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Modeling of EUR-MAD and USD-MAD Exchange Rates

Ahmed Hrifa, (PhD Student)

Laboratory of Research on the Finance, Audit and Governance of Organizations National School of Business and Management of Settat,

Hassan 1er University of Settat, Morocco.

Zineb Bamousse, (PhD Professor)

Laboratory of Research on the Finance, Audit and Governance of Organizations National School of Business and Management of Settat,

Hassan 1er University of Settat, Morocco.

Correspondence address:

ENCG Settat

Km 3, route of Casa BP 658, Settat Hassan 1^er University of Settat Morocco

0523-72-35-77

contactencg@uhp.ac.ma ahmedhrifa00@gmail.com

Disclosure statement: The authors are not aware of any funding, that might be perceived as affecting the objectivity of this study.

Conflicts of interest: The authors reports no conflicts of interest.

Cite this article:

Hrifa, A., & Bamousse, Z. (2021). Modeling of EUR-MAD and USD-MAD Exchange Rates. International Journal of Accounting, Finance, Auditing, Management and Economics, 2(2), 302-323. https://doi.org/10.5281/zenodo.4641517

DOI: 10.5281/zenodo.4641517

Received: 15 January 2021 Published online: March 30, 2021

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Modeling of EUR-MAD and USD-MAD Exchange Rates

Abstract:

Currency Risk can be very problematic for our companies, so it is important for Moroccan companies exposed to currency risk to be able to measure this exposure in order to manage it properly.

The objective of this paper is to estimate the foreign exchange risk EUR / MAD and USD / MAD according to Basel II recommendations on capital requirements for market risks. To quantify the potential capital loss on foreign exchange risk, the experts use the VAR (Value At Risk) model to determine the potential loss that could impact the firm's performance.

We applied the VAR concept to estimate the foreign exchange risk of the Moroccan Dirham against the Euro and then against the US Dollar. Three standard VAR calculation models most used by risk managers (historical method, parametric method and Monte Carlo simulation) were studied and then applied to a historical exchange rate EUR / MAD and USD / MAD over the period from 02/01/2017 to 31/12/2018 and on a database of 520 observations.

Our study revealed that the Monte Carlo simulation method recorded the highest value of the VaR which is established at 281.170 DH against the parametric method which is limited to 204.100 DH. The average of the three m ethods is around 246.170 DH which represents a loss percentage of 0.91% of the total value of the portfolio.

However, to complete this study, we believe that it is wise to use stress testing and backtesting tests which make it possible to validate this VaR model and to meet the standards set by the Basel Committee.

With a confidence level of 99%, we found three exceedances of the losses compared to the values of the average VaR.

For a 95% confidence level, we have 16 exceedances. The maximum loss correspo nds to the date of: 24/08/2017, explained by the sharp drop in the EUR / MAD rate, from 11.235 to 11.156 DH.

Keywords: Currency Risk, Exchange Rates, Value-At-Risk, Stress testing, Backtesting JEL Classification: F, F3, F31.

Paper type: Empirical research.

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1. Introduction:

In recent years, the evolution of the increasingly turbulent and financialized economic environment, the interconnection, the globalization of capital markets and the increase in financial market volatility are all variables that have contributed to increa se the financial risks of companies. One of these risks is the risk created by unpredictable changes in exchange rates. It is generally accepted that returns on financial assets are logical responses to external shocks (A. Schipperijn 2018).

In the context of this international environment still marked by uncertainty and lack of visibility, VaR has emerged as one of the most commonly used measures to assess, on a given date, the importance of financial risks (Y .Bakkar 2018). VAR is the benchmark model adopted by financial institutions to quantify market risk (De Jonghe, 2010; Hurlin, et al., 2007; Szegö, 2002; among others). Indeed, it is a decision-making tool for the investor: if the probable loss is considered too high, he can either take a short pos ition, i.e.

sell part of the securities present in the portfolio, or take measures to hedge the portfolio allowing it to reduce the overall risk it faces.

In this article we are interested in VaR as a risk assessment tool on the Moroccan foreign exchange market. Three standard models for calculating the VAR most used by risk managers (the historical method, the parametric method and the Monte-Carlo simulation) by virtue of their simplicity, were studied and then applied to a historical exchange rate.

EUR / MAD and USD / MAD for the period from 01/02/2017 to 12/31/2018.

In this wake, our work is divided into five sections. The first section briefly introduces the concept of exposure to currency risk, the relationship between exchange rate and yield, the definition and calculation of the VAR. Also, we discuss the three methods of estimating VAR and we will present the retained hypotheses. In the second section, we will present the methodology adopted in order to apply the three approaches to calculating the VAR as well as the main results obtained. The comparison of the results obtained under the three approaches will be dealt with in the third section, and we will devote to the application of the stress testing technique to our sample. In order to validate a VaR calculation model, it is necessary to implement tests on real data: Backtesting. This exercise is essential because it gives, a posteriori, an image of the quality of the risk measurement. The last section is devoted to summary and conclusions.

2. Literature review:

2.1. The concept of exposure to currency risk

The research published on the concept of currency risk reveals its polymorphous and polysemic character, causing an amalgamation with related notions such as exchange rate, exposure, profitability, value, etc.

