optimal level of leverage using numerical methods

8/20/2019 Optimal Level of Leverage using Numerical Methods

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Optimal Level of Leverage using Numerical

Methods

Elton SbruzziUniversity of [email protected]

Steve PhelpsUniversity of [email protected]

Jul 11, 2013

Abstract

Leverage can have a significant impact of stock returns on equity

returns. For example, if a stock return is 5% and the level of leverage

is 5, the equity return is 25%. However, if the level of leverage is -5%,

with the same stock return, the equity return is -25%. From this ex-

ample, notice that what dictates the equity return is the combination

of stock return and the level of leverage. Thus, in order to improve theinvestment performance, the appropriate level of leverage is as crucial

as the appropriate stock. In order to provide a closed-form solution,

previous studies proposed analytical methods to optimise the level of

leverage that assume no fat-tails on the series of stock returns, and

consequenlty, underestimate extreme returns and the risk of default.

In this paper, we introduce numerical methods in order to optimise

the level of leverage. This allows us to relax the assumption of no fat-

tails on the series of stock returns, and consequently, takes extreme

returns and the risk of default into account properly.

JEL Classification : C61Keywords : Optimal level of Leverage, geometric Brownian motion, nu-merical methods.

1 Introduction

Today, leverage plays a very important role in the financial markets. New

instruments designed to facilitate leveraged investments such as Financial

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Spread Betting (FSB) and Contract for Difference (CFD) in conjunction

with technology advances in electronic markets make leverage accessible for

every type of investor, from small individual investors up to big funds. For

example, in 2007, 30% of the volume on the London Stock Exchange was

driven for leveraged investments (FSA, 2007).

Studies considering leverage and individual investment performance are

relevant for two reasons. Firstly, the equity return formula has two compo-

nents: the stock return and the level of leverage. Additionally, the level of

leverage is linear in the equity return formula (Peters, 2011). This means

that leverage has the ability to modify the impact of the stock return on theequity return and, consequently, on an individual’s investment performance.

For example, a positive stock return can lead to negative equity return if the

level of leverage is negative.

Secondly, leverage can lead the investor to ruin. For example, suppose

that an investor is 10 times leveraged and the stock return is -10%, the equity

return is -100%. Therefore, the end of period return is -100%, and then after

the -100% equity return, the investor has no more capital to invest and,

consequently, would not be able to open any position.Leverage has the ability to impact the investment performance. However,

if used on an excessive level, leverage can lead investor to bankruptcy. Thus,

if excessive level of leverage is bad for investments, could leverage be good

on some level? Is there an optimal level of leverage? If yes, could it be

measured?

Kelly Jr (1956) demonstrated that in binomial games with positive expec-

tation, there is one specific value of leverage which maximises the geometric

mean or the end of period wealth. He proposed a model to obtain this value

and this model was called the Kelly criterion. Today, one of the possible

uses of the Kelly criterion is to determine the optimal level of leverage on

investments considering the possible returns on different discrete scenarios.

However, the Kelly criterion is not suitable for determining the optimal level

of leverage in continuous games when the number of possible outcomes is

unlimited, as with financial returns.

Rotando and Thorp (1992) propose the Kelly criterion version for con-

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tinuous gambling games. But the use of the Kelly criterion for continuous

gambling games is not straight forward. Continuous games can assume in-

finite number of outcomes and the Kelly criterion was designed to assume

a finite number of outcomes. Hence, they assumed that the returns in the

financial market are normally distributed and such distribution should be

transformed to quasi-normal distribution by cutting off the tails in order to

limit the number of outcomes and respect the Kelly criterion original as-

sumptions.

In order to obtain a more realistic approach, Peters (2011) introduces a

new model to obtain the optimal level of leverage using geometric Brownianmotion. He demonstrated that the optimal level of leverage is simply the

estimation of log-return divided by the variance of log-return. In this model,

the objective is to estimate the appropriate expected return and variance.

