Hyun Song Shin

English

Research Papers in Economics

Discussion of “The Leverage Cycle”
by John Geanakoplos∗
Hyun Song Shin
Princeton University
June 2009
John Geanakoplos deserves credit for having blazed the trail on the anal-
ysis of leverage and collateral, and their crucial impact on the workings of the
ﬁnancial system. His 1997 paper “Promises, Promises” was a milestone in
the way that it brought together the institutions and practices that underpin
modern capital markets with rigorous general equilibrium theory. It is also
famous as an engrossing autobiography of the twists and turns in the route
by which a theorist became a mortgage hedge fund principal. The discus-
sion reﬂects the authoritative insights of a market professional as well as that
of an economic theorist - a rare combination indeed. He has subsequently
developed and reﬁned the approach in several additional contributions. His
2000 Econometric Society World Congress lecture in Seattle (published as
Geanakoplos (2003)) anticipates many of the ingredients of the current pa-
per, further developed to incorporate shifts in belief and endowments in a
dynamic economy in Fostel and Geanakoplos (2008).
In developing these ideas, he has been way ahead of the pack. Indeed,
until recently, there was no pack. He was so far ahead of the curve that
his early contributions did not attract the attention of the broader profes-
sion (outside the group of general equilibrium theorists) until the real world
caught up with the theory. The ﬁnancial crisis has changed everything, and
h i sw o r ki sn o wc e n t e rs t a g e ,a si td e s e r v e s .
The theoretical framework developed by Geanakoplos reﬂects the intellec-
tual pedigree from general equilibrium theory, especially the ﬁeld of general
∗Discussion prepared for the 2009 NBER Macro Annual
1equilibrium with incomplete markets (GEI). However, as elegant and rigor-
ous as it is, this pedigree is also double-edged. At the same time as serving
as a source of inspiration and a showcase for theoretical rigor, it can also be
an impediment to more robust and intuitive modeling approaches that would
ﬁnd wide adoption and use by an applied audience in corporate ﬁnance and
macroeconomics. The general results are very general, and address issues
such as existence of equilibrium and their constrained eﬃciency. Although
these general results give some guidance on potential sources of ineﬃciency
for welfare analysis purposes, they are not easy to utilize in applied modeling.
The more suggestive applied discussion is mainly weaved through a series of
ingeniously crafted examples that have been constructed to illustrate a par-
ticular point. However, in constrast to the very general theoretical results,
these examples are stark and specialized. They beg the inevitable question
of how robust they are to rough handling in applied modeling.
The theoretical pedigree from GEI analysis also serves to divert attention
away from the crucial role played by banks and other ﬁnancial intermediaries.
This crisis, as with many others, is diﬃcult to explain fully without placing
t h er o l eo fﬁnancial intermediaries at the center. Securitization was meant to
disperse credit risk to those who were better able to absorb losses. Financial
intermediaries were meant to play their role in dispersing credit risk. In fact,
in the current crisis the risks were concentrated in the ﬁnancial intermediary
sector itself. As leveraged institutions, they were the most vulnerable to
losses on their assets, as they were in danger of having their equity wiped
out, as many have found to their cost.
In this commentary, I make two broad points. First, the “narrative by
examples” approach to applied modeling practised in Geanakoplos’s paper
can be highly enlightening but they can fall short of providing robust oﬀ-the-
shelf models that can be used directly by applied researchers. One of the
reasons for this gap is the very feature that makes general equilibrium theory
so appealing - its solid foundations in terms of the preferences and beliefs of
the agents. Alternative staging posts for the analysis that use intermediate
categories such as balance sheet classiﬁcations and institutional “frictions”
may be easier to work with and equally illuminating. I give an example
below.
Secondly, there is still a need for a theoretical framework that gives a
role to ﬁnancial intermediaries that is commensurate with their importance
in practice. In spite of the advances made by John Geanakoplos in this
and in previous papers, much still remains to be done in bringing ﬁnancial
2intermediaries into the analysis of ﬁnancial booms and busts.
Recasting the Main Insights
A key ingredient in the Geanakplos paper is a division of roles where agents
are divided into the natural buyers of an asset (whether of houses or mort-
gages) and those who could potentially hold these assets, but normally end
up as lenders to the natural buyers, instead. The natural buyers are those
with the most optimistic beliefs about the asset’s future value, and they are
enabled to hold a larger position in these assets than they could based on
their own resources by the credit supplied by the less optimstic agents.
The collateral requirment and the discounts (haircuts) arise from the need
to satisfy the less optimistic agents that the loan is safe. But following bad
news for the asset, there is a redistribution of wealth away from the optimists
toward the pessimists. Some of the most overstretched optimists will have
their equity wiped out altogether by the price change. The marginal buyer
is therefore likely to be someone who is less optimistic or less rich than would
have been the case if the asset had not been purchased on margin, and the
wealth redistribution not been so adverse. For all these reasons, the shock
to the asset price is ampliﬁed through changes in the wealth distribution and
the identity of the marginal buyer.
The basic scenario painted above (both on the way up, and on the way
down) could be told in a simpler, static model where everyone has the same
beliefs. Set today’s date to 0. A single risky security is traded today in
anticipation of its realized payoﬀ next period (date 1). The risky security’s
payoﬀ is a random variable ˜ ,w i t he x p e c t e dv a l u e0, and uniformly
distributed over the interval [ −  + ]f o rs m a l l0. The uniform
density enables risk-free debt contracts to be written, as in Geanakoplos’s
paper. The variance of ˜  is 2 = 2
3 . There is also a risk-free security, cash,
that pays an interest rate of zero.
Let  be the price of the risky security. For an investor with equity  who
holds  units of the risky security, the payoﬀ of the portfolio is the random
variable:
 ≡ ˜  +(  − )( 1 )
Now, in the same spirit as in Geanakoplos’s paper, introduce two groups
of investors - passive investors and active investors. The passive investors can
be thought of as non-leveraged investors such as pension funds and mutual
3funds, while the active investors can be interpreted as leveraged institutions
such as banks and securities ﬁrms who manage their balance sheets actively.
Suppose passive investors have mean-variance preferences, and maximize
 =  () − 1
22
 where 0 is the investor’s “risk tolerance” and 2
 is
t h ev a r i a n c eo f. The passive investor chooses  to maximize
 ()= +(  − ) −
1
6

