The E?ect of Leverage on Financial Markets?†
Nathan Hodas,‡ Jacopo Tagliabue, Martin Schmidt, and Jeremy Barofsky
(Dated: September 1, 2009)
When people get excited about their prospects on the stock market, they borrow money from
the bank to invest. This leverage e?ectively couples the bank to the stock market. Thus, interest
rates determine demand for stock, and demand for stock can determine interest rates. Does this
interplay cause traders to naturally ?nd a stable balance of leverage and aggressiveness? How do
the behavioral traits of traders in?uence the stability of these interactions? Are there regulatory
behaviors, such as limiting leverage or slowing margin calls, that would contribute to the overall
health of the market? We present an economy consisting of a banking sector and an equity market,
with traders transferring money between the two. Using an agent-based model, we will be able to
examine how leverage couples the bank to the equities market. Furthermore, we can explore how
di?erent leverage strategies e?ects the stability of these markets.
their maximum leverage and makes a margin call on the
o?ending funds. Thurner et al. showed that this mech-
Leverage, the act of borrowing money for investing, is
anism accelerates the decrease of stock prices, forcing
a powerful force within ?nancial markets. It allows in-
previously safe funds into margin calls, as well, thereby
vestors to extract large gains from small movements in
increasing the volatility. Independent of any built-in non-
underlying prices. It can also force losses of equal magni-
standard rational behavior, the price distribution became
tude. Because those who lend money to others for lever-
heavy tailed and displayed clustered volatility. Conse-
age place this money at risk, these banks and brokerage
quently, system wide risk increases contrary to the inten-
?rms insist that traders maintain a minimum amount
tions of individual leverage requirements.
of equity, called margin. When the value of an invest-
This model touches upon two di?erent, unusual,
ment drops, traders can exceed the maximum allowed
strands of research. First, it relies on exclusively agent-
leverage, leading to a margin call from the lender, which
based modeling – a technique not widely used within the
forces the trader to unwind positions to keep leverage
economics discipline in spite of strong recommendations
within allowed levels. This selling e?ects prices, which
from various researchers to adopt it more widely [2, 3].
can trigger a feedback loop and a downward spiral in
A common critique of agent based models points to the
prices. Conversely, rising prices increase trader’s equity,
plethora of parameters associated with often ad-hoc be-
allowing them to take ever greater amounts of leverage,
haviors . However, the standard rational or random
driving prices up.
behaviors are unable to capture a good deal of observed
Although this cycle may make sense conceptually, it
behavior . The agent based model of Thurner et al.
remains to be fully explored by economists. In a recent
arrived, for the ?rst time, to a heavy tailed and clus-
paper, Thurner, Farmer, and Geanakoplos  presented
tered volatility price distributions by a mechanism that
an agent-based model of leveraged asset purchases with
exclusively relies on margin calls.
Four di?erent types of agents populate
On the other hand, it relates to research on the lever-
their model: noise traders buying and selling randomly
age cycle that John Geanakoplos and others have under-
around the fundamental value of a stock, hedge funds
taken over the last decade (see, for example Geanako-
buying stock when its price is below the fundamental
plos , Fostel & Geanakoplos ).
value and holding cash otherwise, investors investing in
tends to focus on the interest rate that equilibrates de-
hedge funds or holding cash otherwise, and lastly a bank
mand and supply for credit while neglecting the highly
lending money to hedge funds at no interest. The bank
variable leverage ratios, see Figure 1. Interest rates re-
extends their interest free loans whenever a hedge fund
lates to the impatience of borrowers, while the leverage
requests it and does not exceed a predetermined maxi-
relates to the nervousness of lenders. If one examines the
mum leverage, de?ned as (total assets purchased / wealth
time-dependence of leverage ratios and assumes that in-
of borrower). If fully leveraged hedge funds face falling
vestors, due to di?erent degrees of sophistication or risk
stock prices, the bank observe that these funds exceed
aversion, value assets di?erently, one has identi?ed a
mechanism that can create excessive asset price move-
In boom times, when lenders are not worried
about repayment, leverage is readily given, expanding de-
?DRAFT: do not quote.
mand for assets and in?ating prices. During crises, fear
†We would like to the H. Doyne Farmer for guidance and sharing
leads to a contraction of leverage, depressing prices. For
his work with us. We would also like to thank Corinne Teeter and
example, the leverage ratio in the US mortgage market
JP Gonzales for helpful discussions. Of course, many other CSSS
during the years 199-2006 was 20:1, whereas it has fallen
2009 participants lent too many thoughts and ideas to acknowledge
individually, so we thank you all. Especially Tom Carter.
to or even below 5:1 in 2007-2009. This points to the
necessity to limit swings in leverage and the possibility
FIG. 1: Historic leverage ratios of a speci?c hedge fund, Ellington Capital, exemplify the variability of margin rations over
that government regulation might by the only means to
achieve this restraint .
