , s)), the
manager’s compensation is calculated according to (4) and the expectation is calculated
over the four possible states {sS } = {HH, HL, LH, LL}.
PROPOSITION 6. The unique Nash equilibrium policies are functions of x and have the
following properties:
i. At time 0, φ∗ NP
i0 ≤ φi and φ∗ NP
j 0 ≥ φj
ii. If the high returns are realized in period 1 (r1 = rH ) then φN
i
P
≥ φ∗ ∗ ∗ ∗
i1 ≥ φi0 and φj 1 ≥ φj 0
iii. If the low returns are realized in period 1 (r1 = rL ) then φ∗ ∗ ∗ ∗ NP
i1 ≤ φi0 and φj 0 ≥ φj 1 ≥ φj
35
APPENDIX B: NUMERICAL ANALYSIS
In this section I describe the calibration of the three period model and evaluate its quan-
titative implications for portfolio choice and trading strategies. I ﬁrst calibrate the trading
behavior of fund managers to the observed behavior in the Colombian pension industry be-
fore June 2007, and show that the change in behavior implied by the model following an
exogenous change in the formula of the MRG is quantitatively similar to the changes in PFA
behavior observed in the data after June 2007.
Before proceeding with the calibration it is important to keep in mind that I have con-
structed a simple model to highlight the potential impact of an MRG on portfolio choice
with strategic managers. To obtain a parsimonious and tractable set up I have made two
important assumptions. First, managers diﬀer only in their risk aversion. Second, there are
only two securities, a risk-free and a risky asset. With two securities ∆w and d in section
2 are one dimensional objects, so direction as deﬁned in equation (1) can only take three
possible values {-1,0,1}, as the measure is normalized by the magnitude of both ∆w and d.
To give a sense of the magnitude of the adjustment in portfolio weights I redeﬁne direction
in period 1 as follows:
(φ∗ ∗ ∗ ∗
i1 − φi0 ) φj 0 − φi0
directioni1 = 2 (B1)
φN
i
P
− φN
j
P
where the two terms in the numerator are the model analogues for ∆w and d respectively.
The denominator in (B1) normalizes the measure by the maximum distance between the
two portfolios when the managers are choosing their normal strategies. Finally, measuring
direction in period 1 alone might underestimate important adjustments to the portfolio
allocations carried out in period 0. For this period, I introduce an alternative measure of
direction based on deviations from the normal policy, as follows:
36
φ∗ NP
i0 − φ i φN
j
P
− φN
i
P
directioni0 = 2 (B2)
φN
i
P
− φN
j
P
Direction for manager j is obtain by switching subscripts i for j in equations (B1) and
(B2). Here direction ∈ [0, 1] is zero when both managers choose their normal policies in each
period and takes positive values when the manager moves her portfolio towards the other
manager.
A period in the model represents a quarter. The timing of events is as follows: Each
manager chooses their equilibrium portfolio at t = 0. After the ﬁrst quarter, returns for the
risky asset are realized and managers choose their optimal policy at t = 1, changing their
portfolio allocation depending on their relative performance. The risky asset returns for the
second quarter are realized, and the MRG is enforced if a manager’s cumulative returns over
the two quarters are below the benchmark.
Using historic returns for the Colombian stock market calculated with the IGBC index
(see Table 5), I estimate the mean, standard deviation and skewness of these overall returns,
and then solve for p, rH and rL to match these three moments. In a two manager setting, the
Ri +Rj
MRG between January 2004 and June 2007 can be written as M RG = 70% 2
where R
are individual fund 3-year annual returns. The equivalent MRG measure for fund i in terms
Ri +Rj
of the compensation formula (4) deﬁned in the model is calculated as 70% 2
= Rj − x .
In the data, the yearly average PFA return over a three year window is 11%. This would
imply a yearly x = 1.65%. In the model economy, the MRG is applied to the cumulative
returns of two quarters, so the actual measure of MRG for a semester is adjusted by a factor
of two, so that x0 = 3.3%.
