MSc International Economics and Finance

Portfolio Optimization Using the Mean Absolute

Deviation Model

Course: Applied Financial Economics

Prof. Topaloglou N.

Fugert Rrasa

Nikolaos – Alexandros Angelopoulos

Filippas Beteniotis

Introduction

Purpose of this project is the practical implementation of portfolio optimization techniques

on real life data.

As a prerequisite to implementing the minimization of the exposure to financial risk

techniques, we first identify the main sources of risk to investors and institution in the market which make financial optimization relevant and we provide a brief overview of them.

We then consider the problem of formally defining risk and we provide a generic

mathematical framework for optimally allocating financial assets, from which the main risk management models are derived.

After presenting the main risk measures utilized by financial practitioners, we optimize using

the scenario optimization technique. We proceed by using the Mean Absolute Deviation

(MAD) method to generate a portfolio of 25 stocks chosen from the S&P 500 index. To

implement the chosen portfolio optimization method, we used the algebraic modelling language GAMS (General Algebraic Modelling System).

We construct an optimal portfolio of stocks and we conducted on it static tests (constructing the efficient frontier of portfolios) as well as dynamic tests (backtesting for 30 months).

Finally, we provide the concluding remarks of our project.

Financial Risks

Generally, financial risk relates to the probability of having a realization of a random variable

different to the realization preferred by the economic agent. Financial Risk can be analyzed in many different sub-categories:

-Credit Risk

Credit risk, formally defined as ‘the potential that a contractual party will fail to meet its

obligations in accordance with the agreed terms’, is the risk of default or change in the credit

quality of issuers of securities to whom a company has an exposure.

Credit risk is of importance when considering corporate bonds, but it is also a major

influence on corporate money market instruments, bonds issued by sovereigns, etc. It is

generally diversifiable, but difficult to eliminate completely. This is because a portion of the

default risk may, in fact, result from the systematic risk of the market.

Any factors that can influence the value of the counterparty and affect the value of the

transaction go by the generic name of credit triggers or credit events. The five most common credit events are:

1. Failure to meet payment obligations when due.

2. Bankruptcy

3. Repudiation

4. Material adverse debt restructuring

5. Obligation acceleration or obligation default.

The traditional approach to managing credit risk is to evaluate the risk by assessing the borrower’s ability to repay.

-Market Risk

Market risk is a generic term which refers to the sensitivity of an asset or portfolio to overall

price movements that could cause its value to decrease. Depending on which market is being analyzed, market risk has different interpretations.

For example, in the fixed-income market, the standard measure of risk is the interest rate.

Other standard market risk factors are stock indices, commodity prices, foreign exchange

rates, real estate indices etc.

While market risk cannot be completely removed by diversification, it can be reduced by

hedging. For example, the use of interest rate and inflation swaps can produce offsetting positions swaps can produce offsetting positions whereby the risks are hedged.

-Liquidity Risk

Liquidity risk refers to the risk of a funding crisis. It is defined as the possibility that over a specific horizon the bank will become unable to settle obligations with immediacy.

There exist many diverse reasons about what causes liquidity risk and we could constraint

ourselves to say that its causes lie on the departure from the symmetric information

paradigm. In this context, liquidity risk is considered endemic in the financial system, having

the potential to destabilize it. In such cases, emergency liquidity provisions could serve as a

toll to restore balance.

Liquidity risk is particularly important for actively managed portfolios which depend on

frequent trading. If the liquidity of a particular instrument worsens, losses will materialize when selling the security if all other conditions remain constant.

-Sector Risk

A sector is a set of securities sharing some common characteristics. It follows that sector risk

is the danger that the stocks of many of the companies in one sector will fall in price at the same time because of an event that affects the entire industry.

Since sectors share some common attributes, they are likely to be influenced by common risk factors.

- Other Risks

There are certain kinds of risks which are most clearly displayed to certain kinds of financial

instruments. These risks involve the volatility risk and Sharpe risk.

By volatility risk we mean that the more volatile the market, the higher the price of the

affected security. Volatility risk has a major impact in options and securities embedding options although it is also present in the fixed income market.

The Sharpe risk on the other hand is more applicable on the fixed income market. It is the risk caused by changes in the shape of the term structure of the interest rate.

Other kinds of risk involve operational risk, which is defined as ‘’the risk of a change in value

caused by the fact that actual losses, incurred for inadequate or failed internal processes,

people and systems, or from external events (including legal risk), differ from the expected

losses’’, reputational risk which a risk of loss resulting from damages to a firm's reputation,

in lost revenue, etc.

