Limits of Arbitrage: Understanding How Hedge Funds Fail
Mila Getmansky’ and Andrew W. Lo”
March 2, 2005
Abstract
Even if arbitrage opportunities are found in a statistical sense, they might not be
exploitable due to unexpected widening of spreads. This paper models sucha
case in the framework of a hedge fund. Specifically, Long Term Capital
Management is presented as a case study. In particular, we calculate the
likelihood of hedge fund failure and survival given different statistical arbitrage
opportunities and hedge fund risk management decisions. Dynamic relationships
between a hedge fund, dealer, and market (investor) are modeled. The model
explores phenomenon when a fund manager who engages in arbitrage and uses
high leverage might lose all his money before realizing positions at a profit. As
assets go down in value, the firm has to post more collateral or decrease position
exposure. We observe if positions converge before all collateral has been
exhausted, the most profitable strategy is to post more collateral and increase
position exposure. However, if positions diverge beyond the point of remaining
cash, a hedge fund will avoid collapse if it decreases position exposure.
However, we find that a large and visible hedge fund like LTCM that affects asset
prices as it attempts to unwind its positions is not going to escape the collapse by
decreasing position exposure because the effect of its sales will drive the stock
price further down. We propose that given positions are well diversified and not
closely correlated, leverage by itself does not lead to the collapse of a fund.
Correlated positions in the absence of leverage might lead to a loss, but are not
subject to collateral collapse. However, the superimposition of both leverage and
induced high correlation between assets can lead to a collapse. The paper
explores these “flight to quality” and “collateral collapse” dynamics in depth.
‘ Isenberg School of Management, University of Massachusetts, 121 Presidents Drive, Room 308C,
Amherst, MA 01003, (413) 577-3308 (voice), (413) 545-3858 (fax), msherman@ som.umass.edu (email).
? MIT Sloan School of Management, 50 Memorial Drive, E52-432, Cambridge, MA 02142-1347, (617)
253-0920 (voice), (617) 258-5727 (fax), alo@ mit.edu (email).
Table of Contents
1. Introduction
2. Hedge Fund Overview
3. Long Term Capital Management Hedge Fund
4, Brokerage Accounts
5. Regulations - Margin Requirements
6. Broker-Dealer
7. Model Conceptualization
8. Part 1 - Two Agents
Dynamic Hypotheses
Assumptions
Formulations
Results
Necessary Condition for a Hedge Fund to Fail
9. Part 1 - Three Agents
Dynamic Hypotheses
Assumptions
Formulations
Results
10. Implications for Risk Management
11. Discussion and Conclusion
12. References
13. Appendix
101
102
105
106
1. Introduction
This paper presents framework for modeling limits of arbitrage using system dynamics
methodology. Limits of arbitrage are well researched and categorized in finance
literature. However, most of approaches include econometrics, linear extrapolations or
dynamic programming. Although feedback is known to play a critical role in nonlinear
systems such as open market trading, this analysis has not been introduced in previous
works. This paper is one in a series that tries to explain limits of arbitrage using system
dynamics method. In particular, the paper explores how hedge funds fail given arbitrage
opportunities. The collapse of the Long Term Capital Management is used as a case
study.
Limits of Arbitrage
One of the most fundamental notions in finance is arbitrage. Arbitrage is defined as “the
simultaneous purchase and sale of the same, or essentially similar, securities in two
different markets for advantageously different prices” (Sharpe and Alexander, 1990).
This kind of arbitrage requires no capital and does not have any risk in the limit that the
two securities are identical. Given that L and S are identical securities in different
markets trading at different prices, the arbitrageur will be certain to make a profit, and his
future cash flows will be zero. However, according to the efficient market hypothesis
(Samuelson, 1965) and the Law of One Price, the profitable arbitrage cannot exist. The
Law of One Price asserts that any two assets (or positions formed from traded assets)
with the same payoff must have the same value or price. The premise of the efficient
market hypothesis is that stock prices are always “right”; therefore, no one can predict the
market's future direction, which, in turn, must be “random.” For this to hold, prices have
to be set by rational and well informed investors. The hypothesis was developed by
Samuelson (1965) and Harry Roberts and expanded by Eugene Fama and Merton Miller.
As long as there is one investor in the market for whom more is better, such an
investor would take advantage of any arbitrage and scale it up arbitrarily. Such behavior
is inconsistent with economic equilibrium. This conclusion is formalized in the No
Arbitrage Theorem.
A more common strategy implemented by hedge funds is the statistical arbitrage
orrisk arbitrage. In this case, an arbitrageur has a fundamental price in mind. He finds
two securities that have similar payoffs and act essentially the same except that at some
time, one security is overpriced and another is underpriced. The hedge fund manager will
buy an underpriced security and short sell the overpriced one, making the profit, and
hoping that the prices will converge in the future. To borrow a security, a hedge fund has
to post a collateral, long security or cash, for example. However, if positions diverge
before converging, the hedge fund is faced with a margin call. A hedge fund has to either
post more collateral or close the positions. Even though a hedge fund is going to realize
profits once positions converge, in the short run, the hedge fund must carry a loss. A
hedge fund might fail if it does not have enough capital to cover the margin.
While a losing trade may well turn around eventually (assuming, of course, that it
was properly conceived to begin with), the turn could arrive too late to do the trader any
good - meaning, of course, that he might go broke in the interim. John Maynard Keynes
in his famous quote said: “Markets can remain irrational longer than you can remain
solvent.”
Shleifer and Vishny studied the limits of arbitrage and the impact of noise traders
on the arbitrage opportunities (Shleifer and Vishny, 1997). They warned that an arbitrage
firm of Long-Term’s type can collapse if the market is overwhelmed by noise traders
who push prices away from the true value. It might lead to adverse price shock that can
force LTCM to liquidate its positions at low prices.
Leverage
LE, leverage ratio, equals assets, A, divided by equity, E. sa (eq.1)
Retum on equity, R* or retum on capital equals to R® = ao Ey (eq 2)
0
However, return on assets, RA equals to
a_ Av Ay Ly +E, -Ly-Ey _ Ly(l+r)+E,-Ly-E, _ Lor +E, —E,
R* (eq. 3)
Ay E, +L, E,+Ly E, +L,
where L is leverage, and ris the interest rate that has to be paid back on leverage to the
lender bank. For example, if L* = 2 and r=0, then R' = L*R*. A hedge fund will use
leverage in order to increase return on equity.
The Federal Reserve Board, under a statutory provision known as “Regulations
T,” sets a limit on broker loans for stocks, or “margin.” For the past twenty-five years,
the Fed has set the maximum margin loan at 50 percent of the total investment which
translates into the maximum L* = 2.
Leverage is used to buy more securities than available cash allows you to. For
example, say, stock XY Z is going to go up by 20% inayear. A customer puts $100 of
his own money to XYZ. The early return is 20%, so in a year, the customer has $120.
However, let’s say that the customer borrowed another $100 (interest =5 %). Ina year,
the stock XY Z is worth $240. After paying off the broker $105, the customer is left with
$135. Therefore, with leverage, the customer's profit is 35%, which is significantly
better than 20% with margin. However, if a position goes down, than a customer with
margin can lose more money than if he did not use leverage. For example, the stock goes
down by 20% in ayear. Therefore, it would be worth $180 without a margin, or $160
with margin. After paying off $105, the customer is left with $55, which is a 45% loss,
compared to 20% loss in the case a customer does not use leverage.
2. Hedge Fund Overview
The term ‘Hedge Fund’ originated when Alfred Winslow Jones founded a novel
approach to investing in 1949. He discovered an innovative strategy for maximizing
asset retums and minimizing market risk. The strategy was based on “hedging” long
stock positions with short stock positions by using leverage to increase potential of
retums. Jones bought seemingly cheap stocks and sold short overpriced stocks. In
theory, the Jones’s portfolio was “market neutral.” Any market event will increase the
value of one half of his portfolio and depress the second half. His net retum would
depend only on his ability to single out the relative best and worst. In 1966, Carol J.
Loomis’ article in Fortune magazine entitled The J ones Nobody Can Keep Up With,
revealed that by using this double-parameter model, Jones outperformed the highest-
ranking mutual funds of the 1950's and 1960's by over 44%. This breakthrough
technique catalyzed the most lucrative and unregulated financial industry in the history of
economics, a multi-billion-dollar industry consistently attracting smart and wealthy
individuals.
As of October 2003, the size of the global single-manager hedge fund universe
(not including funds of funds) is $650 - $700 billion (Tremont Company). There are
about 5,000 global single-manager hedge funds in the hedge fund universe. There are
about 1,200 - 1,400 funds of funds. There are about 3,000 distinct hedge fund managers
that manage both offshore and domestic accounts. In 1990, there were 610 funds
managing $39 billion. Despite spectacular growth and performance in double digits of
various hedge funds, there have been many horrifying collapses and bankruptcies of
hedge funds such as Granite Capital and LTCM.
Hedge funds differ from mutual funds and other investment vehicles by both
internal structure and investment discipline. Hedge fund managers are not restricted to
any particular type of investments. Hedge funds can buy (long) or sell (short) securities
that they do not own. They are not restricted to common “buy and hold" strategies. Most
U.S. hedge funds are limited partnerships, or limited liability companies, established to
invest in public securities. However, there is no common definition of a hedge fund. U.S.
hedge funds are defined by their freedom from regulatory controls stipulated by the
Investment Company Act of 1940. Before 1996, a hedge fund had a 100 investor limit in
order to qualify as a limited partnership. However, under the National
Securities Markets Improvement Act of 1996, the 100 investor limit was lifted. The
minimum new worth requirement for a qualified investor is $5 million and the minimum
institution capital is $25 million. Companies can also become reporting companies
voluntarily by filing with the SEC. Under the Exchange Act, a company must become a
reporting company if it has at least 500 shareholders and $10 million in assets. The
Exchange Act contains registration and reporting provisions that may apply to hedge
funds.
Depending upon their activities, in addition to complying with the federal
securities laws, hedge funds and their advisers may have to comply with other laws
including the Commodity Exchange Act (“CEA”), rules promulgated by the National
Association of Securities Dealers (“NASD”) and/or provisions of the Employment
Retirement Income Security Act (“ERISA”). In addition, hedge funds may be subject to
certain regulations promulgated by the Department of the Treasury, including rules
relating to the prevention of money laundering. Moreover, hedge fund advisers are
subject to certain state laws.
Offshore hedge funds are typically corporations registered in a tax haven such as
the British Virgin Islands, the Bahamas, Bermuda, the Cayman Islands, Dublin, or
Luxembourg, where tax liabilities to non-U.S. citizens are minimal. In general, the hedge
fund industry is not transparent to regulators unlike the mutual funds industry. Like
mutual funds, hedge funds are actively managed investment portfolios holding positions
in publicly traded securities. However, unlike mutual funds, hedge funds have greater
flexibility in the kind of securities they can invest in. Hedge funds can invest in domestic
and intemational debt and derivative securities. They can take undiversified positions,
sell short, and lever up their portfolios. These alternative investments mainly attract
institutions and wealthy individuals with minimum investments typically in the range of
$250,000 - $1 million. Hedge funds are also characterized by a substantial managerial
investment and strong managerial incentives. On average, hedge fund managers receive
a1% annual management fee and 20% of the annual profits. Most of funds employ a
bonus incentive fee: managers are paid a percentage of the excess of a fund's retum over
some level, commonly called a “high-water mark.” If a hedge fund incurred losses in the
past, its managers can be paid in present period only if return in this period exceeds the
“high-water mark” plus past losses.
Hedge funds seek to generate above-average retums to their investors. For many
investors, hedge funds act as risk managers since their returns are often not correlated
with equities or fixed-income securities. Most hedge funds use the following strategies:
e Short selling. The strategy involves the sales of borrowed securities hoping the price
of these securities will go down. A hedge fund manager should have sufficient skills
and expertise to identify overvalued securities and being able to cost-efficiently
borrow the overpriced stocks.
e Hedging. The strategy involves decreasing risk inherent in hedge fund's portfolio.
The risks might be the following: political, economic, company, interest rate and
market. Hedging can use the combination of derivatives and short sales. Hedge fund
managers should he able to use efficient hedging techniques. For example, it is very
costly and not efficient to hedge by shorting a share of a stock for every share held
long in the portfolio. It might be more economical to short contracts or shares of
different assets which are highly correlated with the underlying asset.
Arbitrage. The strategy involves finding any price inefficiencies or discrepancies
between securities or markets. The strategy is risk-free; however, in current efficient
markets it is very hard to find any price inefficiencies. Even if such inefficiencies are
found, they do not last. Therefore, fund managers tend to use leverage in order to
enhance returns due to such minuscule short-term opportunities.
Leveraging. The strategy involves either borrowing money, to increase the size of the
portfolio; or assigning cash or securities as down payment, collateral, or margin fora
percentage of the position one seeks to establish.
Synthetic positions or derivatives. The strategy involves using derivative contracts to
establish certain positions or strategies in the hedge fund.
There are many hedge fund types. The list of hedge fund types is the following:
Macro funds
Special-situation funds
Pure equity funds
Convertible arbitrage funds
Funds of funds
Market-neutral funds
Commodity trading advisor funds
Private equity funds
Risk arbitrage funds
Long or short funds
Emerging market funds
Event risk funds
Restructured or defaulted security funds
The recent hedge fund collapses and developments in hedge fund industry make
SEC anxious. At the end of July 2002, 55% of hedge funds in the TASS database were
down in asset values for the year. Because of high-water marks, the need to recoup
losses before taking incentive fees on gains, it will be difficult for many hedge funds to
obtain a profit soon. That increases the probability of default for many hedge funds. The
average hedge fund advisor is 35 years old, very young. The average age for a hedge
fund (not included funds of funds) is 46 months with a median of 35 months (Getmansky,
2003). Also, the recent “retailization” of the industry - the introduction of products that
make hedge funds available to investors with as little as $25,000 to invest, makes SEC
worried.
3. Long Term Capital Management Hedge Fund
Long Term Capital Management (LTCM) was started in February, 1994 by the
infamous Salomon Brother ’s arbitrage trader John Meriwether. The beginning of LTCM
was very rocky, having trouble gathering enough investors to trust John Meriwether.
After hard work from its prime broker, Merrill Lynch and its many talented partners, who
included Nobel prize winners Myron Scholes and Robert Merton, LTCM eventually
raised 1.25 billion dollars of assets to launch the hedge fund.
The structure of LTCM was drastically different from other hedge funds. For
example, investment fee paid to the partners was 25% instead of the usual 20%, and
yearly management fee was 2% instead of the usual annual 1%. LTCM also required
investors to invest at least $3 million. Investors were forced to sign a contract of holding
their investments for at least three years. LTCM was also extremely secretive. LTCM
had about 100 investors and 200 employees.
LTCM’s financial strategy concentrated on “relative value” trades in bond
markets. Long-Term would buy underpriced bonds and sell overpriced ones. It would
bet on spreads between pairs of bonds to either converge or diverge. For example, they
bought underpriced off-the-run US treasury bonds (because they are less liquid) and
shorted on-the-run (more liquid) treasuries, betting on the convergence of the two assets.
The government has the same likelihood of paying off off-the-run and on-the-run bonds.
The net risk was minimal because long and short positions were highly correlated. Bonds
usually rise and fall in sync; therefore, spreads don’t move as much as the bonds
themselves.
Another trade example is the following: If interest rates in Italy were
significantly higher than in Germany, making Italian bonds cheaper than German ones,
the hedge fund would invest in Italy and short Germany. The fund would profit if this
differential narrowed. Since most of the spreads discovered by LTCM were very small,
LTCM had to have huge leverage in order to make significant profits. The leverage rate
was about 20 to 30 times the investment. The Federal Reserve Board, under a statutory
provision known as “Regulation T,” sets a limit on broker loans for stocks, or “margin.”
For the past twenty-five years, The Fed has set the maximum margin loan at 50 percent
of the total investment. When LTCM purchased stocks, it was subject to Reg T.
However, the fund rarely purchased stock outright; instead, it entered into derivative
contracts such as swaps, that mimicked the behavior of stocks. LTCM also used highly
complicated mathematic models to achieve elevated returns and control risk. They
utilized swaps options and other derivatives to control their trades.
The firm earned 20% net of fees in 1994. In 1995 it earned 43%, in 1996 - 41%,
and in 1997 - 25% net of fees retum on equity. Including the money from new investors,
the company’s equity capital had, in less than two years, tripled, to a total of $3.6 billion.
The assets also grew to $102 billion. Thus, at the end of 1995, it was leveraged 28 to 1.
Leverage did not include derivatives. The return on total capital was approximately
2.45%. By the spring of 1996, the Long-Term grew to $140 billion in assets. By 1997, it
had more than $5 billion in equity. By 1998, the worst month was the loss of 2.9%.
According to their models, the maximum that they could lose on any single day was $45
million.
By borrowing or selling bonds that were in high demand with a smaller interest
rate and by purchasing bonds that were slightly less in demand and that therefore yielded
a little bit higher interest rate, LTCM was in effect a liquidity provider to capital markets.
Asa bank which earns money on a spread by charging borrowers a slightly higher
interest rate than it paid to depositors, the hedge fund was eaming profit on the spread
between the two assets. LTCM in effect was buying assets that everybody wanted to
10
sell. Therefore, those assets were not totally independent. In case of a mass selling
panic, the fund could default if everybody wanted to sell and nobody wanted to buy.
LTCM also had several brokers lending money to the fund. Brokers involved
were Bear Steams, Goldman Sacks, Morgan Stanley, JP Morgan, Lehman Brothers,
Chase Manhattan, Banker's Trust, Union Bank of Switzerland, UBS Warburg and
Salomon Smith Bamey. Long-Term would place orders of each leg of a trade with a
different broker, so nobody could see the whole trade. LTCM could get rid of the haircut
fee required to be paid to brokers for borrowing money. All of its brokers complied with
the LTCM’s strict requirements, allowing the fund to be the most unregulated hedge fund
during that time.
LTCM disclosed its total assets and liabilities to its banks each quarter and to
investors each month. It also reported those numbers to the Commodity Futures Trading
Commission. It reported its derivative totals only annually. People were aware of high
leverage and exposure; however, nobody thought that it might lead to LTCM failure.
However, LTCM did not disclose details of assets. Banks only knew their own exposure
to Long-Term, but not exposures of others. About 55 banks were doing financing for
LTCM.
The failure of LTCM came on as a thundering shock to the financial world.
When the Russian government defaulted on its debts in August 17, 1998, liquidity
suddenly evaporated from international financial markets. Instead of converging,
LTCM’s positions began to diverge. The partnership knew perfectly well that over the
short term, prices might diverge. But they always calculated the risks and the
consequences of divergence with special statistical “value-at-risk” models. In August
1998, asset prices plummeted. LTCM lost lots of money because it could not liquidate its
assets before the value of its portfolio dropped. LTCM was a victim of “flight to
liquidity.” People wanted to buy less risky Treasuries and get rid of risky bonds. People
were afraid of going short on Treasuries. Only LTCM held short positions on Treasuries
and long positions in riskier bonds. And as Treasuries rallied, spreads between them and
other bonds widened. Mortgage-backed securities jumped from 96 basis points over
Treasurys to 113 points. Corporate bonds rose from 99 to 105, and junk bonds rose from
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224 to 266. Even seemingly safe off-the-run Treasurys climbed from 6 points over to 8
points over. In every market, the spreads widened leading to LTCM losing money.
In June, the fund lost 10%. Ona single day, August 21, the LTCM portfolio lost
$553 million - 15% of its capital. It had started the year with $4.67 billion. Suddenly, it
was down to $2.9 billion. On September 2, 1998 Meriwether sent a letter to his investors
saying that the fund had lost $2.5 billion or 52% of its value that year, $2.1 billion in
August alone. LTCM capital base had shrunk to $2.3 billion. The fund had $125 billion
in assets - 98 % of its prior total and the leverage increased to 55:1 due to the now-
shrunken equity - in addition to the massive leverage in its derivative bets, such as equity
volatility and swap spreads. At that point, leverage was very high, and the fund’s
partners were looking forward to sell some positions and raise more money before the
end of the month. LTCM had a difficulty of reducing its positions with the markets
under the stress. There was no liquidity in the market. Everybody wanted to be out at the
same time - something that models missed. When losses mount, leveraged investors
such as Long-Term are forced to sell, lest their losses overwhelm them. When a firm has
to sell without buyers, prices are very high. In addition, Wall Street players learned more
about the fund’s positions, and went against them. They wanted to “squeeze” as much as
possible from the fund, knowing that if the fund gets help from the government, it would
be able to buy back its shorts. Therefore, anybody who held those securities would make
money.
In September 1998, many banks were exposed to the same positions as LTCM.
Therefore, to cut their losses, they unwound those positions, thus, hurting LTCM.
Therefore, both cutting the losses and predatory trading led to the collapse of the fund.
Also, Long-Term trades were in highly specialized instruments, such as equity volatility.
Only a handful of banks traded them. LTCM was short on the equity volatility, and
sooner or later they would have to buy. The dealers refused to sell, only at very high
prices.
On Thursday, September 10, the firm had lost $145 million; on Friday, $120
million. The next week on Monday it lost $55 million: on Tuesday, $87 million, and on
Wednesday $122 million. LTCM was down to $1.5 billion. Due to the excess leverage
of LTCM, the potential failure of the hedge fund triggered the attention of the Fed. On
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September 20", 1998, the fed representatives visited the office of LTCM in Greenwich,
CN. They were amazed to find that LTCM’s on balance sheet assets totaled around $125
billion, on a capital base of $4 billion, a leverage of about 30 times. But that leverage was
increased tenfold by LTCM's off balance sheet business whose notional principal ran to
around $1 trillion. On September 21, 1998, LTCM had its second biggest loss of $500
million. At that point, the assets were worth $100 billion. Thus, even omitting
derivatives, its leverage was greater than 100 to 1. Now, if LTCM lost 1%, it would be
wiped out. LTCM exposed its books to Peter Fisher of New Y ork Fed. He saw that in all
markets LTCM was badly hurt. All its positions became perfectly correlated in the crisis
period. Fisher was not worried that the markets would go down; he was afraid that they
would not trade at all. Bankruptcy was out of the question because bankruptcy filing
would make all counterparties go after the collateral further depressing the value of the
collateral. Also, nobody wanted to buy the firm and obtain assets such as equity
volatility or sophisticated derivatives. If one bank bought the firm, then it would be in
the same position as LTCM, and given that by now positions of LTCM were exposed,
other banks would try to trade against it. Therefore, the only solution was for all banks to
work together.
