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ESTIMATION OF
MAXIMUM AVERAGE LOSS FOR INVESTMENT POSITION IN FUTURES OF THE ATHENS
By Dr COSTAS KYRITSIS By APOSTOLIS KIOHOS
Department
of Computing and Insurance)
Maths
and
Software Laboratory
of
Abstract
In this paper
we discuss the risk of mark-to-market loss of positions with leverage, of infinite
horizon, in futures. We make the usual assumptions of Lognormal distribution
and geometric Brownian motion, for the underlying as in the Black-Scholes
options pricing model. With these assumptions we estimate the tables of required liquidity for futures on FTSE-20 and
FTSE-40 in the Athens Derivatives Exchange Market and the maximum average Loss
of infinite horizon investment positions in Derivative Exchange Market.
Key words
Derivatives,
Geometric Brownian Motion, Stochastic Differential
Equations, Simulation, Investment, Liquidity
§1 Introduction Since August 1999 the Athens
Derivative Exchange Market (ADEX) introduced for the first time futures on the Index FTSE-20,
and soon afterwards on the Index FTSE-40. The peculiarities and risks of
investing to futures are not quite clear to the present average investor. In a
first publication [Kyritsis C (2001)] we analyzed the required Liquidity of
finite horizon investments in futures. We made use of the conditional
volatility.
In this paper we analyze the
liquidity requirement of infinite horizon investments from the point of view of
average maximum loss. Of course the investments in Derivatives have always an
expiration date. But putting an infinite horizon in the investment makes
calculations simpler and at the same time the real risk is less or equal to the
estimated so it is always safer for more risk averse
decision makers.
The main idea is that it
should always be possible to pay the average maximum loss besides the margin
reservations. So an estimate of an average maximum loss, given a margin
percentage, leads directly to a liquidity percentage. The method of average
maximum loss is simpler to calculate, to understand and apply, than the method
of conditional volatility.
§2
Leverage and bankruptcy of positions in Futures
When investing in positions
on futures we do not pay all the money of the investment.
Instead it is calculated
daily the profit or loss of the investment position (called mark-to-market) and is paid by the
investors and Brokerage Companies to an appropriate clearance bank (Alpha
Credit Bank). In addition it is paid a percentage only of the height of the
position, as much as it is considered it is risked for 1-2 days for ADEX to
close the position, if anything goes wrong (default position). The percentage
is estimated according to the volatility (standard deviation) of the daily
percentage changes of the underlying Index and is called Margin . This percentage at present is 12% for the futures.(March
2001). This makes an advantage for the investor as he must only pay 12% of the
height of a position when he opens it. This is called the leverage of the position and is a multiplier of 1/12%=8.33 times.
Of course not only the rate of return is multiplied with this numbers but also the
Beta (or Elasticity) of the position. The advantage of leverage has also its drowbacks and risks. The profit or loss is paid daily on
100% of the height of the position and in a reverse trend of the market can
easily lead to bankruptcy, something not really possible with investment
positions in securities.
§3
Average maximum Loss of an Investment Position.
In order to estimate the average maximum loss we have to assume a model of the underlying
Index, and the correlation and coupling
of the future with the underlying Index.
We shall proceed in a way
that is standard in the pricing of Derivatives and is also used by ADEX in the
estimation of the percentages of 12% for the margin. We shall assume a neutral
market, neither growing neither decaying, but with a trend equal to the risk
free rate (2% in a year base, at present March 2001). So the model of the
underlying index is, as in the Black-Scholes option
pricing model, a Geometric Brownian Motion (continuous time random compound
interest) of normally distributed rate r and volatility ó. For the definition of the stochastic differential
equations and the geometric Brownian motion see Oksental
p121 Chpt. V p 60 ,exerc.7.9 ,p 121,example 5.1 p 60
The
stochastic differential equation of Brownian motion (Ito interpretation) is:
(1)
The exact interpretation of
the symbols requires the concepts and definitions of stochastic Integrals and
is outside the scope of this paper. For the definitions see Oksental
1995.
The distribution of the prices Xt is Lognormal.
The solution of this
stochastic differential equation is given by the formula:
(2)
where Bt is a
Brownian Motion
The logarithm of this
process Xt/ X0 is an ordinary Brownian motion with
drift:
log(Xt/
X0)=(r-(1/2) s2)t + s Bt . (3)
The average time T that it
reaches X for the first time starting from x0
Is T=log(X/ x0)/(r-(1/2)
s2) (4)
If â is the beta
(elasticity) of the future Yt over the index Xt
we assume that the futures also, follows
the equation
(5)
If l is the leverage of the
investment the value of the investment position on the future follows the equation
(6)
The way to estimate the
average maximum loss of the investment is the following:
We shall make use of a
theorem on the Brownian motion that can be found in
[Karlin
S.-Taylor H.(1975)] Corollary 5.1 Chapter 5 p 361.