These three authors assume that the objective is to find the level of lever-

age that maximises the geometric mean of the historical financial returns:

GM =nt=1

(1 + rt)1

n − 1 (1)

where GM is the geometric mean of returns.

Their models uses an analytical methods to yield a closed-form solution.

In order to develop their models, they assume that the historical financial re-

turns are normally distributed, and ignore the existence of fat-tails. However,

Cont (2001) demonstrates that financial returns are not normally distributed

and exhibit fat-tails. This means that if the models that assume the normal

distribution are used to calculate the optimal level of leverage of a historical

financial returns, then they will underestimate the probability of extreme

returns and, consequently, underestimates the risk of default. The result is

that it could lead investors to use an excessive level of leverage which could

be harmful to the investor’s investment performance. For this reason, we

argue that it is necessary to develop models to optimize the level of leverage

that relax the assumption of normal distribution of the series of historical

financial returns.

In this paper, in order to avoid any discussion about the return charac-

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teristics, we propose to estimate the optimal level of leverage using numerical

methods instead of analytical methods with closed-form solution. The differ-

ence between numerical methods and analytical methods is that numerical

methods propose the intensive use of computing capabilities in order to reach

the solution, while analytical model proposes the use of mathematical for-

mula in order to provide the solution. In our case, as a numerical methods,

we use a line-search method to get the level of leverage that maximizes the

geometric mean of a series of historical returns. The insight for this propo-

sition is that analytical methods proposed in the literature are not able to

capture the presence of the fat tails on the return distribution (Cont, 2001)and consequently, their results could not represent the real optimal level of

leverage.

In order to test the hypothesis that numerical methods yield superior

geometric mean to analytical methods, we calculated the optimal level of

leverage using geometric Brownian motion and using numerical methods of

104 series of daily return of the components of Dow Jones index between

2003 and 2010. In the experiment, the stock, the length of the series and the

period are randomly selected, and the result is that the numerical methodshows superior level of leverage and higher geometric mean to a closed-form

model based on geometric Brownian motion.

This paper is structured as follows. In section 2, we detail the models

to optimize the level of leverage proposed in the literature. In section 3,

we introduce the model to optimize the level of leverage using numerical

methods. In section 4, we present the results of the experiment that compares

two different methods to calculate the optimal level of leverage, geometric

Brownian motion and numerical methods. In section 5, we discuss the results

of the experiment. Finally, in section 6, we conclude by presenting some final

remarks and possible future work.

2 Optimal Leverage Models

In this section, we present the three existing models to optimize the level of

leverage. The first model is the Kelly criterion which assumes normal dis-

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tribution of the returns and discrete number of possible outcomes (Kelly Jr,

1956). The second model is the Rotando and Thorp model which assumes

normal distribution of the returns and continuous number of possible out-

comes (Rotando and Thorp, 1992). Finally, the third model is the Peters

model which assumes normal distribution of the returns and that price fol-

lows a geometric Brownian motion (Peters, 2011).

2.1 The Kelly Criterion

In this subsection, we describe the Kelly criterion. The importance of studieson leverage is not recent. Kelly Jr (1956) demonstrates that there is an

optimal level of leverage for binomial games such as coin tossing games.

He assumes that the main objective of the player is to maximise the trial’s

expected value and, consequently, the end of period cumulative return. He

also demonstrates that if you have a positive trial’s expected value, the size of

your leverage depends upon two factors: the value of your expected value and

the probability of ruin. This model is called the Kelly criterion. Originally,

the Kelly criterion consists of a model to maximise returns on a binomial

game similar to coin tossing. The model is described below.