The Fed convinced all the LTCM’s major brokers to bail out the fund’s losses,
believing that if LTCM was allowed to fail, the world financial market would be at risk.
If Long-Term defaulted, all of the banks that lent to LTCM would be left holding one
side of a contract for which the other side no longer existed. Undoubtedly, there would
be a frenzy as every bank rushed to escape its now one-sided obligations and tried to sell
its collateral from Long-Term. LTCM had lots of derivatives which were relatively new.
Officials were afraid that the financial system could crash. The consortium of 14 banks
got $3.65 billion in exchange of 90% of the equity in the fund. The LTCM’s existing
investors would retain the rest 10%. On July 6, 1999, LTCM repaid $300 million to its
original investors. It also paid out $3.65 billion to the 14 consortium members. LTCM
met all margin calls. All of its debts to creditors were repaid in full. Through April
1998, the value of a dollar invested in Long-Term quadrupled to $4.11. By the time of
the bailout, only five months later, 33 cents were remained. After fees, each invested
dollar has grown to $2.85 and then shrank to 23 cents. In net terms, LTCM lost 77%.
13
There are many speculations of major reasons why the hedge fund failed. Many
believed that it wasn’t going to fail at all. In fact, the position taken by LTCM was
simply going to take time to recover and eventually make a profit for the firm. There are
other reasons for the collapse besides LTCM’s strategy. First, the “value-at-risk” model
used by LTCM did not anticipate the “flight to liquidity” taken place in A ugust and
September of 1998. Second, there were other hedge funds and major investment banks
that mimicked strategy used by LTCM in convergence arbitrage. Third, LTCM partners
lost faith in the strategy and started closing positions using the firm’s assets. Fearing the
failure, they made it inevitable by draining the firm of its remaining capital. Fourth,
LTCM had about 8% of its book exposure to Russia, which could come to about $10
billion exposure. Fifth, LTCM took speculative positions in takeover stocks, such as
Tellabs whose share price fell over 40% when it failed to take over Ciena. Sixth, LTCM
was exposed to mortgage-backed securities, which experienced a downturn in 1998.
4. Brokerage Accounts
Brokerage houses offer clients numerous types of accounts. The most common
ones are cash and margin accounts. These accounts represent different levels of credit
and trustworthiness of the account holder as evaluated by brokerage houses.
A cash account is generally called “Type 1” account. A customer who has a cash
account can make trades in that account, but he has to pay in full for all purchases by the
settlement date. Again, different brokerage houses have different rules depending on the
relationships with an entity. Several brokerage houses would allow a customer to execute
buy or sell orders before cash is deposited in the account. The requirements to open a
cash account are very minimal. However, several brokerage houses may require a
significant deposit, of as much as $10,000, before customers can open the account.
A margin account is a type of brokerage accounts that allows customers to borrow
money against securities they own. This account is sometimes called a “Type 2” account.
In order to obtain this account, an entity must pass a security and background check.
Short sales also occur in a margin account. Having a margin account makes it possible to
take a margin load. A customer can buy securities or short sell securities on margin, or
he can extract cash from an equity position without having to sell it (thus avoiding the
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taxes on selling positions or the chance of missing a run-up). In order to sell-short a
stock, a broker needs to lend the stock to sell. The broker goes to another client’s
account or to his own account and borrows those shares to lend it to the customer for the
short-sale. Interestingly, when a customer borrows money from the brokerage firm, the
customer has to pay a fee to the dealer. However, when a broker lends a security for
short-sell, he is not going to pay any interest on the proceeds from the short. There are
exceptions to this rule: really big funds can negotiate a full or partial payment of interest
on short sales funds provided there is sufficient collateral and the dealer does not want to
lose the client.
5. Regulations - Margin Requirements
The basic rules for margin requirements are set by the Federal Reserve Board, the
New Y ork Stock Exchange, and the National Association of Securities Dealers. Every
broker must apply the minimum rules to customers, but a broker is free to apply more
stringent requirements. The amount an entity can borrow from a broker is closely
regulated. The Federal Reserve Board’s Regulation T states how much money an entity
can borrow to establish a new position. The NY SE’s Rule 431 and the NASD’s Rule
2520 both state how much money an entity can continue to borrow to hold an open
position. Federal requirement is 50% for long positions and 150% for short positions.
Note, that the first 100% of the short sale can be satisfied by the proceeds from the short
sale, leaving just 50% for the customer to maintain in margin. In the end, it looks similar
to maintaining along position. For example, $100,000 of cash can be used to buy
$200,000 worth of stock. Maintenance margins can change based on brokerage houses
and client relationships.
Here are few examples how to account for margin requirements. For simplicity,
we assume a conservative estimate that the house margin requirement equals the
Regulation T margin requirement of 50%.
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Accounting for Cash, Long and Short Positions.
Case 1.
A hedge funds has $50 in cash and $100 in long positions. Therefore, Equity = $50 +
$100=$150.
Assets Liability
$50 - Cash $0
$100 - Long position
Equity
$150
Table 1_1: Balance Sheet for Case 1
Case 2.
A hedge fund has $50 in cash and $100 in long positions. Given 50% margin
requirements, a hedge fund can at most carry ($50+$100)*2=$300 positions. It already
has $100 long positions; therefore, can carry $200 short positions. In order to carry $200
short and $100 long positions, the equity in the account should be at least
($200+$100)/2=$150. The hedge fund has exactly $150 in equity.
Assets Liability
$50 - Cash $200 - Short position
$100 - Long position
$200 - Proceeds from Short position Equity
$150
Table 1_2: Balance Sheet for Case 2
For a more detailed picture, in the margin account, the value of the long position
is $100. Debit =$50. Therefore, equity = market value of the long position - debit =
$50. Therefore, the maintenance margin = equity/value of the long position = $50/$100
= 0.5 is within the allowed limit and no margin calls are issued. In the short account, the
value of the short position = $200. Credit = proceeds from the short position + credit
from the margin account + cash = $200+$50+$50=$300. Equity =credit - market value
16
of the short position = $300-$200 = $100. The maintenance margin = equity/value of the
short position = $100/$200 = 0.5, which is within the allowed limit and no margin calls
are issued.
Case 3.
Given case 2, the short position went up to $220. The new balance sheet looks as
following:
Assets Liability
$50 - Cash $220 - Short position
$100 - Long position
$200 - Proceeds from Short position Equity
$130
Table 1_3: Balance Sheet for Case 3
The equity in the account is $100+$50-$20=$130. However, equity needed is
($220+$100)/2=$160. Therefore, extra margin needed is $30. The hedge fund has a
choice: either to come up with more cash- at least $30, or sell $60 worth of long
positions.
Analyzing more in detail, the equity in the margin account = value of the long
position - debit = $100 - $50 = $50. Therefore, maintenance margin = $50/$100 =0.5.
The maintenance requirement is met. In the short account, the value of the short position
= $220. Credit = proceeds from the short position + debit from the margin account +
cash = $200 + $50 + $50 = $300. Therefore, equity in the short account = credit - value
of the short account = $300-$220 = $80. Note, that the sum of equity in margin and short
accounts equals to the total equity of $130 that is obtained earlier. Maintenance margin
for the short account = equity in the short account/market value of the short stock =
$80/$220 = 0.36, which is less than 0.5, the required margin. The margin call equals to
Market value of the short * ( 1+ Maintenance margin) - credit = $220*1.5-$300 = $30.
As obtained above, the extra margin needed is $30. The hedge fund has a choice: either
to come up with more cash- at least $30, or sell $60 worth of long positions.
17
Generally, given changes in margin and short accounts, margin call equals to:
Margin Call = $/°P,° * (1+m)-$/""P,"(1—m) + Debit, — Credit, (eq. 4)
where m is the required maintenance margin.
Case 4.
Given case 3, if a hedge fund decides to come up with $30 of cash to cover margin, then
new equity will be $160. New equity = $50+$30+$100-$20=$160. This is exactly what
is needed. The new balance sheet looks as following:
Assets Liability
$80 - Cash $220 - Short position
$100 - Long position
$200 - Proceeds from Short position Equity
$160
Table 1_4: Balance Sheet for Case 4
Case 5.
Given case 3, if a hedge fund decides to sell $60 worth of long security to cover margin,
then new equity will be $130. New equity = $50+$60-$20+$40=$130. New equity
needed = ($40+$220)/2=$130. Therefore, extra margin is 0. The balance sheet looks as
following:
Assets Liability
$110 - Cash $220 - Short position
$40 - Long position
$200 - Proceeds from Short position Equity
$130
Table 1_5: Balance Sheet for Case 5
18
Case 6.
Given case 2, now the short position becomes worth $180 instead of $200. The balance
sheet looks as follows:
Assets Liability
$50 - Cash $180 - Short position
$100 - Long position
$200 - Proceeds from Short position Equity
$170
Table 1_6: Balance Sheet for Case 6
Equity is worth $170=$50+$100+$20. However, equity needed = ($180+$100)/2=$140.
Therefore, $170-$140=$30 is an excess equity.
Case 7.
Given case 6, there is $30 excess equity. A hedge fund can decide to free up $30 worth
of cash and use this cash to borrow $30*2=$60 worth more short security. The balance
sheet will look as follows:
Assets Liability
$50 - Cash $240 - Short position
$100 - Long position
$260 - Proceeds from Short position Equity
$170
Equity needed is ($100+$240)/2=$170. Current equity = $50+$100+$20=$170.
Table 1_7: Balance Sheet for Case 7
19
Case 8.
Given case 6, there is $30 excess equity. A hedge fund can decide to free up $60 worth
of long positions, and use it to borrow more money = $60. The balance sheet will look as
follows:
Assets Liability
$50 - Cash $240 - Short position
$100 - Long position
$260 - Proceeds from Short position Equity
$170
Table 1_8: Balance Sheet for Case 8
Equity needed is ($100+$240)/2=$170. Current equity = $50+$100+$20=$170.
Case 9.
Given case 1, say both long and short positions change. The value of the long position
went down to $80 from $100, and the value of the short position went up to $220.
Assets Liability
$50 - Cash $220 - Short position
$80 - Long position
$200 - Proceeds from Short position Equity
$110
Table 1_9: Balance Sheet for Case 9
Current equity = $50+$80+$200-$220 = $110. Equity needed is ($80+$220)/2 = $150.
Therefore, margin call is $40.
To look in detail in margin and short accounts, the value of the long position in
the margin account equals to $80. Debit = $50. Therefore, equity = $30. Margin =
$30/$80 = 0.38, which is less than the required 0.5. Therefore, margin needed = value of
20
the long position * margin - equity = $80*0.5-$30 = $10. In the short account, the value
of the short position is $220. Credit = proceeds from the short position + debit from the
margin account + cash = $200+ $50 + $50 =$300. Therefore, the equity in the short
account = credit - value of the short account = $300-$220 = $80. Note, that the sum of
equity in margin and short accounts equals to the total equity of $110 that is obtained
earlier. Maintenance margin for the short account = equity in the short account/market
value of the short stock = $80/$220 = 0.36, which is less than 0.5, the required margin.
Margin call equals to Market value of the short * ( 1+ Maintenance margin) - credit =
$220*1.5-$300 = $30. As obtained above, extra margin needed is $40 = $10 (from
margin account) + $30 (from short account). The hedge fund has a choice: either to
come up with more cash- at least $40, or sell $80 worth of long positions.
6. Broker-Dealer
Broker-dealers (dealers) are intermediaries between a client and a bank. They
purchase, sell, and short-sell securities for a client, borrow securities from one client or
form their own inventory and lend them to another client for short-selling. Dealers
collect fees on all transactions and the amount borrowed. Dealers are responsible to
monitor trading and credit risks. They are responsible to do background checks on their
counterparties and establish credit limits. They are responsible for monitoring collateral
which they pledge with a bank (lender). The dealer’s balance sheet looks the following
with respect to short-sells of a hedge fund. Note, we are omitting assets and liabilities
from other accounts.
Assets Liability
Cash Proceeds from Short position
$100 $100
Table 2: Balance Sheet for a Dealer
21
Proceeds from Short position is the credit balance payable. For example, a hedge fund
short sells $100 worth of securities. The $100 proceeds from short position are recorded
on the Assets side of the hedge-fund balance sheet, to be given to a hedge fund once it
decides to buy back the short positions. Therefore, the Proceeds from Short position are
recorded on the Liability side of the Dealer’s balance sheet. This $100 cash is deposited
in a bank under the dealer’s name.
7. Model C onceptualization
Model Purpose
Even if arbitrage opportunities are found in a statistical sense, they might not be
exploitable. Moreover, a fund manager who engages in such arbitrage might lose all his
money before realizing the positions at a profit. For example, lots of hedge funds find
arbitrage opportunities that are usually very miniscule considering almost efficient
markets, and leverage up the positions in order to make high profit margins. As assets go
down in value, the firm has to post more collateral or unwind positions. If itis
unavailable, this often leads to a hedge fund collapse.
However, given that positions are well diversified and not closely correlated,
leverage by itself, does not lead to the collapse of afund. Correlated positions in the
absence of leverage might lead to a loss, but are not subject to collateral collapse. Given
diversified positions in a fund, a price drop in one asset does not necessarily correspond
to a price drop in another asset, even less likely there is a possibility of a cascade in drop
in prices of all assets. However, the superimposition of both leverage and induced high
correlation between assets can lead to a collapse. This is something that sophisticated
hedge funds like LTCM did not take into equation in determining risk exposure. Their
decisions were bounded rational. The managers separately managed leverage and
diversification of positions, not thinking that two can feed on each other during a period
of a crisis.
Unlike other financial institutions such as mutual funds and banks, hedge fund
can get exposed to various kinds of assets and borrow on margin. Therefore, the
dynamics of “flight to liquidity” and “collateral collapse” can be best studied in the
framework of a hedge fund. Even if a hedge fund has great positions that guarantee a
22
statistical arbitrage, the hedge fund might collapse before these positions converge and
make a profit.
Model Boundary
The model has a hedge fund, a dealer, and market (investors). It has both
financing functions as well as psychological ones such as “flight to quality” and “flight to
liquidity” feedbacks. Balance sheets of a hedge fund and a dealer are modeled as well as
decisions of a hedge fund on taking leverage and how to deal with a margin call. In the
model, a hedge fund can either decide to post more collateral from available cash or close
out positions once a dealer imposes a margin call. The dealer is modeled. During the
“liquidity crunch,” a dealer is risk-averse and is not willing to hold a lot of inventory of
illiquid asset. Investors are modeled. Investors can decide to hold cash, liquid and
illiquid assets. Different types of investors are modeled: momentum, imitators and
noise. Price is endogenously determined in the model based on demand supply balance.
Time Horizon
The time horizon is 100 days, or over 3 1/3 years for the model. I am using the
data for Long Term Capital Management case that has data for four years, from inception
of the fund from June, 1994 to its collapse, September, 1998. “Liquidity crunch” is
modeled over 10 days, which is similar to what is found in the LTCM case.
23
Reference Mode
Price Price
Ps Ps
PL PL
t b t t bh
Time Time
Figure 1. Successful Arbitrage Figure 2. Collapse Before Profits Are Realized
A hedge fund will engage in the arbitrage position at time t. The profit will be realized
at time t, when prices of L and S assets converge. However, it is possible that before
converging prices diverge at time t,. Given hedge fund’ exposure, leverage decisions and
hedge fund manager's decision how to deal with margin calls, a hedge fund can either fail
or realize long-term profits while carrying short-term losses. The goal of the paper is to
understand which decisions and effects lead to the collapse of a hedge fund, and whether
itis possible to prevent the collapse. The collapse is measured by the negative total
equity of a hedge fund.
8. Part 1 - Two Agents
Dynamic Hypotheses
“If you aren’t in debt, you can’t go broke and can’t be made to sell, in which case
“liquidity” is irrelevant. But a leveraged firm may be forced to sell, lest fast
accumulating losses put it out of business. Leverage always gives rise to this same brutal
dynamic, and its dangers cannot be stressed too often.” (Lowenstein, 2000).
Let’s consider the following financial entities: a hedge fund, a dealer, and a bank.
A hedge fund is interested in obtaining leverage. It goes to a dealer, and can borrow
money from a dealer at the maximum 50% margin (value of leverage divided by the total
value of positions). Therefore, for example, if a hedge fund has $10 million worth of
security L, it can borrow another $10 million worth of security S from a broker and sell it
24
to the market. The dealer earns the transaction fee as well as charges the hedge fund for
the loan. Now, the dealer delivers $10 million of security L to the bank and sells $10
million of security S to the market. If the value of security S goes up in value, the hedge
fund has to put more collateral or sell security L and vice versa. A dealer is in the
business of extending credit. The dealer will require more collateral if a value of a long
position goes down or the value of the short position goes up, even if the trade might be
profitable in the future. Say, the value of security S goes up by 20% from $100 to $120.
The value of security L is maintained at $100. Therefore, to maintain this position, the
hedge fund should have equity of $110, and it only has $80. Therefore, a hedge fund has
to put an additional $30 worth of collateral.
Let, L be the value of leverage, A is the total value under management of a hedge
fund, its assets. E is the equity of a hedge fund. Therefore,
A=E+L (eq. 5)
L/A should be at most 50%. In this case, E is the same as collateral. So, if A decreases
to Ai, then the new collateral to be posted is L/0.5- Ai.
The collateral to be posted is max(0, L/0.5- A;) (eq. 6)
Dealers do not win if the value of positions hedge fund is holding goes up;
however, they lose if the value of positions goes down. They might end up responsible
for the value to be paid back to the banks. A broker dealer makes money by providing
credit. He does not want to lose money. The stock and flow diagram of the interactions
between a hedge fund, a dealer, a commercial bank, and a market is shown in Figure 3.
25
Market
7 Money for
Securities Securities
Purchased
Hedge Dealer
Fund Collateral Securities Lent
Interest Paid on Money Borrowed
Money
Borrowed from
the Dealer
Interest Paid on Money
Borrowed from the Bank
Figure 3. Hedge fund, Dealer, Commercial Bank, and Market Interactions
If a dealer calls a hedge fund with a margin call, a hedge fund can either sell the assets,
thus reducing the margin, as depicted by the balancing loop B1 Cover Margin By Closing
Positions, or by using additional cash (proceeds from other trades) to cover the margin, as
depicted by the balancing loop B2 Cover Margin By Available Cash in Figure 4.
26
‘Shares L. .
5 a ena Maximum
Leverage Ratio
» Price L a
a
Value £ + ‘Shares $ +
+
. B1 Cover Value S
Sell Positions Margin By rs +
“ Closing + Rauity £
Positions Equity Needed
Available Price S
+
Extra Margin
Needed
Available
Cash re +
Gap Between Cas Ba Cover Pressure to Cover
Needed And Cash’ the Margin
Available GSD
Cash Needed to
Cover Margin _+
Figure 4. Hedge Fund Decisions How to Deal With Margin
27
Hedge Fund Strategy
A hedge fund maximizes its profits from the statistical arbitrage strategy:
Max(-S/75P° -S?*P")
subject to:
1) The prices P,S and P," will converge at time
2) Cash >0
3) Meeting Margin Calls
Assumptions
1;
A hedge fund's strategy is a statistical arbitrage. Therefore, a hedge fund tries to
buy an undervalued security (long) and short sell an overvalued security (short).
The hedge fund is trying to have the same amount of shares of both long and short
securities.
A typical hedge fund does not typically have cash and will invest all its available
cash into long securities. It will use long securities as a collateral to borrow short
securities.
A dealer has unlimited inventory and will always take the other side of the hedge
fund order.
There is no price impact - selling and buying a security does not change the price
of the security.
A hedge fund starts with no exposure to the long and short positions.
When arbitrage opportunities go away, a hedge fund liquidates both short and
long positions. Therefore, in this model, both position forming and unwinding are
modeled.
7. Federal and house requirements are 50%.
No minimum dollar requirements under regulation T, house and NY SE
requirements.
No interest charge in a margin account, and no interest credit in a short account.
28
Variable Value Units Description
u™* 2 Dmnl Maximum Allowable Leverage
m 0.5 Dmnl House Margin Requirement
m 05 Dmnl Regulation T Margin
Requirement
a 0.8 Dmnl Fraction of Free Cash Invested
Cash;! 40,000 $ Initial Hedge Fund Cash
BES 100 $/Share Initial Price of Security S is
Fundamental Price of Security S
prt 100 $/Share Initial Price of Security L is
Fundamental Price of Security
L
See 0 Share Initial Number of Shares of
Security L a Hedge Fund Has
srs 0 Share Initial Number of Shares of
Security S a Hedge Fund Has
spe 400 Share Initial Number of Shares of
Security L a Dealer Has
gps 400 Share Initial Number of Shares of
Security S a Dealer Has
Su 400 Share Initial Number of Shares of
Security L an Investor Has
gis 400 Share Initial Number of Shares of
Security S an Investor Has
Cash} 40,000 $ Initial Cash an Investor Has
wis 0.5 Dmnl Actual Equity Weight of
Security S by Investor
WwW A L 0.5 Dmnl Actual Equity Weight of
Security L by Investor
Table 3. Assumptions and Initial Conditions
29
Formulations
Hedge Fund Cash
Cash, =] (Income from Other Investments; + Cash Increase; - Cash Decrease;") dt
(eq. 7)
Cash Increase," =Sell Rate; * P,' +Fraction Reinvested* Sell Rate,S* PS (eq. 8)
Cash Decrease;"=Buy Rates * P' +Buy Rates?S *( PS- PS) (eq. 9)
Total Cost Basis
Total Cost Basis?" = j (Increase in Total Value;""- Decrease in Total Value? )dt
(eq. 10)
Increase in Total Value?" = Buy Rate! * BY (eq. 11)
Decrease in Total Value;#* Sell Rate;*“* P,- (eq. 12)
«HLL
_ Total Cost Basis, IESE 40
PY = 5; (eq. 13)
0 If S?* =0
Total Cost Basis?#> = J (Increase in Total Value,”’- Decrease in Total Value??s)dt
(eq. 14)
Increase in Total Value”’> = Sell Rate?> * o (eq. 15)
Decrease in Total Value?*= Buy Rateg!'S* PS (eq. 16)
4 ofS
— Total Cost Basse IFS** 20
Pp? = -S (eq. 17)
0 Ifse =0
Margin and Available Cash
Hedge fund cash is divided into a free cash Cash," that can be used to buy new assets
and into a cash pledged as a collateral in order to cover a margin call Cash;"”.