Finally I calibrate the risk aversion parameters σi and σj to match the average direction
of trading and the average cross-section dispersion of portfolio returns across managers prior
to June 2007. Average direction in the model is calculated as the average of directioni0
and directioni1 for both managers and across all states. Cross-section dispersion of returns
is calculated for the observed returns of both managers in both periods 1 and 2. The
37
model moments implied by this calibration are presented in Table 6. Compared to the data,
the model generates too strong of a negative correlation between relative performance and
trading towards ones peers prior to June 2007. Table 6 also presents the implied model
moments generated by the change in the MRG formula to x = 2.6% starting in July 2007
(in terms of the model analogue this represents a change in the MRG formula to x = 5.2%).
After the change in regulation, model managers trade less towards their peers and exhibit
a less negative correlation between relative performance and the direction of trading. In
addition, the cross section dispersion of returns increases. All of these moment changes are
consistent with the data.
Figures 6a and 6b present the equilibrium strategies for each manager for the calibrated
model. At period 0, if the MRG is strict enough, manager j optimally shifts her portfolio
towards i, while i plays her normal policy. If the stock yields high returns in period 1,
manager j ﬁnds herself behind the other manager and then moves her portfolio closer to i
(φ∗ ∗
j 1 ≥ φj 0 ), buying more shares in the stock, while manager i plays her normal strategy
again and doesn’t rebalance her portfolio. If the low returns are realized, manager j is
overperforming and can play her normal policy given that she is now not constrained by the
MRG, potentially rebalancing her portfolio by increasing her participation in the risk-free
asset (φ∗ ∗
j 1 ≤ φj 0 ), moving away from the other manager. Manager i, underperforming in
this state, ﬁnds it optimal to shift her strategy towards j to avoid the underperformance
penalty, and does so by increasing her participation in the risk-free asset as well (φ∗ ∗
i1 ≤ φi0 ).
Aggregate trading behavior is procyclical in this partial equilibrium model, increasing the
risky asset share after good returns and decreasing it after bad returns. However, managers
are not chasing returns, but are instead chasing each other, setting their portfolio to avoid
being below the MRG.
Figures 6c and 6d display the average of direction across funds for both periods 0 and 1
and the correlation between direction and relative performance for diﬀerent values of x. As
expected, a tighter MRG constraint (small x) results in more shifting towards one’s peers.
38
The U-shape of the correlation measure can be explained as follows. With a tight MRG,
both managers set their portfolio close to each other. Hence, the cross-section dispersion of
returns is small, and subsequent portfolio adjustments are small, even for underperforming
managers. As the MRG is loosened, cross-section dispersion of returns increases, but the
MRG might still bind, and portfolio adjustments also increase after returns are realized to
avoid the penalty. In the limit, with a loose MRG (large x), managers can simply play their
normal policies, the cross-section dispersion of returns is larger, but no portfolio adjustment
is required.
39
APPENDIX C: PROOFS
Proof of Proposition 1: In the absence of an underperformance penalty, by deﬁnition
the normal policy yields a higher expected utility than any other strategy. With the under-
perfomance penalty in place, for a given portfolio choice by the other manager φj 0 , if the
normal policy can be implemented without triggering the penalty in any state, the manager
optimally chooses her normal policy as if there were no relative performance concerns. Note
that if this is the case, the compensation for the manager in each state is βWiT , which is the
H
WiH Wj
same as without the penalty. The maximum allowed shortfall for each state, Wi0
≥ Wj 0
−x
L
WiL Wj
and Wi0
≥ Wj 0
− x, can be written in terms of the share in the risky asset φ using (3) as
x x
φj + B −r L
≥ φi ≥ φj − r H −B
. Hence if φN
i
P
is in this region the manager’s optimal policy is
the normal policy.