More broadly, we could use the generic term residual risk to describe all other risk factors in the market.

Risk Measures

A Formal Definition of Risk - Coherence

For a risk measure to be well constructed it has to be based on a rigorous definition of risk.

However, since risk is a subjective notion interpreted differently by each investor, it follows

that its definitions are very lax.

The absence of a strict definition of risk gives rise to many problems for the financial

engineer and the financial risk manager since it becomes hard to find a generic framework

which would conform to the subjective notions of risk of the investors and of the institutions.

Faced with this problem, in a seminal paper, Artz and Al (1999) established an axiomatic

theory of risk measures, by formally defining certain attributes that anyone should

reasonably require from a risk measure. The risk measures that share these attributes are called coherent, and they have important practical implications.

The co-authors of the paper argued that since risk is related to the variability of the future

values of a position, it is best to consider risk in future values only, instead of defining in terms of changes in values between two dates.

After defining “acceptable” future random net worth and providing a set of axioms about

the set of acceptable future net worth, the authors state the axioms on measures of risk and

relate them to the axioms on acceptance sets, arguing that these axioms should hold for any risk measure which is to be used to effectively regulate or manage risks.

To provide a mathematical definition, we let 𝑀 denote the space of random variables

representing portfolio losses over some fixed time interval, 𝛥. Then, a coherent risk

measure is a real-valued function, 𝑝 ∶ 𝑀 → 𝑅, that satisfies the desirable properties listed below 1-5:

1)Normalized

𝑝(0) = 0

There is no risk in taking no position.

2)Sub-additivity

𝑝(𝐴1 + 𝐴2) ≤ 𝑝(𝐴1) + 𝑝(𝐴2)

This property expresses the fact that a portfolio made of sub-portfolios will risk an amount which is at most the sum of the separate amounts risked by its sub-portfolios.

While this has been the most debated of the risk axioms, it allows for the decentralization of

risk management: If separate risk limits are given to different “desks”, then the risk of the aggregate position is bounded by the sum of the individual risk limits.

3)Positive Homogeneity

𝑝(𝜆𝐴) = 𝜆𝑝(𝐴)

This axiom reflects the fact that there are no diversification benefits when we hold multiples of the same portfolio, A.

It is also somewhat controversial and has been criticized for not penalizing concentration of

risk and any associated liquidity problems. In particular, if λ> 0 is very large, then some people claim that we should require 𝑝(𝜆𝐴) > 𝜆𝑝(𝐴).

4)Translation Invariance

𝑝(𝑍 + 𝐴) = 𝑝(𝑍) − 𝑎

The portfolio A is adding cash A to the portfolio Z. By adding cash to the portfolio, risk is

reduced.

5)Monotonicity

If 𝐴 ≤ 𝑍 then 𝑝(𝐴) ≥ 𝑝(𝑍)

If portfolio Z has better values than portfolio A, then under almost all conditions the risk of Z

should be less than the risk of A. The monotonicity property provides us with the possibility of ordering our investment.

General problem of optimal allocation and consistency of the risk measure

Portfolio theory bases the concept of risk in strong connection with the investor’s risk and

their utility function. In other words, the optimal investment decision always corresponds to

the solution of an expected utility maximization problem. The link be tween utility theory

and the risk of some investment is generally represented by the consistency of the risk measure with some stochastic order.

To construct a generic framework, we consider the problem of optimal allocation between 𝑛

assets with a vector of returns 𝑟 = [𝑟1 , 𝑟2, … . , 𝑟𝑛] where 𝑟𝑖 =𝑃𝑖,𝑡+1−𝑃𝑖,𝑡

𝑃𝑖,𝑡. We do not allow for

short selling in the model, that is, wealth 𝑦𝑖 invested in the i-th asset is non-negative for

every 𝑖 = 1,2, … 𝑛. Therefore, considering an initial wealth 𝑊0, we are facing the following optimization problem:

min𝑦

𝑝(𝑊0 + 𝑦′𝑟)

∑ 𝑦𝑖

𝑛

𝑖=1

= 𝑊0, 𝑦𝑖 ≥ 0, 𝑖 = 1,2, . . 𝑛

𝐸(𝑊0 + 𝑦′𝑟) ≥ 𝜇𝑦

which is equivalent to maximizing the expected utility 𝐸(𝑈(𝑊𝑦)) of future wealth invested

in a portfolio of assets, where 𝑊𝑦 = 𝑊0 + 𝑦′𝑟.