It should be noted that despite the avalanche of the term currency risk in the speeches of managers and consultants and the vast literature attached to it, it remains vague and covers a large body of definitions. There is no one-size-fits-all definition of currency risk, and its measurement poses enormous problems. The concept of currency risk presents itself as a very problematic notion.

Hekman (1983) defined exchange rate exposure as the sensitivity of its economic value, or price, to changes in exchange rates.

Adler and Dumas (1984) define exposure as the relationship between excess returns and exchange rate variation. In other words, exposure to the exchange rate is defined globally as the sensitivity of the value of the company to changes in exchange rates. Exposure arises when unexpected changes in the exchange rate change the market value of the business by changing the expected cash flows in its national currency.

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Doukas et al. (2003) confirm that exposure to the exchange rate can directly affect companies involved in international trade. Domestic firms can also be affected indirectly through a mechanism whereby exchange rate exposure impacts aggregate demand as well as the competitiveness and concentration of the sector.

According to Bergen (2010), exposure to exchange rate fluctuations is generally explained by an impact on:

- The value of net monetary assets with fixed nominal gains and;

- The value of real assets held by the business.

According to Jorion (2011), fluctuations in the exchange rate can affect the current and future cash flows of companies and, finally, stock prices.

Kadouri (2016) defines exposure to foreign exchange risk as the possibility of loss to which economic agents who carry out transactions in foreign currencies are subject to unfavorable variations in the exchange rates of these currencies against the national currency.

2.2. The relationship between exchange rate and yield

Theoretically, the relationship between the exchange rate and stock returns can be postulated as positive (currency depreciation makes local firms more competitive, which leads to increased exports as long as stock prices rise) or negative (if production is dependent on the cost of production which would increase due to currency depreciation, thus reducing profitability and a consequent decline in stock returns), and a weak or no relation (the price of a company-oriented export increases with currency depreciation, since the cost of entry is also affected by this currency depreciation where the effect would be canceled to some extent due to the increase in the cost of production).

Several theories have given great importance to the notion of exchange rate and its relationship with the profitability of companies, among the best known:

- Purchasing Power Parity Theory: this theory is based on the law of one price, according to which the price of a tradable good is the same everywhere, once it is converted into a common currency.

- Stakeholder Theory: this theory offers a participatory approach to strategy design.

Rather than considering the strategy only in the single dimension of the fight against competition, the stakeholder theory advocates the integration of all the partners in the process.

- Market Timing Theory: according to this theory, companies issue shares when share prices are high and / or the stock market environment is favorable, then they issue debt and buy back shares when prices are falling and / or the stock market is depressed

- Agency Theory: in this theory, business leaders who act as agents of shareholders have an obligation to manage the business in accordance with the interests of the shareholders.

- Hedging and Maximizing Shareholder Value: according to this theory, hedging against exchange rate risk maximizes the performance of the firm, by reducing taxes and costs relating to market imperfections, in particular costs related to financial distress, debt and underinvestment. In fact, hedging makes it possible to reduce the volatility of cash flows, which allows the firm to reduce the cost of financial distress and on the other hand to minimize its dependence on external financing and avoid underinvestment.

2.3. Definition and calculation of VAR

Value at Risk, better known as Value-at-Risk or VaR, is a measure of the potential loss that can occur as a result of adverse movements in market prices (Jorion, 2007). In finance,

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VaR corresponds to '' the maximum potential loss that a portfolio can suffer, under normal market conditions, during a period of time fixed with a given confidence threshold '' (Hull, 2007).

Statistically, the VaR can be determined according to a quantile of the distribution of theoretical losses, resulting from the possible movements of risk factors, over a fixed time horizon h.

The Value-at-Risk (VaR) of the level of risk associated with the risk factor is given by the following formula: αX

𝑉𝑎𝑅(𝑋, 𝛼) = inf {𝑥|Pr (𝑋 ≤ 𝑥) ≥ 𝛼}

with :

• 𝛼 represents the level of risk associated with the measurement 𝑉𝑎𝑅

• 𝑉𝑎𝑅(𝑋, 𝛼) is the amount that will cover the loss generated by the risk according to the probability.𝑋𝛼

• It should also be noted that where denotes the quantile function of the law of.𝑉𝑎𝑅(𝑋, 𝛼) = 𝐹_𝑋⁻¹(𝛼)𝐹_𝑋⁻¹(𝛼)𝑋

• the is not consistent because it is not sub-additive𝑉𝑎𝑅

In financial circles, VAR is calculated at a confidence level ranging from 90% to 99% and over a time horizon of 10 days.

2.4. Standard Methods of VAR Estimation - Parametric (or analytical) VAR

The calculation of the parametric VaR is based on two main assumptions:

• The normality of the return on the asset portfolio. This normality can be reflected in the multi-normality of the vector of asset price returns or of the risk factors making up this portfolio.

• Each asset can be broken down linearly into risk factors. This breakdown facilitates the valuation of the portfolio of assets considered.

- Historical VAR (or non-parametric)

One of nonparametric methods for calculating the VaR is Historical simulation. In this method there is no need to know distribution of data. In fact, VaR is computed by attention of an assumptive time series of returns and supposition that changes of future data are based on historical changes. The convenience of this method is no variance and covariance need to calculate. This method believes that behavior of returns is the same as before (Shokoofeh B. & Navidi S., 2017).