If the manager cannot implement her normal policy without avoiding the underperfor-
mance penalty, the compensation (4) needs to be calculated accordingly with a penalty in
that state. Also, noting that a manager cannot be a loser in both states, to solve the opti-
mization problem I split the problem into two regions, one where the manager is a loser in
the low state and another where the manager is a loser in the high state.
For the manager playing to lose in the low state the problem is transformed into a
constrained maximization as follows (loser in the low state):
x
max EU FiS (φi0 , φj 0 ) + µ φi − φj − (C1)
φi0 B − rL
The Kuhn-Tucker conditions are
(1 − p)(β + γ )B − rL U FiL (φi0 , φj 0 ) + β (rH − B )pU FiH (φi0 , φj 0 ) + µ = 0 (C2)
40
x
µ φi − φj − = 0, µ ≥ 0 (C3)
B − rL
which imply that the best response in this region is:
ϕ i0 x
ϕi0 > φj 0 + (interior solution)
ˆi0 = B −rL
φ (C4)
φ + x
j0 B −r L
otherwise (corner solution)
where ϕi0 solves (C2) with µ = 0. Solving for ϕi0 , the best response in this region can
ˆi0 = a + b φj 0 + c x, where the coeﬃcients
be written as a linear function of φj 0 and x as φ i i i
take the following values:
Interior solution
ai = (β + i γ )Ai B ; bi = i γ (rH − B )Ai ; ci = i γAi
where
−1
Ai = i (β + γ )(rH − B ) + β (B − rL )
and
−1/σi
(β +γ )p(rH −B )
i = β (1−p)(B −rL )
Corner solution
1
ai = 0; bi = 1; ci = rH −B
To solve the optimal portfolio in the region where manager i pays the underperformance
penalty in the high state (loser in the high state), I proceed in similar fashion. The best re-
ˆi0 = ai + bi φj 0 − ci x,
sponse function of manager i in this region is again linear in x and φj 0 as φ
with the coeﬃcients deﬁned as follows:
Interior solution
ai = (1 − i )βAi B ; bi = γ (B − rL )Ai ; ci = γAi
41
where
H −1
Ai = i β (r − B ) + (β + γ )(B − rL )
and
−1/σi
βp(rH −B )
i = (β +γ )(1−p)(B −rL )
Corner solution
1
ai = 0; bi = 1; ci = B −r L
Q.E.D
Proof of Corollary 2: The assertion follows directly from the corner solutions in Propo-
sition 1 that eﬀectively determine the boundaries of the shift.
Q.E.D
Proof of Corollary 3: The optimality condition for an interior solution for a manager
playing to lose in the low state reads
(1 − p)(B − rL ) u (F L ) β
H H
= →0 (C5)
p(r − B ) u (F ) β+γ
In order for the marginal rate of substitution between the compensation in the low and
high state to go to zero, the manager must end up with all the wealth in the low state, which
x
implies ϕ → −∞. Hence the boundary condition ϕi0 ≤ φj 0 + B − rL
in (C4) is always satisﬁed
ˆi0 = φj 0 +
and the manager’s best response is φ x
.
B −r L
The optimality condition for an interior solution for a manager playing to lose in the high
state reads
(1 − p)(B − rL ) u (F L ) β+γ
H H
= →∞ (C6)
p(r − B ) u (F ) β
42
In order for the marginal rate of substitution between the compensation in the low and
high state to go to inﬁnity, the manager must end up with all the wealth in the high state,
x
which implies ϕ → ∞. Hence the boundary condition ϕi0 ≥ φj 0 − rH −B
is always satisﬁed
ˆi0 = φj 0 −
and the manager’s best response is φ x
.
r H −B
Q.E.D
43
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46
Notes
1
See for instance Turner and Rajnes (2001) for a review on these systems. Castaneda and Rudolph (2010)
present a theoretical analysis of portfolio choice under peer-based and index-based MRG.
2
In June 2008 some of the limits were: (i) Maximum 50% in domestic government debt. (ii) Maximum
40% in equity securities. (iii) Maximum 40% in foreign securities.