A risk measure is said to be consistent with an order relation if 𝐸(𝑈(𝑊𝑥 )) ≥ 𝐸(𝑈(𝑊𝑦)) if it

implies that 𝑝(𝑊𝑥 ) ≤ 𝑝(𝑊𝑦) for all future wealths 𝑊𝑥 and 𝑊𝑦, that is, if 𝑊𝑥 first and second

order stochastically dominates 𝑊𝑦.

Consistency is absolutely necessary for a risk measure to make sense. It ensures us that we

can characterize the set of all the optimal choices when either wealth distributions or

expected utility depend on a finite number of parameters

As a consequence of consistency, all the best investments of a given category of investors (non-satiable, risk-averse, non-satiable and risk-averse) are among the less risky ones.

Risk Management Models

Mean Variance

Mean Variance models assume that portfolios can be completely characteri zed by their mean return and variance (or risk).

Generally speaking, therefore, an investor’s optimal portfolio could be best described by

performing as (r, σ), where r is a desired average rate of return, and 𝜎2 the minimal variance

possible for this given return.

Mathematically, for a portfolio of n risky assets, we want to find a solution to:

𝑚𝑖𝑛∑ ∑ 𝜎𝑖𝑗𝑛𝑗=1 𝑥𝑖𝑥𝑗

𝑛𝑖=1 ,

Subject to:

∑𝑟𝑗

𝑛

𝑗=1

𝑥𝑗 = 𝜌𝑀0

∑ 𝑥𝑗

𝑛

𝑗=1

= 𝑀0

𝑥𝑗 ≤ 𝑢𝑗

The constraints require that the total sum of the returns of each asset times the amount

invested in that asset is equal to the minimum rate of return the investor wants times the

total amount of money being invested, where 𝑟𝑗is the average daily return of asset j, ρ is

the minimal rate of return required by the investor which is not portfolio dependent, 𝑀0is the total amount of money being invested which is constant, and 𝑢𝑗is the maximum amount

the investor wishes to place in a single stock.

The optimal portfolios for this investor are those that achieve the highest expected return

for a given level of variance and the smallest possible variance for a given level of return. Such portfolios are called mean-variance portfolios.

Value at Risk (VaR)

The Value at Risk (VaR) model measures the potential loss in value of a risky asset or

portfolio over a defined period for a given confidence interval. Its focus is clearly on

downside risk and potential losses. Unlike standard deviation, VaR focuses on a specif ic part of the distribution specified by the confidence level. It is estimated with parametric models.

The model, therefore, has 3 key elements: a specified level of loss in value, a fixed time period over which risk is assessed and a confidence interval.

Mathematically (for continuous distributions), VaR is defined as:

𝑉𝑎𝑅(𝒙, 𝑎) = 𝑚𝑖𝑛{𝑢: 𝐹(𝑥, 𝑢) ≥ 1 − 𝑎}

where 𝒙 is the vector denoting asset allocation 𝒙 = (𝑥1,𝑥2,… . 𝑥𝑛)𝑇and F is a distribution

function such that 𝐹(𝑥, 𝑢) = 𝑃{𝑅(𝒙, �̃�} ≤ 𝑢.

What the equation says is that the returns of a certain portfolio 𝒙 will be below 𝑉𝑎𝑅(𝒙, 𝑎) with a likelihood of (1 − 𝑎) ∙ 100%.

One of the most important properties of VaR is the stability of the estimation procedures.

Because VaR disregards the tail, it is not affected by very high tail losses, which are usually difficult to measure.

However, this also means that VaR does not account for properties of the distribution

beyond the confidence level. Its disregard of the tail of the distribution, which may be quite

long, could lead to the unintentional bearing of high risk, since it is providing no information about the magnitude of the loss exceeding the quantile.

Perhaps the principal criticism made to VaR when compared to other risk measures is that it

fails to be sub-additive, and consequently fails to be coherent. Since, as we saw, sub-

additivity allows the decentralized calculation of the risks from different positions in the firm

and the sum of these risks will be an overestimation of the total risk in the portfolio, the fact

that VaR lacks this property may lead to considerable problems, such as that portfolio

diversification might actually increase the risk.

Conditional Value at Risk (CVaR)

A risk measure related to VaR is CVaR (Conditional Value at Risk) or Expected Shortfall (ES). It

is defined as the expected loss conditional on the fact that the loss exceeds the VaR at a

given confidence level. What CVaR therefore does, is to provide us with information about

the expected loss in the tail.