Suppose that on each trial the win probability is p > 1/2 and, conse-

quently, the probability of loss is q = 1 − p < 1/2. Once the outcomeprobability is defined, the question is to decide the amount of capital Bi to

bet on each trial with the objective to maximise the expected value of the

end of trials wealth, E (X n). Let T i = 1 if the ith trial is a win and T i = −1,if the ith trial is a loss. Furthermore, letting X 0 be the initial capital, then

X i = X i−1 + T iBi, for i = 1,...,n and, consequently X n = X 0 +ni=1 T iBi,

then

E (X n) = X 0 +ni=1

( p − q )E (Bi) > 0 (2)

Assuming p > 1/2 > q and, consequently the expected value of the end

of trials wealth E (X n) is positive, the objective is to find the amount of the

available capital which should be bet, Bi, and the amount of capital that

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should be saved for later trials, X i−1. On one hand, if the player decides to

bet all the capital, this increases the probability of ruin. On the other hand,

if he decides to bet the minimal capital, the player reduces the probability

to maximise the value of the end of trial’s wealth. Therefore, there is some

optimal level of leverage or optimal fraction, l , which balances the objective to

maximise the expected value with the restriction to minimise the probability

of ruin.

In the coin-tossing game, since the gambling probability and the payoff at

each trial are the same, it is clear that the optimal level of leverage is the same

for all trials. This assumption of fixed leverage helps us to comprehend thepurpose of the Kelly criterion. Maximising the expected end of trial wealth

is similar to maximising the expected value of the growth rate coefficient or

the geometric mean, GM k(l):

GM k(l) = E [log X nX 0

]1

n = p log(1 + l) + q log(1− l) (3)

GM k(l) = p

1 + l −

q

1 − l (4)

GM k (l) = −l2 + 2l( p − q ) − 1

(1 − l2)2 (5)

where GM k(l) is the geometric mean approximation on the Kelly criterion

and 0 ≤ l < 1. From Eq. (5), the function GM k(l) is concave in l , henceit can be maximised. Furthermore, solving Eq. (4), the result is that the

optimal level of leverage is l∗ = p − q .

2.2 The Rotando and Thorp Model

In this subsection, we describe the Rotando and Thorp model. Rotando and

Thorp (1992) argue that Kelly criterion cannot be directly used on invest-

ments. The main reason is that with games like coin tossing there are a

discrete number of possible outcomes, whereas with investments, the num-

ber of possible outcomes is continuous. In order to use the Kelly criterion

in financial markets, the original model proposed in Eq. (3) should be mod-

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ified and adapted to represent continuous outcomes. This modification is

proposed by Rotando and Thorp. They propose a new model incorporating

continuous outcomes, but respecting the Kelly’s original insight: the defini-

tion of the optimal level of leverage in order to maximise the end of period

wealth or the end of period return, E [log(Xn/X 0)].

According to Rotando and Thorp, trading financial securities can be con-

sidered a continuous game (Rotando and Thorp, 1992). Thus, in order to

model the possible outcomes, they assume that financial returns are normally

distributed. However, the simple use of the unaltered normal probability dis-

tribution is inadequate because this distribution allows an infinite range of possible returns. Therefore, they modify the standard normal curve using a

correction term for ”chopping off the tails”. This results in new parameters,

h and α, which serves to maintain the mean and the standard deviation at

values similar to those of the standard normal distribution.

Similar to binomial games, the objective in this model is to find out the

level of leverage that maximises function GM rt(l), which in this case is:

GM rt(l) = BA log(1 + rl)dN (r) (6)

where GM rt(l) is the geometric mean approximation on the Rotando and

Thorp model.

GM rt(l) =

BA

log(1 + rl)[h + 1√

2πα2e−(r−µ)

2/2α2 ] (7)

where A = µ − 3σ and B = µ + 3σ.The demonstration of first-order and the second order conditions of Eq.

(7) is complicated. However, numerical methods can be used to obtain thevalue of l that maximises the function GM rt(l). For example, simulating the

model proposed for the 59 year period from 1926 to 1984, with the µ = 0.058

and σ = 0.216 by using numerical methods, Rotando and Thorp (1992) found

that the optimal level of leverage is l∗ = 1.17.