3
Other smaller fees apply in special cases (e.g. there is a fee when the worker changes PFA, as well as a
proportional fee on the value of the account when the worker has not made a contribution for six consecutive
months.)
4
A similar provision is in place in other countries, including Chile, Peru, the Dominican Republic and
Uruguay.
5
As an example of how this new formula loosened the MRG constraint (increased the maximum allowed
shortfall) consider the date December 31, 2009. Between December 31, 2006 and December 31, 2009 the
industry annual average returns were 6.01%. With the new formula in place, the MRG was 3.41%, instead
of the 4.20% that would have occurred under the older formula.
6
In the 15 year history of the private pension system (between 1996 and 2010), no PFA ever yielded
returns below the MRG. Even in the turmoil of October 2008, the PFA with the lowest returns managed to
have returns 118bps above the MRG (this is the closest any PFA was to the MRG in the sample period)
7
The only major restriction that these funds share is the short selling ban.
8
A similar measure of direction was ﬁrst introduced by Koch (2012). Here I deﬁne the angle between the
active change in weights and the initial distance to the peer benchmark, as opposed to the angle between
the active change in weights and the peer benchmark active change in weights as in Koch (2012).
9
Selling past losers can also be explained by window dressing. For US pension funds see Lakonishok et al.
(1991)
10
See Grinblatt, Titman and Wermers (1995), Grinblatt and Keloharju (2000) among many others. Rad-
datz and Schmukler (2013) also document the presence of momentum trading for Chilean PFAs
11 i
To reduce the eﬀect of outliers ∆wst is winsorized at 1% each tail.
12
Alternatively to the adjusted weight change, a PFA might opt for a “pasive” rebalancing of its portfolio
by accounting for changes in security prices. For this reason, I estimate equation (2) using the unadjusted
i i i
change in weights, i.e. ∆wst = wst+1 − wst .
13
According to ASOFONDOS, four of the six PFAs had only one CEO replacement each during the sample
period, occurring on the following dates: October 2006, February 2008, October 2008 and May 2010. The
other two PFAs changed their CEO four times each between January 2004 and December 2010.
47
14 s s
One should note that if both managers have the same returns to start the period (Ri 1 = Rj 1 ) the
equilibrium strategies are exactly the same as in the two period model.
15
These not need to be the ones calculated for the two period economy. I call them φ0 just to be consistent
with the notation.
48
Table 1
Summary statistics for Colombian pension funds and voluntary fund holdings
Key statistics are provided below (at two-year intervals) for the Colombian pension funds and voluntary
funds. For each column, statistics are shown for the portfolios reported by June 30 of each year, except as
noted. The database, made available by the Association of Pension Fund Administrators (ASOFONDOS),
includes monthly portfolio holdings of each security in every pension fund and voluntary fund from January
31, 2004 to December 31, 2010. Panel A documents the total number of funds, the total assets under
management and the share invested in stocks traded publicly in the domestic capital market. Panel B shows
the average number of stocks held per fund at each date, the number of diﬀerent stocks held by all six
pension funds as a group and the number of stocks in the IGBC index, which is a major stock index for
the Colombian stock market. Panel B also provides trading data, inferred from the diﬀerence in portfolio
holding between May 31 and June 30 of each year. Panel C shows key statistics on relative performance
between funds and portfolio diﬀerences between each fund and the peer portfolio and between each fund
and the market portfolio. Relative performance is measured as the diﬀerence between the peer returns and
individual fund returns, for annual returns measured over a three year rolling window. Distance measures
a fund’s exposure to each stock relative to the benchmark: di i i
st = πst − wst . Panels D and E present key
statistics for voluntary funds.