2
2 (2)
The optimal holding of the risky security is
 =
⎧
⎪ ⎨
⎪ ⎩
3
2 ( − )i f 
0o t h e r w i s e
(3)
These linear demands can be summed to give the aggregate demand. If 
is the risk tolerance of the th investor and  =
P
 , then (3) gives the
aggregate demand of the passive sector as a whole.
Now turn to the portfolio decision of the active (leveraged) investors.
These active investors are risk-neutral but face a Value-at-Risk (VaR) con-
straint, as is commonly the case for banks and other leveraged institutions.
The general VaR constraint is that the capital cushion be large enough that
the default probability is kept below some benchmark level. Consider the
special case where that benchmark level is zero. Then, the VaR constraint
boils down to the conditiion that leveraged investors issue only risk-free debt,
as in Geanakoplos’s model.
Denote by VaR the Value-at-Risk of the leveraged investor. The con-
straint is that the investor’s capital (equity)  be large enough to cover this
Value-at-Risk. The optimization problem is:
max
  ()s u b j e c t t o V a R ≤  (4)
If the price is too high (i.e. when  ) the investor holds no risky securities.
When  ,t h e n () is strictly increasing in , and so the Value-at-Risk
constraint binds. The optimal holding of the risky security can be obtained
by solving VaR = . To solve this equation, write out the balance sheet of
the leveraged trader as
4Assets Liabilities
equity, 
securities, 
debt,  − 
The Value-at-Risk constraint stipulates that the debt issued by the investor
be risk-free. For each unit of the security, the minimum payoﬀ is  − .I n
order for the investor’s debt to be risk-free,  should satisfy − ≤ ( − ),
or
 − ( − ) ≤  (5)
The left hand side is the Value-at-Risk (the worst possible loss), which must
be met by equity . Since the constraint binds, the optimal holding of the
risky securities for the leveraged investor is
 =