- Monte-Carlo simulation method

Monte Carlo VaR assumes that each of the market risk factors follows a known parametric law (generally the normal law), the parameters of which are estimated on the basis of historical data.

The Monte-Carlo VaR is based on the simulation of market factors whose distribution law is given a priori, preferably admissible with history. We can then value the portfolio with the simulated factors, by generating a large number of scenarios; we can then determine the simulated variations of the portfolio, which we assimilate to potential gains or losses. It is then sufficient to calculate the corresponding quantile just as for the historical method. The two methods are therefore very similar. The only difference is the simulation of the factors.

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2.5. Selected hypothesis

In theory, the Monte Carlo simulation method does not require specific models and can be easily adjusted to economic forecasts. The results can be improved by considering a large number of simulated scenarios. For non-linear instruments and especially options, they can be easily included in a portfolio. From these elements, the first hypothesis of our research can be formulated as follows:

- H1: the Monte-Carlo method provides more relevant results than the other VAR assessment methods.

VaR by definition does not provide any forecast indication of the extent of any losses that may result from adverse variations in market factors. It is for this reason that the regulatory authorities require financial institutions to conduct disaster scenario analyzes (stress testing) in addition to the VaR forecast. The stress test or stress test instrument makes it possible to measure the risks that the VaR cannot capture such as scenarios on the future and extreme market movements, to assess the capacity of the bank's equity to cope at times of high losses possible by considering a sufficiently large amount of equity (Saber M., 2018). From these elements, the second hypothesis to be tested is as follows:

- H2: Stress testing is a complementary method to VAR.

Risk managers use a technique known as backtesting to determine the accuracy of a VaR model. Backtesting involves the comparison of the calculated VaR measure to the actual losses (or gains) achieved on the portfolio. Therefore, the third hypothesis of our research can be formulated as follows:

- H3: By applying the “backtesting” method, there are excess losses in relation to the values of the VAR.

3. Methodology 3.1. Data presentation

In order to apply the VaR formulas, we have chosen a database of 520 observations spanning the period 01/02/2017 - 12/31/2018.

Before starting the calculations, we tested the representativeness of our statistical sample.

According to the theory of statistical studies, the sample size is calculated with the following formula:

With:

• n: Minimum sample size for obtaining significant results for an event and a fixed level of risk;

• t: Confidence level (the standard value for the 95% confidence level will be 1.96);

• p: estimated proportion of the population that presents the characteristic, in our study p is 0.5 (We have given the same chance of having a factor or not)

• m: Margin of error, we set it at 5% (recommended).

So, taking a confidence level of 95% and a margin of error of 5%, the sample size should be:

n = 1.96² × 0.5 × 0.5 / 0.05² = 384.16 or 385 individuals 𝑛 =𝑡² ∗ 𝑝 ∗ (1 − 𝑝)

𝑚²

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It should be remembered that the size of our sample (520) satisfies the requested threshold, which is 385.

Below is an extract from the database:

Table 1: Extract from the database

USD/MAD

Last exchange rate Opening 9,5583

9,5485 9,5375 9,5703 9,5632 9,5461 9,5535

…

… 10,0878 10,1515 10,2005

9,551 9,5379

9,569 9,562 9,5445 9,5475 9,516

…

… 10,1554

10,206 10,1535

Source: Database, developed by the authors.

3.2. Modeling VAR portfolio : - Parametric (or analytical) VAR The VAR can be written:

But we have:

Thus, the VAR can be defined as follows:

With: Zα: the centile of order 100α, of the standard normal distribution.

- Historical VAR (or non-parametric)

N P i

P 

_i

) =  = (

Prob

_P_iis the value to be determined.

We distinguish between two cases:

1) i=N* Is an integer: the VAR will be the absolute value of the (Νхα)th smallest value. For example, if we have 100 historical data and the confidence level is 99%, we should take the first value.

EUR/MAD

Date Last exchange rate Opening 31/12/2018

28/12/2018 27/12/2018 26/12/2018 25/12/2018 24/12/2018 21/12/2018

…

… 05/01/2017 04/01/2017 03/01/2017

10,9638 10,9263 10,9024 10,8651 10,8662 10,8789 10,8647

…

… 10,7001 10,6479 10,6164

10,2968 10,9015 10,8651 10,8652 10,883 10,8617 10,8929

…

… 10,648 10,6136 10,6196

 



_ ₌



−

 −



+ −

) .

. . .

. . ( .

Pr ^,

T T

tT t

tT tT

t t

t h

h t

V V

V VaR

V V

ob R

T

T t

t t

h

V V V

VaR

__,

= 

_

( .  . ) −  .

) 1 , 0 (

~ .

.

. 



−

+

tT t

tT tT

V V

R^t ^h 

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2) i=N* Is not an integer, it will then be necessary to calculate the VAR by linear interpolation using the following formula :

) )(

*

(

* *

* 1

*

,h

P

n

N n P

n

P

n

VaR

_

= +  −

₊

−

With n* the integer closest to the value N x. - Monte-Carlo simulation method

Let us consider a portfolio with K risk factors correlated with one another denoted F1, F2 …FK.