Year
2004 2006 2008 2010
Panel A. Pension fund count, assets and asset allocation
Number of funds 6 6 6 6
Total assets ($billions) 8.2 13.8 27.8 44.1
Net ﬂows (contributions minus withdraws $billions) 0.8 1.5 2.4 1.7
Percent invested in domestic stocks 5.0 12.6 22.4 32.1
Largest fund share (percentage over the pension industry) 27.1 26.6 27.2 27.2
Smallest fund share (percentage over the pension industry) 2.9 3.8 4.5 4.8
Panel B. Pension funds domestic stock count and trading statistics
Average number of stocks held per fund 16.2 21.2 26.3 30.0
Number of distinct stocks held by all pension funds 41 50 44 47
Number of stocks in the market index 26 33 27 32
Average stocks traded per fund 7.2 5.2 8.3 7.0
Proportion of trades that are buy (percent) 65.1 61.3 80.0 54.8
Total buys ($millions) 14.1 20.3 82.7 50.7
Total sells ($millions) 4.5 16.3 23.9 80.0
Average yearly sells (percentage of sell volume over total trades) 27.4 25.4 29.2 65.4
Panel C. Pension funds performance and portfolio diﬀerences (standard deviation in parenthesis)
Average relative returns (percent) -0.10 0.13 0.27 0.29
(0.49) (0.79) (1.30) (1.30)
Average peer distance (percent) 0.02 0.00 0.00 0.00
(3.83) (2.01) (1.23) (2.09)
Average market distance (percent) 0.06 0.00 -0.00 0.00
(5.39) (3.10) (2.94) (3.08)
Panel D. Voluntary funds count, assets and asset allocation
Total assets ($billions) 1.2 2.1 4.8 4.3
Percent invested in domestic stocks 1.8 4.5 9.2 14.5
Largest fund share (percentage over the pension industry) 38.1 28.0 47.0 36.1
Smallest fund share (percentage over the pension industry) 1.9 3.5 2.1 4.7
Panel E. Voluntary funds domestic stock count and trading statistics
Average number of stocks held per fund 5.2 13.8 17.8 20.3
Number of distinct stocks held by all pension funds 23 30 36 34
Average stocks traded per fund 0.8 6.5 10.2 12.8
Proportion of trades that are buy (percent) 79.5 38.5 39.4 59.2
Total buys ($millions) 1.9 10.9 12.8 34.3
Total sells ($millions) 49 0.5 7.75 32.9 77.8
Average yearly sells (percentage of sell volume over total trades) 19.8 41.5 71.9 69.4
Table 2
Direction of portfolio weight changes
∆wi ˙i
t dt
The direction measure, directioni
t , for a given fund i at some month t equals ||∆wi i , where ∆wi
t is
t || ||dt ||
the active change in portfolio weights between t and t + 1 adjusted by stock individual returns. di t is the
distance in month t between fund i’s portfolio and the peer portfolio. Statistics are calculated for measures
of direction across funds (directioni
t ). Direction captures whether each fund is moving towards or away from
the peer benchmark.
Mean Median Min Max std dev
Panel A. Statistics for direction
Pension Funds
Before June 2007 0.32 0.32 -0.21 0.71 0.19
After June 2007 0.14 0.16 -0.68 0.50 0.18
Voluntary Funds
Before June 2007 0.13 0.10 -0.73 0.99 0.38
After June 2007 0.13 0.16 -0.93 0.96 0.42
Panel B. Statistics for pension funds relative performance
Relative Returns (in bps)
Before June 2007 0.07 0.13 -3.63 3.71 1.29
After June 2007 -0.07 -0.35 -4.98 5.45 1.86
Panel C. Correlation between direction and relative performance for pension funds
Before June 2007 = -0.31***
After June 2007 = 0.08
50
Table 3
Linear regression for adjusted weight changes
i
The dependent variable is the change in weight ∆wst+1 for stock s between period t and t + 1 for fund i
i i wi ×ret
st
adjusted by the stock returns as follows: ∆wst +1 = wst+1 − , where retst are the gross returns
st
i
s wst ×retst
for stock s between t and t + 1. The unit of observation is a month. “MRG” is a dummy variable, equal
to one for dates prior June 2007 and zero thereafter. “Peer (Market) Distance” is the diﬀerence between
the weight of stock s in the peer (market) portfolio and the weight of s in fund i’s portfolio. The market
portfolio is the IGBC, a major index in the Colombian stock market. “Relative Performance” is the diﬀerence
in returns between manager i and the overall pension industry, measured over the previous 36 months for
each date. “Size” is the share of assets under management of pension fund i as a percentage of the entire
pension industry. Standard errors in parenthesis are adjusted for within-stock clustering in (1) and (2) and
adjusted for within-funds clustering in (3) and (4). Note: ***/**/* indicate that the coeﬃcient estimates
are signicantly diﬀerent from zero at the 1%/5%/10% level.