This paper builds on the Thurner, Farmer, and
Here we document the details of our model, and the
Geanakoplos  model of leveraged asset purchases with
source code will be available on the CSSS 2009 website.
margin calls. It relies on the agent-based approach and
aims to test the robustness of the Thurner et al. results
by adding features that bring the model closer to reality.
The features are independent of each other and, there-
fore, do not obfuscate the analysis. The features of more
Each trader belongs to one of multiple behavioral
realistic models can be compared to the baseline model,
classes. We have deadheads, momentum traders, value
although this as not been done in detail yet. The more
traders, and risk-neutral traders. Deadheads buy, sell,
realistic features are:
and borrow randomly. Momentum traders estimate the
return of the equities market by extrapolating from past
• a greater variety of trading strategies, includ-
Value traders know the fundamental mean-
ing trend reinforcing momentum traders and risk-
reverting price, and use leverage to aggressively invest
when the price falls below this fundamental. Risk-neutral
traders only compare the instantaneous return on invest-
ments between money saved in the bank and money in
• adaptive behavior of hedge funds where more suc-
Many other types of investors could ex-
cessful hedge funds are imitated by less successful
ist, and the simulation allows for extending the family
At the beginning of the simulation, each trader is en-
• endogenously determined interest rates
dowed with a ?xed amount of cash and shares. All cash is
kept in the bank and receives daily interest at the posted
annual rate. The interest payment per day is the an-
• a more realistic banking sector with a network of
nual rate divided by the number of trading days per year
various competing banks
(250), times the amount of deposited cash. Similarly, the
interest payments the bank charges for loans is the annual
We have currently implemented, to some degree, all of
deposit interest rate plus a spread, divided by the num-
these items using NetLogo 4.0.4. However, we have
ber of trading days, multiplied by the size of the agent’s
not implemented all of the features of Thurner et al.
Speci?cally, we do not have independent investors that
Loan interest charged by the bank is deducted from the
pile into successful funds. Without these, we cannot fully
deposited cash. If the trader doesn’t have su?cient cash,
observe market domination that is observed by . This
the bank sells some of the trader’s shares. If the trader
appears to be an essential feature to market crashes in
still does not have enough to pay the interest, the trader
must declare bankruptcy. Once bankrupt, a new trader
rises to take its place starting with a new endowment.
Bu?ett and Ben Graham claim their goal is not to es-
Notably, no trader is able to sell short.
timate a break-even price, but to classify a company as
grossly over- or under-valued. On the other hand, invest-
ment analysts frequently quote “target prices” for stocks,
which can have signi?cant in?uence on short-term price
Deadheads serve as explicit sources of randomness for
In our model, value traders do know the precise mean-
the bank. These traders act without any coherent strat-
reverting price of the equity. The amount to buy is then
egy. At each turn, the agent randomly determines to buy
determined by the potential gain resulting when the price
or sell, and to borrow or to repay. If the trader chooses
returns to the fundamental.
The actual algorithm is
to buy, it spends a random fraction of its available cash.
copied as closely as possible from .
If the trader chooses to sell, it sells a random fraction of
In many ways, value traders are the compliment to
its shares. If desired by the user, the deadhead can also
momentum traders. Value traders attempt to follow the
use leverage by randomly choosing each turn to borrow
maxim of buying high and selling low. Because momen-
or repay. If the trader chooses to borrow, it borrows a
tum traders sell as the price drops, they are frequently
random fraction up to the maximum possible leverage.
the counter-party to trades with value traders. Momen-
If it chooses to repay, it repays a random fraction of its
tum traders, as a whole, tend to buy high and sell low.