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2.3 Geometric Brownian Motion

In this subsection, we describe the Peters model that uses the geometric

Brownian motion (Peters, 2011). Assuming that the price follows a geo-

metric Brownian motion, and consequently, that returns are log-normally

distributed (Hull, 2009), Peters proposes a different model to optimize the

level of leverage (Peters, 2011). Suppose that the price process is:

p(t) = p0[exp (µ −σ2

2 )t + σ ∗ W (t)] (8)

where the Wiener process W(t) is normally distributed.Introducing the leverage, the estimated leveraged log-return and its vari-

ance are respectively:

µl = lµ (9)

σ2l = l2σ2 (10)

where l is the level of leverage.

The estimated log-return of the leveraged investment to the next period

is:

GM p(l) = E [log( pt+1)] − log( pt) = (µl −σ2l2

) (11)

where GM p(l) is the geometric mean approximation on the Peters model.

Substituting Eq. (9) and Eq. (10) into Eq. (11) gives:

GM p(l) = (lµ −l2σ2

2 ) (12)

In order to obtain the optimal level of leverage, Eq. (12) is differentiated

with respect to l and the result is set to zero. Then, the optimal level of

leverage is:

l∗ = µ

σ2 (13)

where l∗ is the optimal level of leverage.

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According to the Peters model, the optimal level of leverage is the esti-

mated log-return divided by its variance.

One of the drawbacks of these models is that they assumes that the finan-

cial returns is normally distributed and, consequently, ignores the presence of

fat-tails (Cont, 2001). With the objective of relax this assumption, in order

to optimize the level of leverage, we propose numerical methods directly in

the original model of geometric mean as described in Eq. (1).

3 Numerical Methods

In this section, we describe the model in which the optimal level of lever-

age is calculated using numerical methods. In general, the models reviewed

in previous section, present the following three characteristics of leverage.

Firstly, the geometric mean is concave in leverage, and consequently, there is

an optimal level of leverage that maximises the geoemtric mean. Secondly,

the optimal level of leverage is achieved analytically using those models.

Finally, thirdly, these models assume normal distributions of returns and,

consequently, ignore the presence of fat tails.The proposal of this section is that, instead of using proxies and analytical

methods with closed-form solution to obtain the optimal level of leverage, we

should return to the original model of geometric mean as described in Eq.

(1) and use numerical methods, in this case line-search, in order to obtain the

optimal level of leverage. By so doing I am able to relax the assumption of

normal distribution of returns, and take the presence of fat-tails into account.

In the line-search method, we randomly select two levels of leverage, l1

and l2, in which each point is on a different side of the maximum value.Next, we select two new levels of leverage, l3 and l4 (Kelley, 1999). This will

increase the geometric mean value. This procedure is performed iteratively

to obtain the specific level of leverage that no other level of leverage results

in a superior value of geometric mean. The resulting level of leverage is

considered the optimal.

In order to use the numerical method,we transform Eq. (1) and introduce

the term leverage in the geometric mean formula:

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GM (l) = [nt=1

(1 + rtl)]1

n − 1 (14)

where GM (l) is the geometric mean formula with the introduction of the

term leverage l, and rtl is the leveraged return at the time t . Iteratively, we

use the line-search method, in order to obtain the value of l that maximises

Eq. (14):

Maxl

GM (l) = Maxl

[n

t=1

(1 + rtl)]1

n

−1 (15)

Numerical methods have three advantages. Firstly, it is not necessary to

make any assumptions about the statistical characteristics of the data which

helps us to avoid any further discussion about the realism of the distribution

selected to represent the data. Secondly, the method is free of approximations

or simplifications, which also helps to make the methods the more realistic

as possible. Finally, thirdly, it is easy to test using real data.