Within stocks Within funds
clustering clustering
Dependent variable (1) (2) (3) (4)
MRG x Peer Distance 0.0515** 0.0403* 0.0515** 0.0403**
(0.0255) (0.0220) (0.0216) (0.0171)
MRG x Relative Performance 0.0024 0.0024 0.0024 0.0024
(0.0027) (0.0027) (0.0076) (0.0074)
MRG x Peer Distance x Relative Performance -1.5505* -1.9248* -1.5505* -1.9248***
(0.9447) (1.0123) (0.9647) (0.6105)
MRG x Size 0.0001 -0.0001 0.0001 -0.0001
(0.0006) (0.0006) (0.0015) (0.0014)
MRG x Size x Peer Distance -0.2201* -0.2464* -0.2201** -0.2464***
(0.1218) (0.1336) (0.0926) (0.0908)
MRG 0.001 0.0001 0.001 0.0001
(0.0001) (0.0001) (0.0001) (0.0001)
Peer Distance 0.1068*** 0.1040*** 0.1068*** 0.1040***
(0.0166) (0.0177) (0.0168) (0.0153)
Relative Performance -0.0035* -0.0034* -0.0035 -0.0034
(0.0019) (0.0019) (0.0086) (0.0085)
Peer Distance x Relative Performance 0.8529 0.7604 0.8529 0.7604
(0.8067) (0.8053) (1.0410) (0.9031)
Size -0.0017*** -0.0017*** -0.0017* -0.0017*
(0.0005) (0.0004) (0.0009) (0.0009)
Size x Peer Distance -0.3464*** -0.3478*** -0.3464*** -0.3478***
(0.0875) (0.0872) (0.0780) (0.0771)
MRG x Market Distance 0.0198 0.0198
(0.0153) (0.0121)
MRG x Market Distance x Relative Performance 0.0519 0.0519
(0.4751) (0.6943)
Market Distance 0.0032 0.0032
(0.0056) (0.0023)
Market Distance x Relative Performance 0.0742 0.0742
(0.2499) (0.2698)
Controls YES YES YES YES
Pension fund ﬁxed eﬀects YES YES YES YES
Number of observations 18960 18960 18960 18960
51
Table 4
Probit regression of buying or selling a stock
i i
The dependent variable is the dummy variable buyst +1 or sellst+1 , which indicates whether a given fund i
in period t + 1 increases or decreases the number of shares in stock s. The unit of observation is a month.
“MRG” is a dummy variable equal to one for dates prior June 2007 and zero thereafter. “Peer (Market)
Distance” is the diﬀerence between the weight of stock s in the peer (market) portfolio and the weight of
s in fund i’s portfolio. The market portfolio is the IGBC, a major index in the Colombian stock market.
“Relative Performance” is the diﬀerence in returns between manager i and the overall pension industry,
measured over the previous 36 months for each date. “Size” is the share of assets under management of
pension fund i as a percentage of the entire pension industry. Standard errors in parenthesis are adjusted
for within-stock clustering. Note: ***/**/* indicate that the coeﬃcient estimates are signicantly diﬀerent
from zero at the 1%/5%/10% level.