30
Cash!'* =| (Income from Other Investments?!" + Cash Increase?!" - Cash
Decreases?” + Margin Refund, - Margin Call? )dt (eq. 18)
Cash Increase," = Sell Rate?" * Pe 4Fraction Reinvested* Sell Rates?'* Pe (eq. 19)
Cash Decrease,"*= Buy Rate" * P' +Buy Rate; *( P.S- BS) (eq. 20)
Cash Decision = 1 if a hedge fund decides to finance Margin Needed; with money made
from other trades.
Cash Decision = 0 if a hedge fund decides to finance Margin Needed ; by closing long
position.
Margin Call
. HF
Cash Decision * MIN( — Matyi Nese, nae Gat, 5
Time to Cover Margin’ Minimum Payment Time,
= If Margin Needed, > 0 (eq. 21)
0 If Margin Needed, <0
Margin Refund;"*
4 HP
Cash Decision* MIN(- — Margin Needed, =) Cash; -
Time to Cover Margin’ Minimum Payment Time,
= If Margin Needed, < 0 (eq. 22)
0 If Margin Needed, > 0
Margin Needed, = Margin Required;-Margin, (eq. 23)
Margin Required, = -(1+m)* S/"°P (eq. 24)
Margin, = $"P: *(1-m) + Total Cost Basis,"® + Cash?” (eq. 25)
31
Hedge Fund Balance Sheet
Equity" - Cash! + $"P‘ +Total Cost Basis," - S$" SPS (eq. 26)
Profits’ = Equity" - Equity! (eq. 27)
Desired Shares
Si? is the desired long shares by a hedge fund. S,""* is the desired short shares by a
hedge fund.
1 If Pi <Ps
Decision to Get Into Arbitrage= poe eg. 28)
7 f If PE > PS (eq
If Decision to Get Into Arbitrage=0 THEN Sf to (eq. 29)
If Decision to Get Into Arbitrage=1 AND Extra Margin Needed; <0 THEN
HF (4 _ ¢F Cash HLF
(Lash, (1 “I )Cashy ,0),0) (eq. 30)
Sf?" = MAX (S" + MAX
If Decision to Get Into Arbitrage=1 AND Extra Margin Needed; >0 THEN
HF max z
Si" = Cash Decision* MAX a +Sft. L at Needed, ,0) + (1— Cash Decision ) *
t t
MAX (SH — L iw Needed, 0)
t
(eq. 31)
In this model, the desired amount of short stock has two different formulations.
First, given the maximum allowed leverage provided by a dealer for a hedge fund, a
hedge fund can borrow the stock S and short sell it:
L™® & set * Pt _ set * pt
SH? S = —MIN ( >
5 SA) (eq. 32)
However, in order to execute a perfect arbitrage (buying x shares of stock L and short
selling x shares of stock S), a hedge fund can desire to have as much shares of stock S as
it desires of having a stock L.
SHDS — gH DL (eq. 33)
32
Results
Scenario 1_0. Equilibrium
The model is put into equilibrium when prices of long and short securities equal their
fundamental prices.
Fundamental prices of short and long security = P* * = P** =100 $/share.
Under these conditions, a hedge fund is not going to get into the arbitrage position.
Scenario 1_1. Successful arbitrage - Keep Within Margin Requirements
Fundamental prices of short and long security = P** = P* + =100 $/share.
Prices are exogenous, no feedback. A hedge fund manager sells short shares of security
S such that he is within margin requirements.
PF+ = 100+ STEP(-20,20)+STEP(20,30) $/share
P*S = 100+STEP(20,20)-STEP(20,30) $/share
Price L Price S
Cr eT) 0 10 2 30 4050 O70 8100
Time (Day) Time (Day)
Price L: S1_cd1. ——————_________—— $/share Price $ : S1_ed1 ————__—_________— $/share
Price L : Eq sishare Price $: Eq S/share
Figure C1_1_ 1. Price of Security L Figure C1_1_2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days, and then prices of both assets converge
again.
33
In this scenario, a hedge fund decides to short-sell as much as allowed within margin
requirements, so the hedge fund manager will not face a margin call.
S1_cd1big - arun with Cash Decision=1 and Income from Other Investments =
STEP(1000,10)-STEP(1000,20)
S1_cdi - arun with Cash Decision=1
S1_cd0 - amun with Cash Decision=0
Income from Other Investments Decision to Get Into Arbitrage
1,000 1
750 0.75
500 05
250 0.25
0 0
0 10 6200 300 40 50 6070 B09 100 i} 10 20 300 «40 (5060 70 8D 100
Time (Day) Time (Day)
Income from Other Investments : $1_cdlbig ——————————._$/Day Decision to Get Into Arbitrage : $1_odibig ————————— fraction
Income from Other Investments : S1_cd0 ‘$iDay Decision to Get Into Arbitrage : S1_od1 fraction
Income from Other Investments : S1_ed1 ——-----------------------_ $/Day Decision to Get Into Arbitrage : $1_cdQ. —--------------——-_ fraction
Income from Other Investments : Eq s/Day Decision to Get Into Arbitrage : Eq fraction
Figure C1_1_3. Income from Other Figure C1_1_4. Decision to Get
Investments Into Arbitrage
Shares Security L
400
300
200
100
0
0 10 50 ~—-60 70 80 90 100
Time (Day)
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Figure C1_1_5. Shares of Security L
34
Shares Security S
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Secutity S[Hf] : S1_cdibig shares
Shares Secutity S[Hf] :S1_cd1 shares
Shares Security S| - shares
Shares Security S| shares
Shares Security S| ~ shares
Shares Security S| shares
Shares Security S| shares
Shares Security S| shares
Figure C1_1_6. Shares of Security S
Free Cash Cash Contrib to Cover Margin
80,000 02
60,000 0.15
40,000 OL
20,000 0.05
0 0
0 0 2 30 4 50 60 7 80 90 100 0 1 20 30 50 80 100
Time (Day) ‘Time (Day)
Free Cash{Hf] : $1 cdibig $ Cash Contrib to Cover Margin: $1_cdLbig s
Free Cash{Hf] : $1 cdi $ Cash Contrib to Cover Margin: $1_cdl $
Free Cash{Hf] : S$1_cd0. $ Cash Contrib to Cover Margin: $1_ dQ). ——-------------------~ $
§ Cash Contrib to Cover Margin : Eq iN
Free Cash( Hi]: Eq
Figure C1_1_7. Free Cash
Figure C1_1_8. Cash Contribution to Cover
Margin
35
Margin Needed Equity
10,000 20,000
0 10 20 30 40 «450 60) 670) 680) (90100 0 10 20 30 40 «5060 7 «68069100
Time (Day) Time (Day)
‘Margin Needed : S1_cdtbig $ Equity{Hf] : S1_cdibig ——__________________ ¢
Margin Needed : S1_cdi $ Equity{Hf] : $1_cd1 $
‘Margin Needed : $1), ae § BquGtEN ¢S15oN0 eg
‘Margin Needed : Eq $ Eqguity[Hf] : Eq $
Figure C1_1_9. Margin Needed Figure C1_1_ 10. Equity
Profit
40,000
30,000
20,000
10,000
-0.02
0 10 20 30 40 50 60 70 980 90 100
Time (Day)
Profit{Hf] : S1 cdibig $
Profit{Hf] : S1_cd1 $
Profit{Hf] :S1_cd0 — $
Profit{Hf] : Eq $
Figure C1_1_11. Profit
In all runs, after prices for long and short securities converge at time = 30 days, profits
become positive. Between time = 20 and 30 days, inruns S1_cd1 and S1_cd0 profits are
not realized; however, for run $1_cd1big, profits are positive ($10,000 due to
accumulation of Income from Other Investments). In all runs, decisions to short are
made within margin requirements. Therefore, a dealer never makes a margin call to the
hedge fund. Asa result, behavior of S1cd1 and S1cd0 is identical. The difference in
36
behavior of runs Sicd1 and S1cd1big is intuitive. If a hedge fund has more cash
available, it will be able to buy more long shares and short more short shares, thus
making more profit. Therefore, the profit for the run S1cd1big is higher than for other
two runs. Final equity is also higher for this run due to the income from other
investments.
Scenario 2. Successful arbitrage - Short as Much as Long
Fundamental prices of short and long security = P** = P* =100 $/share.
Prices are exogenous, no feedback.
PF+ = 100+ STEP(-20,20)+STEP(20,30) $/share
P*S = 100+STEP(20,20)-STEP(20,30) $/share
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days, and then prices of both assets converge
again.
Price L Price S
100 200
% 170
80 140
7 10
60 80
o 0 0 0 0 0 0 10 8 9 100 obo 0 0 6 7 00 9 10
Time (Day) Time (Day)
Price $2 cd] ————________— qjshare Price: $2 oj] ———____________ gare
Price: $2 ed0 sistare Price: $2 odd stare
PHOS Eee ener es SSE PHDE SEY eeecercrrerenerecre hat
Figure C1_2_1. Price of Security LL Figure C1_2_2. Price of Security S
In this scenario, a hedge fund decides to short-sell as many shares as he desires to long, in
order to make a perfect arbitrage.
$2_cd1big - arun with Cash Decision=1 and Income from Other Investments =
STEP(1000,10)-STEP(1000,20)
37
$2_cd1 - arun with Cash Decision=1
$2_cd0 - arun with Cash Decision=0
Income from Other Investments Decision to Get Into Arbitrage
1,000 1
730 0.75
500 05
50 0.25
0 0
o 10 © 3% 4 50 6 7 8 % 100 0 10 20 30 40 5060 7 80 «90 6100
‘Time (Day) ‘Time (Day)
‘[ncome from Other Investments : $2 odibig ——————————-_ Day Decision to Get Into Arbitrage : $2 cdibig —————————— fraction
Income from Other Investments : $2_od1 ‘Day Decision to Get Into Arbitrage : S2_cd1 fraction
Income from Other Investments : $2 od0. ————— ——— $Day Decision to Get Into Arbitrage : S2_cd0 — fraction
‘Income from Other Investment : Eq siDay Decision to Get Into Arbitrage : Eq fraction
Figure C1_2_3. Income from Other Figure C1_2_4. Decision to Get Into
Investments Arbitrage
An arbitrage window exists from time = 20 to 30 days.
Shares Security L
400
300
Shares Secutity L|
Shares Secutity L|
Figure C1_2_5. Shares of Security L
38
Shares Security S
0 10 20 30 40 50 60 70 80 90 ~=100
Time (Day)
shares
shares
- shares
- shares
shares
shares
shares
shares
Shares Secutity S|
Shares Secutity S|
Figure C1_2_6. Shares of Security S
Free Cash Cash Contrib to Cover Margin
80,000 6,000
60,000 ral 4,500 i
40,000 r f ai 3,000
] i]
20,000 \ J 1,500 } \
I}
0 0
Oo 10 2 30 40 50 60 70 8 690 6100 0 0 20 3 40 50 6 70 80 90 100
Time (Day) ‘Time (Day)
Free Cas{Hf] : S2_cdlbig $ Cash Contrib to Cover Maryin : $2 cdibig $
Free Cast Hf] : $2. cdi $ Cash Contrib to Cover Margin : S2_cdl $
Free Cash Hf] :$2_¢€0) —————— § Cash Contrib to Cover Margin : $20) §
Free CasHi] : Eq $ Cash Contrib to Cover Margin: Eq $
Figure C1_2_7. Free Cash Figure C1_2_8. Cash Contribution to Cover
Margin
39
Margin Needed Equity
2000 K- 80,000
3500 tl 65,000
9000 50,000 Ly
14500 35,000
200000 20,000
o 0 0 3% 40 5 60 7 80 9 100 o 0 2 3 4 50 6 7 8 90 100
Time (Day) Time (Day)
Margin Needed : $2.cdthig —————————— §_Bmuity[Hf]: S2_cdtbig $
Martin Neoded $2-cdl $ — BuityfHfl: $2 cdl $
Margin Needed : $2.40. $§ Equi Hf] :$2_0d0 ~$§
Margin Neoded : Eq § — Eguitf Hl: Eq $
Figure C1_2_9. Margin Needed Figure C1_2_ 10. Equity
Profit
40,000
30,000
20,000
10,000
-0.004
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Profit{Hf] : S2 cdibig $
Profit{Hf] : S2_cd1 $
Profit{Hf] : $2 cd0 — $
Profit{Hf] : Eq sesso $
Figure C1_2_11. Profit
Due to the price difference between long and short securities, the hedge fund manager
faces a margin call by a broker. In case of S2_cd1 and S2_cd1big, margin is covered by
available cash. However, in case of S2_cd0, margin is covered by selling long shares.
Therefore, the amount of long and short shares in S2_cd0 is much smaller than in other
tuns. At time = 30 days, the hedge fund becomes lucky and positions converge. Due toa
40
relative small amount of short and long securities in S2_cd0, the profits in this run are
much smaller than for S2_cd1 run ($39 compared to $8889). Also, in S2_cd1big run,
profits ($20,840) are larger than in S2_cd1 run because of the extra income that is used to
buy additional long shares and borrow short shares that would be realized later at a profit.
Profit Margin Needed
10,000 2,000 N
7300 1,000 He
f
5000 4,000 t
i
209 7,000
as -10,000
oo 0 0 50 om a) 90 100 o 1 2 30 a 50 60 70 @0 90 100
Time (Day Time (Day)
Profit] : $2 cd]. 5 ‘Margin Needed : $2_cd1
Profil]: Steal 4
Profi{H] : Eq ————————— =
Figure C1_2_12. Profits Compared,
Cash Decision = 1
Shares Security L
Margin Needed : S1_ed1
Menpin Needed» Eq ore
Figure C1_2_13. Margin Needed Compared
Cash Decision = 1
Shares Security S
o 0 0 % 1 5 6 70 8
Time (Day)
90 100
o 1 2 3% 4 5 6 70 & 90 100
Time (Day)
Figure C1_2 14. Shares of Security L
Compared, Cash Decision = 1
Figure C1_2_15. Shares of Security S
Compared, Cash Decision =1
The above graphs compare behavior in Scenario 1 with behavior Scenario 2 for the case
where cash decision = 1. In both cases profits are always positive as the arbitrage
strategy was properly conceived and was realized at time = 30 days. However, in case
$2_cd1, a hedge fund manager was not constrained to operate within margin
requirements by selling as many shares of expensive security S as the number of shares
of a cheaper security L.
41
Profit Margin Needed
10,000 200
7,500 1,850
5,000 3,900
2,499 5,950
0.02 8,000
0 i 2 0 4 50 6 70 a 90 100 ow 2 30 40 50
‘Time (Day) Time (Day)
Profi{Hf]: $2. eq ————__________________ 5 Margin Needed : $2 cdg. ————________________ ¢
Profi{Hf]: S1_cd0 s ‘Maruin Needed : S1_cd0 8
PooBiQ HES ype ee Margin Needed : Eq $
6 708) «80 «100
Figure C1_2_16. Profits Compared, Figure C1_2_17. Margin Needed Compared
Cash Decision =0 Cash Decision = 0
Shares Security L
400 T Fa
Shares Security S
0 10 20 630 (40 50D] o 0 20 3
Time (Day)
Figure C1_2_18. Shares of Security L Figure C1_2_19. Shares of Security S
Compared, Cash Decision =0 Compared, Cash Decision = 0
The above graphs compare behavior in Scenario 1 with behavior Scenario 2 for the case
where cash decision =0. In S2_cd0 case, due to the decision of a hedge fund manager to
keep as many shares of security S as security L, Margin Needed is positive, compared to
run S1_cd0. Therefore, in S2_cd0, the hedge fund has to dispose of the both long and
short securities compared to S1_cd0, which leads to a much smaller long term profit for
run S2_cd0, compared to run S1_cd0. In this case, it pays to stay longer in the position
by constraining to keep positions within margin limits, waiting for the arbitrage
opportunities to realize.
Also, as can be seen from scenarios 1 and 2, given that positions will converge at
some time, it is better to use money from available cash and finance the margin calls than
selling securities.
42
Scenario 3. Successful arbitrage - Keep Within Margin Requirements
Fundamental prices of short and long security = P'S = P* + =100 $/share.
Prices are exogenous, no feedback.
P** = 100 $/share
P*S = 100+STEP(40,20)-STEP(40,30) $/share
Price L Price S
100 200
95 170
90 140
85 no
80 80
o. 0 2 30 4 50 oo 70 0 99 100 o 0 2 3 4 50 6 7 8 90 100
‘Time (Day) Time (Day)
Price L: $3. cd S/share Price § :$3_e0) share
Price L: Eq Sishare Price S : Eq Sishare
Figure C1_3_1. Price of Security L
Figure C1_3_2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days, and then prices of both assets converge
again.
In this scenario, a hedge fund manager decides to short-sell as much as allowed within
margin requirements, so he would not have a margin call.
$3_cd0 - arun with Cash Decision=0
Equity Profit
60,000 10,000
50,000 7,500
40,000 {I 5,000
30,000 2,499
20,000 0.02
o 0 0 3 4 50 6 70 8 90 100 o 0 2 3 40 50 6 7 8 90 100
Time (Day) Time (Day)
Equity(H] : $3 edo —___________________ 5 Profit] :$3.ed9 —_____________________ 5
Equity Hf] : St edo $ Profit]: S1_cdo s
Equity) § Eq) anna § Profil) # Bq) a
Figure C1_3 3. Equity
Figure C1_3_4. Profit
43
Here, scenarios 1 and 3 are compared, cash decision = 0. In both scenarios, between time
= 20 and 30 days, the spread between a short and a long position equals 40 $/share.
However, in the scenario 1, the spread is distributed equally between long and short
positions; whereas, in the scenario 3, the price of security L stays at its fundamental
value, and the price of security S increases by 40% between time = 20 and 30 days. In
the absence of margin requirements and laws regulating leverage, the results from two
scenarios should be exactly the same. However, due to leverage requirements and
proportional changes in prices, the results are different.
The total equilibrium profits in S1_cd0 are larger than in S3_cd0. In Sicd0,
profits come from both short and long positions; whereas, in S3_cd0, profits only come
from the convergence of the short position to the equilibrium value. Due to the
maximum leverage ratio of 2, the amount of borrowed short positions (in $ amount) can
be no more than the amount of long positions (in $ amount).
Proof for these scenarios: Assume that shares are bought and disposed of
instantaneously. In Scenario 1, given x dollars of cash available to invest in long assets
. . ae x 35 .
and the maximum leverage ratio of 2, profit is: 302" + i927 = ba x. In Scenario 3, the
profit is: a 40 = a x, where profit in Scenario 1 is always bigger than profit in
Scenario 3 for any x. The intuition for the result is the following: It is better to be
exposed to an asset that will have a bigger proportional increase (in the case of long) or
decrease (in the case of short).
Scenario 4. Successful arbitrage - Short as Much as Long
Fundamental prices of short and long security = P"* = P** =100 $/share.
Prices are exogenous, no feedback.
PF — 100 $/share
PS = 100+STEP(40,20)-STEP(40,30) $/share
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days, and then prices of both assets converge
again.
44
Price L Price $
100 200
5 170
90 140
85 110
80 Ey
0 1 2 3 40 50 6 70 8 90 100 0 0 2 3 4 50 60 7 8 9 100
Time (Day) Time (Day)
Price L: S4_ cd). ————_____________ share Price S : $4 cdl share
Price L: Eq Sishare Price S : Eq Sishare
Figure C1_4 1. Price of Security L
Figure C1_4 2. Price of Security S
In this scenario, a hedge fund decides to short-sell as many shares as he desires to long, in
order to make a perfect arbitrage.
S4_cd1 - arun with Cash Decision=1
Shares Security L Shares Security S
400 0
\ f
300 100 rq
NG
200 200
100 300
0 ! 400
o 0 2% 3% 4 50 6 7 8 9% 100 o 0 0 3% 40 50 60 70 8 90 100
Time (Day) Time (Day)
Shares Security L[HA]: $4 cdl. ———______— stares Shares Security S{HE] : $4 edt shares
Shares Security LH]: S2_ dl stares Shares Security S{Hf] : $2_ed shares
Shares SecusityL[HA]: Eq) —~ stares Shares Security S{HE] : Eq ———— ~ shares
Figure C1_4 3. Shares of Security L
Figure C1_4 4. Shares of Security S
Equity Profit
60,000 10,000
30,000 7500
40,000 5,000
30,000 2,500
20,000 0.008
o 1 2% 30 a 50 60 70 80 90 10 ow 2% 3 4 50 60 70 80 90 100
Time (Day) Time (Day)
Equity(Hf] : $441. —————______________ ¢ Profit{Hf] : $4.¢d1); ———______________ 5
Equity(Hf] : S2-cd1 $ Profit{Hf] : S2_cd1 $
Equity} : Eq’. ————— tt § Profit{Hf] : Eq $
Figure C1_4 5. Equity
45
Figure C1_4 6. Profit
Profits for scenario 4 are a little bit smaller than profits in scenario 2. This is due to
different amount of shares of both securities L and S in both scenarios. In scenario 4,
security L is more expensive (100 $/share) than security L in scenario 2 (80 $/share);
therefore, given the same amount of cash, a hedge fund can afford to buy less of security
L, and therefore, short less of security S leading to smaller final arbitrage profits. Note,
that if the number of long and short securities were the same in both scenarios, profits
would be the same. For the proof, look at the section: Necessary Condition for a Hedge
Fund to Fail.
Scenario 5. Collapse - Keep Within Margin Requirements
Fundamental prices of short and long security = P'S = P*" =100 $/share.
Prices are exogenous, no feedback.