Arbitrage is a major force in determining prices. In
Momentum traders, like many of us, are trend follow-
this model, arbitrage takes place between the return of
ers. They attempt to catch the market’s up-swings and
the market and the interest from the banks. Unlike mo-
avoid the market’s down-swings. They possess su?cient
mentum traders, which estimate returns using extrap-
reason to compare their expected returns in the market
olation, the model’s risk-neutral traders know the true
with their expected returns by keeping their money in
annual return on the fundamental price. To achieve the
the bank. They calculate the expected return by ?rst
best possible investment gains, the risk-neutral trader
calculating the market price return over a trader-speci?c
moves money into the best investment vehicle.
time-window. This return is then multiplied by an exag-
geration factor, which is also speci?c to each trader. The
Each turn, the trader compares the interest on deposits
exaggeration represents the psychological hysteria that
with the return of the market. If the interest for deposits
causes people to over estimate ups and downs . If the
exceeds the fundamental return, the trader moves all of
expected return exceeds the deposit-rate o?ered by the
its assets into the bank, selling all held shares. Alter-
bank, the momentum trader takes all of its money out
natively, when the interest rate is smaller than the fun-
of the bank and puts it in the market. The variation
damental return, the trader uses its cash to buy shares.
in parameters for estimating the return means that, al-
When the return on equity exceeds even the interest on
though any one trader may over-react, putting their life
a loan, the risk neutral trader uses leverage to put as
savings in the market, the group of momentum traders as
much money as possible into the stock market. To avoid
a whole will act more progressively. When the expected
monolithic behavior, each risk neutral trader estimates
return exceeds the rate for borrowing a loan (the deposit-
the true fundamental return with a small error.
rate plus the spread), the momentum trader will borrow
In this way, the risk neutral trader helps to synchronize
money from the bank to invest. The amount borrowed
the market return with interest rates. When the bank
equals the maximum possible loan times a scaling factor,
has an excess of cash, and interest rates are low, the risk
neutral traders will borrow, bringing up interest rates.
If interest rates are excessively high, risk neutral traders
expected return ? current borrow rate .
are an explicit mechanism from moving cash from the
market into the bank, depressing interest rates.
Momentum traders can cause a self-ful?lling prophecy by
pushing more money into the market as it goes up and
pulling money out of the market as it declines.
The current simulation is also a ?rst step towards a
full-?edged implementation of “evolutionary” dynamics
among agents. In particular, we want to investigate how
A value trader is someone who buys an equity because
di?erent leverage choices give rise to di?erent global out-
they believe it is underpriced. In reality, this is an imper-
comes in the market. Thurner et al. observed that high
fect estimation, at best. Value investors such as Warren
leverage led to an increase in volatility and crashes, while
no leverage also increased volatility . Evolutionary Dy-
feature of banking regulations, the minimum reserve re-
namics may help uncover an ”optimal” leverage level.
quirement . This is, banks tune interest rates to stay
The current model features imitation dynamics and
as close as possible to the minimum reserve level.
replicator dynamics. The former is a function that tries
The minimum reserve is a set fraction of deposits the
to model the spread of good strategies in the market.
bank must keep on hand (10%). A bank in our model
Agents are allowed to look into their neighborhood and
obtains its revenue from interest payments on loans. A
copy, with a certain probability, agents that prove to be
bank can maximize its pro?ts by lending as much as pos-
more successful in the market. The latter dynamic is
sible while still maintaining the needed cash minimum.
the well-known standard dynamics of evolutionary game
To determine interest rates that maximize pro?t, instead
theory, where the proportion at t + 1 of a given breed,
of solving a simultaneous equation to balance demand
A, in a population, P , is given by the proportion at time
and supply of cash between the traders and the bank,
t times the ratio between B’s ?tness and P ’s average
banks can use the interest rate as a control parameter to
attract loans or deposits.
Although further developments are still needed to crit-
Multiple times each turn, the bank adjusts its rate to
ically assess the importance of evolutionary processes in
stay as close as possible to the target reserve level via an
the market, strategic interactions among breeds appear
open feedback loop. When called upon, the bank adjusts
to be a particularly promising line of enquiry. For ex-
ample, it is well known that mutual funds that advertise
di?erent strategies frequent hold very similar portfolios
?r = ? r(?Deposits ? Cash on Hand)
because fund managers fear underperforming their com-
, where r is the interest rate the bank pays for deposits,
?r is the change in the rate, ? is an arbitrary sensitivity
parameter, and ? is the reserve fraction, 10%.
The factor of
r was chosen to draw analogy with the
Cox-Ingersoll-Ross model of interest rate ?uctuations.