Dependent variable Probability of buying Probability of selling
conditional on ownership
(1) (2) (3) (4)
MRG x Peer Distance 86.51*** 79.64*** -25.51 -16.39
(26.14) (25.64) (30.02) (39.92)
MRG x Relative Performance 6.97 7.25 0.51 1.30
(4.61) (4.94) (5.67) (5.96)
MRG x Peer Distance x Relative Performance -2105.06* -3980.69** 155.80 -1566.10
(1259.69) (1629.30) (849.30) (1369.78)
MRG x Size 2.27*** 2.28*** 5.55* 5.57*
(0.74) (0.86) (3.11) (3.13)
MRG x Size x Peer Distance -427.09** -445.82** 77.49 71.79
(191.90) (195.08) (165.83) (154.14)
MRG 0.00 0.00 0.01 0.01
(0.01) (0.01) (0.02) (0.02)
Peer Distance -54.59* -45.18 -70.44*** -73.80***
(28.37) (31.86) (14.67) (15.46)
Relative Performance -8.68*** -8.16** -8.77** -8.89**
(3.21) (3.49) (3.94) (3.95)
Peer Distance x Relative Performance 1491.73 809.40 -703.27 -556.61
(1043.67) (1130.83) (632.17) (699.52)
Size -4.41*** -4.48*** -6.03*** -6.10***
(0.51) (0.53) (1.50) (1.48)
Size x Peer Distance 216.03 210.56 276.35*** 276.18***
(145.99) (144.45) (84.24) (84.69)
MRG x Market Distance -0.39 -15.24
(36.22) (22.34)
MRG x Market Distance x Relative Performance 1587.83 1565.04
(1870.28) (1709.24)
Market Distance -8.45 3.26
(12.80) (3.47)
Market Distance x Relative Performance 663.93 -131.84
(415.18) (262.43)
Constant 35.15* 35.45* 42.82*** 43.25***
(19.22) (19.13) (14.59) (14.65)
Pension fund ﬁxed eﬀects YES YES YES YES
Number of observations 18960 18960 11299 11299
52
Table 5
Calibration
The return process of the risky asset is calibrated according to the Colombian stock market historical returns
(Data available after 1987). The risk aversion parameters σi and σj are calibrated to match the average
direction of trading and the average cross-section dispersion of returns prior to June 2007.
Value Source/Target
Panel A. Colombian stock market historical returns
Risk Premium 12.5%
Standard Deviation 19%
Skewness -0.33
Panel B. Model parameters
Probability of high state p=0.55 Mean, variance, skewness historical returns
High returns rH = 25% Mean, variance, skewness historical returns
Low returns rL = −10% Mean, variance, skewness historical returns
Minimum Return Guarantee x0 = 3.3% Before June 2007
x1 = 5.2% After June 2007
γ=1 Deﬁned by Colombian Government
PFA management fee β = 0.8% Management fees
Panel C. Calibration
Risk Aversion σi = 2.75, σj = 2.21 Direction and cross-section
variation of returns before June 2007
Table 6
Eﬀects from the change in the Minimum Return Guarantee formula
Empirical and model-implied moments before and after the change in the MRG formula in June 2007. The
parameters are set according to the calibration in Table 5.
Before June 07 After June 07
Data Model Data Model
Mean Direction 0.32 0.32 0.14 0.21
Std. dev. of relative returns 1.19% 1.08% 1.86% 1.54%
Corr(direction, rel) -0.31 -0.90 0.08 -0.65
53
Figure 1. Economy with three assets. This ﬁgure presents two examples of changes in the portfolio
composition in the space of weights for stocks A and B. The portfolio of fund i moves from wi i i
t to wt + ∆wt .
i i i
The distance vector is dt = Πt − wt , which represents the initial diﬀerence between fund i and the peer
portfolio at the beginning of the period t. θ is the angle formed between the change in the portfolio of
manager i and the distance vector. Direction is deﬁned as cos θ. When manager i moves towards the peer
portfolio, θ is smaller and direction is closer to 1, as in panel (a). In panel (b) the manager moves away from
the peer benchmark and direction takes smaller values as the angle increases.