The Theorem goes like this:
Let X(t) be a Brownian
motion process with drift ì >0. Let
W=max(X(0)-X(t)). For all t>=0. (7)
The W has exponential
distribution
Pr(W>w)=exp(-ëw), w>=0 (8)
Where ë=(2|ì|)/(ó2) (9)
The formula (3) above shows that the logarithm of the prices of the future
follows a Brownian motion with (let us say positive) drift
(r-(1/2) s2)
Therefore the
average maximum downward deviation of the logarithm is
s2 /(2*|(r-(1/2) s2)|)
(10)
As the logarithm
is a monotonous function (respecting order) and the average of a logarithm is
the logarithm of the average, the average downward deviation of the price of
the underlying in other words the average maximum loss of the underlying is
(referring to formula 2 above)
X0exp(s2 /(2*|(r-(1/2) s2)|)) (11)
Thus we get
the next statement
Theorem A
The average maximum loss as a percentage is
|1-
exp(s2 /(2*|(r-(1/2) s2)|))| (12)
§4
Tables of Average Maximum Loss.
In the next tables
we have calculated the rate ,volatility
for a 30 days sample of the index
ftse-20 and consequently the average maximum loss and liquidity percentage (or
percentage to invest) for Long or short positions for a period of 43 days
during 2001. (We assume â=1 for the
futures on them).In the calculations we do not consider the leverage of the
positions in the futures. The percentage to invest is defined by the assumption
that the average maximum loss should be kept in cash . So if the average
maximum loss is say 40%, as the margin
now (2001) is a 12% , then the
percentage to keep in cash is 40%/(12% +40%) and the percentage to invest
therefore is 60%/(12%+40%).
|
Date |
Unsystematic
Risk |
BETA for 30
days |
Average rate
for 30 days |
Variance of General index |
Daily Variance of Ftse |
AverageMaximumLoss |
Percentage to Invest |
Year Volatility of Ftse |
|
3/1/2001 |
0,0002 |
1,1952 |
-0,026 |
0,0005 |
0,000854 |
37,8118% |
24,09% |
0,461946 |
|
4/1/2001 |
0,0002 |
1,1885 |
-0,027 |
0,0005 |
0,000835 |
37,6409% |
24,17% |
0,456838 |
|
5/1/2001 |
0,0002 |
1,1885 |
-0,027 |
0,0005 |
0,000835 |
37,6409% |
24,17% |
0,456838 |
|
8/1/2001 |
0,0002 |
1,2338 |
-0,028 |
0,0005 |
0,000944 |
45,5163% |
20,86% |
0,485736 |
|
9/1/2001 |
0,0002 |
1,2509 |
-0,029 |
0,0005 |
0,000975 |
50,2929% |
19,26% |
0,493766 |
|
10/1/2001 |
0,0002 |
1,257 |
-0,03 |
0,0005 |
0,000954 |
49,9603% |
19,37% |
0,488253 |
|
11/1/2001 |
0,0003 |
1,1246 |
-0,03 |
0,0006 |
0,001084 |
59,1191% |
16,87% |
0,520627 |
|
12/1/2001 |
0,0003 |
1,1065 |
-0,03 |
0,0006 |
0,001059 |
56,6757% |
17,47% |
0,51456 |
|
15/1/2001 |
0,0003 |
1,1506 |
-0,03 |
0,0007 |
0,001224 |
68,4562% |
14,91% |
0,553205 |
|
16/1/2001 |
0,0003 |
1,1606 |
-0,029 |
0,0007 |
0,00124 |
69,2779% |
14,76% |
0,556877 |
|
17/1/2001 |
0,0003 |
1,1309 |
-0,029 |
0,0006 |
0,001048 |
55,6878% |
17,73% |
0,511829 |
|
18/1/2001 |
0,0002 |
1,1485 |
-0,031 |
0,0006 |
0,00091 |
48,9313% |
19,69% |