P** = 100 $/share
P*S = 100+ STEP(20,20)+STEP(260,30)-STEP(280,60) $/share
Price L Price $
100 400
95 300
90 200
85 20. |. 11S
80 0
0 10 20 30 40 50.60 70 80 90 = 100 0 10 20 30 40) (50 60 70 BO 90100
‘Time (Day) Time (Day)
Price L: $5_cd). ——HH4H44H4H4H444H444_ $/share Price S : $5_ cdl. ———————————______—. $/share
Price L: Eq ‘$/share Price S: Eq $S/share
Figure C1_5_1. Price of Security L Figure C1_5_2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days. From time 30 to time 60 days, instead of
converging, prices diverge even more before converging at time 60 days.
In this scenario, a hedge fund decides to short-sell as much as allowed within margin
requirements, so a hedge fund manager would not have a margin call.
$5_cdibig - a run with Cash Decision=1 and Income from Other Investments =
STEP(1000,10)-STEP(1000,20)
46
S5_cdi - arun with Cash Decision=1
S5_cd0 - arun with Cash Decision=0
Shares Security L
400
300
200
100
0 :
0 10 2 30 40 50 60 70 80 90 100
Time (Day)
Shares Security L{Hf] : $5_cdbig shares
Shares Security L{Hf] : $5_cd1 shares
Shares Security L{Hf] :$5_cd0 ~~ - shares
Shares Security L{Hf] : Eq ------ -- shares
Shares Security L[D] : $5_cd1big -+ shares
Shares Security L[D] :$5_cd1 shares
Shares Security L[D] :$5_cd0 shares
Shares Security L[D] a -- shares
Figure C1_5_3. Shares of Security L
In S5_cd1big run, more shares can be bought because of higher available cash.
47
Shares Security S
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Figure C1_5 4. Shares of Security S
From time = 20 to 30 days, Margin Needed is negative; therefore, for runs S5_cd1 and
S5_cd0, the number of shares sold short and long is the same. However, from time = 30
days, the Margin Needed becomes positive. Therefore, in run S5_cd0, to cover the
margin, long shares are forced to be sold. In order not to increase the margin, shorts have
to be covered. On the other hand, in run S5_cd1, the fund chooses to cover margin call
with available cash. However, as soon as the available cash runs out, the hedge fund
manager is forced to sell long shares and cover the shorts. However, in this simulation,
cash runs out before all shares of security S can be bought off. There is a lag for
unwinding the positions in the run S5_cd1 compared to S5_cd0 because in the first case,
the manager covers the margin with cash before unwinding the shares, and in the second
case, the manager unwinds the shares as soon as possible.
48
Free Cash Cash Contrib to Cover Margin
60,000 60,000
IE 45,000
/
f 390,000 ; |
i
15,000 \
_ o
0 50 60 70 «80 90 (100 o 0 2 50 60 70 8 9 100
Time (Day) Time (Day)
Free CasifHf]: $5, cdlbig Cash Contrib to Cover Margin: $5.cdbig
Free Casif Hf]: $5_cd Cash Contrib to Cover Margin: $5.cdl
Free Cast]: S5-cd0. — Cash Contrib to Cover Margin: $5.cd0. —~
Free Cash( Hf]: Eq Cash Contrib to Cover Margin: Eq
Figure C1_5 5. Free Cash
Figure C1_5_6. Cash Contribution to
Cover Margin
Margin Needed Equity
60,000
44,900
7 29,800
-40,000 14,700
-80,000 400 1
0 10 6200 30 4050 0 70 B80 90 100 0 10 200 300 40 50 60 70 80 100
Time (Day) Time (Day)
‘Margin Needed : $5 cd1big $ Equity{Hf] : $5_cdibig $
‘Margin Needed : $5 cdl. $ Equity(HE] : $5 edt $
Margin Needed : S5_cd0. — = $ Equity(H] : $5. cd. ——~ $
Margin Needed : Eq $ Equity(Hé] : Eq $
Figure C1_5_7. Margin Needed
Figure C1_5_8. Equity
49
Profit
10,000
-7,500
-25,000
-42,500 ee ee ee ee eee ee es oo ee
-60,000
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Profit{Hf] : S5_cd1big $
Profit{Hf] : $5 cd1 $
Profit Hf] : S5_cd0 $
Profit{Hf] : Eq -----~- $
Figure C1_5 9. Profit
For runs S5_cd1 and S5_cd0, we have seen a collapse in the hedge fund, as indicated by
negative equity. In scenario S5_cdi1big, a hedge fund has an outside stream of cash to
make sure that equity does not fall below zero. The profits for cases S5_cd1 and S5_cd0
are the same before time = 60 days because the number of both long and short sells is the
same at time = 30 days before the divergence in prices. At time=60 days, positions
converge, thus making profits fora hedge fund. By time = 60 days, almost no positions
are opened in scenario S5_cd0 compared to scenario S5_cd1. Compared to other funds, a
hedge fund has an outside income of cash in scenario S5_cd1big, thus, a hedge fund does
not fail in this case.
However, note, that formulation of P' is chosen as to be near a break-point for a
hedge fund to have positive to negative equity. The break-point for an increase of price
of security S is 258 $/share, and here 260 $/share is chosen.
PF = 100 $/share
P*S = 100+ STEP(20,20)+STEP(260,30)-STEP(280,60) $/share
50
In this case, for a hedge fund to fail, the following condition should hold: Cash Hf 9 +
L
se (Ax + ay <0 (for derivation, look at Necessary Condition for a Hedge Fund to
0
Fail section of the paper). In this case, Cash Hf » = $40,000. S/'"=200 shares. Ax =0.
Sj'° =-155 shares. Therefore, Ay =258 $/share.
In a separate simulation, not shown here, the same conditions as depicted in this
scenario are used except that
PS = 100+ STEP(20,20)+STEP(500,30)-STEP(520,60) $/share
A new price fora short asset is much higher than the break-point price. For this case, the
losses for the run S5_cd1big are much bigger than losses for the run S5_cd1 because of a
greater exposure of the hedge fund to short shares in that case. Starting time = 30 days, a
hedge fund has to cover the margin and unwind the positions at a loss.
Scenario 6. Collapse - Short As Much as Long.
Fundamental prices of short and long security = P* * = P** =100 $/share.
Prices are exogenous, no feedback.
PF+ = 100 $/share
P*S = 100+ STEP(20,20)+STEP(215,30)-STEP(235,60) $/share
Price L
Price S
10 400
9 300
0 200
85 100 L____I
80 0
0 10 200 300 4050 O70 809100 0 1 20 30 «40 «650 6070 80 90100
Time (Day) ‘Time (Day)
Price L : $6 cd), ———__________. $/share Price $ : $6_cdl1 ———_________________ $/share
PriceL : Eq s/share Price S: Eq S/share
Figure C1_6_ 1. Price of Security L Figure C1_6_2. Price of Security S
51
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days. From time 30 to time 60 days, instead of
converging, prices diverge even more before converging at time 60 days.
In this scenario, a hedge fund decides to short-sell as many shares as he desires to long, in
order to make a perfect arbitrage.
S6_cdibig - a run with Cash Decision=1 and Income from Other Investments =
STEP(1000,10)-STEP(1000,20)
S6_cdi - arun with Cash Decision=1
S6_cd0 - arun with Cash Decision=0
Shares Security L
400
300
0 10 20 30 40 «50 60 70 80 90 100
Time (Day)
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Hf] : $6_cdbig shares
Hf] : S6_cd1 shares
:$6cd0 ~
:Eq
--- shares
shares
~+ shares
shares
shares
- shares
Figure C1_6 3. Shares of Security L
52
Shares Security S
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Secutity S|
Shares Secutity S|
Shares Secutity S
Shares Secutity S|
Shares Secutity S
Shares Secutity S|
Figure C1_6_4. Shares of Security S
Free Cash Cash Contrib to Cover Margin
60,000 60,000
45,000 45,000 i
i |
3e.000 | 30,000 fs \
i Al
| f \
15,000 | 15,000 f \
\
’ i ; |
0 10 2 30 40 50 6 70 8 90 100 0 10 20 30 «40 «650 660) 670) 80 90100
Time (Day) Time (Day)
Free Cast{ Hf] : $6_cd1big ——————_—__________ § Cash Contrib to Cover Margin : S6 cdibig $
Free Cast{Hf] : S6_cd1 $ Cash Contrib to Cover Margin : S6_cd1 $
Free Cast{Hf] : S6_ cd. $ Cash Contrib to Cover Margin : S6_ cd) mn §
Free Cash Hf] : Eq $ Cash Contrib to Cover Margin : Eq $
Figure C1_6_5. Free Cash Figure C1_6_6. Cash Contribution to Cover
Margin
53
Margin Needed Eauity
80,000 60,000
40,000 Nott 44,750
0 r 29,500 |
40,000 14,250 |
80,000 -1,000
0 wm 2 3 4 50 6 70 8 0 100 om 2 20 70 80 90 100
405060
‘Time (Day) ‘Time (Day)
Margin Needed: $6_clbig HHH =
Margin Needed : S6_cdl
Margin Needed : S6_cd)
Margin Needed : Eq
Equiy[Hf]: S6_cdibig
Equity{H] : S6_cdi
EquityfHf] : S6_cd0. —
EquitylHf] : Eq
Figure C1_6_7. Margin Needed Figure C1_6_8. Equity
Profit
10,000
-7,500
-25,000
-42,500
-60,000
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Profit{Hf] : S6_cdlbig
Profit{Hf] : S6 cd1
Profit{Hf] : S6_cd0 —
Profit{Hf] : Eq --------
PAA
Figure C1_6 9. Profit
In this scenario, a hedge fund manager would like to have as many shares of asset S as of
asset L. Due to price differences, Margin Needed is greater than 0 in all three runs from
time = 30 to 60 days. Profits and Final Equity are negative for run S6_cd1, small and
positive for run S6_cdlbig and large and positive for run S6_cd0. In S6_cd0, the
manager is precautious, and is not throwing “good money after bad.” In this scenario, he
minimizes potential losses in case of position divergence (this is what exactly happened
54
in the Scenario 6) compared with position convergence (what a hedge fund manager
counts on). In this case, compared to previous scenarios, it pays off to be cautious.
Most of smaller hedge funds, not LTCM, close out their positions as soon as they
are faced with a margin call. Therefore, as in 1998, lots of smaller funds managed to
close off their positions and survive. In the case of large hedge funds like LTCM which
had available cash from other positions, these hedge funds decided to finance the margin
calls with cash before closing their positions. Only after it had no cash left, LTCM was
forced to sell its positions at a loss due to divergence in the prices of long and short
assets.
In the case of a future position convergence, it is better to use cash to optimize
exposure to the position that due to future position convergence will become profitable.
However, even if positions eventually converge, they might diverge, as in the case of
Scenario 6 and lead to a collateral collapse.
In this case, P** = 100+ STEP(20,20)+STEP(215,30)-STEP(235,60) $/share,
where 215 $/share is very close to a break-point of 210 $/share. The break-point fora
hedge fund to collapse is calculated in the section: Necessary Condition for a Hedge
Fund to Fail. For this case, in order for a hedge fund to fail, the following condition
should hold:
Cash Hf + AxSj'" — AySj'S =Cash Hf + (Ax + Ay)Sj'* <0
Cash Hf - $40,000. Ax =0, and S;'" = —S,'* =190.39 shares. Therefore, Ay should be
at least 210 $/share.
Note, if a price of a short asset S is going to go much higher than the break-point,
then both equity and profits are negative for run S6_cd1big. The losses are augmented in
run S6_cdlbig compared to run S6_cd1 as a hedge fund manager is using more available
cash to buy more assets and cover margin with cash.
Scenario 7. Success: Use Short Proceeds to Obtain More Positions
Fundamental prices of short and long security = P*’* = P** =100 $/share.
Prices are exogenous, no feedback.
PF+ = 100+ STEP(-20,20)+STEP(20,30) $/share
55
P*S = 100+STEP(20,20)-STEP(20,30) $/share
Price L Price S
100 200
60 0
. 0 0 0 © 0 6 7 8 0 100 co 0 20 0 4 50 6 7 8 9% 100
Time (Day) Time (Day)
Price L : $7_¢d, ———____________. /share Price $ : $7_ed);$ ———_____________. share
Price L: Eq: /share Price $ : Eq $ishare
Figure C1_7_1. Price of Security L Figure C1_7_2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days, and then prices of both assets converge
again.
In this scenario, a hedge fund decides to short-sell as many shares as he desires to
long, in order to make a perfect arbitrage.
Also, in this scenario, proceeds from short selling are used to increase hedge
fund’s cash, and therefore, can be used to obtain more shares. In the previous scenarios,
that cash was set aside and could not be used to buy more of security L.
Cash Increase," =Sell Rate," * P," + Sell Rate, * PS *Fraction Reinvested
Fraction Reinvested = 0.5
The fact that proceeds from the sale of the short security can be used to buy along
security produces a positive feedback loop.
56
Proceeds from
Short Selling
+
Short Sell +
Shares S
Figure 5. Using Short Proceeds to Go Long
Hedge fund’s cash is used to buy shares of L. Due to margin requirements, each dollar of
security L can be used as a collateral to short sell a dollar of security S. If the proceeds
from short selling are transferred back into the cash position of the hedge fund, more
shares of L can be bought, more shares of S can be short sold, and so on.
S7_cd1big - arun with Cash Decision=1 and Income from Other Investments =
STEP(1000,10)-STEP(1000,20)
S7_cd1 - arun with Cash Decision=1
S7_cd0 - arun with Cash Decision=0
Shares Security L Shares Security S
ow 20 3 40 50 6 7 8 9 100 0 10 2 30
Time (Day)
1: SD) B
Figure C1_7 3. Shares of Security L Figure C1_7_4. Shares of Security S
57
Free Cash
o 0 » 30 40 50 60
Time (Day)
Free Cash{Hf]: $7 ed). ————_______________ ¢
Free Cast{Hif] : S2_cd $
Hive Cash Hi] iq $
Figure C1_7_5. Free Cash
Cash Contrib to Cover Margin
8,000 |
6,000 |
ia |
4,000 ry |
2,000 fi
1)
y
o 1 2% 30 40 50 60 7 80 90 100
Time (Day)
‘Cash Contrib to Cover Margin: $7 cdl 5
‘Cash Contrib to Cover Margin : S2_cdl $
‘Cash Contib to Cover Margin: Eq ~ $
Figure C1_7_6. Cash Contribution to Cover
Margin
Margin Needed Equity
2,000 nel 80,000
3,500 f 65,000
!
9,000 ; 50,000 ff
t
14,500 35,000
20,000 20,000
0 1 20 3% 40 6 70 8 «90 100
30
Time (Day)
Margin Needed : $7 cdl
Marvin Needed : Sci
Margin Needed : Eq
Figure C1_7_7. Margin Needed
Equity Hf] : $7 cdl
Equity Hf] : S2_cdl
Equity Hf]: Bq
Figure C1_7_8. Equity
58
Profit
40,000
30,000
20,000
10,000
-0.004
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Profit{Hf] : S7 cd1
Profit{ Hf] : S2_cd1
Profit{Hf] : Eq
PAR
Figure C1_7_9. Profit
Here scenarios 2 and 7 are compared. The scenarios are the same except the reinforcing
loop “Use Short to Buy Long” is absent from the scenario 2 and is present in the scenario
7. As can be seen from figures above, due to the reinforcing loop, the hedge fund is more
exposed to both long and short positions. The hedge fund manager uses free cash to
cover the margin calls when the positions diverge. However, due to a lucky position
convergence, the hedge fund is making money in both scenarios, more in scenario 7
where it has more position exposure. Therefore, the reinforcing loop amplifies the gains.
The analogy for the difference in behavior in runs S7_cd1big and S2_cd1big is similar to
the differences in behavior in runs S7_cd1 and $2_cd1. The behavior in runs S2_cd0 and
S7_cd0 is virtually the same due to a very small position exposure in both cases.
According to securities regulations in the USA, it is illegal to reinvest any
proceeds from the short sale. Therefore, in the USA, the Fraction Reinvested = 0.
Scenario 8. Failure: Use Short Proceeds to Obtain More Positions
Fundamental prices of short and long security = P** = P** = 100 $/share.
59
Prices are exogenous, no feedback.
P** = 100 $/share
P*S = 100+ STEP(20,20)+STEP(215,30)-STEP(235,60) $/share
Price L Price $
100 400
95 300
90 200
85 100
80 0
0 10 20 30 40 «50 60 670) 680) 90100 0 10 20 30 40 50 60 70 80 90 100
Time (Day) Time (Day)
Price L : $8. ed). —————__________—+——- $/share Price S : $8_cd1. —————————__________—- $/share
Price L: Eq /share Price S : Eq $/share
Figure C1_8 1. Price of Security L Figure C1_8 2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days. From time 30 to time 60 days, instead of
converging, prices diverge even more before converging at time 60 days.
In this scenario, a hedge fund decides to short-sell as many shares as he desires to long, in
order to make a perfect arbitrage.
Also, in this scenario, like in scenario 7, proceeds from short selling are used to increase
hedge fund’s cash, and therefore, can be used to obtain more shares. In previous
scenarios, that cash was set aside and could not be used to buy more of security L.
Cash Increase;"* =Sell Rate?" * P.' + Sell Rate; * PS *Fraction Reinvested
Fraction Reinvested = 0.5
$8 _cdibig - a run with Cash Decision=1 and Income from Other Investments =
STEP(1000,10)-STEP(1000,20)
$8 cdi - arun with Cash Decision=1
S8_cd0 - arun with Cash Decision=0
60
Shares Security L Shares Security S
400 T Py 800
300 400 7
200 0
100 400 if
0 ‘ -800
0 0 2 3% 4 5 6 70 80 9% 100 o 0 0 3 4 50 60 70 8 90 100
Time (Day) Time (Day)
Stan Say LS Stans Sams Seat tS $$$ stars
Stam Sey LI Sime Shae Seay Sol ——— = Ss
: Sam Say SH Stas
Sam Sey SD] 9 ot Sa
Sten Sey UD] Se Sins Same Sey DS ol ae
SierSee U9]: SaeSey SOL ey Ss
Figure C1_8 3. Shares of Security L Figure C1_8 4. Shares of Security S
Figure C1_8 7. Margin Needed
Free Cash Cash Contrib to Cover Margin
60,000 60,000
45,000 45,000 +>
\ / -
30,000 Ne v 30,000 FE
\ \
15,000 | 15,000 4
\
0 SE 0
oO 10 20 30) «40 (50 O70 BO 0100 oO 10 6200 30040 50 0 70 B89 100
Time (Day) Time (Day)
Free Cash{Hf] : $8_cdl. ————_____________ g Cash Contrib to Cover Margin : $8_cdl1. —————_________
Free Cash{Hf] : S6_cdl $ Cash Contrib to Cover Margin : S6_cdi $
Free Cash{Hf] : Eq $ Cash Contrib to Cover Margin : Eq —§
Figure C1_8_5. Free Cash Figure C1_8 6. Cash Contribution to Cover
Margin
Margin Needed Equity
100,000 60,000 mI
50,000 CS 44,500 Al
0 29,000
50,000 13,500
100,000 2,000
0 10 20 30 40 50 60 6-70) 680) 90100 0 10 2 630 40 50 6070 BH 100
‘Time (Day) ‘Time (Day)
‘Margin Needed : $8_ ed) Equity(Hq] : $8_cd). ————______________—- 5
‘Margin Needed : 86 cdl Equity{Hf] : $6 cdl 8
‘Margin Needed : Eq. Equity( Hf]: Eq $
Figure C1_8 8. Equity
61
Profit
0 10 20 30 40 50 60 #70 80 90 100
Time (Day)
Profit{Hf] : S8 cd1
Profit{ Hf] : S6_cd1
Profit{Hf] : Eq
PAR
Figure C1_8 9. Profit
Scenarios 6 and 8 are compared. The scenarios are the same except for the reinforcing
loop “Use Short to Buy Long” that is absent from scenario 6 and is present in scenario 8.
In both cases, we observe the failure of a hedge fund. However, the process is reinforced
in scenario 8, as the proceeds are used to buy more security L and short sell more security
S. Therefore, the total position exposure in scenario 8 is bigger than in scenario 6. In
both cases, instead of converging, the positions greatly diverge. Free cash in both cases
is 0. In both cases, the fund collapses, but the collapse is more pronounced in the
scenario 8 compared to scenario 6 due to a higher position exposure (note, in this
scenario, the difference is not too big due to price of security S being close to break-point
for both cases). The analogy for the difference in behavior in runs S8_cd1big and
S6_cdibig is similar to the differences in behavior in runs S8_cd1 and S6_cdl. The
behavior for runs S8_cd0 and S6_cd0 is virtually the same due to a very small position
exposure in both cases.
According to securities regulations in the USA, it is illegal to reinvest any
proceeds from the short sale. Therefore, in the USA, the Fraction Reinvested = 0.
62
Scenario 9. Divergence Before Convergence
Fundamental prices of short and long security = P'S = P*“ =100 $/share.
Prices are exogenous, no feedback.
PF = 100+STEP(-20,30)+STEP(+20,60) $/share
P*S = 100+STEP(20,20)-STEP(20,30)+STEP(40,30)-STEP(40,60) $/share
Price L Price S
100 200
60 80
. 0 20 3 40 oo 70 a 90 100 ow 2 30
30 a 50 60 70 a0 90 100
Time (Day) ‘Time (Day)
Price L : $9. ed); ——_____________ share Price §: $9 ed), —_____________ g/share
Price L: Eq slshare Price $: Eq Sfshare
Figure C1_9 1. Price of Security L Figure C1_9 2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days. From time 30 to time 60 days, instead of
converging, prices diverge even more before converging at time 60 days. Note, the
divergence is not as extreme as in the scenarios 5,6 and 8.
In this scenario, a hedge fund manager decides to short-sell as much as allowed within
margin requirements, so he would not have a margin call.
Fraction Reinvested = 0
S9_cdibig - a run with Cash Decision=1 and Income from Other Investments =
STEP(1000,10)-STEP(1000,20)
S9_ cdi - arun with Cash Decision=1
S9_cd0 - arun with Cash Decision=0
63
Shares Security L
400
300
0 10 20. 30 40 «50. 60 70 80 90 100
Time (Day)
Shares Security L[Hf] : $9_cdibig shares
Shares Security L[Hf}: $9_cd1 shares
Shares Security Lf 1 edd shares
Shares Security Lf shares
Shares Security L{D]: $9. cdtbig shares
0
[D.