4 Experiment

In this section, we present the experiment that compares two different meth-

ods to calculate the optimal level of leverage, analytical methods, represented

by the geometric Brownian motion and numerical methods. This section is

divided in two subsections. In the first subsection, we describe the methodol-

ogy of the experiment. And, in the second subsection, we present the results

of the experiment.

4.1 Methodology

In this subsection, we describe the methodology of the experiment in which

we test the ability of numerical methods to achieve the level of leverage that

optimises the geometric mean of the series of returns superior to the Peters

model based on the geometric Brownian motion (see section 2). As empirical

evidence, we use the series of daily returns of the components of the Dow

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Jones index between 2003 and 2010. The data series contains 1024 days of 30

different stocks. We assume that the empirical data is sufficient to represent

financial returns.

The experiment consists of a random selection of a window of a series of

prices over 104 iterations. In order to obtain the most representative data

series during different periods of time, we randomly select three parameters

on each iteration: the stock, the length of the series, and the window period.

For example, we randomly select the series of returns of the stock of General

Electric Company (ticker: GE) between 01/02/2005 and 19/04/2007, as well

as the series of returns of of the stock of Pfizer Inc. (ticker: PFE) between23/08/2008 and 30/11/2010.

In each iteration, we calculate two different parameters: the optimal level

of leverage and the optimal leveraged geometric mean. We use two different

methods: the optimal level of leverage formula, as described by Eq. (13),

and the numerical methods as described by Eq. (15). This generates four

different data series each containing 104 data points: 1) the series of optimal

level of leverage based on the geometric Brownian motion, 2) the series of

optimal level of leverage using numerical method, 3) the series of optimal thegeometric mean of the historical returns using numerical method, and 4) the

series of optimal the geometric mean of the historical returns based on the

geometric Brownian motion.

4.2 Results

In this subsection, we present the results of the experiment. In order to

compare the methods and their results, we test two different null hypothe-

ses. Firstly, we test the null hypothesis that the optimal level of leverageusing numerical method is similar to the optimal level of leverage using on

geometric Brownian motion. Secondly, we test the null hypothesis that the

leveraged geometric mean using numerical methods is similar to the leveraged

geometric mean using on geometric Brownian motion.

4.2.1 Geometric Mean

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Item GM NM GM GBM

Mean 3.9004x10−4 3.8932x10−4Median 2.0925x10−4 2.0922x10−4

Minimum 0 -3.9415x10−4

Maximum 0.0073 0.0073Standard Deviation 5.6597x10−4 5.6463x10−4

Skewness 3.8271 3.8207Kurtosis 25.5245 25.4607Size 10−4 10−4

Table 1: Statistical analysis of the geometric mean of using the geometric

Brownian motion approach and the numerical methods to calculate the op-timal level of leverage.

From Table 1, the mean and the median of the series of geometric means

calculated using the numerical methods are superior to the series of geometric

means using the geometric Brownian motion. It indicates that the numerical

methods could yield to a higher value of geometric mean than the geometric

Brownian motion.

Item GM NM −

GM GBM Confidence Interval(P≤95%)

[1.66x10−4;2.16x10−4]

Null Hypothesis Rejected

Table 2: Comparison between geometric Brownian motion approach andnumerical methods of the geometric mean using the optimal level of leverage.

From Table 2, the confidence interval of the difference between the series

of geometric means calculated using the numerical methods and the series

of geometric means using the geometric Brownian motion is positive. This

demonstrates that numerical methods yield to a higher value of geometric

mean than the geometric Brownian motion with a probability superior to

97.5%.

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0 2000 4000 6000 8000 100000

2

4

6

8 x 10

−3

G e o m e t r i c M e a n

Numerical Methods

0 2000 4000 6000 8000 10000−2

0

2

4

6

8x 10

−3

Iterations


Geometric Brownian Motion

Figure 1: The plot of the geometric means calculated using numerical meth-ods and geometric Brownian motion.

From Figure 1, there is no negative value of the geometric mean of the

historical returns of the use of numerical methods. However, there is one

negative value for the geometric mean of the use of the geometric Brownian

motion.