We calculate the historical returns R_i^t of each risk factor i between t and t+h. these yields are given by the following relation:

)

ln( _t

i h t h i

t

i F

R F

+ = +

With:

R

^t⁺^h

= ( R

₁^t⁺^h

, R

₂^t⁺^h

.... R

_n^t⁺^h

)

the vector of the returns of the K risk factors.

We have: R_i^t⁺^h ~(,) with the vector average returns (₁,₂,...,_K)

And :













=

I J, J,1

I 2, 2,2 2,1

I 1, 1,2 1,1

...

σ

. ...

.

. ....

. .

...σ σ

σ

...σ σ

σ

Σ



The matrix of variances covariance’s of returns.

The Monte Carlo simulation is done in five steps which we will describe as follows:

• Step1: Generation of random vectors from the chosen law. This step consists in the generation of K independent random variables that we will represent by the vector Z = (Z1, Z2...Zk). Each generated Z line vector is called a simulated scenario.

• Step 2 : Cholesky decomposition Proposal :

Any semi-definite positive matrix D can be factorized in the form: D =AA^T

Where A is a square matrix, which can be considered to be a «Square Root Matrix» of D. the matrix A is not unique. Thus, several factorizations of a given matrix D are possible.

These factorizations are known as Cholesky factorizations.

When D is a positive definite matrix then the matrix A is unique.

The variance covariance matrix Σ has the same properties as D. We then perform Cholesky decomposition such that Σ =AA^T: With:

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











=

I J, J,1

I 2, 2,2

2,1

I 1, 1,2

1,1

...

. ...

.

. ....

. .

...

A

a a

a

a a

a

The coefficients of the matrix A are given by the following relations:













−

=

−



=



−

=

−

= 1 i

1 k

jk ik ji

ii

1 i

1 k

2 jk ii

i j pour

) a a a (σ

1

i j pour

σ a

i j pour

0

aji

With :



_ij

= cov( R

_i

, R

_j

)

• Step 3 : Calculate the yield scenarios

Let: r = (r1, r2, ...rk) be the vector of simulated returns over h days. Each simulated vector represents a yield scenario. These scenarios are calculated as follows: we seek to generate random vectors resulting from the random vectors Z generated at the level of the first step (simulated scenarios). This transformation is done as follows : r = AZ^T + µ.

With A the Cholesky factorization matrix of the variance-covariance matrix .

• Step 4 : Calculation of risk factor scenarios

Let: f= (f1 ... fK) be the vector of the simulated returns. The inverse expression of the calculation of returns is used to determine the simulated risk factors. This expression is as follows:

With F_i^réf is the reference risk factor i used to calculate the risk factor scenarios. We generally take F_i^réf as those of the day of the VAR calculation.

• Step 5: calculation of the VAR.

From the simulated risk factor scenarios, the portfolio value is calculated for each scenario. Then, we calculate the changes in the value of the portfolio and we order them, then we just have to take the percentile of order A of this distribution. This percentile corresponds to the value of the VAR at horizon h and to the α % confidence threshold.

In short and for reasons of simplification, we have chosen to work with logarithmic returns where:

𝑅_𝑙𝑜𝑔^𝑖 (𝑡) = ln (𝑆_𝑖,𝑡 𝑆_𝑖,0)

The method consists in simulating a large number of times the trajectories of the returns of each security on the horizon of 1 day.

In our case, we have set the number of simulations to 5000, because we first need to simulate 5000 reduced normal centered random variables. Next, we perform the Cholesky decomposition of the portfolio's exchange rate return correlation matrix. This step is

) exp(

. _i

réf i

i F r

f =

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necessary since it makes it possible to take into account the correlation between the various securities making up the portfolio. Then, we generate 5000 possible scenarios of the 2 exchange rates. And finally, we calculate 5000 possible returns of the portfolio and we proceed to estimate the VaR as for the historical method, from the generated sample.

4. Results and discussion

4.1. Application of historical VAR 4.1.1 Foreign exchange portfolio:

The historical method is a very simple approach. It consists of being based on the empirical distribution of historical data of portfolio returns. This involves calculating the daily returns of the portfolio at each historical date and ranking them in ascending order in order to choose the return equivalent to N × (1-α), where N is the number of returns and α is the level of trust.

The portfolio return is the sum of the weighted returns of the securities in the portfolio.

We have taken a portfolio of a Moroccan company which mainly consists of two EUR and USD exchange rates, as shown in the following table:

Table 1-1: Extract from the database

Motto Position %

USD 1,040,000.00 40%

EUR 1,560,000.00 60%

Source: Authors

4.1.2 Verification of stationarity:

The historical method is based on the assumption of stationarity which states that the past will be reproduced in the future. To do this, we used the Dickey Fuller Augmented Test (DFA). This test makes it possible to highlight the stationary character or not of a chronicle by the determination of a deterministic or stochastic tendency.

The principle of this test is simple, we test the null hypothesis H0: existence of a unit root against the alternative hypothesis H1: no unit root.

If the hypothesis H0 is true, the chronicle is not stationary

The DFA statistic establishes tables analogous to the tables of Student's t.