[D.
Shares Security L[D sa cdl shares
Shares Security L{D]: S shares
Shares Security L[D]: E shares
Figure C1_9 3. Shares of Security L
Shares Security S
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Security S[HE] :
Shares Secutity S
Shares Security S[HE]
shares
shares
shares
$9_cdtbig
y edd
Shares Security S[HE] : shares
~~ shares
Shares Secutity S[D] : $9 cd1 shares
Shares Security S[D]:$9_cd0 shares
L q
Shares Security S[D] : $9_cdibig
1
L
L
Shares Security S[D]: Eq. ~ -- shares
Figure C1_9 4. Shares of Security S
64
Free Cash Cash Contrib to Cover Margin
6000
4500
3,000 te
1,500 |
0
ot 2% 3% 4 5 6 7 8 9% 100
Time (Day)
Free Cash{Hi] : $9 cdlbig
Free Cash{Hf]: $9 cd
Free Cash{Hi] : $9 cd0 —
Free Cash{Hf] : Eq
Figure C1_9 5. Free Cash
o 1 2 2% 40 6 70 «8 «690 100
50
Time (Day)
Cash Contrib to Cover Margin: $9 cdtbig
Cash Contrib to Cover Margin: $9 cdl
Cash Contrib to Cover Margin : Sed. —~
Cash Contib to Cover Margin : Eq
Figure C1_9 6. Cash Contribution to Cover
Margin
Margin Needed Equity
20,000 60,000
10,000 50,000
0 40,000 ZL nnn
20,000 20,000
0 10 200 300 40 50 6070 BO 9 100 0 10 6200 300 40 50. 80 90100
‘Time (Day) ‘Time (Day)
‘Margin Needed : $9 cdtbig Equity{Hf] : $9_cdibig $
‘Margin Needed : S9-cd Equity Hf] : S9 cdi $
‘Margin Needed : $9 cd0 —~ Equity[Hf] : $9 cd0 cs $
‘Margin Needed : Eq Equity Hf] : Eq $
Figure C1_9 7. Margin Needed
Figure C1_9 8. Equity
65
Profit
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Profit{Hf] : S9_cdibig
Profit{Hf] : S9 cd1
Profit{ Hf] : S9_cd0
Profit{Hf] : Eq --------
Figure C1_9 9. Profit
As prices diverge before converging after time = 30 days, a hedge fund manager uses
available cash to cover margin and buys more of the long asset in S9_cd1 and S9_cdibig
runs. In comparison, in S9_cd0, a hedge fund manager unwinds both long and short
positions in order to reduce the margin. As a result, final profits and equity in the
S9_cd1big are larger than in S9_cd1 which are in retum larger than profits in the S9_cd0
case. However, in all three cases, the hedge fund survives.
Scenario 10. Divergence Before Convergence: Different Strategies Compared
Fundamental prices of short and long security = P** = P** =100 $/share.
Prices are exogenous, no feedback.
PF+ = 100+STEP(-20,30)+STEP(+20,60) $/share
P*S = 100+STEP(20,20)-STEP(20,30)+STEP(40,30)-STEP(40,60) $/share
66
Price L Price S
100 200
» 170
80 140
60 80
0 0 20 3 40. 50) 60) 670 80100 0 10 20 30 40 50 60 70 80 90 = (100
Time (Day) Time (Day)
Price L: $10_edQ. ———_HH4H4H4H4H4HH4H4H444H4H4H44H44H4H4H4HHHH—_ $/share Price S :$10_ci). ——_________________. #/sinre,
Price L: Eq /share Price $ : Eq $/share
Figure C1_10_1. Price of Security L Figure C1_10 2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days. From time 30 to time 60 days, instead of
converging, prices diverge even more before converging at time 60 days. Note, the
divergence is not as extreme as in the scenarios 5,6 and 8.
In this scenario, a hedge fund manager decides to short-sell as many shares as it goes
long. This scenario is compared to scenario 9 where a hedge fund manager decides to
short-sell as much as allowed within margin requirements, so he would not have a margin
call.
$10_cdi - arun with Cash Decision=1
$10_cd0 - arun with Cash Decision=0
67
Shares Security L
400
200
0 ~~,
0 10 20 30 40 50 ~—-«60 70 80 90 100
Time (Day)
Shares Security L[Hf] :$10_cd0 shares
Shares Security L{Hf] :$9_cd0 shares
Shares Security L[Hf] : S10 cd1 shares
Shares Security L[Hf] - s - ~ shares
Shares Security L[Hf] shares
Shares Security L[ shares
Shares Security L[ shares
Shares Security L[ shares
Shares Security L[ shares
Shares Security L[ shares
Figure C1_10_3. Shares of Security L
Shares Security S
600
0
-600
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Security S[Hf] :$10_cd0 shares
Shares Security S[Hf] :S9_cd0 shares
Shares Security S[Hf]:$10_cd1 ~~ shares
Shares Security S[HE] : - os - aoe ee ~ shares
Shares Security S[HE] : shares
Shares Secutity Sf shares
Shares Secutity S[ shares
Shares Secutity S shares
Shares Secutity S[D] :$9_ shares
Shares Security S[D]:Eq ~~ shares
Figure C1_10 4. Shares of Security S
68
Free Cash Cash Contrib to Cover Margin
60,000 10,000
45,000 Saree 7500 i
ee 4
30,000 eae i 5,000
15,000 I | 2500
0 0 an \
o 0 0 3% 4 5 6 7 8 9 100 o 0 0 0 4 0 0 7 8 % 100
Time (Day) Time (Daj)
Free Cast Hi] :S10_ od s ‘Cash Contrib to Cover Margin: $10 cd) —————___________________ §
Free Cast] $9 ci $ —CashConttto Cover Margin: $9 od $
Free Cast{Hif]|:$10 ofl — 8 ‘Cash Contmb to Cover Margin: $10. od! =:
Free Cast] :$9 oil $ —CashContnbto Cover Nargin:$9 dL $
Free Cast Hf] : Eq. s ‘Cash Contnb to Cover Margin : Eq —§
Figure C1_10_5. Free Cash
Figure C1_10_6. Cash Contribution to
Cover Margin
Margin Needed Equity
20,000 60,000
10,000 50,000
0 40,000 iia
10,000 30,000
20,000 20,000
o m0 0 % 4 50 6 7 8 % 100 o 0 0 9 © 50 6 7 o % 100
Time (Day) ‘Time (Day)
Marja Needed 10. cid $B. S10_otd $
Marj Need Seid $ Eun :89 cd $
Mt | «| RRM el :
Marga Newie 80d $a 89 ot §
‘Manin Needed : Eq 8 Egquay(tf): Eq $
Figure C1_10_7. Margin Needed
Figure C1_10_8. Equity
69
Profit
0 10 20 30 40 50 = 60 70 +80 90 100
Time (Day)
Profit{Hf] :S10_cd0
Profit{Hf] :S9_cd0
Profit{Hf] :S10_ cdl ---
Profit{Hf] :S9_cd1
Profit(Hf] : Eq
Figure C1_10_9. Profit
Here $10_cd0 run is compared to S9_cd0 run, and S10_cdi runis compared to S9_cd1
run. As prices diverge even more at time = 30 days before converging at time = 60 days,
a hedge fund manager tries to cover new margin either by selling security L or by using
free money. A hedge fund does not increase its positions as the spread widens further. In
this case, a hedge fund is better off by using a strategy of shorting as many shares of S as
going long on L.
Scenario 11. An Increase in Maximum Allowed Leverage
Fundamental prices of short and long security = P** = P** =100 $/share.
Prices are exogenous, no feedback.
PF+ = 100+STEP(-20,30)+STEP(+20,60) $/share
P*S = 100+STEP(20,20)-STEP(20,30)+STEP(40,30)-STEP(40,60) $/share
70
Price L Price S
100 200
90 170
80 140
0 110
0 10 6200 «3040 50 60 7080 90 100 i} 10 2 30 40) «(506070 80 9 100
Time (Day) Time (Day)
Price L: S11_ ed). ——————————______—— $/share Price S : $11_cd1. ———————————_____—- $/share
Price L : Eq $/share Price S: Eq — $/share
Figure C1_11_1. Price of Security L Figure C1_11_2. Price of Security S
Therefore, from time = 0 to time = 20 days, prices are the same, then an arbitrage
opportunity window exists from 20 to 30 days. From time 30 to time 60 days, instead of
converging, prices diverge even more before converging at time 60 days. Note, the
divergence is not as extreme as in the scenarios 5,6 and 8.
Maximum Allowed Leverage = 10 (instead of the usual 2)
In this scenario, a hedge fund manager decides to short-sell as much as allowed within
margin requirements, so he would not have a margin call.
$11_cdi - arun with Cash Decision=1
71
Shares Security L
400
300
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Security L|
Shares Secutity L|
Shares Secutity L|
[Hf] :S11_cd1 shares
[Hf] : a cdl shares
HE] : Ec ~~ shares
Shares Security L[D] : sil cdl ~ shares
Shares Security L[D] : i cdl = ~- shares
Shares Security L[D] : Ec shares
Figure C1_11_3. Shares of Security L
Shares Security S
800
400
0
-400
-800
0 10 2 30 40 50 60 70 80 90 100
Time (Day)
Shares Security S[Hf] : S11_cd1 shares
Shares Security S[Hf] : S9_cd1 shares
Shares Security S[Hf] : Eq. —-- shares
Shares Security S[D]:S11_cd1 - -- shares
Shares Security S[D]:S9 cdi ---- see shares
Shares Security S[D] : Eq shares
Figure C1_11_4. Shares of Security S
72
Margin Needed Equity
6000 60,000
ies
5,500 Moo ; 50,000
\ /
-17,000 = i 40,000
28,500 30,000
40,000 20,000
o 0 2% 3 4 50 60 70 80 9 100 o 1 2 3% 4 50 @ 7 8 9% 100
Time (Day) Time (Day)
Margin Needed : $11 edl. —————__________ 5 Equity(H] : S11 cdl
Margin Needed : $9. cdl a — $ Equi{Hf] : $9 cdl
Margin Needed : Eq soseenennnnnnnnnnnnnnnnnnnin § Eqqity(H] Eq)
Figure C1_11_5. Margin Needed Figure C1_11_6. Equity
Profit
Profit{Hf] : S11 cdl $
Profit{ Hf] : S9_cd1 $
Profit{Hf] : Eq
Figure C1_11_7. Profit
Given a higher Maximum A llowed Leverage, a hedge fund can actually borrow more of
an asset S and obtain higher Equity and Profits as can be seen by comparing runs
$10_cdi and S9_cd1. The runs are exactly the same except for the amount of Maximum
Allowed Leverage. In S10_cd1, the Maximum Allowed Leverage is 10, and in S9_cd1,
the Maximum Allowed Leverage is 2.
73
Necessary C ondition for a Hedge Fund to Fail
The necessary condition for a hedge fund to fail if its equity becomes negative.
Mathematically, for a hedge fund to fail, the following inequalities must hold:
Equity = Assets + Liabilities < 0 (eq. 34)
Cash Hf; + Market Value of L Position ; + Credit Balance of S Position ,- Market Value
of S Position ; (eq. 35)
Market Value of L Position; = P,'S!* (eq. 36)
Credit Balance of S Position ; = Total Basis Security S = — P,°S;* (eq. 37)
Market Value of S$ Position; = —P,°S/"* (eq. 38)
Therefore, for a hedge fund to fail, the following inequality should hold:
Cash Hf; + P'S? —PSsis + PSShS <0 (eq. 39)
Cash Hf; + (Pj +Ax)S* — PY SiS + (P~ —Ay)S2*<0 (eq. 40)
Arhitrage Window
Here a special case for a necessary condition for a hedge fund to fail is explored. In this
case, it is assumed that either a spread converged or diverged at time no earlier than
time=t.
Assume S,'" = Sj)" and $;"° = Sj'S, where initial time is when an arbitrage position is
conceived. This assumption makes sense, given no exogenous cash inflows or outflows,
a hedge fund is not going to unwind it position unless at time t, it is hit with a margin call
or the spread converged. This assumption follows directly from the main assumption in
this case that either a spread converged or diverged at time no earlier than time=t.
Therefore, for a hedge fund to fail:
Cash Hf, + (Pj +Ax)Sf*—P,/ SiS + (PY —Ay)Sj'° <0 (eq. 41)
Cash Hf; + (Pj +Ax)S#" — AySi'% <0 (eq. 42)
Assume a hedge fund uses its cash only to buy security L. A hedge fund does not have
any other expenditures and is not buying back security S. This assumption follows
directly from the main assumption in this case that either a spread either converged or
diverged at time no earlier than time=t.
74
Therefore, for a hedge fund to fail:
Cash Hf) —P/ Si" + (Py + Ax)Sj'" — AyS;'* <0 (eq. 43)
Cash Hf + AxS!" — AySiS <0 (eq. 44)
Case 1: Short as Much as Long
In this case,
Sits. sis (eq. 45)
Cash Hf 9 + AxSj'" — AyS;'* =Cash Hf) + (Ax + Ay)Sj'" <0 (eq. 46)
So, for example, final equity and profits are the same for scenarios 2 and 4 depicted
above. In scenario 2, P,’ =80 $/share, P,° =120 $/share. In scenario 4, P,’ =100 $/share,
P, =140 $/share, and in both scenarios, P, =P,6 =100 $/share. In scenario 2, Ax =20
$/share, and Ay =20 $/share. In scenario 4, Ax =0 $/share, and Ay =40 $/share.
Therefore, Cash Hf) + AxS/'" — AyS#'S =Cash Hf + (Ax + Ay)S,)"is the same for two
scenarios. In both scenarios, hedge funds have the same equity and final profits.
Case 2: Keep Within Margin Requirements
In this case,
plight
- 8S = os (eq. 47)
0
HL HS ut, AyPy So” _
Cash Hf + AxS;" — AyS;'* =Cash Hf) + AxS; +—ps = Cash Hf o+
0
L
Si(ax +s )<0 (eq. 48)
0
So, for example, final equity and profits are different for scenarios 1 and 3 depicted
above. In scenario 1, P,’ =80 $/share, P,° =120 $/share. In scenario 3, P,’ =100 $/share,
P,° =140 $/share, and in both scenarios, P," =P,° =100 $/share. In scenario 1, Ax =20
$/share, and Ay =20 $/share. In scenario 3, Ax =0 $/share, and Ay =40 $/share.
75
A : HLL AyPy° So""Po
Therefore, in scenario 1, Cash Hf) + Sy“ (Ax + ps ) =Cash Hf) +20 ot
0 0
29 Su “Po
Pe
L H,LpL
In scenario 3, Cash Hf 9 + Sj'"(Ax + ats ) =Cash Hf) +40 se Py
0 0
Since P,° >P,', equity and final profits in scenario 1 will always be higher than those in
scenario 3, even though Ax + Ay is the same for both scenarios. The result is due to an
understanding that only proportional, and not absolute increases and decreases in prices
of underlying assets matter in calculation of equity and profits.
76
9. Part 1 - Three Agents
Dynamic Hypotheses
If arbitrage spreads widen, as happened in May, 1998 for LTCM, people start liquidating,
therefore, further depressing the price of an illiquid asset L and increasing the price of a
liquid asset S. In this case, a preference for liquidity increases. As preference for
liquidity increases, that forces more people to liquidate illiquid positions and buy liquid
ones. People were willing to buy Treasuries at any price as long as they got out of the
risky bonds and obtained the less risky instruments. Everybody on the street started
talking about “flight to quality” or buying Treasury bonds. That lead to losses of LTCM.
Owing to its loss of capital, Long-Term’ s leverage had become very high, because losses
accumulate faster as leverage increases. Therefore, they wanted to sell something. At
that point, leverage was very high, and the fund's partners were looking forward to sell
several positions and raise more money before the end of the month. LTCM knew it had
to reduce its positions, but couldn’t with markets under the stress. There was no liquidity
in the market. Everybody wanted to be out at the same time - something that models
missed. When losses mount, leveraged investors such as Long-Term are forced to sell,
lest their losses overwhelm them. When a firm has to sell without buyers, prices are very
low. In addition, Wall Street players learned more about the fund's positions, and went
against them. They wanted to “squeeze” as much as possible from the fund, knowing
that if the fund would get help from the government, it will be able to buy back its shorts.
Therefore, anybody who held those securities would make money. In September 1998,
many banks were exposed to the same positions as LTCM. Therefore, to cut their losses,
they unwound those positions, thus, hurting LTCM. Therefore, both cutting the losses
and predatory trading led to the collapse of the fund.
As price of illiquid security L goes down, therefore, the net asset value of a hedge
fund goes down. That in turn leads for the collateral value to decrease. Lenders either
require more collateral, or in case of many leveraged hedge funds, they pressure the
hedge fund (through the dealer) to sell the assets. As it usually happens, hedge funds use
their own assets as collateral, which leads to this vicious loop: R1 Collateral Collapse
described in Figure 6. Note, large hedge funds like LTCM are considered in this causal
diagram as only large hedge funds can have market impact.
77
Shares of Illiquid
Asset L Owned by
ie Net Asset HF
Value ~~~ :
+ Equity
Price of Iliquid -
Asset L
+
R1 Leverage
Net Buy/Sell Collateral
Balance Collapse ty -
- Collateral
Value
Shares of Illiquid
Asset L Sold by
HF
Figure 6. Collateral Collapse Feedback
As is described in the case, the pressure to sell usually decreases price, leading to
an increase in volatility (negative trend of an illiquid security L) of an asset leading to
more pressure to sell by lenders. This dynamic is described in Figure 7.
78
_ Shares of Hiquid
~ Asset L Owned by
- pitaede HF
ge * Value Se, :
+ Equity
Price of liquid \
Asset L
rage
Negative Trend in R2 li f+ RI Lever \
Pre of Mig Asset (* Flight A NetBuvisel Collateral . \
Liquidity Balance Collapse th
\ \ oy Dealer 4 Collateral |
Value
Shares of liquid
AssetL Sold by, Pressure to
Beale
Preference to Own
Asset L by Dealer;
Figure 7: Flight to Liquidity by Dealer Feedback
Dealers do not want to be in a position to be left with an inventory of illiquid security L
as prices of that security continue to go down. Therefore, they prefer to be buy less of
security L on their own account, further exacerbating the Net Buy/Sell Balance. As is
depicted in Figure 7, as price of security L goes down and a hedge fund is forced to sell
more shares, preference to owning those shares by a dealer goes down.
Net Assets (A) of a hedge funds equals the sum of Price (P) multiplied by Shares
(S) for each position in a fund: A = xs, PB (eq. 49)
i=l
Equity (E) equals to net Assets (A) minus Leverage (L): E = A-—L (eq. 50)
Collateral Value (C) equals to Assets minus Leverage: C =A-L. (eq. 51)
In this case, E is the same as collateral. So, if A decreases to Aj, then the new collateral
to be posted is L/0.5- Ai.
The collateral to be posted is max(0, L/0.5- A;). (eq. 52)
Price is assumed to take the following form: P, =a, — £,AS,. (eq. 53)
So price is anchored to some fundamental value @,, and is adjusted according to £,AS, ,
where £, is a illiquidity proxy for the asset, and AS, is the volume of net stock sold.
Therefore, if #, is high, then the price impact of a sell is very large. We expect that to
happen for illiquid stocks or during “liquidity crunch.” The “liquidity crunch” or “flight
79
to quality” by a dealer is depicted in R2 Flight to Liquidity by Dealer reinforcing loop
and by a momentum investor in R3 Flight to Liquidity by Momentum Investors depicted
in Figure 8.
— Shares of Iliquid
— Asset L Owned by
—e Net Asset
Value ~~. .
3 Flight ; Baily
Liqudity A Price of liquid
‘ orem Asset L
Negative Trend in veSIDIS, ft | a
Price of Miqud Asset Net Buy/Sell B.) =]
Pressure to Sell by ee cies
| ~~4 Momentum investors”
watt,
si J"
(eye Tea» oY
Liquidity ssetL Sold by eee
by Dealer, Dealer
Preference to Own. 7 NS
Asset L by Dealer; =
Figure 8: Flight to Liquidity by Momentum Investors
Momentum investors are investors that make decisions on buying and selling stock by
following a price of the asset. If price is going down, they extrapolate the trend and
decide that the price would go down even more, thus, these investors prefer to own less
of that asset. The reverse is true. However, it is important to note that momentum
strategy is not solely responsible for collapse of a hedge fund. In LTCM case, there were
several types of players in the market: 1) momentum (analyzed in Figure 8), 2) imitators
(analyzed in Figure 9), and 3) predators (actively trying to bankrupt LTCM).
Figure 9 depicts a reinforcing loop that leads to a collapse of a hedge fund where
investors are imitators. They follow and imitate buy and sell orders of the hedge fund.
80
Shares of Iliquid
Asset L Owned by
Net Asset! HE
R2 Flight Value ae
{ to ; + Equity
Liquidity Price of liquid ;
i \ orden Asset L
Negative Trend in ———_ f+ Leverage
Price of Illiquid Asset Net Buy/Sell
L
* Balance Collapse 4 ¥ |
: Collateral
RA lig J”
Pressure To Sell 4 Liquidity Shares of Mig
e id
Preference to Own by Imitatn '
Paleo b Ov P \qmitlos/ ASeTL Soldby 4 Presueto
Figure 9: Flight to Liquidity by Imitators
As more shares of illiquid security L are sold by a hedge fund, more shares of security L
are sold by imitators, thus leading to a lower price, and leading to collateral collapse.
Note, that for analysis, it is important to differentiate between large and small
relative to the marketplace hedge funds. For example, LTCM was a large hedge fund,
and its buying and selling significantly contributed to swings in prices in those assets.
However, a small hedge fund can decide or even be forced by a dealer to sell its
securities, and will have virtually no impact on prices of these securities. For smaller
hedge funds the reinforcing loops R1-R4 are not that strong due to a minimal price
impact. For smaller hedge funds, mostly balancing loops in Figure 4: Hedge Fund
Decisions How to Deal With Margin are in place.
Assumptions
1. A hedge fund’s strategy is a statistical arbitrage. Therefore, a hedge fund tries to
buy an undervalued security (long) and short sell an overvalued security (short).