 − ( − )
(6)
and the balance sheet is
Assets Liabilities
equity, 
securities, 
debt, ( − )
(7)
Since (6) is linear in , aggregate demand of the leveraged sector has the
same form as (6), when  is the aggregate capital of the leveraged sector as
a whole. Denoting by  the holding of the risky securities by the active
investors and by  the holding by the passive investors, the market clearing
condition is
 +  =  (8)
where  is the total endowment of the risky securities. Figure 1 illustrates
the equilibrium for a ﬁxed value of aggregate capital .F o r t h e p a s s i v e
investors, their demand is linear with the intercept at . The demand of the
leveraged sector can be read oﬀ from (6). The solution is fully determined as
a function of . In a dynamic model,  can be treated as the state variable
(Danielsson, et al. (2009)).
Now consider a possible scenario following an improvement in the funda-
mentals of the security where the expected payoﬀ of the risky securities rises
5p
0 S
demand of 
passive investors
demand of 
VaR-constrained
investors
q q
Figure 1: Market Clearing Price
p
0 S
q
' q ' q
' p
Figure 2: Ampliﬁed response to improvement in fundamentals 
6Initial 
balance sheet After q shock
Final
balance sheet
debt
equity
assets
increase 
in equity
equity
assets
debt
assets
increase in
value of
securities
equity
debt
new 
borrowing
new 
purchase of
securities
Figure 3: Balance sheet expansion from  shock
from from  to 0. Figure 2 illustrates the scenario. The improvement in the
fundamentals of the risky security pushes up the demand curves for both the
passive and active investors, as illustrated in Figure 2. However, there is an
ampliﬁed response from the leveraged sector as a result of mark-to-market
gains on their balance sheets.
From (7), denote by 0 the new equity level of the leveraged investors that
incorporates the capital gain when the price rises to 0. The initial amount
of debt was ( − ). Since the new asset value is 0, the new equity level
0 is

0 =(  + 
0 − ) (9)
Figure 3 breaks out the steps in the balance sheet expansion. The initial
balance sheet is on the left, where the total asset value is . The middle
balance sheet shows the eﬀect of an improvement in fundamentals that comes
from an increase in , but before any adjustment in the risky security holding.
There is an increase in the value of the securities without any change in
the debt value, since the debt was already risk-free to begin with. So,
the increase in asset value ﬂows through entirely to an increase in equity.
Equation (9) expresses the new value of equity 0 i nt h em i d d l eb a l a n c es h e e t
in Figure 3.
The increase in equity relaxes the Value-at-Risk constraint, and the lever-
7aged sector can increase its holding of risky securities. The new holding 0 is
larger, and is enough to make the VaR constraint bind at the higher equity
level, with a higher fundamental value 0.T h a t i s ,

0 =(  + 
0 − 
0)
0 (10)
After the  shock, the investor’s balance sheet has strengthened, in that
capital has increased without any change in debt value. There has been an
erosion of leverage, and excess capacity appears on the balance sheet. Equity
is now larger than is necessary to meet the Value-at-Risk. In order to utilize
the slack in balance sheet capacity, the investor takes on additional debt
to purchase additional risky securities. The demand response is upward-
sloping. The new holding of securities is now 0, and the total asset value is
00. Equation (10) expresses the new value of equity 0 in terms of the new
higher holding 0 in the right hand side balance sheet in Figure 3. From (9)
and (10), we can write the new holding 0 of the risky security as

0 = 
µ
1+
0 − 
 + 0 − 0
¶
(11)
From the demand of passive investors (3) and market clearing,