The excess reserve term multiplying the
r is therefore
In our model, interest rates frequently determine the
analogous to the source of noise in the CIR model, and ?
actions of traders who weigh returns in the stock market
becomes a volatility. More speci?cally, ? determines how
with interest from deposits. Intuitively, rising interest
severely the bank should adjust interest rates, and it is
rates depress market prices by increasing the cost of bor-
usually a very small number. In Figure 2, we show how
rowing and competing for investing money. Conversely,
overreaction leads to instability in interest rates and, by
low interest rates encourage loans and make risky invest-
extension, the stock market. For the same population
ments more attractive.
of traders (1000 risk-neutral, 100 deadheads, 1000 value,
One way to create a model economy is to ?x interest
1000 momentums), the choice of ? determines the stabil-
rates, allowing the experimenter to observe causal e?ects
ity of interest rates. The di?erence between ? = 0.001
of interest rate perturbations. However, when demand
(Fig. 2(a)) and ? = 0.002 (Fig. 2(b)) is su?cient to desta-
for stock is high, so is demand for loans. Fixing the in-
bilize interest rates.
terest rate could lead to fundamental imbalances. Buy-
The feedback interest rate model is su?cient to prod-
ing frenzies that encourage borrowing deplete banks of
uct a stable equilibrium between the fundamental return
cash reserves, driving up interest rates. Some equilib-
of the market and deposit interest rates, when the trader
rium between the market and the banks should emerge,
population is dominated by risk-neutral traders. No ex-
if it exists.
cess returns are possible. However, this validation alone
Thus, the central question in integrating a banking sec-
is not su?cient for banks to survive.
tor is how to faithfully determine interest rates endoge-
In reality, the return on risk-free government
A bank must pay interest on deposits from income from
bonds and interbank lender rates play a large role. Fur-
loans. If the bank were to charge the same interest rate
thermore, there is not one unique interest rate; they vary
for loans and deposits, a bank can only sustainably oper-
both from bank to bank and depend on loan terms. Ul-
ate when loans happen to exceed deposits. To ensure the
timately, many of the factors that determine these rates
bank can meet its obligations, it charges borrowers an
are political or ideological, such as “breaking the back
additional spread on top of the base deposit rate. In our
of in?ations.” These would be beyond the scope of our
model, spread is calculated by determining the amount
the bank needs to charge its borrowers to cover its ex-
penses, and the bank gradually adjusts the spread up or
down to cover expenses:
(Expenses ? Income)
How can we endogenously determine interest rates
without predetermining some of the most common inter-
Because an increase in spread discourages loans, it has
est rate driving forces? In our model, we use a common
the knock-on e?ect of depressing the deposit rate. As
(a) ? = 0.001
(b) ? = 0.002
FIG. 2: Excess sensitivity, ?, can cause banks to over-adjust the interest rate. This causes traders to correspondingly over-react
to the large change in rate, so the bank over-reacts in the other direction.
Figure 2(a) shows, deposit rates can e?ectively be driven
may try its best to lend out no more than regulations al-
to zero, while the borrowing rate can be a more common
low, but unexpected withdrawals can cause it to suddenly
number, say 4 or 5%. This disparity in interest rates
be in violation of the minimum reserve. In such a case,
should be familiar to anyone with a checking or margin
banks must borrow from other banks on the overnight
account, especially in recent months.
market. This money is intended to be repaid the next
Various variations of the spread and deposit rate pro-
day, along with a nominal interest fee.
cesses can alter the character of the interest rate dynam-
Our model has been extended to include multiple
ics, but that does not mean any single model is correct or
banks, but at the moment it is restricted only one. When
incorrect. Comparing a simulated single spot rate with
a bank needs to borrow, it borrows from the equivalent of
a real-world spot-rate remains to be done. Furthermore,
the Federal Reserve at a user-set rate. Although the user
implementing a term-structure would add signi?cantly
chooses this Federal Funds Rate, the underlying interest
to the complexity of any agent based model, because it
rate is determined by market dynamics, endogenously.
requires agents to distinguish expectations of returns at
NetLogo allows this rate to be changed mid-simulation,
di?erent times in the future and develop a strategy that
so we can understand how the target rate e?ects interest
accounts for money being sequestered in CD’s, bonds,
etc. for di?erent, often overlapping, periods.
Along with cash from deposits, the bank has cash as-
sets that it starts with and earns from loan interest pay-
ments. This cash counts toward the reserve requirements,
and has the e?ect of depressing interest rates. The bank
can borrow money to invest in loans, but it must pay in-
Just like individuals, banks frequently borrow to in-
terest at the target rate. This interest is deducted from
vest or cover short-term liabilities.