The hedge fund is trying to have the same amount of shares of both long and short
securities.
2. A typical hedge fund does not typically have a lot of cash and will invest most of
its available cash into long securities. It will invest most of its available cash
81
(Fraction of Free Cash Invested = 0.8). It will use long securities and cash as a
collateral to borrow short securities.
3. A dealer will always take the other side of the hedge fund order unless there is a
“liquidity crunch” time.
4. There are three agents: a hedge fund, a dealer and an investor
5. An investor can take several strategies: momentum, imitator, and noise.
6. Price is endogenous. Fora large hedge fund, there is a price impact - selling and
buying a security changes the price of the security.
7. A hedge fund starts with no exposure to the long and short positions.
8. When arbitrage opportunities go away, a hedge fund liquidates both short and
long positions. Therefore, in this model, both position forming and unwinding are
modeled.
9. Federal and house requirements are 50%.
10. No minimum dollar requirements under regulation T, house and NY SE
requirements.
11. No interest charge in a margin account, and no interest credit in a short account.
Formulations
Investor Cash
Cash; =) (Income; - Consumption; + Cash Increase! - Cash Decrease;)dt
(eq. 54)
Cash Increase! = Sell Rate;'>* P,S + Sell Rate" * P, (eq. 55)
Cash Decrease;'=Buy Rate; * P,' +Buy Rate’ * PS Buy Rate!" * P," (eq. 56)
si Sps
w's= (ant eq. 57
' SP he SSBF (a, 59)
sitpl
wits 1 tt eq. 58
O° Sophy sp fo 28)
82
Pricing
Pp‘ = P*** Effect of demand supply balance on price Security S (eq. 59)
pes =| (Change in Expected Price S,)dt (eq. 60)
pS — pes
Ch inE ted Price S,= . 61
ange eee mice st Time to Adjust Expected Price Security S ea G1)
P= P!+* Effect of demand supply balance on price Security L (eq. 62)
pe =| (Change in Expected Price L,)dt (eq. 63)
pi _ pet
Change in Expected Price L,= : : (eq. 64)
Time to Adjust Expected Price Security L
Momentum Investor
wes = 1. we (eq. 65)
we" =]* Table for Desired Equity Weight L(F orecast Price Relative to Current Price L;)
(eq. 66)
where | is preference for liquidity fraction
Forecast Price Relative to Current Price L, =Forecast Price L/Perceived Price L;
(eq. 67)
Forecast Price Lt = Perceived Price L;*(1+Trend in Price L;*(Price Forecast
Horizon+Time to perceive price)) (eq. 68)
Perceived Price L, = j (Change in Perceived Price L,)dt (eq. 69)
LL . .
Change in Perceived Price Lr= Ey — Perceived Price L,. (eq. 70)
Time to Perceive Price L
Imitator Investor
Desired Sale Rate," = Desired Sale Rates?" (eq. 71)
Desired Sale Rate;’> = Desired Sale Rate;'® (eq. 72)
Desired Buy Rate," = Desired Buy Rate," (eq. 73)
Desired Buy Rate,’’ = Desired Buy Rates (eq. 74)
83
Results
Scenario 1. Large Hedge Fund and Momentum Investor. Prices Diverge Instead of
Converging
There are three players present: investor, dealer and a hedge fund. An investoris a
momentum investor. A dealer has a preference for liquidity when the price of illiquid
asset L has a high negative trend, and a hedge fund is forced to sell due to margin call.
Prices of illiquid security L and liquid security S are endogenous.
A model starts in equilibrium. P,' = P,° = 100 $/share
S}’" = Sy" =400 shares. Sj'* = 0 shares. S?* = S;’S =400 shares.
Sj’ = 0 shares. Cash j° = $40,000 dollars
Inv1_cd1 is a simulation where Cash Decision = 1
Inv1_cd0 is a simulation where Cash Decision = 0
Desired Equity Weight L Inv = Investor Fraction* (Preference for Liquidity
Fraction*Table for Desired Equity Weight L(Forecast Price Relative to Current Price
L))
Investor Fraction=1
Preference for Liquidity Fraction= 1-STEP(0.1,10)+STEP(0.1,20)-
STEP(0.8,20)+STEP(0.8,30) is an exogenous input
84
Preference for Liquidity Fraction
4 [|
0.75
0.5
0.25
0
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Preference for Liquidity Fraction: Invl cdl. ———————__L Dm
Preference for Liquidity Fraction : Inv1_cd0 Dmnl
Preference for Liquidity Fraction : Eq
Figure 2_1_1. Preference for Liquidity Fraction
Price L. Price S
200 800
150 re 600
fh
100 Hs } 400 |
} ae |
} } | ~\
50 YU 200 iL Kore wa at \ TY
cl ALIS
0 0
o 0 20 3% 4 50 6 7 80 90 100 o 0 20 0 © 5 6 7 & 9% 100
Time (Day) Time (Day)
Price L :Inv1 ed S/share Price S:Invl_ od $ishare
Price L: Inv1_ ed S/share Price S: Inv odd Sishare
PriceL: Eq —— — §fshare Paige §: Eq) meena ———— s/share
Figure 2_1_2. Price of Security L
85
Figure 2_1_3. Price of Security S
Shares Security L
Time (Day)
Figure 2_1_4. Shares of Security L
Shares Security S
-600
0 10 20 30 40 50 60 70 80 90 ~=100
Time (Day)
Shares Security S[Inv] : Invi_cd1 shares
Shares Security S[Inv] : Inv shares
Shares Security [Inv] : Eq shares
Shares Security S[Hf] : ines shares
i
i
Shares Security S[Hf] : Inv1_cd0 shares
Shares Security S[Hf] : Eq shares
Figure 2_1_5. Shares of Security S
86
Free Cash Cash Contrib to Cover Margin
40,000 | 40,000
30,000 " 30,000
\ Al h
20,000 { fh 20,000 aa
SIT AN TL lA
10,000 t | | i iH ey 10,000 i \ /
\ \ \
‘ ULL oLL LIA LN
0 0 20 30 40 50 60 70 80 90 100 0 10 22 30 40 50 60 70 80 90 100
Time (Day) Time (Day)
Free Cash{Hf] : Invi cdl $ ‘Cash Contrib to Cover Margin: Inv1_cd1 $
Free Cash{ Hf] : Inv1_cd0 aaeen — § ‘Cash Contrib to Cover Margin : Inv1_cd0 $
roe CaniQHi]: Rg) © ef Cash Contrib to Cover Margin : Eq: =——————————-—— §
Figure 2_1_6. Free Cash Figure 2_1_7. Cash Contribution to Cover
Margin
Equity
60,000
30,000
0
-30,000
-60,000
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Equity[Hf] : Inv1_cd1 $
Equity[Hf] : Inv1_cd0 $
Equity[Hf] : Eq a $
Figure 2_1_8. Equity
87
Profit
4,000 =
\/
-17,000
-38,000
-59,000
-80,000
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Profit{Hf] : Invi cd1 $
Profit{Hf] : Invl_cd0 $
Profit{ Hf] : Bq. --n--nnn $
Figure 2_1_9. Profit
From time = 0 to 10 days, no arbitrage opportunities exist. From time = 10 to 20 days,
arbitrage opportunities exist for a hedge fund as there is a spread between a cheaper
illiquid security L and liquid security S. From time = 20 to 30 days, instead of
converging, the spread diverges even more, thus, a hedge fund is hit with a margin call.
Asa hedge fund is hit with a margin call, hedge fund has to sell the illiquid security L
(long) at further depressed prices. In this case, it is better for a hedge fund to unwind its
positions as quick as possible instead of “throwing good money after bad,” as LTCM did.
This scenario depicts exactly what happened to LTCM. LTCM was a large fund, and by
unwinding its positions, it contributed to a further decrease in prices of security L.
Investors were afraid that the drop of prices was going to continue and started selling
more of security L, thus further reducing the prices. LTCM did not have enough cash
from other positions to finance the ever increasing margin calls from brokers. However,
it tried to hang on to the positions without selling them. At LTCM hedge fund managers
would double up the exposure of a position when spreads widen even further as they
88
firmly believed in their strategy that spreads would narrow in the near future
(Lowenstein, 2000).
As can be seen in this scenario, a hedge fund runs out of all available cash by time
= 25 days, before spreads converge. After that, the hedge fund manager has to sell
securities at even lower prices in order to cover a margin call leading to lower profits, and
collapse of a hedge fund. As a result, if a large hedge fund does not have enough funding
to finance margin calls through widening of spreads, it is always better for the hedge fund
to unwind its positions as soon as a margin call is issued. As a hedge fund is selling its
positions, thus decreasing the price, momentum investors demand less of the security.
Also, in the times of liquidity crunch, as it happened in the 10 day window, a dealer
prefers not to buy the illiquid securities as he is afraid of being left with an inventory that
is spiraling down in value.
Scenario 2. Small Hedge Fund and Momentum Investor. Prices Diverge Instead of
Converging
There are three players present: investor, dealer and a hedge fund. An investoris a
momentum investor. A dealer has a preference for liquidity when price of illiquid asset L
has high negative trend, and a hedge fund is forced to sell due to margin call. Prices of
illiquid security L and liquid security S are endogenous.
A model starts in equilibrium. P,' = P,° = 100 $/share
S)’" = S;" =400 shares. Sj" = 0 shares. S?* = S;'* =400 shares.
Sj'* = Oshares. Cash ;’ = $1,000 dollars. The difference between Simulation 2 and
Simulation 1 is that in simulation 1, a hedge fund is large compared to other market
players, and in simulation 2, a hedge fund is small.
Invs_cd1 is a simulation where Cash Decision = 1
Invs_cd0 is a simulation where Cash Decision = 0
Desired Equity Weight L Inv = Investor Fraction* (Preference for Liquidity
Fraction*Table for Desired Equity Weight L(Forecast Price Relative to Current Price
L))
Investor Fraction=1
89
Preference for Liquidity Fraction= 1-STEP(0.1,10)+STEP(0.1,20)-
STEP(0.8,20)+STEP(0.8,30) is an exogenous input
Preference for Liquidity Fraction
1 L{ |
0.75
0.5
0.25
0
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Preference for Liquidity Fraction : Invs_cd1 Dmnl
Preference for Liquidity Fraction : Invs_cd0 Dmol
Preference for Liquidity Fraction : Eqs Dmol
Figure 2_2_1. Preference for Liquidity Fraction
Price L Price $
ir
0 0 +
0 1 0 3 4 50 6 0 8 9% 100 0 0 20 3 4 50 60 6 (8D 100
Tine (Day) Time (Day)
Price L: Invs od S/share Price : Invs edl S/share
Price L : Invs odd $/share Price : Invs cd S/share
Pile Lig Rg a sseeessssessesrssesrecetseeessceeecemenn Seto Dice §..iqs sss S/share
Figure 2_2_2. Price of Security L
Figure 2_2_3. Price of Security S
90
Shares Security L
600
450
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Security L[Inv] : Invs_cd1 shares
Shares Security L[Inv] : Invs_cd0 shares
Shares Security L[Inv
Eqs shares
Hf]: Invs_cd1 - -- shares
Hf] :Invs_cd0 ~~ -- ~~ shares
THE] : Eqs shares
Shares Secutity L|
Shares Secutity L|
Shares Secutity L|
Figure 2_2_4. Shares of Security L
Shares Security S
600
ce BO
297
145.5
-6
0 10 = 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Security S[Inv] : Invs_cd1 shares
Shares Security S[Inv] : Invs_cd0 shares
Shares Security S[Inv
Shares Security S[HE] :Invs_cd1 ---
Shares Security S[Hf] :Invs_cd0 ~~ + shares
Shares Security S[Hf] : Eqs shares
-- shares
[
[
[Inv]: Eqs —~ shares
i Seoetnewe
[
L
Figure 2_2_5. Shares of Security S
91
Cash Contrib to Cover Margin
Free Cash
4,000 600
i
3,000 A 450 i\
|
i | I\
2,000 ++ 300 ++
1 al |
AWN I} \
1,000 150 f
a Ht j
AN | TAN TA TT
0 at ae! wt 0 !
0 10 200 300 40 50 7 100 0 1 2 3 40 50 6 7 8 % 100
‘Time (Day) ‘Time (Day)
Free Cash{Hf] : Invs cdi $ Cash Contrib to Cover Margin: Invs od1 $s
Free Cash{Hf] : Invs_cd0 $ Cash Contrib to Cover Margin: Invs cd s
—$ Cash Contrib to Cover Margin: Eqs°——————————~§
Free Cast] ¢ Bg
Figure 2_2_6. Free Cash
Figure 2_2_7. Cash Contribution to Cover
Margin
Equity
0 10 20 30 40 50 = 60 70 ~=80 90 100
Time (Day)
Equity[Hf] : Invs cd1
Equity[Hf] : Invs_cd0
Figure 2_2_8. Equity
92
Profit
0 10 20 30 40 50 60 70 ~80 90 100
Time (Day)
Profit{Hf] : Invs cd1 $
Profit{Hf] : Invs_cd0 $
Profit{He] : EQS) <-------nnnnn $
Figure 2_2_9. Profit
From time =0 to 10 days, no arbitrage opportunities exist. Starting time = 10 days,
arbitrage opportunities exist for a hedge fund as there is a spread between a cheaper
illiquid security L and liquid security S. From time = 20 to 30 days, instead of
converging, the spread diverges even more, thus, a hedge fund is hit with a margin call.
Asa hedge fund is hit with a margin call, hedge fund has to sell the illiquid security L
(long) at further depressing prices. Compared to scenario 1, where a large hedge fund is
present, selling security L does not impact the price alot. Investors are adjusting their
desired equity weights for securities L and S. As price of L goes down, the desired
equity weight of security L by investors goes down, and the actual equity weight of
security L goes down too. Therefore, investors are not selling as much of L as in
scenario 1.
Generally, by time = 30 when preference for liquidity went away and spreads
converge, a hedge fund is in a similar position in both runs (Invs_cd1 and Invs cd0). A
hedge fund is a little bit better in Invs_cd0 run. Therefore, for a small hedge fund, once it
is hit with a margin call, it is better to unwind its positions rather than finance it with
93
available cash (throwing good money after bad), given that a hedge fund does not have
enough cash to finance all its margin calls.
Scenario 3. Large Hedge Fund and Imitator Investor. Prices Diverge Instead of
Converging
There are three players present: investor, dealer and a hedge fund. An investor is an
imitator. A hedge fund is large compared to other players in the market. A dealer has a
preference for liquidity when price of illiquid asset L has high negative trend, and a
hedge fund is forced to sell due to margin call. Prices of illiquid security L and liquid
security S are endogenous.
A model starts in equilibrium. P,' = P,° = 100 $/share
S)’" = S;" =400 shares. Sj’ = 0 shares. S?’* = S;'° =400 shares.
Sj’ = 0 shares. Cash .' = $40,000 dollars.
Invf_cd1 is a simulation where Cash Decision = 1
Invf_cd0 is a simulation where Cash Decision =0
P,: =Expected Price L*Effect of demand supply balance on price Security L*(1-
STEP (0.1,10)+STEP(0.1,20)-STEP(0.8,20)+STEP(0.8,30))
Price L Price $
400 4000
300 3,000
1
0 LI ‘ 0 - ae
n 0 3
0 4 5 6 670) 8) 90 100 0 10 20 3 40 50 6 7 8 90 100
Time (Day) Time (Day)
Price: Inf cl ————————————— sta Price: vf of) ——————H §/suare
Price: lf ci Sishae Price S: vod sishare
Bt: Ep ee ~ Stare Prioe $3 Eq) ——_$ $$$ §are
Figure 2_3_1. Price of Security L Figure 2_3 2. Price of Security S
94
Shares Security L
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
shares
shares
shares
shares
shares
shares
Figure 2_3 3. Shares of Security L
Shares Security S
0 10 20 30 40 50 60 70 80 90 ~=100
Time (Day)
Shares Security S[Inv] : Invf_cd1 shares
Shares Security S[Inv] : Invf_cd0 shares
Shares Security S[Inv] : Eq -- shares
[
L
Shares Security S[HE] : Inve. cdl — -- shares
Shares Security S[Hf] : Invf_cd0 wee noes shares
Shares Security S[Hf] : Eq shares
Figure 2.3 4. Shares of Security S
95
Free Cash Cash Contrib to Cover Margin
60,000 4,000 ij
|
\
45,000 3,000 i)
|
30,000 Nd 2,000 | |
1
15,000 NL 1,000 |
||
0 0
0 10 20 30 «40 «650 660) «670 = 8090100 0 1 2 30 40 50 60 70 8 90 100
Time (Day) Time (Day)
Free Cash{ Hf] : Invf_ cdl. ——————___________— § Cash Contrib to Cover Margin : Invf cdl ————————___—-. §
Free Cash{ Hf] : Invf_cd0 $ Cash Contrib to Cover Margin : Invf_cd0 $
Free Cash{Hf] : Eq. mene § Cash Contib to Cover Margin : Eq) ——————————--— §
Figure 2_3_5. Free Cash Figure 2_3_6. Cash Contribution to Cover
Margin
Equity Profit
200,000 0000
150,000 50,000 errr |
100,000 2,000 |
50,000 10000 {TI
AI
0 -40,000
0. 0 0 3% 4 5 6 7 8 9% 100 00 0 0» 0 3 6 7 8 9 100
Time (Day) Time (Day)
Equi]: vt od § Profi]: ivf cdl $
Euity( Hf]: Ivf odd $ Profi]: ivf odd §
Equiy(Hif): Eq $Y s Eq ea §
Figure 2_3_7. Equity Figure 2_3_8. Profit
From time = 0 to 10 days, no arbitrage opportunities exist. Starting time = 10 days,
arbitrage opportunities exist for a hedge fund as there is a spread between a cheaper
illiquid security L and liquid security S. From time = 20 to 30 days, instead of
converging, the spread diverges even more, thus, a hedge fund is hit with a margin call.
Asa hedge fund is hit with a margin call, the hedge fund has to sell the illiquid security L
(long) at further depressing prices. Compared to scenario 1, here a hedge fund has
enough cash to finance margin calls before spreads narrow. In this scenario, Free Cash is
96
always positive. In this case, it is better to use cash, as equity during the times of spread
divergence and final equity are higher for a case Invf_cd1 where a hedge fund actually
uses cash to finance its margin compared to case Invf_cd0 where a hedge fund unwinds
its positions once hit with a margin call.
Scenario 4. Large Hedge Fund and Imitator Investor. Prices Diverge and Never
Converge
There are three players present: investor, dealer and a hedge fund. An investor is an
imitator. A hedge fund is large compared to other players in the market. A dealer has a
preference for liquidity when price of illiquid asset L has high negative trend, and a
hedge fund is forced to sell due to margin call. Prices of illiquid security L and liquid
security S are endogenous.
A model starts in equilibrium. P,' = P,° = 100 $/share
S)’" = S;" =400 shares. Sj'* = 0 shares. S)’* = S;'° =400 shares.
S,'* = 0 shares. Cash ,' = $40,000 dollars.
Invf_cd1 is a simulation where Cash Decision = 1
Invf_cd0 is a simulation where Cash Decision = 0
P' =Expected Price L*Effect of demand supply balance on price Security L*(1-
STEP (0.4,10)+STEP(0.4,20)-STEP(0.8,20)+STEP(0.8,30))
Price L Price $
i a [ett
0 0 0 3 4 50 6 7 8 0 100 0 10 2 30 40 50 60 70 80 90 100
Time (Day) Time (Day)
Price L : Invf cdl test ————————————_—— /share Price S : Invf cd1_ test! ————__________— $/share
Price L: Inf cd0_test share Price S: Invf_cd0_testh Sshare
Price Ls Eq mene Shane Price § : Eg) neeeeeeeeeennnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn sishare
Figure 2_4 1. Price of Security L Figure 2_4 2. Price of Security S
97
Shares Security L
200 VA
A
0 / ofan
0 10 20 30 40 50 60 70 80 90 100
Time (Day)
Shares Security L{Inv] : Invf_cd1_testl shares
Shares Security L{Inv] : Invf_cd0_te shares
Shares Security L{Inv] : Eq. -- shares
shares
shares
shares
Figure 2_4 3. Shares of Security L
Price S
Time (Day)
Price S : Invf_cd1_testl $/share
Price S : Invf_cd0_test1 $/share
PriceS: Eq —
Figure 2_4 4. Shares of Security S
98
Free Cash Cash Contrib to Cover Margin
40,000 [7] 40,000
\
30,000 \ 330,000
20,000 N 20,000 ber
| {
10,000 \ 10,000 {
\
0 \ 0 |
0 0 20 3 40 50 6 7 8 9%
0 0 20 30 4 5 6 7 8%
Time (Day) Time (Day)
Free Cash{Hf]: Inf cdl test) ———————— § __CashContrib to Cover Margin: Invf cdl test! ——_—_——— 5
Free Cash{ Hf]: Inf cd test $ Cash Conti to Cover Margin Inv cd tetl $
Free Cast(Hi]: Eq — a ———- § _CashContib to Cover Margin: Eg —__________—. §
Figure 2_4 5. Free Cash
Figure 2_4 6. Cash Contribution to Cover
Margin
Equity Profit
60,000 00
‘aI LS anaaeee
5,000 ee $9,400
-70,000 199,600
135,000 ‘| am
-200,000 J 400,000
. 0 0 0 4 3 a 70 60 9) 100 . 0 0 9 4 5 6 70 a) 00 100
Time (Day) Time (Day)
Equity(Hf]: Inf cdl test) ————————— Profi]: inf cil test). _____________-
uit]: lv cd testt § Profi]: vf cites $
Equiy(Hf]: Eq § Profi]: Eq)
Figure 2_4 7. Equity
Figure 2_4 8. Profit
From time = 0 to 10 days, no arbitrage opportunities exist. Starting time = 10 days,
arbitrage opportunities exist for a hedge fund as there is a spread between a cheaper
illiquid security L and liquid security S. From time = 20 days, instead of converging, the
spread diverges even more, thus, a hedge fund is hit with a margin call. As a hedge fund
is hit with a margin call, hedge fund has to sell the illiquid security L (long) at further
depressing prices. Compared to scenario 3, here a hedge fund does not have enough cash
to finance margin calls before spreads narrow. In this case, spreads never narrow. As
can be seen from the runs, a hedge fund fails if it uses a strategy of using available cash
99
to finance margin calls (Invf_cd1_test1). It survives if once a hedge fund is hit with a
margin call, it unwinds its positions. It is true that by closing its positions, investors
imitate the hedge fund and close positions, thus depressing prices even more. However,
this effect is present in both runs : Invf_cd1_testl and Invf_cd1_test0. By financing
margin with available cash, a hedge fund only introduces a lag in this relationship, and
will have to unwind its positions at a later time at lower prices.