CVaR is generally considered to have more attractive properties than VaR. In particular, it is

a more consistent risk measure since it is sub-additive and convex.

This property of CVaR, that it accounts for risks beyond VaR makes it generally more conservative than VaR.

For continuous distributions, CVaR is defined as follows:

𝐶𝑉𝑎𝑅(𝒙,𝑎) = 𝐸[𝑅(𝒙, �̃�)|𝑅(𝒙,�̃�) ≤ 𝑉𝑎𝑅(𝒙, 𝑎)]

The conditional expectation presented above is the basis for the name of conditional value

at risk,

For discrete distributions, such as scenario optimization, however, we have to consider a different formulation:

𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 −1

1 − 𝑎∑𝑃𝑠

𝑆

𝑠=1

𝑦𝑠+

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥 ∈ 𝑋, 𝑧 ∈ 𝑅

𝑥𝑇 �̅� ≥ 𝜇

𝑦𝑠+ ≥ 𝑧 − 𝑥𝑇𝑟𝑠 𝑠 = 1,2, … . . , 𝑛

𝑦𝑠+ ≥ 0 𝑠 = 1,2, … , 𝑛

Where we define the following: 𝑧 = 𝑉𝑎𝑅(𝒙, 𝑎) and an auxiliary variable for every scenario 𝑠 ∈ 𝑆, 𝑦𝑠

+ = max [0, 𝑧 − 𝑅(𝑥, 𝑟𝑠)].

Put-Call Efficient Frontiers

Put-Call efficient frontier models provide a framework to select portfolios whose risk

measure is given by the expected value of the downside deviations from a given target, and the reward measure is the expected value of the upside deviations from the same target.

This model trades off the portfolio risk against the portfolio reward. The upside potential has

identical payoffs to a call option on the future portfolio return relative to the target. When

the portfolio return is below the target there is zero upside potential, and the call option is

out-of-the-money. When the portfolio return exceeds the target the upside potential is

precisely the payoff of a call that is in the-money.

Similarly, the downside payoffs are identical to those of a short position in a put option on

the future portfolio return relative to the target. The portfolio call value is the expected

upside and the portfolio put value is the expected downside. Portfolios that achieve the

higher call for a given put are called put/call efficient.

The deviations of the portfolio return from the random target (g) are expressed using

auxiliary variables y and y as 0, ,( ] [ )y max R x r g % % %and 0,[ ( ], ) y max g R x r % % % .

y measures the upside potential of the portfolio to outperform the target. y measures

the downside risk and has the same payoff as a short position in a European put option.

To trace the efficient frontier, we formulate a linear program:

1

1

       

. .     

        [ ( , ) 

       

] 0,

, 0, 

s

s

s

s

S

s

s

s

s s

S

s

s s

R

Maximize p y

S t p y

y y r s

y

x

y

g

s

Mean Absolute Deviation (M.A.D)

The MAD model was originally proposed as an alternative to the classical Mean-Variance

model by Konno and Yamazaki (1992). The main reasons cited in their paper that led them

develop MAD were the computational burden for the practitioner that arises due to the

quadratic nature of the Markowitz model, the inadequacy of the standard deviation to

convince researchers that is a valid measure of risk(more specifically the assumption that

perceptions are symmetric around the mean) and because of the transaction costs that used

to arise by maintaining a large number of portfolios, that quadratic programs used to find as optimal solution to portfolio optimization problems.

In their paper, they proved that if the returns are multivariate normally distributed, then

utilizing the mean absolute deviation as a measure of risk, is essentially the same as the

utilization of standard deviation. Mathematically, this means that minimizing the w(x) function

𝑤(𝑥) = 𝐸[|∑ 𝑅𝑗𝑛𝑗=1 𝑥𝑗 − 𝐸[∑ 𝑅𝑗

𝑛𝑗=1 𝑥𝑗]|],

is equivalent to minimizing

𝜎(𝑥) = √∑ ∑𝜎𝑖𝑗

𝑛

𝑗=1

𝑛

𝑖=1

𝑥𝑖𝑥𝑗

We can therefore state the minimization problem is mathematically formulated as follows

min 𝐸[|∑ 𝑅𝑗

𝑛

𝑗=1

𝑥𝑗 − 𝐸 [∑ 𝑅𝑗

𝑛

𝑗=1

𝑥𝑗]|]

Subject to:

∑ 𝐸[𝑅𝑗

𝑛

𝑗=1

𝑥𝑗] ≥ 𝜌𝑀0

∑ 𝑥𝑗

𝑛

𝑗=1

= 𝑀0

0 ≤ 𝑥𝑗 ≤ 𝑢𝑗, 𝑗 = 1, … … . , 𝑛

where, as before, 𝑢𝑗 is the maximum amount the investor wishes to place in a single stock,

ρ is the minimum rate of return required by the investor which is not portfolio dependent

and 𝑀0 is the total amount of money being invested which is constant.