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0 2000 4000 6000 8000 10000−2

0

2

4

6

8

10

x 10−4

Iterations

D i f f e r e n c e o f G e o m e t r i c M e a n

Figure 2: The plot of the difference of the geometric means between numericalmethods and geometric Brownian motion.

From Figure 2, there is no negative difference between the series of geo-

metric means calculated using the numerical methods and the series of geo-metric means using the geometric Brownian motion. This demonstrates that

numerical methods yield a value of geometric mean superior to geometric

Brownian motion.

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−5

0

5

10

15x 10

−4

Numerical Methods Geometric Brownian Motion


Figure 3: The boxplot of the geometric mean using numerical methods andbased on the geometric Brownian motion.

From Figure 3, the difference between the series of geometric means cal-

culated using the numerical methods and the series of geometric means using

the geometric Brownian motion is not clear. However, the bottom value of

the numerical methods is superior to the minimum value of the geometric

Brownian motion.

4.2.2 Level of Leverage

Item LNM LGBM Mean 0.9621 0.9639Median 0.7903 0.7924Minimum -4.7141 -4.7611Maximum 8.0387 9.0770Standard Deviation 1.0202 1.0292Skewness 1.6113 1.6790

Kurtosis 8.7796 9.2992Size 10−4 10−4

Table 3: Statistical analysis of the optimal level of leverage of the use of thegeometric Brownian motion approach and the numerical methods.

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From Table 3, the mean and the median of the series of the optimal level

of leverage calculated using the numerical methods are inferior to the series

of the optimal level of leverage using the geometric Brownian motion. This

indicates that numerical methods could yield a lower value of optimal level

of leverage than geometric Brownian motion.

Item LNM − LGBM Confidence Interval(P≤95%)

[0.1934; 0.2428]

Null Hypothesis Rejected

Table 4: Comparison between geometric Brownian motion approach andnumerical methods of the optimal level of leverage.

From Table 4, the confidence interval of the difference between the series

of optimal level of leverage calculated using numerical methods and the series

of optimal levels of leverage using geometric Brownian motion is positive.

This demonstrates that numerical methods yield to a lower value of the

optimal level of leverage than geometric Brownian motion with a probability

superior to 97.5%.

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0 2000 4000 6000 8000 10000−5

0

5

10


Numerical Methods

0 2000 4000 6000 8000 10000−5

0

5

10

Iterations


Geometric Brownian Motion

Figure 4: The plot of the level of leverage calculated using numerical methodsand geometric Brownian motion.

From Figure 4, the range of the possible values of the optimal level of

leverage varies between -5 and 10 in both approaches, numerical methodsand geometric Brownian motion. This demonstrates that the excessive levels

of leverage, e.g. tenfold, are not rational because it has not been between the

range of the level of leverage that maximises the historical equity returns.

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0 2000 4000 6000 8000 10000−2

−1.5

−1

−0.5

0

0.5

Iterations

D i f f e r e n c e o f L e v e l o f L e v e r a g e

Figure 5: The plot of the difference of the level of leverages between numericalmethods and geometric Brownian motion.

From Figure 5, the difference between the series of the optimal level

of leverage calculated using numerical methods and the series of geometricmeans using geometric Brownian motion is distributed among positive and

negative values. Hence, visually, it is not possible to distinguish the optimal

level of leverage using numerical methods to the optimal level of leverage

using geometric Brownian motion.

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−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Numerical Methods Geometric Brownian Motion

L e v e l o f L e v e r a g e

Figure 6: The boxplot of the level of leverage using numerical methods andbased on the geometric Brownian motion.

From Figure 6, the difference between the series of geometric means cal-

culated using the numerical methods and the series of geometric means usingthe geometric Brownian motion is not clear.