If then we accept the hypothesis H0; there is a unit root, so the process is not stationary.𝑡_{𝑠𝑡𝑎𝑡𝑖𝑐} > 𝑡_{𝑡𝑎𝑏𝑢𝑙é}

The results of this test carried out by the software EVIEWS showed that only two series of exchange rates: EUR and USD which are stationary.

The test result applied to the EUR is presented below.

Table 2: ADF test applied to the EUR / MAD exchange rate

Null Hypothesis: EUR has a unit root Exogenous: Constant

Lag Length: 5 (Automatic based on SIC, MAXLAG = 21)

t-Statistic Prob.

Augmented Dickey-Fuller test statistic -3.234002 0.0184 Critical values

test:

1% level -3.436154 5% level -2.863991 10% level -2.568126

Source: Database, developed by the authors

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This table informs us about the Student t-Statistic statistics associated with the EUR variable. This is equal to -3.234. To test the hypothesis H0, we then use the thresholds tabulated by Dickey and Fuller and for an infinite sample size, we find for a threshold of 5%

tabulé = -2.863991. Thus, in this case:, we reject the null hypothesis of unit root. The EUR / MAD exchange rate is then stationary.𝑡_{𝑠𝑡𝑎𝑡𝑖𝑐} > 𝑡_{𝑡𝑎𝑏𝑢𝑙é}

Likewise, we carried out the ADF test for the USD / MAD exchange rate, of which we found the following result:

Table 3: ADF test applied to the USD / MAD exchange rate

Null Hypothesis: DKK has a unit root Exogenous: Constant

t-Statistic Prob.

Augmented Dickey-Fuller test statistic -3.062644 0.0298 Critical values test: 1% level -3.436154

5% level -2.863991

10% level -2.568126

Out of sixteen currencies only two are stationary. This result may lead to a bias in the VaR estimate, but given that the share of EUR and USD in our portfolio exceeds 80%, the impact of other currencies will be negligible.

4.1.3 Calculation of historical VAR

For the foreign exchange portfolio, the calculation of the historical VaR only requires the history of the exchange rates, to calculate it we proceed as follows:

First of all, we calculate the daily value of the portfolio in dirhams, from 01/02/2017 to 12/31/2018, such that the portfolio value on day d is equal to the sum of the values of the positions in dirhams obtained by the multiplication of currency positions by the exchange rate of the same day.

Then, we calculate the daily variations of these values between time j and j-1, these variations represent a history of losses or profits (P&L). These are given by the following table:

Finally, and from a vector of variations composed of 520 observations and using the percentile function (matrix;), we determine the historical VaR, which is nothing other than the quantile corresponding to the confidence threshold of 1%.

The VaR is thus given at a confidence level of 99% and for a horizon of 24 hours; the results found are represented by the figure below:

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Table 4: Calculation of historical VaR at 12/31/2018

Descending ranking

Date Portfolio value P & L 1

…

… 513 514 515 516 517 518 519 520

27/06/2017

…

… 28/12/2018 16/01/2017 16/08/2017 13/12/2018 16/10/2017 12/12/2018 24/08/2017 26/06/2017

27 440 036

…

… 26 975 468 27 111 708 27 146 236 26 860 756 27 139 112 26 726 700 27 140 100 27 201 408

217 880

…

… - 68 692 - 73 736 - 80 080 - 85 332 - 93 548 - 134 056 - 185 692 - 238 628

VAR 24h (99%) - 80 080

CVAR 24h (99%) - 136 223 VAR 10 days (99%) - 253 235 CVAR 10 days (99%) - 430 774

By applying the following formula:VaR__,₁₀_j = 10*VaR__,₂₄_h, it is possible to determine the 10-day VaR which stands at - 253,235 dirhams, it represents only 0.94% of the total value of the foreign exchange portfolio.

The 10-day RBAF is-430 774 dirhams, it represents only 1.59% of the total value of the foreign exchange portfolio.

4.2 Applying the parametric VAR

This method, also called the variance-covariance method, is based on two assumptions: the normality of the returns of the securities in the portfolio and the linearity of the risk profile of the same portfolio.

First, we draw the normal QQ Plot (Quantile-Quantile Plot) which allows us to represent the quantiles of the distribution of returns as a function of the quantiles of the normal distribution N (0,1). As shown in the following figures:

Figure 1:Normality test for EUR / MAD and USD/MAD returns

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According to the figures above, the QQ Plot is confused with the line of the normal distribution. Therefore, the distribution of EUR / MAD and USD / MAD returns are normal.

After checking the normality of the two returns, it is possible to calculate the parametric VaR, so it is necessary to estimate the two parameters of the portfolio namely: the average return and the standard deviation.

𝑉𝑎𝑅_{𝛼%,1𝑗𝑜𝑢𝑟} = 𝜇 + 𝜎𝑍_𝛼 The results are:

Portfolio value on 31/12/2018 27 044 160

Average yield 0,0015%

Standard deviation of returns 0,1019%

We find that the portfolio average is low. This is because portfolio returns are stationary and oscillate around 0.