Scenario Optimization

Implementing MAD

For the implementation of MAD, we use data from the S&P 500 index, a stock market index

based on the market capitalization of 500 large companies having common stock listed on

the NYSE or NASDAQ.

From the index historical data from December 31, 1999 to November 30, 2016, we

calculated the average rate of return for each stock. From the data, we chose the 25 stocks

that yielded the highest return rate during the period under examination. We excluded from

our analysis shares of companies that were not in the American Stock Exchange before 01/01/2000, such as Facebook, Phillip Morris, Netflix, etc.

Utilizing the algebraic modelling language GAMS to create a portfolio with MAD, we plugged

in the data from our stocks of choice and we generated the minimum MAD portfolio. We

ended up with a portfolio that has 12 stocks, Expected Return=0.025% and MAD=0.035%.

Our optimal portfolio has the following stocks:

Monster Beverage

Carmax

Tractor Supply

Alexion Pharms.

Apple

Reynolds American

Gilead Sciences

Ventas

Flir Systems

Range Res.

Humana

Pioneer Ntrl Res.

Testing the Portfolio

To test the efficiency of our portfolio, we conduct on it both static and dynamic tests. The

static test is implemented by graphing the efficient frontier generated by MAD, that is,

graphing the set of portfolios which satisfy the condition that no other portfolio exists with a higher expected return but with the same mean absolute deviation of return.

After ensuring that our portfolio lies on the efficient frontier, we conduct a dynamic test on it. This is done via backtesitng our strategy for 30 periods, to ensure its viability.

Static Test - Efficient Frontier

To create an efficient frontier we modified the GAMS code that yielded a MAD portfolio by creating a for loop. The for loop works as follows:

We create 2 variables denoting the minimun and the maximum return in the universe of

stocks. We then create a variable “step” which is equal to the difference between the

maximum and the minimum return and we divide it by the number of portfolios we want

our frontier to contain. We then set the target expected return equal to the minimum

possible return in the universe. We utilize these variables in the for loop. Starting with the

target return equal to the minimum possible return, for each incrementation of the target return by ‘step’, a portfolio is generated until the maximum return is reached.

The GAMS code we used is the following:

Running the model, we create a MAD efficient fronter, made by 11 portfolios. The results

the code yielded are the following (all the following numbers are in percentages):

The weights needed to generate each of these portfolios are:

Finally, graphing the efficient frontier, we get the following result:

From the above results, we obseve that the almost all of the efficient portfolios consist of stocks of the same companies. The only thing that varies is their weights.

Having constructed the efficient frontier, we can now derive the minimum variance portfolio

which would be chosen by the most risk averse investor, or the MVP portofolio:

0.02530.3888

0.75621.1236

1.4911.8583

2.22572.5931

2.96053.3279

3.6953

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2 4 6 8 10

Expe

cted

Ret

urn

MAD

Efficient Frontier

Going upwards on the efficient frontier line, from the MVP portofolio to the portfolio of the

risk lover investor, we observe that the risk and the expected return of the portfolio increase

disproportionally, but however very close to linealry, to one another.

Dynamic Test-Backtesting

To test our findings dynamically, we conduct a backtest. To estimate the efficiency of our

strategy against different strategies, we are going to implement the dynamic test on both

the risk averse type of investor as well as the risk loving type. What the backtest is going to

show us is how a strategy would have performed if it had been employed in a past period.

Our general backtesting strategy is as follows: we backtest our strategy 30 months back, this

means that we start from May 2014. We use the historical data from January 2000 to May

2014 and we plug them in our GAMS code, to generate an optimal portfolio. From the

generated portfolio, we keep the weights of the stocks and we multiply them with the actual

corresponding real returns of the stocks for the period under examination. The result we get is the return our portfolio would have yielded, had we adopted our strategy.

We repeat the process for 29 more periods, each time rolling over one month as the staring

period, i.e. from February 2000 to June 2014, from March 2000 to July 2014 etc.