5 Discussion

The results of the experiment show that numerical methods allows us to cal-

culate the optimal level of leverage that results in superior estimation of the

geometric mean of the historical returns compared with the use of geomet-

ric Brownian motion. The confidence interval of the difference of geometricmean of the daily historical returns is between 1.66x10−4 and 2.16x10−4.

In one hand, analytical methods such as geometric Brownian motion as-

sume that the returns are normally distributed and ignores the presence of

fat-tails. Consequently, this underestimates extreme returns. In the other

hand, numerical methods relax the assumption of normal distribution of the

returns, and do not require a closed-form solution. This makes the calcula-

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tion of the optimal level of leverage more reliable, and allows us to obtain the

correct level of leverage that maximises the geometric mean of the historical

return.

6 Summary

In this paper, we introduced a new model that uses numerical method in

order to find the level of leverage that maximises the geometric mean of the

series of historical equity returns. Our method uses line-search to numerically

search for the optimal level of leverage. To the best of our knowledge this

is the first algorithm for calculating the optimal level of leverage using the

geometric mean approach which does not assume that returns are normally

distributed or that the price process follows geometric Brownian motion.

Furthermore, we tested the ability of numerical methods to yield superior

geometric mean to analytical methods, represented by the geometric Brow-

nian motion. In order to do that, we calculated the optimal level of leverage

using numerical methods and using geometric Brownian motion of 104 series

of daily returns of the components of the Dow Jones Index between 2003 and2010. The data series contains 1024 days of 30 different stocks. We assume

that the empirical data is sufficient to represent financial returns. In the

experiment, the stock, the length of the series and the window period are

randomly selected. We found that the numerical methods yield superior ge-

ometric mean of the series of historical equity returns to geometric Brownian

motion, even though, when it shows inferior level of leverage.

The contribution of this paper is the introduction of numerical methods

in order to calculate the optimal level of leverage. Previous studies haveproposed analytical methods in order to calculate the optimal level of lever-

age. However, these models assume that returns are normally distributed

(Kelly Jr, 1956; Peters, 2011; Rotando and Thorp, 1992). By using numeri-

cal methods in order to optimise the level of leverage, we are able to avoid

this assumption and hence avoid under-estimating extreme returns (Cont,

2001).

The results of the experiments in this paper show the advances caused

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by the use of numerical methods in the leverage models. For this reason,

we argue that the proxy and the analytical methods cannot be directly used

to calculate the leveraged equity returns because the proxy and the models

assume that the returns are normally distributed, and ignore the presence

of fat-tails and, consequently, underestimate extreme returns present on the

leveraged investments.

We proposed a numerical methods to optimize the level of leverage of

an investment in a single stock. For future work, it could be interesting to

investigate and experiment the possibility of the use of numerical methods

to optimize the level of leverage of a portfolio of stocks. Estrada (2009)proposes an analytical method to calculate the optimal weight of a single

stock in a portfolio of stocks. However, his model also assumes the normal

distribution of the returns. Thus, this could be interesting to experiment

numerical methods in order to optimize the portfolio of stocks in terms of

geometric mean, that relax the assumption of normal distribution of the

returns.

ReferencesCont, R. (2001). Empirical properties of asset returns: stylized facts and

statistical issues. Quantitative Finance 1, 223–236.

Estrada, J. (2009). Geometric mean maximization: an overlooked portfolioapproach? Available at SSRN 1421232 .

FSA (2007). Disclosure of contracts for difference.

Hull, J. C. (2009). Options, futures, and other derivatives.

Kelley, C. T. (1999). Iterative methods for optimization , Volume 18. Siam.

Kelly Jr, J. (1956). A new interpretation of information rate. Information Theory, IRE Transactions on 2 (3), 185–189.

Peters, O. (2011). Optimal leverage from non-ergodicity. Quantitative Fi-nance 11 (11), 1593–1602.

Rotando, L. M. and E. O. Thorp (1992). The kelly criterion and the stockmarket. The American Mathematical Monthly 99 (10), 922–931.

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