The minimum re-
the cash at hand, so it can threaten the reserve balance of
serve requirement applies to cash in a “vault,” whether
the bank. Hence, borrowing for making loans can in?ate
that money belongs to the bank or not . A bank
the interest rate.
The bank’s leverage is capped, and it can extend its
is updated once per turn. This fundamental price can
lending power by increasing its assets or obtaining more
gradually grow over time, giving the risk-neutral traders
deposits. Formally, the bank’s leverage is
something to compare to deposit rates.
Because our model has only one agent trading at a
time, the demand of all other agents is constant of the
assets + ? deposits
course of a trade. As a result, for a purchase initiated
trade, the agent sets the amount of money they are going
The bank can not borrow to exceed this leverage. How-
to spend (as opposed to buying a ?xed number of shares).
ever, in the case when the bank has insu?cient cash to
The new price is
cover a withdrawal, the bank is forced to borrow to en-
sure customers can recover their deposits, no matter
?t + $ ,
the leverage required. If a bank does not have su?cient
N ? D
cash to pay interest on its deposits, it will attempt to
where $ is the cash amount the trader is committing, and
borrow the necessary funds. In the case when the bank
D is that total shares held by all other agents.
is unable to borrow this money, it does not pay the in-
When an agent initiates a sell order, they put up a ?xed
terest. It alters its published deposit rate to re?ect the
number of shares (as opposed to selling a ?xed dollar
amount it is able to pay in deposits. Unlike withdrawing
amount). Again, no agents trade simultaneously, so the
deposits, interest payments are not guaranteed.
new price is
The bank is not subject to margin calls, because there
is no mechanism for the bank to force traders to repay
their loans. The bank attempts to repay its debts as
N ? D + S
quickly as possible, in line with the notion of an overnight
where S is the number of shares to be sold. The larger
market. If it can not a?ord to repay its debts, interest
the total available shares, N , the smaller the price im-
continues to accrue and compound, but the bank does
pact of any one trader, making the choice of N an impor-
not declare bankruptcy. However, the bank may not be
tant parameter in the model. The pricing algorithm has
able to make new loans or pay interest on deposits; when
not been thoroughly checked to ensure zero-sum trades,
the bank exceeds maximum leverage due to poor ?nancial
meaning the amount of money in the market may not be
performance it may become a “zombie” bank. How this
constant. It seems to work well enough.
situation plays out in the simulation is not always clear,
but we have certainly experienced similar credit freezes
The ?rst stage in our investigation is to simply validate
the model and understand its components. This takes
two forms, if we hope to draw analogy with real market
Our equity market contains only a single stock.
dynamics, we must explicitly compare known features of
this way, it represents the equity sector as a whole. The
real markets with those produced by our model. Second,
dynamics of the market are modeled after , with a
we have modeled much of our work after Thurner et al, so
number of modi?cations to account for the diversity of
we should recreate their behavior as a check. In fact, as
strategies by the agents.
mentioned earlier, the latter is not quite possible, because
Because we do not employ an order book to determine
we have not implemented independent investors who put
prices, agents do not explicitly trade with each other.
their money directly into the trading agents, a key feature
Instead, the counterparty to every trade is an implicit
of Thurner et al.
noise trader, which soaks up excess demand. As with
Here is a brief list of notable results gained so far:
Thurner et al., the total number of shares is split between
the agents and the noise traders. Because all shares are
1. Risk-neutral traders are by far the most vulnerable
held either by an agent or an implicit noise trader, we
2. Margin calls actually decrease bankruptcy rates, es-
pecially among risk-neutral traders.
Di(P ) + Dnt(P ) = N,
3. Margin calls decrease interest rates, See Figure 3.
where Di is the demand of trader i given price P , and
4. Momentum traders cause interest rates to be corre-
Dnt is the demand held by the noise traders. To set the
lated with equity price returns, not anti-correlated
price, we de?ne, as with Thurner et al.
as normally assumed.
5. Value traders, as expected, are the most likely to
nt(P ) = ?t/P,
earn pro?ts, but are rarely leveraged more than 2 or
where ?t is a mean reverting geometric brownian mo-
3 times. In good conditions, their earnings decrease
tion, which weakly reverts to a fundamental price and
panding it into more complex realms. For example, we
also have a credit network between multiple banks, but
this increased complexity slows down the simulation sig-
ni?cantly. It also increases the di?culty in debugging and
uncovering accounting errors. We will be better able to
add traders who execute more complex behavioral strate-
gies, which will allow us to investigate the e?ects of psy-
chology on market behavior.