The calculation of the VaR by the parametric method gives the following results:

Table 5: Calculation of the parametric VAR

VAR (99%) VAR (95%) VAR (90%)

% 0,239% 0,169% 0,132%

Portfolio value on 31/12/2018 27 044 160

VAR 24h 64 543 45 755 35 739

VAR 10 days 204 104 144 691 113 018

We find that the VaR is an increasing function of the confidence threshold. The greatest loss is recorded for the highest threshold which is 99%. In addition, there is a 1% chance of losing more than 0.239% of the value of the portfolio over a one day horizon and a 10%

chance of losing a value greater than 0.132% of the same portfolio over the same horizon.

4.3 Application of the Monte Carlo VAR:

• Generation of 5000 random vectors according to the MN law (0,1)

This involves generating 5000 independent random vectors according to the MN (0,1) law.

Each vector is of size (2.1) since the portfolio is made up of 2 exchange rates. Indeed, each scenario predicts the future evolution of the two prices at the same time, and therefore of the portfolio.

Table 6: Simulation of EUR and USD rates against MAD

Number Alea 1 Alea 2 EUR exchange

rate simulation

USD exchange rate simulation 1

2 3 4 5 6 7 8 9 10

…

… 4999 5000

0,474485 0,0872082 0,8088506 0,508308 0,7752979 0,3100569 0,9463072 0,2811415 0,1007257 0,7384537

…

… 0,5831782 0,6271943

0,5113603 0,6067204 0,3127407 0,318662 0,854702 0,2216512

0,653571 0,387095 0,7032409

0,003007

…

… 0,7206353 0,5130725

11,07 10,81 11,26 11,09 11,24 10,99 11,41 10,97 10,83 11,21

…

… 11,13 11,15

9,60 9,67 9,47 9,47 9,87 9,40 9,70 9,52 9,73 8,91

…

… 9,75 9,60 Source: Database, developed by the authors

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Table 7: EUR and USD yield simulation

Number EUR exchange rate simulation

USD exchange rate simulation 1

2 3 4 5 6 7 8 9 10

…

… 4999 5000

-2,016 % 1,000 % 5,044 % -1,596 % 1,531 % 2,506 % 0,338 % 0,303 % 1,186 % 0,033 %

…

… -1,705 % -1,022 %

1,661 % 0,949 % 1,022 % 0,423 % 3,488 % -2,506 % -1,116 % 1,497 % 5,258 % 2,107 %

…

… 5,974 % 0,139 %

• Cholesky decomposition of the correlation matrix

This step is essential in the approach to Monte-Carlo. This factorization makes it possible to take into account the correlations between the two securities in the portfolio. Cholesky factorization allows for a positive definite matrix A, to determine a lower triangular matrix L such that:

𝐴 = 𝐿𝐿^𝑇

The Cholesky decomposition performed on the correlation matrix, gave rise to the lower triangular matrix below:

• Multiplication of the vectors generated by the matrix resulting from the factorization of Cholesky

After having obtained the lower triangular matrix via the Cholesky decomposition which we will denote by L, one proceeds to the multiplication of the random vectors generated at the beginning by this matrix. This makes it possible to have new random vectors whose components take into account the correlation between the two titles. It is these latter vectors which will be used subsequently. Table 8, below shows the results of this calculation:

Table 8: Simulation of portfolio return EUR yield simulation with

correlation

USD yield simulation with correlation

Portfolio yield

1,450%

1,945%

0,891%

1,967%

2,189%

1,380%

-1,305%

1,604%

3,254%

1,450%

1,945%

0,891%

1,967%

2,189%

1,380%

-1,305%

1,604%

3,254%

1,450%

1,945%

0,891%

1,967%

2,189%

1,380%

-1,305%

1,604%

3,254%

1 0,0000

0,09225 1

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1,806%

1,731%

2,215%

1,194%

1,132%

-1,000%

0,248%

1,806%

1,731%

2,215%

1,194%

1,132%

-1,000%

0,248%

1,806%

1,731%

2,215%

1,194%

1,132%

-1,000%

0,248%

Source: Database, developed by the authors Figure 2: Simulation of USD yield with correlation

Figure 3: Simulation of EUR yield with correlation

• Determination of the portfolio VaR

After having simulated the trajectories of the portfolio's return, we proceed in a manner similar to the historical method. Indeed, the simulated returns are classified in decreasing order. Therefore, the VaR for a confidence level α corresponds to the return of the position N

× (1- α).

The results of the Monte-Carlo VaR calculation are as follows:

-4,000%

-3,000%

-2,000%

-1,000%

0,000%

1,000%

2,000%

3,000%

4,000%

5,000%

6,000%

7,000%

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Simulation Rendement USD avec la corrélation

-4,000%

-2,000%

0,000%

2,000%

4,000%

6,000%

8,000%

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Simulation Rendement EUR avec la corrélation

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Table 9: Calculation of the Monte-Carlo VAR

VAR (99%)

VAR (95%) VAR (90%)

% -0,329% -0,187% -0,124%

VAR 24h -88 913 -50 704 -33 527

VAR 10 days -281 168 -160 340 -106 021

4.4 Comparisons of results:

The table below represents a summary of the results found for the value at risk for a time horizon of 10 days and for a confidence level of 99%.