Scenario 5. Large Hedge Fund and Noise Investor. Prices Diverge Instead of
Converging
There are three players present: investor, dealer and a hedge fund. An investor is a noise
trader. A hedge fund is large compared to other players in the market. A dealer has a
preference for liquidity when price of illiquid asset L has a high negative trend, and a
hedge fund is forced to sell due to margin call. Prices of illiquid security L and liquid
security S are endogenous.
A model starts in equilibrium. P,' = P,° = 100 $/share
Sp?" =$) =400 shares. Sj'' = 0 shares. S$? = Sj'° =400 shares.
Sj’ = 0 shares. Cash ,° = $40,000 dollars.
In this case (to be shown in the next draft), a hedge fund is better off using cash to
finance off its margin call instead of unwinding the positions. As hedge fund sells
illiquid positions, the price of those positions goes down due to a price impact of the
large hedge fund. Therefore, if a hedge fund has enough cash to finance margin before
positions converge, a hedge fund is better off following this strategy.
100
10. Implications for Risk Management
Using Value at Risk Analysis (VAR) explained below, VAR =1.65A6 , where 5
is the standard deviation of the hedge fund performance, and is the squared root of the
variance: 5? = }'P’:S457 + )°)°o,,P,S,57P,S,5), where o, is the correlation
i=l i=l jal
between assets (i) and (j) held by a hedge fund. It is important to note that during
“liquidity crunch” positions that were previously not correlated, become dependent. In
the context of this model, as £, or liquidity preference become high for hedge fund
positions, then com becomes higher, therefore, inflating the variance of the hedge fund.
Hence, Value at Risk of the hedge fund increases.
Value At Risk Analysis (VAR)
VAR describes how risky a stock is.
VAR is the maximum expected loss over a given horizon period at a given level of
confidence C i.e., the maximum likely loss.
VAR depends upon two arbitrarily chosen parameters: the horizon period (daily, weekly,
monthly, quarterly, etc.) and the level of confidence (90%, 95%, 99%, 99.9%, etc.).
Origins: October 1994, J.P. Morgan, RiskMetric
For example: Results show that 99% quarterly VAR is $.767 million. It means
that the probability of losing more than $767,000 over a quarter is less than or equal to
01.
Calculation of VAR: VAR=Market Cap* Standard Deviation* 1.65 where
Market Cap ($Billion)
Standard Deviation (s.d. of monthly returns)
1.65 is the one side 5% point (Prob(z<-1.65)=0.05 if z obeys standard normal)
VAR ($Billion, 5%, 1 month)
101
11. Discussion and C onclusion
A hedge fund manager will conceive an arbitrage trade if the manager sees mispricing in
illiquid and liquid assets. The manager would buy a cheaper illiquid asset (security L)
and will short a more expensive liquid asset (security S). Typically, a hedge fund
manager has two ways of getting into the arbitrage trade. Of course, any combinations of
these two options are also present in hedge fund strategies. In the first case, a hedge fund
manager would max out on security S and security L positions, subject to being within a
limit of an allowable margin. For example, if a hedge fund spent all of its cash and
bought $100,000 worth of security L, a hedge fund manager can at most sell $100,000
worth of security S in the case of maximum leverage equals to two. In the second case, a
hedge fund manager will short sell as many shares of security S as it would go long on
security L. For example, if a hedge fund buys 1,000 shares of security L for $80
($80,000 total value), a hedge fund manager will go 1,000 shares short on security S that
sells for $120 ($120,000). Given a hedge fund cash is 0, a hedge fund manager is going
to be hit with a margin call from a dealer.
Once a margin call is issued, a hedge fund has two options: liquidate the
conceived positions or use money from other trades to finance this position. In the paper,
itis shown that a hedge fund that is not constrained to keep within margin requirements
and has cash from other accounts at its disposal, will earn more profits compared to a
similar hedge fund that is constrained to keep within margin requirements, given that
arbitrage strategy is successful. However, the opposite is true in a case where a hedge
fund does not have an access to cash from other accounts. In this case, a hedge fund that
keeps within margin requirements will earn more profits compared to a hedge fund that is
not constrained to keep within margin requirements, given that arbitrage strategy is
successful. In this case, a hedge fund that is not constrained to operate within margin
requirements, is hit with a margin call that a hedge fund manager will only be able to
fulfill by unwinding the positions. In the end, the value of the position is much smaller
compared to the case where a hedge fund manager is constrained to stay within hedge
fund margin requirements, resulting in smaller profits.
In a case when a spread widens before narrowing without causing a hedge fund
collapse in the interim, a hedge fund is better off by following a strategy of shorting as
102
many shares as it has long compared to a strategy of keeping within margin requirements.
This assumes that a hedge fund does not increase its exposure to both S and L securities
as spread unexpectedly widen up. This is true for small hedge funds; however, in
LTCM, many traders followed an approach that if spreads widen up, they would double
up their exposure. If a hedge fund follows a strategy of keeping within margin
requirements, a hedge fund is better off by using cash from other accounts to finance a
margin due to widening spreads.
In a case that an arbitrage position (strategy: short as much as long) goes against
a hedge fund manager, i.e. the spread between positions L and S widen, a hedge fund
manager prevents a collapse by unwinding positions as soon as margin call is issued. If a
hedge fund manager decides to finance the margin with other money by “throwing good
money after bad,” a hedge fund will collapse. This conclusion is applicable to all small
hedge funds that are not price-makers in the marketplace. If a hedge fund decides to keep
within margin requirements, then the likelihood of a hedge fund collapse given that
margin is financed with new money or by unwinding positions is the same.
If a large hedge fund does not have enough funding to finance margin calls
through widening of spreads, it is always better for the hedge fund to unwind its positions
as soon as a margin call is issued. As a hedge fund is selling its positions, thus
decreasing the price, momentum investors demand less of the security. Also, in the times
of liquidity crunch, a dealer prefers not to buy the illiquid securities as he is afraid of
being left with an inventory that is spiraling down in value.
If a large hedge fund has enough cash to finance margin calls before spreads
narrow, it is better to use cash to cover margins compared to unwinding positions.
Fora small hedge fund, once it is hit with a margin call, it is better to unwind its
positions rather than finance it with available cash (throwing good money after bad),
given that a hedge fund does not have enough cash to finance all its margin calls.
In conclusion, even if arbitrage opportunities are found in a statistical sense, they
might not be exploitable. Moreover, a fund manager who engages in such risk arbitrage
might lose all his money before realizing the positions at a profit. The hedge fund
collapse happens due to an unmet margin call that arises due to leverage effects. A hedge
fund manager can decide how to deal with a margin call. As managers at most of small
103
hedge funds do, they usually decrease the position exposure or close out the position
entirely. This will assure that a hedge fund will not lose any money given further
position divergence instead of an intended divergence. However, if positions converge, a
hedge fund is likely to earn less profit compared to hedge funds that put available cash to
cover margins. Large hedge funds, like LTCM, which can use cash from some
profitable positions to finance margins in other positions, are likely to use available cash
to cover margins instead of closing out the positions. In case of position divergence, they
are likely to make more money than other hedge funds; however, they are also more
likely to suffer a severe collapse in case of further prolonged position divergence. This is
what happened in the LTCM case. The managers were “throwing good money after bad”
leading to the collapse of the fund. Meanwhile, lots of smaller hedge funds that followed
similar strategies to LTCM, survived due to a timely closure of their levered positions.
As assets go down in value, the firm has to post more collateral. If it is
unavailable, this often leads to a hedge fund collapse. However, given that positions are
well diversified and not closely correlated, leverage by itself, does not lead to the collapse
of afund. Correlated positions in the absence of leverage might lead to a loss, but are not
subject to collateral collapse. Given diversified positions in a fund, a price drop in one
asset does not necessarily correspond to a price drop in another asset, even less likely
there is a possibility of a cascade in drop in prices of all assets. However, the
superimposition of both leverage and induced high correlation between assets can lead to
acollapse. This is something that sophisticated hedge funds like LTCM did not take into
equation in determining risk exposure. Their decisions were bounded rational. The
managers separately managed leverage and diversification of positions, not thinking that
two can feed on each other during a period of a crisis leading to the “flight to quality” and
“collateral collapse” dynamics.
104
12. References
I.
13.
Getmansky, M., 2003, “The Life Cycle of Hedge Funds: Fund Flows, Size and
Performance,” MIT Sloan School of Management Working Paper.
Kramer, D, Weiss, P, Rifkind, Wharton & Garrison, October, 2002. “Preventing
Fraud in the Hedge Fund Industry. What Regulation Makes Sense?” Borsa
Italiana’ s Second Hedge Fund Conference.
Liu, J. and Francis A. Longstaff, 2001, Losing Money On Arbitrage: Optimal
Dynamic Portfolio Choice in Markets with A rhitrage Opportunities,” Working
Paper, The Anderson School at UCLA.
Loomis, Carol J. article in Fortune magazine, 1966 “The Jones Nobody Can Keep
Up With.”
Lowenstein, R. “When Genious Failed: The Rise and Fall of Long-Term Capital
Management” 2000. Random House, New Y ork.
Perold, Andre, 1999. Long-Term Capital Management, L.P. Harvard Business
School Cases A-D.
Samuelson, P., 1965, “Proof that Properly Anticipated Prices Fluctuate
Randomly,” Industrial Management Review 6, 41-49.
Schleifer, Andrei and Robert W. Vishny, 1997, “The Limits of Arbitrage,”
Journal of Finance 52, 35-55.
Sharpe, William and Gordon Alexander, 1990, Investments, 4" edition. Prentice
Hall, Englewood Cliffs, NJ.
. Sterman, John. Business Dynamics: Systems Thinking and Modeling fora
Complex World. Irwin McGraw-Hill. 2000.
. Tremont Company, distributor of TASS Database.
. Conversations with Mike Epstein, Veteran trader and dealer at NY SE and Cowen
and Company.
Conversations with Chris Petherick, a trader and a margin clerk at Starks
Investments.
http://invest-faq.com/articles/regul-margin.html#
http://www.nasd.com/Investor/T rading/Margin/
105
13. Appendix
Model Formulation Equations are available upon request.
106
13. Appendix
Model Formulation
Initial Free Cash= INITIAL(
Free Cash[Hf])
~ $
Fraction of Free Cash Invested=
0.8
~ Dmnl
Demand Supply table Security S1(
[(0,0.5)-(3,4)],(0,0.5),(0.2,0.55),(0.4,0.62),(0.6,0.7),(0.8,0.84),(1,1),(1.2,1.5),(\
1.4,1.6),(1.6,1.7),(1.8,1.8),(2,2),(3,3))
= Dmnl
Desired Equity Weight L=
Investor Fraction* (Preference for Liquidity Fraction*Table for Desired Equity
Weight L\
(Forecast Price Relative to Current Price L
~ Dmnol
~ Dmnol
Effect of demand supply balance on price Security S=
STEP(Demand Supply table Security S2(Perceived Demand Supply Balance
Security S),0)-\
STEP(Demand Supply table Security S2(Perceived Demand Supply
Balance Security S),20\
)+STEP(Demand Supply table Security S1(Perceived Demand Supply
Balance Security S),\
20)-STEP(Demand Supply table Security $1(Perceived Demand Supply
Balance Security S\
),30)+STEP(Demand Supply table Security S2(Perceived Demand Supply
Balance Security S\
),30)+ 0*Demand Supply table Security S2(Perceived Demand Supply
Balance Security S\
)
~ fraction
~ Effect of demand/supply balance on price. It adjusts expected price to \
the real price
|
Investor Fraction=
1
~ Dmnl
= |
Maintenance Margin=
Federal Regulation Fraction
~ Dmnl
# |
Margin=
Shares Security L[Hf]*Price L*(1-Federal Regulation Fraction) +T otal Basis
Security S\
[Hf]+Cash Contrib to Cover Margin
~ $
Margin Needed=
Margin Required-Margin
~ $
Desired Equity Weight S=
(1-Desired Equity Weight L)*Investor Fraction
~ Dmnl
Cash Increase Hf=
Sell Rate Security L[Hf]*Price L+Fraction Reinvested* Sell Rate Security
S[Hf}* Price S
~ $/Day
~ |
Cash Decrease Hf=
Buy Rate Security S[Hf]* (Price S-Average Basis for Security S[Hf])+Buy Rate
Security L\
[Hf]*Price L
~ $/Day
~ Decrease in cash due to buying of stocks
Total Desired Buy Rate Security L=
Desired Buy Rate Security L[Hf]+Desired Buy Rate Security L[D]+Desired Buy
Rate Security L\
[Inv]* Investor Fraction
~ shares/Day
~ The sum of total desired buy rates for both fundamental and momentum \
investors
|
Desired Shares By Hf Security L=
(IF THEN ELSE(Decision to Get Into Arbitrage=1,IF THEN ELSE(Margin
Needed>0, Cash Decision\
*MAX (Free Cash[Hf]/Price L+Shares Security L
[Hf]-Maximum Allowed Leverage* Margin Needed/Price L,0) + (1-Cash Decision
)*MAX (Shares Security L[Hf]-Maximum Allowed Leverage* Margin
Needed/Price L,0) , MAX (\
Shares Security L[Hf]+MA X ((Free Cash[Hf]-(1-Fraction of Free Cash
Invested)* Initial Free Cash\
)/Price L,0),0)),0))
~ shares
Total Desired Sell Rate Security L=
Desired Sell Rate Security L[Hf]+Desired Sell Rate Security L[D]+Desired Sell
Rate Security L\
[Inv]* Investor Fraction
~ shares/Day
~ The sum of total desired sell rates for both fundamental and momentum \
investors
|
Total Desired Sell Rate Security S=
Desired Sell Rate Security S[Hf]+Desired Sell Rate Security S[D]+Desired Sell
Rate Security S\
[Inv]* Investor Fraction
~ shares/Day
~ The sum of total desired sell rates for both fundamental and momentum \
investors
|
Consumption Inv=
Total Wealth Inv* Fraction of Wealth Spent on Consumption* Investor Fraction
~ $/Day
~ Amount of dollars spent on consumption each day.
Total Income Inv=
87.6* Investor Fraction
~ $/Day
~ Total income earned by both fundamental and momentum investors.
Initial Shares Security L[Inv]=
Desired Equity Weight L*Total Income Inv/(Fraction of Wealth Spent on
Consumption*
Initial Price L)*0+278*0+400* Investor Fraction ~~|
Initial Shares Security L[Hf]=
0~-|
Initial Shares Security L[D]=
400+722*0
~ shares
~ Initial number of shares available in market
Initial Shares Security S[Inv]=
Desired Equity Weight S*Total Income Inv/(Fraction of Wealth Spent on
Consumption*
Initial Price S)*0+578*0+400* Investor Fraction ~~|
Initial Shares Security S[Hf]=
0~-|
Initial Shares Security S[D]=
400+422*0
~ shares
~ Initial number of shares available in market
Initial Total Cash Inv=
Investor Fraction* (Total Income Inv/(Fraction of Wealth Spent on Consumption)-
Market Value of Equity Invested Inv\
)
~ $
~ Initial total amount of cash. When calculated, the initial amount of cash \
is $40,000, and each person has 400 shares in each.
|
Total Desired Buy Rate Security S=
Desired Buy Rate Security S[Hf]+Desired Buy Rate Security S[D]+Desired Buy
Rate Security S\
[Inv]}* Investor Fraction
~ shares/Day
~ The sum of total desired buy rates for both fundamental and momentum \
investors
|
Gap Shares By Hf Security S=
Desired Shares By Hf Security S-Shares Security S[Hf]
~ shares
Gap Shares by Hf Security L=
Desired Shares By Hf Security L-Shares Security L[Hf]
~ shares
Actual Equity Weight Security L Inv=
ZIDZ(Shares Security L[Inv]*Price L,Market Value of Equity Invested Inv)
~ fraction
Actual Equity Weight Security S Inv=
ZIDZ(Shares Security S[Inv]* Price S,Market V alue of Equity Invested Inv)
~ fraction
~ Fraction of equity held by investor
Preference for Liquidity Fraction=
(1-STEP(0.1,10)+STEP(0.1,20)-STEP(0.8,20)+STEP(0.8,30))*0+1
= Dmnl
Maximum Allowed Leverage=
1/Federal Regulation Fraction
= fraction
Desired Shares By Hf Security S=
-1*MIN((Cash Contrib to Cover Margin*2*0+Maximum Allowed
Leverage* Shares Security L[\
Hf]*Price L-Shares Security L[Hf]* Price L
)/Price S,Desired Shares By Hf Security L)-0*Desired Shares By Hf Security L
2 shares
~ Usually, I put a MIN function.
|
Demand Supply table Security L2(
[(0,0.5)-(3,1.5)1,(0,0.5),(0.2,0.55),(0.4,0.62),(0.6,0.7),(0.8,0.84),(1,1),(1.2,1.16\
),(1.4,1.3),(1.6,1.38),(1.8,1.45),(2,1.47),(3,1.5))
~ Dmnl
Desired Portfolio Rebalancing Security S Inv=
(Desired Equity Weight S-Actual Equity Weight Security S Inv)/Normal Portfolio
Rebalancing Time Security S
[Inv]
~ 1/Day
~ Daily portfolio rebalancing fraction desired by investors
Change in Perceived Price L=
(Price L-Perceived Price L)/Time to perceive price
~ $/(Day*share)
|
Desired Buy Rate Security S[Inv]=
MAX (0,(Cash Inv/Price S)*Normal Daily Tumover Security S[Inv]+(Cash
Inv/Price S
)*Desired Portfolio Rebalancing Security S Inv) ~~|
Desired Buy Rate Security S[Hf]=
MAX(0,MIN(Gap Shares By Hf Security S/Normal Portfolio Rebalancing Time
Security S[Hf\
],(Free Cash[Hf]/Price S)/Normal Portfolio Rebalancing Time Security S
[Hf])) ~~]
Desired Buy Rate Security S[D]=
Desired Sell Rate Security S[Hf]*(1+STEP(0.8,20)-STEP(0.8,30))
~ shares/Day
~ Desired buy rate by each type of the investors.
Total Assets[Hf]=
Cash Hf+Credit Balance of S Position[Hf]+Market Value of L Position[Hf]
~ $
Historical Price L= INTEG (
Change in Historical Price L,
Initial Price L)
7 $/share
Desired Sell Rate Security S[Inv]=
MAX (0,Shares Security S[Inv]*Normal Daily Tumover Security S[Inv]-Shares
Security S\
[Inv]*Desired Portfolio Rebalancing Security S Inv
~
Desired Sell Rate Security S[Hf]=
MAX(0,-Gap Shares By Hf Security S/Normal Portfolio Rebalancing Time
Security S[Hf])\
Desired Sell Rate Security S[D]=
Desired Buy Rate Security S[Hf]
~ shares/Day
~ Desired sell rate by each type of investors
Decrease in Total Value Security S[Hf]=
Buy Rate Security S[Hf]*A verage Basis for Security S[Hf]
~ $/Day
Effect of demand supply balance on price Security L=
(STEP(Demand Supply table Security L2(Perceived Demand Supply Balance
Security L),0)\
-STEP(Demand Supply table Security L2(Perceived Demand Supply
Balance Security L),20\
)+STEP(Demand Supply table Security L1(Perceived Demand Supply
Balance Security L),\
20)-STEP(Demand Supply table Security L1(Perceived Demand Supply
Balance Security L\
),30)+STEP(Demand Supply table Security L2(Perceived Demand Supply
Balance Security L\
),30))*0+Demand Supply table Security L2(Perceived Demand Supply
Balance Security L\
)
~ fraction
~ Effect of demand/supply balance on price. It adjusts expected price to \
the real price
|
Perceived Price L= INTEG (
Change in Perceived Price L,
Initial Price L)
2 $/share
Demand Supply table Security L1(
[(0,0.5)-(3,1.5)],(0,0.5),(0.2,0.55),(0.4,0.62),(0.6,0.7),(0.8,0.84),(1,0.84),(1.2,0.84\
),(1.4,0.84),(1.6,0.84),(1.8,0.84),(2,0.84),(3,0.84))
~ Dmnl
~ |
Spread=
Price S-Price L
= $/share
Desired Buy Rate Security L[Inv]=
MAX (0,(Cash Inv/Price L)*Normal Daily Turnover Security L[Inv]+{Cash
Inv/Price L
)*Desired Portfolio Rebalancing Security L Inv) ~~|
Desired Buy Rate Security L[Hf]=
MIN(Desired Buy Rate Security L for Rebalancing, Maximum Share Purchase
Rate[Hf]) ~~|
Desired Buy Rate Security L[D]=
Desired Sell Rate Security L[Hf]}*(1-STEP(0.8,20)+STEP(0.8,30))
~ shares/Day
~ Desired buy rate by each type of the investors.