0,476949 |
|
19/1/2001 |
0,0002 |
1,1549 |
-0,031 |
0,0006 |
0,000907 |
49,5285% |
19,50% |
0,47631 |
|
22/1/2001 |
0,0013 |
0,0576 |
-0,032 |
0,0006 |
0,001275 |
78,7285% |
13,23% |
0,564572 |
|
23/1/2001 |
7E-05 |
1,3764 |
-0,032 |
0,0004 |
0,000872 |
49,1165% |
19,63% |
0,466904 |
|
24/1/2001 |
7E-05 |
1,3704 |
-0,035 |
0,0004 |
0,000857 |
52,6611% |
18,56% |
0,462762 |
|
25/1/2001 |
0,0001 |
1,3017 |
-0,035 |
0,0004 |
0,00085 |
52,8936% |
18,49% |
0,460912 |
|
26/1/2001 |
0,0001 |
1,2687 |
-0,036 |
0,0004 |
0,000834 |
53,1531% |
18,42% |
0,456526 |
|
29/1/2001 |
0,0002 |
1,2655 |
-0,037 |
0,0004 |
0,000816 |
53,7830% |
18,24% |
0,451718 |
|
30/1/2001 |
0,0002 |
1,2636 |
-0,038 |
0,0004 |
0,000843 |
57,4241% |
17,29% |
0,459009 |
|
31/1/2001 |
0,0002 |
1,1922 |
-0,039 |
0,0005 |
0,000902 |
64,1609% |
15,76% |
0,474985 |
|
1/2/2001 |
0,0002 |
1,1714 |
-0,039 |
0,0005 |
0,000879 |
62,3608% |
16,14% |
0,468833 |
|
2/2/2001 |
0,0002 |
1,1765 |
-0,04 |
0,0005 |
0,000898 |
65,9084% |
15,40% |
0,473774 |
|
5/2/2001 |
0,0002 |
1,2137 |
-0,039 |
0,0005 |
0,000937 |
68,0976% |
14,98% |
0,484022 |
|
6/2/2001 |
0,0002 |
1,201 |
-0,038 |
0,0005 |
0,000969 |
69,0006% |
14,81% |
0,492119 |
|
7/2/2001 |
0,0002 |
1,207 |
-0,038 |
0,0005 |
0,000973 |
69,1451% |
14,79% |
0,493107 |
|
8/2/2001 |
0,0003 |
1,1904 |
-0,039 |
0,0005 |
0,000979 |
71,1249% |
14,44% |
0,494666 |
|
9/2/2001 |
0,0003 |
1,1763 |
-0,04 |
0,0005 |
0,000979 |
73,9884% |
13,96% |
0,494813 |
|
12/2/2001 |
0,0003 |
1,173 |
-0,04 |
0,0005 |
0,001047 |
81,2309% |
12,87% |
0,511608 |
|
13/2/2001 |
0,0003 |
1,168 |
-0,04 |
0,0005 |
0,001042 |
81,8545% |
12,79% |
0,510414 |
|
14/2/2001 |
0,0004 |
1,1669 |
-0,041 |
0,0005 |
0,001083 |
87,6833% |
12,04% |
0,520409 |
|
15/2/2001 |
0,0004 |
1,1851 |
-0,04 |
0,0005 |
0,001078 |
85,4915% |
12,31% |
0,519044 |
|
16/2/2001 |
0,0004 |
1,15 |
-0,041 |
0,0005 |
0,000996 |
78,0027% |
13,33% |
0,499067 |
|
19/2/2001 |
0,0004 |
1,1568 |
-0,025 |
0,0005 |
0,00098 |
42,1816% |
22,15% |
0,495096 |
|
20/2/2001 |
0,0004 |
1,1319 |
-0,053 |
0,0005 |
0,00099 |
110,0591% |
9,83% |
0,497466 |
|
21/2/2001 |
0,0004 |
1,3202 |
-0,053 |
0,0003 |
0,000898 |
95,8285% |
11,13% |
0,473785 |
|
22/2/2001 |
0,0003 |
1,3257 |
-0,053 |
0,0003 |
0,000829 |
86,7790% |
12,15% |
0,455127 |
|
23/2/2001 |
0,0003 |
1,3502 |
-0,053 |
0,0002 |
0,000732 |
74,0986% |
13,94% |
0,427887 |
|
27/2/2001 |
0,0003 |
1,3861 |
-0,054 |
0,0002 |
0,000783 |
81,4854% |
12,84% |
0,442465 |
|
28/2/2001 |
0,0003 |
1,3574 |
-0,054 |
0,0002 |
0,000731 |
75,3731% |
13,73% |
0,427452 |
|
1/3/2001 |
0,0003 |
1,3592 |
-0,055 |
0,0002 |
0,000726 |
76,2499% |
13,60% |
0,426166 |
|
2/3/2001 |
0,0003 |
1,4426 |
-0,055 |
0,0002 |
0,000762 |
80,4376% |
12,98% |
0,43642 |
|
5/3/2001 |
0,0003 |
1,5486 |
-0,056 |
0,0002 |
0,000777 |
85,2209% |
12,34% |
0,440694 |
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