Figure 4: Results of the estimation of the three standard approaches to VAR

From the table we see that the Monte Carlo method recorded the highest value of the VaR which is established at 281,168 DH against the parametric method which is limited to 204,104 DH. The average of the three methods is around 246,169 DH, which represents a loss percentage 0.91% of the total value of the portfolio.

Table 10: Number of VAR exceedances of the three approaches

Number of historical VAR overruns

Number of para VAR overruns

Number of mounted VAR overruns

6 1,2%

11 2,1%

1 0,2%

1%

253 235

204 104

281 168

246 169

0 50 000 100 000 150 000 200 000 250 000 300 000

VaR historique VaR paramétrique VaR Monte Carlo Moyenne

Comparaison des résultats

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Figure 5: Graphic presentation of the result of the estimation of the three standard approaches to VAR

From Table 11 and Figure 7, it is clear that the Monte Carlo VaR is the simulation that provides more relevant results to assess the real risk of the MAD-EUR and MAD-USD exchange rate.

4.5 Validation techniques and Stress-testing:

The use of stress tests is one of the basic principles for an internal model to be validated in order to know the amount of potential losses in the event of dangerous and significant fluctuations in the market. The aim of these simulations is then to complete the risk measurement system, and in particular the VaR measurements. Stress tests, sometimes called stress scenarios, and more commonly known as stress testing, should be tools for understanding the exposure of a company's foreign exchange portfolio to a serious crisis.

Unlike VaR, they must make it possible to answer the following question:

How much loss does the business face in the next crisis if the trading book does not change?

To analyze the impact of a stressful situation on a foreign exchange portfolio, we have chosen the approach based on the Theory of Extreme Values (TVE)¹. This technique consists in estimating the potential loss by subjecting the model to extreme variations of the parameters, corresponding to a financial catastrophe scenario.

The stress testing technique is generally done by disturbing the variance covariance matrix while keeping the semi-definite property positive in other words we have increased the coefficients of the variance covariance matrix so that they do not exceed the value 1;

After calculating the covariance matrix, we use it to obtain the Cholesky matrix denoted by C such that: V = CTC

Once C obtained we make the product CTZ = Y where Z is the vector of simulated random numbers according to the standard normal distribution.

The following graphs illustrate the changes in the fluctuations of returns depending on the disturbance factor applied to the variance-covariance matrix:

1 TVE concerns probabilistic and statistical questions with very high or very low values in sequences of random variables and in stochastic processes.

-300 000 -200 000 -100 000 - 100 000 200 000 300 000

1 32 63 94 125

156

187 218 249 280 311 342 373 404

435

466 497

P&L Var His VaR Mont carlo

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Figure 6: Simulation of the EUR / MAD price according to a stress factor of 1

Source: Database, developed by the authors Figure 7: Simulation of the EUR / MAD price according to a stress factor of 2

Source: Database, developed by the authors Figure 8: Simulation of the EUR / MAD price according to a stress factor of 3

1 Facteur de stress

9,50 10,00 10,50 11,00 11,50 12,00 12,50 13,00 13,50

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241 251 261 271 281 291

Titre de l'axe

Simulation cours EUR/MAD

9,50 10,00 10,50 11,00 11,50 12,00 12,50 13,00 13,50

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241 251 261 271 281 291

Titre de l'axe

3 Facteur de stress

9,50 10,00 10,50 11,00 11,50 12,00 12,50 13,00 13,50

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241 251 261 271 281 291

Titre de l'axe

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After choosing a stressor 2 as recommended by the experts, we proceeded to the same steps to calculate the VaR, the results are given below:

Table 11: Estimated VAR according to the stressor of 2

VAR (99%)

VAR (95%) VAR (90%)

% -0,690% -0,471% -0,347%

VAR 24h -186 644 -127 312 -93 788

VAR 10 days -590 219 -402 596 -296 583

Source: Database, developed by the authors

With the stress factor of 2, we see that the value of the 10-day VaR increased from 246,170 to 590,220 DH for a confidence level of 99%, with an increase of 240%.

It is for this reason that experts recommend that stress testing should be performed regularly to deal with any significant market variation.

4.6 Backtesting:

In order to validate a VaR calculation model, it is necessary to implement tests on real data: Backtesting. This exercise is essential because it gives, a posteriori, an image of the quality of the risk measurement. It is based on the notion of the number of exceptions, that is to say the number of times the actual loss observed is greater than the amount forecast and calculated by the internal model.

For a history of n days, it is therefore a matter of comparing for each day t, the VaR calculated for a confidence level of 99% over a one-day horizon, with the P&L distribution.

This comparison must be made over a minimum period of 250 days.

Recall that P&L of a day t is the difference between the real value of the portfolio at t and its value at t-1.

Under normal market conditions, a daily VaR with a confidence level of 1% results in a loss level which will be exceeded on average once in every hundred (100) days.

Depending on the number of exceptions recorded obtained by comparing the daily profits and losses (P&L) on the portfolio and the corresponding VaRs, the Basel committee then defines three zones to assess the results of the ex post controls and to apply any additional multiplier.