From this process, we get our hypothesized return, which are the returns we would have generated had we invested in the portfolio and held it for the given period

Now, we consider the different strategies we want to backtest. Specifically, to run the

backtest for the risk averse portfolio, we run the GAMS code, setting the target return equal

to 0 and minimizing MAD, for 30 periods as described above. We obtain the weights of the

minimum MAD portfolios for the last 30 months and we multiply the weight of each

portfolio with the realized returns of the index. This should yield the monthly returns had we

started with the risk averse strategy and kept it until the end of the period under

examination.

Similarly, we follow the same procedure to obtain the risk loving portfolio. The only

difference with the procedure described above is that we change the GAMS code such that it maximizes the total return we should expect from it.

Finally, to check the efficiency of the different strategies, we test the performance of the

portfolios against the benchmark index. Graphing the performance of the portfolio strategies against the index yields the following results:

We now consider the diagrmatical analysis of the two strategies:

We first assess the risk loving portfolio. As regards the positive returns, we can clearly see

that the risk loving portfolio significantly outperforms the index, generating higher returns.

The same is true however for its negative retuens which are clearly magnified compared to the index.

Regarding the minimum MAD portfolio, it appears from the diagram that most of the times

it outperforms the portfolio on the positive side, though not at such magnitude as the risk

loving portfolio. Regarding the negative returns, we see that it very closely mirrors the negavite movements of the index.

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

0 5 10 15 20 25 30 35

Risk Lover Index Risk Averse Series1Series2

We now proceed to the numerical analysis of our strategies:

A look on the returns confirms our diagramatical analysis. The risk lover clearly ‘bleeds’

more than the index regarding the negative returns, while we see that the returns of the risk

averse returns are, most of the times, considerably close to the index.

For a more rigorous analysis we calculated the beta of the portfolios, their variance and their correlation with the index:

We see that the beta of the risk averse portfolio (the beta has been calculated with the

variance of the index for the last 30 months under examination), is 0.84, which generally

indicates that the risk averse portfolio is less volatile than the index. Its correlation with the index is signifacantly high at 0.61, which our confirms our diagramatical analysis.

On the other hand, the risk lover portfolio with a beta of 1.32 has much more systematic risk

than the risk averse portfolio, which is a result that adheres to the intuition. Its correlation

with the index is significantly low, at a level of 0.13.

Conclusions

In this project, we created a portfolio out of 25 stocks chosen from the S&P 500 index. Using

the scenario optimization technique and implementing the MAD lineal optimization model

with GAMS, we generated an optimal portfolio. The resulting minimum MAD portfolio consists of 12 shares.

To test the efficiency of our portfolio we conducted on it both static and dynamic tests. Our

minimum MAD portfolio was on the MAD efficient frontier, as we demonstrated with our

GAMS algorithm, and it performed, if not optimally, significantly well against the benchmark index when we backtested it.

For the dynamic test, we considered the opposite strategy to the minimum MAD, that is , we

created a risk loving portfolio. We backtested our minimum MAD portfolio against both the

index and the risk loving one, and we procceded with both a diagramatical and a numerical assesment of both strategies.

References -Hiroshi Konno, Hiroaki Yamazaki. Mean Absolute Deviation and Its Applications to Tokyo Stock Market. Management Science, Vol. 37, No. 5, 519-531, 1991

- Wojtek Michalowski, Wlodzimierz Ogryczak. Extending the MAD Portfolio Optimization Model to Incorporate Downside RiskAversion. IR-98-041. June 1998

- Roy H. Kwon, Stephen J. Stoyan. Mean–Absolute Deviation PortfolioModels with Discrete Choice Constraints. Algorithmic Operations Research Vol.6. 2011

- Stavros A. Zenios (Eds.). Financial Optimization. Cambridge University Press. 1993

-Andrea Consiglio, Soren S. Nielsen, Stavros A. Zenios. Practical Financial Optimization, A

Library of GAMS Models.John Wiley & Sons. 2009

-Paul Embrechts, Hansjorg Furrer, Roger Kaufmann. Different Kinds of Risk.

- Artzner, P., Delbaen, F., Eber, J.M. and Heath, D.: Coherent measures of risk. Mathematical Finance 9, 203-228, (1999)

- Sergio Ortobellia, Svetlozar T. Rachevb, Stoyan Stoyanovc, Frank J. Fabozzid, Almira Biglovae. The Proper Use of Risk Measures in Portfolio Theory.