(a) (b)
B B
1 1
1 A 1 A
Figure 2. Average direction of portfolio change. The direction measure, directioni
t , for a given fund i
i ˙i
∆w t dt i
at some month t equals i || ||di || ,
||∆wt
where ∆wt is the active change in portfolio weights between t and t + 1
t
adjusted by stock individual returns. di
t is the distance between fund i’s portfolio and the peer portfolio as of
month t. The ﬁgure reports the monthly value of direction averaged across the six PFAs, for pension funds
(solid line) and voluntary funds (dotted line).
0.45
Pension Funds
0.4
Voluntary Funds
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Change in MRG
‐0.05
Jun‐04 Jun‐05 Jun‐06 Jun‐07 Jun‐08 Jun‐09 Jun‐10
54
Figure 3. Marginal eﬀects. Diﬀerence in marginal eﬀects of distance on adjustments in portfolios weights
before and after the policy change ∂ ∆w(M
∂d
RG=1)
− ∂ ∆w(M
∂d
RG=0)
with 90% conﬁdence intervals.
0.15
−−−− 90% Confidence interval
0.1
0.05
0
−0.05
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02
Relative Performance
Figure 4. Utility maximization for manager i for a given portfolio choice by j φj . When the ﬁnal
Ws Ws
wealth of manager i WiT i0
is below the peer benchmark WjT j0
− x for either the high or low state she pays the
underperfomance penalty, thus reducing the net compensation. In panel (a) the manager’s best response is
to play her normal policy φ ˆi = φN P . Here, she dosn’t pay the underperformance penalty in any state. In
i
panel (b), the manager’s best response is a portfolio with a lower share in the risky asset than in her normal
ˆi < φN P , to reduce the underperformance penalty that is paid if the low state of returns is realized.
policy φ i
(a) (b)
55
Figure 5. Nash Equilibrium Portfolios φ∗ ∗
i , φj . Best responses by managers i (solid line) and j (dash-
dotted line). In panel (a) both managers optimally shift their portfolio towards their peer. In panel (b)
manager j does all the shift, while manager i plays her normal policy.
(a) (b)
ϕ ϕ
∗ ∗
,
∗ ∗
,
ϕ
ϕ
56
Figure 6. Nash Equilibria and trading behavior in a calibrated three period economy for
diﬀerent values of x. Panels (a) and (b) present the Nash equilibrium policies in period 0 and period 1 for
(φ∗ −φN P )(φN P
−φN P
)
managers i and j respectively. Direction is plotted in panel (c) using the formula i0 iN P N j
2
i
for
(φi −φj ) P
(φ∗ −φ∗ ∗ ∗
i0 )(φj 0 −φi0 )
period 0 and i1 N P NP 2
for period 1. Direction is calculated for each manager and averaged across
( i
φ − φj )
i and j for periods 0 and 1. The correlation between relative performance and direction presented in panel
(d) is calculated for period 1 for each pair of direction and relative returns for both managers. Parameters
are set according to the calibration in Table 5.
(a) (b)
Manager i Equilibrium Strategies Manager j Equilibrium Strategies
0.55 0.55
After June−07
Before June−07
After June−07
Before June−07
φ*
i0
φ*
j0
0.5 0.5
φ*
i1
(H) φ*
j1
(H)
0.45 φ*
i1
(L) 0.45 φ*
j1
(L)
Share in the risky asset
Share in the risky asset
0.4 0.4
0.35 0.35
0.3 0.3
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
x x
(c)
(d)
Average Direction
1 Correlation(Direction, Relative Performance)
After June−07
Before June−07
0.1
After June−07
Before June−07
0.9
0
0.8
−0.1
0.7 −0.2
0.6 −0.3
0.5 −0.4
−0.5
0.4
−0.6
0.3
−0.7
0.2
−0.8
0.1 −0.9
0 −1
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
x x
57