To fully compare our model with Thurner et al., we
need to incorporate independent investors who place
FIG. 3: Margin calls decrease interest rates. By turning o?
their money into the accounts of traders who are perform-
margin calls (orange arrows), we see the system relaxes to a
ing well. This increases the market representation of that
higher interest rate. When margin calls are turned back on
particular strategy. If the independent investors switch
(blue arrows), the sudden wave of margin calls brings more
funds, this can cause a sell-o?, furthering any downward
cash back to the bank, immediately lowering interest rates.
price movements. We must also fully validate our mar-
ket dynamics by comparing our price and interest-rate
statistics to those observed in reality. We do not expect
6. Insu?cient maximum leverage increases the inter-
to be able to reproduce reality exactly, but we hope to
est spread, because the bank can not make enough
observe gross features.
loans to cover deposits. It also decreases the de-
posit interest rate, so both depositors and lenders
We have constructed an agent-based model of an eq-
Our research into evolutionary dynamics is still at its
uities market that endogenously determines both stock
beginning. We have yet to really fully integrate the com-
price and interest rates. We have the ability to observe
plete system within the model. We look forward to ob-
the e?ects of di?erent trading strategies on price and in-
serving how market dynamics are e?ected by changes in
terest rates. Although interest rates are in?uenced by
strategy. We will try di?erent measures of “?tness,” such
many factors, of which only a small part is demand for
as trying to minimize volatility, maximize Sharpe ratio,
leverage, we have a system that can explore how banks
or perhaps another metric of excess return. Our cur-
are coupled to the stock-market.
rent implementation of evolutionary dynamics involves
With this tool, and a few more modi?cations, we can
competition between deadheads and momentum traders.
follow the lead of Thurner et al., and investigate the ef-
In this, we see that the ?nal population is path depen-
fect of leverage on market stability. Our evolutionary
dent, with either all deadheads or all momentums. This
dynamic will allow us to search for an ideal level of lever-
suggests this research may yield some very interesting,
age, which a question of great importance in the wake of
context dependent, results.
the recent credit crisis. A great deal of ?nger pointing
We hope to migrate our code from NetLogo into a
and recrimination has emerged after the credit-crisis. Be-
faster language. NetLogo has been very e?ective at pro-
cause our agents play pure strategies, and not greed, ob-
totyping and inspecting our market, but lack of speed,
fuscation, or cheating, we have a window into how credit
and particularly debugging features, prevent us from ex-
shocks can form and, possibly, how they can be avoided.
 S. Thurner, J. D. Farmer, J. Geanakoplos “Lever-
 J. Geanakoplos, “End the obsession with interest.” Na-
ture 457 (2009) 963.
 J. D. Farmer, D. Foley “The economy needs agent-based
 D. A. Moore, T. R. Kurtzberg , C. R. Fox, M. H. Baz-
modeling,” Nature, 460, (2009) 485-486.
erman “Positive Illusions and Forecasting Errors in Mu-
 E. Bonabeau. “Agent-based modeling:
tual Fund Investment Decisions” Organ Behav Hum De-
techniques for simulating human systems” PNAS, 99,
cis Process 79 (1999) 95-114.
 J. Doyne Farmer and John Geanakoplos, “The virtues
wiki/Reserve requirement Aug. 29, 2009.
and vices of equilibrium and the future of ?nancial eco-
 this assumption is contrary to the standard economic the-
nomics” Cowles Foundation Discussion Paper No. 1647
ory that ascribes the same fundamental value to everyone
 J. Geanakoplos, “The Leverage Cycle”, Cowles Founda-
 Most features, such as momentum traders comparing the
tion Discussioon Paper No. 1715. (2009).
market return to savings returns, can easily be modi?ed
 A. Fostel, J. Geanakoplos, “Leverage Cycles and the
to understand complete e?ects of a behavior, with and
Anxious Economy”, American Economic Review, 98,
with-out speci?c sub-behaviors
 Success could be high returns, low volatility, or high
Sharpe ratio, etc. This is for future work.
 Think of this as an FDIC insurance. We do not allow the
 Technically, all deposits are liabilities. Cash is frequently
bank to entirely fail
apt to be called away. Thus, the reserve requirement can
be met by keeping cash on hand or by short term bor-