Arbitrage Profit Per Combined Trade=
Initial Price S-Initial Price L+Expected Price L-Expected Price S
~ $/share
Desired Portfolio Rebalancing Security L Inv=
(Desired Equity Weight L-Actual Equity Weight Security L Inv)/Normal
Portfolio Rebalancing Time Security L
[Inv]
~ 1/Day
~ Daily portfolio rebalancing fraction desired by investors
Desired Sell Rate Security L[Inv]=
MAX (0,Shares Security L[Inv]}*Normal Daily Turnover Security L[Inv]-Shares
Security L\
[Inv]*Desired Portfolio Rebalancing Security L Inv
Desired Sell Rate Security L[Hf]=
MAX (0,-Gap Shares by Hf Security L/Normal Portfolio Rebalancing Time
Security L[Hf])\
Desired Sell Rate Security L[D]=
Desired Buy Rate Security L[Hf]
~ shares/Day
~ Desired sell rate by each type of investors
Change in expected price L=
(Price L-Expected Price L)/Time to A djust expected price Security L
= $/(share* Day)
~ The change in expected price
Price L=
Expected Price L* Effect of demand supply balance on price Security
L+0*Fundamental Price Security L\
+0*Test2*0+T est1* 0+0*Test22
2 $/share
Table for Desired Equity Weight L(
[(0.5,0.1)-(1.5,1)],(0.5,0.1),(0.6,0.103),(0.7,0.125),(0.8,0.18),(0.9,0.3),(1,0.5),(\
1.1,0.7),(1.2,0.82),(1.3,0.92),(1.4,0.97),(1.5,1))
~ Dmnl
Forecast Price Relative to Current Price L=
Forecast Price L/Perceived Price L
~ Dmnl
Change in Historical Price L=
(Perceived Price L-Historical Price L)/Duration over which to calculate price
trend
= $/(Day*share)
Price Forecast Horizon=
2
2 Day
Duration over which to calculate price trend=
5
2 Day
Forecast Price L=
Perceived Price L*(1+Trend in Price L*(Price Forecast Horizon+Time to perceive
price\
)
~ $/share
Time to perceive price=
~ Day
Trend in Price L=
(Perceived Price L-Historical Price L)/(Historical Price L* Duration over which to
calculate price trend\
)
~ 1/Day
Cash Decrease D=
Buy Rate Security L[Hf]*A verage Basis for Security S[Hf]
~ $/Day
~ Decrease in cash due to buying of stocks
Total Basis Security S[Hf]=INTEG (
+Increase in Total Value Security S[Hf]-Decrease in Total Value Security S[Hf],
-Initial Price S* Initial Shares Security S[Hf])
~ $
Expected Price S= INTEG (
Change in expected price S,
Initial Price S)
= $/share
~ Price expected by investor. The expected price is adjusted to the real \
price with some time lag
|
Cash D=INTEG (
Cash Increase D-Cash Decrease D,
-Initial Price S* Initial Shares Security S[Hf])
2 $
Past Price S= INTEG (
Change in Past Price S,
10
Initial Price S)
= $/share
Minimum Risk Premium=
0.001
~ 1/Day
~ The minimum risk premium investors require.
Risk Premium=
Volatility Switch*A verage Risk Premium*MA X (Minimum Risk
Premium, V olatility of Return\
)/Average SD of Return
+(1-Volatility Switch)*A verage Risk Premium
~ 1/Day
~ Risk premium desired by fundamental investors.
Demand Supply Balance Security S=
XIDZ(Total Desired Buy Rate Security S,Total Desired Sell Rate Security S, 1)
~ fraction
~ The ratio of total desired buy rate to total desired sell rate. The ratio \
measures the balance between the supply and demand of the stock
|
Past Price L=INTEG (
Change in Past Price L,
Initial Price L)
~ $/share
Change in Past Price L=
(Price L-Past Price L)/Time to Change Past Price L
~ $/(Day*share)
Change in Past Price S=
(Price S-Past Price S)/Time to Change Past Price S
~ $/(Day*share)
|
Initial Equity[Hf]= INITIAL(
Equiyl Ht)
11
Desired Buy Rate Security L for Rebalancing=
MAX(0,Gap Shares by Hf Security L/Normal Portfolio Rebalancing Time
Security L[Hf])
~ shares/Day
Maximum Share Purchase Rate[Hf]=
(Free Cash{Hf]/Minimum payment time[Hf])/Price L
~ shares/Day
Profit{Hf]=
Equity[Hf]-Initial Equity[Hf]
2 $
Credit Balance of S Position[Hf]=
Total Basis Security S[Hf]
2 $
Market Value of L Position{Hf]=
Price L*Shares Security L[Hf]
$
Market Value of S Position[Hf]=
-Price S*Shares Security S[Hf]
~ $
Total Liabilities[Hf]=
Market Value of S Position[Hf]
~ $
Equity[Hf]=
Total Assets[Hf]-T otal Liabilities[Hf]
~ $
Decrease in Total Value Security L[Hf]=
Sell Rate Security L[Hf]*A verage Basis for Security L[Hf]
~ $/Day
12
Margin Required=
-(1+Maintenance Margin)* Price S* Shares Security S[Hf]
o $
Average Basis for Security L[Hf]=
ZIDZ( Total Basis Security L[Hf] , Shares Security L[Hf] )
~ $/share
Average Basis for Security S[Hf]=
ZIDZ(Total Basis Security S[Hf],-Shares Security S[Hf])
~ $/share
Total Basis Security L[Hf]=INTEG (
+Increase in Total Value Security L[Hf]-Decrease in Total Value Security L[Hf],
Initial Price L* Initial Shares Security L[Hf])
2 $
Retum=
(Price L-Past Price L)/Past Price L/Duration Over Which Retum is Calculated
~ 1/Day
~ Daily return of an asset.
Federal Regulation Fraction=
0.5
~ Dmnl
Increase in Total V alue Security L[Hf]=
Buy Rate Security L[Hf]*Price L
~ $/Day
Increase in Total Value Security S[Hf]=
Sell Rate Security S[Hf]* Price S
~ $/Day
= |
Margin Call=
Cash Decision* (IF THEN ELSE(Margin Needed>0,MIN(Margin Needed/Time to
Cover Margin, \
13
Free Cash[{Hf]/Minimum payment time[Hf]),0))
~ $/Day
Fraction Reinvested=
0
~ fraction
~ |
Test4=
STEP(40,20)-STEP(40,30)
= $/share
~ |
Price S=
Expected Price S*Effect of demand supply balance on price Security
S+0*Fundamental Price Security S\
+Test3*0+0*Test4
~ $/share
Test22=
STEP(-80,20)+STEP(80,30)
~ $/share
Margin Refund=
Cash Decision* (IF THEN ELSE(Margin Needed<=0,MIN(-
Margin Needed/Time to Cover Margin, Cash Contrib to Cover Margin
/Minimum payment time[Hf] ),0))
~ $/Day
Test3=
STEP(20,20)-STEP(20,30)+STEP(40,30)-STEP(40,60)
~ $/share
Test2=
STEP(-20,30)+STEP(+20,60)
~ $/share
= |
Cash L=INTEG (
Cash Increase L+Margin Call-Margin Refund,
9.6e+007)
14
~ $
Cash Contrib to Cover Margin=INTEG (
Margin Call-Margin Refund,
0)
= $
Free Cash[Hf]=INTEG (
Cash Increase Hf 1+Income from Other Investments 1+Margin Refund-Cash
Decrease Hf 1-\
Margin Call,
Initial Total Cash Hf)
2 $
Cash Decrease Hf 1=
Cash Decrease Hf
~ $/Day
Income from Other Investments 1=
Income from Other Investments
~ $/Day
Cash Increase Hf 1=
Cash Increase Hf
~ $/Day
Change in Smoothed Shares S=
(Desired Shares By Hf Security S-Smoothed Desired Shares By Hf Security
S)/Time to Update Shares S
~ shares/Day
Smoothed Desired Shares By Hf Security S= INTEG (
Change in Smoothed Shares S,
0)
2 shares
Time to Update Shares S=
1
15
~ Day
Cash Decision=
1
~ fraction
~ Cash Decision=1 if a Hedge Fund decides to cover margin by using cash.
If\
Cash Decision=0, then a Hedge Fund decides to cover margin by buying
back \
and selling securities.
Time to Change Past Price S=
1
~ Day
Cash Hf=INTEG (
Income from Other Investments+Cash Increase Hf-Cash Decrease Hf,
Initial Total Cash Hf)
~ $
Income from Other Investments=
0*STEP(1000,20)-0*STEP(1000,30)
~ $/Day
Cash Increase Inv=
Sell Rate Security S[Inv]*Price S+Sell Rate Security L[Inv]* Price L
= $/Day
~ Increase in cash due to selling of stocks
Demand Supply Balance Security L=
XIDZ(Total Desired Buy Rate Security L, Total Desired Sell Rate Security L,
1)*(1+Noise in Demand Supply Balance Security L\
)
~ fraction
~ The ratio of total desired buy rate to total desired sell rate. The ratio \
measures the balance between the supply and demand of the stock
|
Decision to Get Into Arbitrage=
IF THEN ELSE(Price L<Price S, 1, 0)
16
~ fraction
Change in Smoothed Shares=
(Desired Shares By Hf Security L-Smoothed Desired Shares By Hf Security
L)/Time to Update Shares
~ shares/Day
~ |
Time to Update Shares=
1
= Day
Smoothed Desired Shares By Hf Security L=INTEG (
Change in Smoothed Shares,
0)
~ shares
Cash Increase L=
Sell Rate Security S[Hf]* Price S+Sell Rate Security L[Hf]*Price L
~ $/Day
Minimum payment time[Hf]=
~ Day
* |
Testl=
STEP(-20,30)+STEP(+20,60)
~ $/share
Trading Volume Security L=
MIN(Total Desired Buy Rate Security L,Total Desired Sell Rate Security L)
~ shares/Day
~ The actual volume of shares traded yearly
Cash Increase D=
Sell Rate Security S[Hf]* Price S
~ $/Day
~ Increase in cash due to selling of stocks
17
Time to Cover Margin=
1
~ Day
Noise in Demand Supply Balance Security L=
Pink Noise*Switch for Noise*STEP(1, Noise Start Time)
~ fraction
2 |
Switch for Noise=
0
~ fraction
Fundamental Price Security L=
100
~ $/share
2 |
Initial Total Cash Hf=
96000*0+40000+60000*0+1000*0
~ $
~ Initial total amount of cash
|
Initial Price L=
80*0+100
~ $/share
~ Initial price
Buy Rate Security L[Types of investors]=
IF THEN ELSE(Total Desired Buy Rate Security L=0,0,Trading Volume Security
L* Desired Buy Rate Security L\
[Types of investors]/Total Desired Buy Rate Security L)
~ shares/Day
~ The rate of acquiring shares
Cash Decrease Inv=
Buy Rate Security S[Inv]*Price S+Buy Rate Security L[Inv]* Price L
= $/Day
~ Decrease in cash due to buying of stocks
18
Change in demand supply balance Security L=
(Demand Supply Balance Security L-Perceived Demand Supply Balance Security
L)/Time to perceive demand supply balance Security L
~ 1/Day
~ Change in demand/supply balance
|
Change in expected price S=
(Price S-Expected Price S)/Time to Adjust expected price Security S
= $/(share* Day)
~ The change in expected price
Normal Daily Turnover Security L[Types of investors]=
0.003
~ 1/Day
~ The fraction of equity sold/bought during normal conditions
Income Inv=
Total Income Inv
~ $/Day
~ Amount of dollars earned each day by an investor
Normal Portfolio Rebalancing Time Security L[Inv]=
10 ~+|
Normal Portfolio Rebalancing Time Security L[Hf]=
3
~ Day
Perceived Demand Supply Balance Security L= INTEG (
Change in demand supply balance Security L,
1)
- fraction
~ Perceived demand/supply balance by investors
Fundamental Price Security S=
100
~ $/share
Expected Price L=INTEG (
19
Change in expected price L,
Initial Price L)
~ $/share
~ Price expected by investor. The expected price is adjusted to the real \
price with some time lag
|
Noise Start Time=
5
2 Day
Sell Rate Security L[Types of investors]=
IF THEN ELSE(Total Desired Sell Rate Security L=0,0,Trading Volume Security
L* Desired Sell Rate Security L\
[Types of investors]/Total Desired Sell Rate Security L)
~ shares/Day
~ The rate of selling shares
Demand Supply table Security L(
[(0,0.5)-(3,1.5)],(0,0.5),(0.2,0.55),(0.4,0.62),(0.6,0.7),(0.8,0.84),(1,1),(1.2,1.16\
),(1.4,1.3),(1.6,1.38),(1.8,1.45),(2,1.47),(3,1.5))
~ fraction
~ Table which depicts an effect of demand/supply balance on price\!\!\!
Shares Security L[Types of investors]=INTEG (
Buy Rate Security L[Types of investors]-Sell Rate Security L[Types of investors],
Initial Shares Security L[Types of investors])
~ shares
~ The number of shares held by a specific type of investors.
Market Value of Equity Invested Inv=
Shares Security S[Inv]*Price S+Shares Security L[Inv]}*Price L
~ Market value of equity investment for each investor
Time to perceive demand supply balance Security L=
1
~ Day
~ Time to perceive demand/supply balance
20
Time to Adjust expected price Security L=
14
~ Day
~ Time to adjust expected price
Total Shares Security L=
SUM(Shares Security L[Types of investors! ])
~ shares
~ Total shares in the market. The model assumes that no shares are issued.
\
Therefore, this amount should be conserved and equal to the Initial
Number \
Initial Earnings0=
0.015
~ $/(Day*share)
~ 1/63
|
Time to Perceive Earnings EA =
1
~ Day
~ Perception time for earnings. It takes into account how long it takes to \
publish and see the earnings.
|
Earnings Forecast Horizon EA=
63
= Day
~ Time horizon for the calculation of earnings in the future.
Duration Over Which to Calculate Earnings Trend EA=
126
~ Day
~ Time over which trend for earnings is calculated
Table for Desired Equity W eight UA(
[(0.5,0.1)-(1.5,1)],(0.5,0.1),(0.6,0.103),(0.7,0.125),(0.8,0.18),(0.9,0.3),(1,0.5),(\
1.1,0.7),(1.2,0.82),(1.3,0.92),(1.4,0.97),(1.5,1))
~ Dmnl
of Shares.
21
Table for the effect of earnings growth on discount rate EA (
[(-0.4,0)-(0.4,0.4)],(-0.1,0.0001),(-
0.001,0.0001),(0.0001,0.0001),(0.0002,0.0002),(\
0.0005,0.0005),(0.001,0.001),(0.002,0.002),(0.3,0.3))
~ fraction
~ Table for the effect of k-g on indicated fundamental value of an \
asset.\!\!\!
|
Table for Desired Equity Weight EA (
[(0,0)-(4,1)],(0,1),(0.2,0.97),(0.4,0.9),(0.6,0.83),(0.8,0.7),(1,0.5),(1.2,0.3),(1.4\
,0.16),(1.6,0.11),(1.8,0.07),(2,0.04),(2.2,0.03),(2.4,0.02),(2.6,0.01),(2.8,0.005),\
(3,0),(4,0))
~ fraction
~ Table which calculates the desired equity fraction for fundamental \
investor\!\!\!
|
Time to perceive value EA=
1
~ Day
~ Time to perceive value
|
Switch for Step=
0
~ Dmnl
~ |
Eaming Noise Start Time=
= Day
Noise in Earnings=
Pink Noise*STEP(1, Earning Noise Start Time)
~ Dmnl
~ |
Switch for White Noise=
1
~ Dmnl
22
White Noise2=
Mean+(Standard Deviation’2*(2-(TIME STEP/Correlation Time))/(TIME
STEP/Correlation Time\
))0.5*RANDOM NORMAL(-100, 100, 0, 1, Noise Seed )
~ Dmnol
Change in Pink Noise=
(White Noise-Pink Noise)/Correlation Time
~ 1/Day
White Noise=
MIN(MAX (Switch for White Noise*W hite Noisel +(1-Switch for White
Noise)*W hite Noise2,\
0),2)
~ Dmnol
White Noise1=
Mean+Standard D eviation*((24*C orrelation Time/TIME STEP)0.5)*RANDOM
UNIFORM(-0.5,0.5\
Noise Seed)
~ Dmnl
Standard Deviation=
0.15
~ Dmnl
Switch for Pink Noise=
STEP(1,20)
~ Dmnl
Noise Seed=
3
~ Dmnl
~ Originally, had 2.
|
23
Pink Noise=INTEG (
Change in Pink Noise,
Mean)
~ Dmnol
~ |
Correlation Time=
1
2 Day
~ 1/4 of the year
Volatility Switch=
1
~ Dmnl
Average SD of Retum=
0.0008
~ 1/Day
~ 20.39% /year.
|
Normal Portfolio Rebalancing Time Security S[Inv]=
5
Normal Portfolio Rebalancing Time Security S [Hf]=
3
= Day
Initial Earnings=
0.015
~ $/(Day*share)
~ 1/63
|
Fraction of Wealth Spent on Consumption=
0.00073
~ 1/Day
~ Fraction of wealth spent on consumption by investors daily
wealth spent on consumption is 18.4% /year.
|
Duration Over Which Change in Consumption is Calculated=
5
24
. Fractional \
~ Day
Time to Change Past Consumption=
10
~ Day
Time to Update Moving Average of Change in Consumption=
10
~ Day
~ Time over which the moving average of change in consumption is
calculated.
Average Risk Premium=
0.00035
~ 1/Day
~ Market risk premium = beta of an asset multiplied by the difference of the
expected market return and the risk-free rate. 0.0874/252
|
Volatility of Retum=
sqrt((Retum-Moving Average of Returm)*(Return-Moving Average of Return))
~ 1/Day
~ Volatility of a stock
|
Time to Update Moving Average of Retum=
200
= Day
~ Time over which the moving average of return is calculated.
Change in Moving Average of Retum=
(Return-Moving Average of Return)/Time to Update Moving Average of Return
~ 1/(Day* Day)
~ Rate of the increase in moving average of retum
Earnings Forecast Horizon=
63
= Day
~ Time horizon for the calculation of earnings in the future.
25
Moving Average of Retum=INTEG (
Change in Moving Average of Retum,
0)
~ 1/Day
~ This belief is a weighted average of the current value of return and past \
belief.
|
Shares Security S[Types of investors]= INTEG (
Buy Rate Security S[Types of investors]-Sell Rate Security S[Types of investors],
Initial Shares Security S[Types of investors])
~ shares
~ The number of shares held by a specific type of investors.
Duration Over Which Return is Calculated=
1
~ Day
~ Time duration over which retum is calculated. In this model, daily retun \
is assumed.
|
Time to Change Past Price L=
1
~ Day
~ Time to change past price. It is assumed that price is changed every day.
Perceived Demand Supply Balance Security S= INTEG (
Change in demand supply balance Security S,
1)
~ fraction
~ Perceived demand/supply balance by investors
Riskless Rate=
0.00015
~ 1/Day
~ Current forward real interest rate or another proxy.0.0376/252
|
Cost of Equity=
Riskless Rate+Risk Premium
~ 1/Day
~ Required rate of return or cost of equity.
26
Total Shares Security S=
SUM(Shares Security S[Types of investors!])
~ shares
~ Total shares in the market. The model assumes that no shares are issued.
\
Therefore, this amount should be conserved and equal to the Initial
Number \
Table for the effect of earnings growth on discount rate(
[(-0.4,0)-(0.4,0.4)],(-0.1,0.0001),(-
0.001,0.0001),(0.0001,0.0001),(0.0002,0.0002),(\
0.0005,0.0005),(0.001,0.001),(0.002,0.002),(0.3,0.3))
~ fraction
~ Table for the effect of k-g on indicated fundamental value of an \
asset.\!\!\!
|
Normal Daily Turnover Security S[Types of investors]=
0.003
~ 1/Day
~ The fraction of equity sold/bought during normal conditions
|
Cash Inv=INTEG (
Cash Increase Inv-+Income Inv-Cash Decrease Inv-Consumption Inv,
Initial Total Cash Inv)
~ $
~ Amount of cash held by investor
Sell Rate Security S[Types of investors]=
IF THEN ELSE(Total Desired Sell Rate Security S=0,0,Trading Volume Security
S*Desired Sell Rate Security S\
[Types of investors]/Total Desired Sell Rate Security S)
~ shares/Day
~ The rate of selling shares
Buy Rate Security S[Types of investors]=
IF THEN ELSE(Total Desired Buy Rate Security S=0,0,Trading Volume Security
S*Desired Buy Rate Security S\
[Types of investors]/Total Desired Buy Rate Security S)
of Shares.
27
~ shares/Day
~ The rate of acquiring shares
Time to Adjust expected price Security S=
14
= Day
~ Time to adjust expected price
Change in demand supply balance Security S=
(Demand Supply Balance Security S-Perceived Demand Supply Balance Security
S)/Time to perceive demand supply balance Security S
~ 1/Day
~ Change in demand/supply balance
|
Demand Supply table Security S2(
[(0,0.5)-(3,1.5)],(0,0.5),(0.2,0.55),(0.4,0.62),(0.6,0.7),(0.8,0.84),(1,1),(1.2,1.16\
),(1.4,1.3),(1.6,1.38),(1.8,1.45),(2,1.47),(3,1.5))
~ fraction
~ Table which depicts an effect of demand/supply balance on price\!\!\!
Time to perceive demand supply balance Security S=
1
= Day
~ Time to perceive demand/supply balance
Table for Desired Equity W eight F(
[(0,0)-(4,1)],(0,1),(0.2,0.97),(0.4,0.9),(0.6,0.83),(0.8,0.7),(1,0.5),(1.2,0.3),(1.4\
,0.16),(1.6,0.11),(1.8,0.07),(2,0.04),(2.2,0.03),(2.4,0.02),(2.6,0.01),(2.8,0.005),\
(3,0),(4,0))
~ fraction
~ Table which calculates the desired equity fraction for fundamental \
investor\!\!\!
|
Total Wealth Inv=
Cash Inv+Market Value of Equity Invested Inv
~ $
~ Sum of cash and equity holdings for each investor
28
Time to Perceive Earnings=
60
~ Day
~ Perception time for earnings. It takes into account how long it takes to \
publish and see the earnings.
|
Time to perceive value=
2
2 Day
~ Time to perceive value
|
Duration Over Which to Calculate Earnings Trend=
126
~ Day
~ Time over which trend for earnings is calculated
|
Initial Price S=
120*0+100
= $/share
~ Initial price
Trading V olume Security S=
MIN(Total Desired Buy Rate Security S,Total Desired Sell Rate Security S)
~ shares/Day
~ The actual volume of shares traded yearly
Types of investors:
Inv, Hf, D
a CR RK RK KKK RRR KKK
.Control
SC RR RR RRR
Simulation Control Paramaters
FINAL TIME =100
= Day
~ The final time for the simulation.
29
INITIAL TIME =0
~ Day
~ The initial time for the simulation.
SAVEPER =
TIME STEP
~ Day
~ The frequency with which output is stored.
|
TIME STEP =0.03125
~ Day
~ The time step for the simulation.
30