0 − 
0 =
2
3
(
0 − )
Substituting into (11),

0 = 
Ã
1+
0 − 
 + 2
3 (0 − )
!
(12)
This deﬁnes a quadratic equation in 0. The solution is where the right
hand side of (12) cuts the 45 degree line. The leveraged sector ampliﬁes
booms and busts if 0 −  has the same sign as 0 − . Then, any shift in
fundamentals gets ampliﬁed by the portfolio decisions of the leveraged sector.
The condition for ampliﬁcation is that the denominator in the second term
of (12) is positive. But this condition is guaranteed from (11) and the fact
that 0  0 − (i.e. that the price is higher than the worst possible realized
outcome).
Ampliﬁcation is increasing in leverage, seen from the fact that 0 −  is
larger when  is small. Recall that  is the fundamental risk. When 
8is small, the associated Value-at-Risk is also small, allowing the leveraged
sector to maintain high leverage. The higher is the leverage, the greater is
the marked-to-market capital gains and losses. Ampliﬁcation is large when
the leveraged sector itself is large relative to the total economy. Finally, note
that the ampliﬁcation is more likely when the passive sector’s risk tolerance
 is high.
The amplifying mechanism works exactly in reverse on the way down. A
negative shock to the fundamentals of the risky security drives down its price,
which erodes the marked-to-market capital of the leveraged sector. The
erosion of capital induces the sector to shed assets so as to reduce leverage
down to a level that is consistent with the VaR constraint. Risk premium
increases when the leveraged sector suﬀers losses, since  − increases. The
two circular ﬁgures below depict the feedback from prices to actions to back
to prices, both on the “way up” and on the “way down”.1
Stronger
balance sheets Increase
B/S size
Adjust leverage
Asset price boom
Feedback “on the way up”
Weaker
balance sheets Reduce
B/S size
Adjust leverage
Asset price decline
Feedback “on the way down”
This example illustrates the amplifying eﬀect of fundamental changes
to asset values in the same spirit as the examples in Geanakoplos’s paper.
What the construction shows is how the main insights can be formalized fairly
robustly by invoking the intermediate modeling device of balance sheet man-
agement. There is no need to rely on diﬀerences in beliefs or speciﬁc features
of the time tree. Instead, the job is done by the upward-sloping demand
responses of the leveraged traders, who operate with warped incentives, or
constraints that reﬂect the anticipation of those warped incentives. Gromb
and Vayanos (2002) examined balance sheet eﬀects with leveraged traders,
and Xiong’s (2001) “wealth eﬀects” have similar consequences to the mark-
to-market capital gains of VaR-constrained investors. Brunnermeier and
1Adrian and Shin (2007) discuss the empirical consequences of such balance sheet
dynamics for the ﬁnancial system as a whole.
9Pedersen (2009) have examined the feedback through “margin spirals” of
leveraged investors.
These alternative approaches (as well as my example above) have two pos-
sible advantages over Geanakoplos’s more explicitly micro-founded approach.
First, the active investors with warped incentives are very reminiscent of the
banks and ﬁnancial intermediaries who operate with risk constraints. The
constraints need to be explained, but once they are assumed to be in place,
many steps in the argument become more transparent. The constraints
themselves come from outside the simple model. The most natural way to
explain such constraints would be through agency frictions (see Adrian and
Shin (2008)). Second, by illustrating the equilibrium both “on the way up”
and “on the way down” in the same model, the mechanisms invoked are less
tailored to the paricular scenario to be painted. The price to be paid is that
we lose the rigor of the foundational elements of GEI general equilibrium
theory. But for applied modeling purposes, some short-cuts may prove to
be an advantage.
References
[1] Adrian, Tobias and Hyun Song Shin (2007) “Liquidity and Leverage”
forthcoming in the Journal of Financial Intermediation
[2] Adrian, Tobias and Hyun Song Shin (2008) “Financial Intermediary
Leverage and Value at Risk” Federal Reserve Bank of New York Staﬀ
Reports, 338, July 2008
[3] Brunnermeier, Markus and Lasse Pedersen (2009) “Market Liquidity and
Funding Liquidity” Review of Financial Studies 22, 2201-2238
[4] Danielsson, Jon, Hyun Song Shin and Jean-Pierre Zigrand (2009) “En-
dogenous Risk and Risk Appetite” working paper
[5] Fostel, Ana and John Geanakoplos (2008) “Leverage Cycles and the Anx-
ious Economy” American Economic Review, 98:4, 1211—1244
[6] Geanakoplos, John. (1997) “Promises, Promises.” In The Economy as an
Evolving Complex System II,e d .W .B r i a nA r t h u r ,S t e v e nD u r l a u f ,a n d
David Lane, 285—320. Reading, MA: Addison-Wesley.
10[7] Geanakoplos, John. (2003) “Liquidity, Default, and Crashes: Endoge-
nous Contracts in General Equilibrium.” In Advances in Economics and
Econometrics: Theory and Applications, Eighth World Conference,V o l .
2, 170—205. Econometric Society Monographs.
[8] Gromb, Denis and Dimitri Vayanos (2002) “Equilibrium and Welfare in
Markets with Financially Constrained Arbitrageurs,” Journal of Finan-
cial Economics 66, pp. 361-407.
[9] Xiong, Wei (2001) “Convergence Trading with Wealth Eﬀects: An Am-
pliﬁcation Mechanism in Financial Markets”, Journal of Financial Eco-
nomics 62, 247-292
11

Convergence Trading with Wealth Eﬀects: An Ampliﬁcation Mechanism in Financial Markets”,

Equilibrium and Welfare in Markets with Financially Constrained Arbitrageurs,”

Financial Intermediary Leverage and Value at Risk” Federal Reserve Bank of New York Staﬀ Reports,

Hyun Song Shin and Jean-Pierre Zigrand

Liquidity and Leverage” forthcoming in the

Market Liquidity and Funding Liquidity”

Promises, Promises.” In The Economy as an Evolving Complex System II,e d .W .B r i a nA r t h u r ,S t e v e nD u r l a u f ,a n d David Lane,

NBER Macroeconomics Annual 2009, Volume 24

NBER Macroeconomics Annual 2009, Volume 24

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