The total number of exceptions can be seen as a random variable following a binomial distribution, the probability of "success" (exception or exceeding of the VaR) is equal to p = 0.01 for a VaR of 1% and the number d The observations of this law is exactly n (the number of days of the Backtesting (520 for example)). The average number of exceptions is therefore n * p = 5.2 and the variance of the exceptional losses is n * p * (1 - p) = 5.148 which corresponds to a standard deviation of 2.27.

Using the fact that the binomial distribution converges to the normal distribution for a large n and a small p. The 99% confidence interval of the number of exceptions for a valid VaR model is:

[(np - Zα * ); (np + Zα * )]

With Zα the quantile of order α = 1% of the standard normal distribution. This interval is equal in our example to:

[5.2 - 2.326 * 2.27; 2.5 + 2.326 * 2.27] = [-0.08; 10.48]

This means that for 520 observations, we are 99% sure to observe a number of exceptions between 0 and 10 if our model for calculating the VaR is valid.

p) - (1 p

n n p(1-p)

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This example shows us that Backtesting is not 100% clear as to the validity of the model since it depends on the period considered and the level of confidence adopted.

For our portfolio, we compare the daily losses (P&L) with the VaR at 99% and 95% over a 24-hour horizon. This daily comparison is carried out from 11/09/2011 to 03/30/2012, that is to say 100 working days.

Backtesting applied to the parametric and historical VaR gives the results illustrated by the following graph:

Figure 9: Backtesting applied to the average VaR

With a 99% confidence level, we see three exceedances of the losses over the values of the average VaR, as shown in the graph above, which gives a percentage of exceedance of 0.58%

less than the risk of 1%.

For a confidence level of 95%, we observe 16 exceedances which represent 3.08%.

The maximum loss is around 185,690 DH, corresponds to the date of 08/24/2017, explained by the sharp decrease in the EUR / MAD rate, from 11.235 to 11.156 DH.

5. Summary and conclusions:

In order to reduce the risk, it is now clear that several techniques must be used to measure and quantify the risks. VaR is one of them that fulfills its mission in normal market times, but this must be complemented by stress analyzes and scenarios for crisis environments. This paper has been devoted to the practical implementation of VaR according to the three methods. We tried to verify three hypotheses (2.5). In order to apply the VaR formulas, we have chosen a database of 520 observations spanning the period 01/02/2017 - 12/31/2018.

Next, we presented the VaR calculation results for each section. And finally, we proceeded to compare the results found by the three methods.

The empirical study carried out on the risk of variations in EUR / MAD and USD / MAD exchange rates shows that the Monte Carlo method recorded the highest value of the VaR which is established at 281,170 DH against the parametric method which is limited to 204,100 DH. The average of the three methods is around 246,170 DH, which represents a loss percentage 0.91% of the total value of the portfolio.

The comparison of the results provided by the three VaR methods shows that the Monte Carlo VaR is the simulation that provides more relevant results to assess the real risk of the MAD- EUR and MAD-USD exchange rate. Our results join those of Jordan & Mackay (1997) and Duffie & Pan (1997) who concluded that the Monte Carlo simulation technique is by far the

-250 000 -200 000 -150 000 -100 000 -50 000 - 50 000 100 000 150 000 200 000 250 000

1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210 221 232 243 254 265 276 287 298 309 320 331 342 353 364 375 386 397 408 419 430 441 452 463 474 485 496 507 518

Perte potentielle

Backtesting

P&L VaR,99%

VaR,95%

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most precise and the most flexible, but it involves very long computation. However, Coronado et al. (2010) concluded that the empirical superiority of the Monte Carlo simulation method does not turn out to be as slow as considered. Soufia A. (2002) also showed that the Monte Carlo method gives very good results, but it requires significant power and computing time.

Finally, to complete this study, we believe that it is wise to use stress testing and backtesting tests which allow this VaR model to be validated and to comply with the standards imposed by the Basel Committee. With a confidence level of 99%, we found three exceedances of the losses compared to the values of the average VaR. For a 95% confidence level, we observed 16 exceedances. The maximum loss corresponds to the date of:

08/24/2017, explained by the sharp drop in the EUR / MAD rate, from 11.235 to 11.156 DH.

For future studies, other risk measure can be compared to find the best one. References:

(1) Adler M. & Dumas B. (1984). «Exposure to currency risk: definition and measurement», Financial management, Summer, pp 41-50.

(2) Bergen, J. (2010). «6 Factors That Influence Exchange Rates», Investopedia, http://www.investopedia.com/articles/basics/04/050704.asp#axzz288Smcag.

(3) Coronado & al. (2010). « Giving with one hand » International Journal of Sociology and Social Policy, vol. 30, no 11/12, p. 666-682.

(4) De Jonghe, (2010). «Back to the basics in banking? A micro-analysis of banking system stability », Journal of Financial Intermediation, Elsevier, vol. 19(3), pages 387- 417, July.

(5) Doukas, J., Hall, P.H. & Lang, L.H.P. (2003). «Exchange Rate Exposure at the Firm and Industry Levet ». Financial Markets, Institutions & Instruments, Vol. 12, p. 291- 346.

(6) Duffie D. & Pan J. (1997). « An overview of Value-At-Risk ». The journal of derivatives, printemps, pp.7-49, ISSN, 1074-1240.

(7) Jorion, P. (2011). «The pricing of exchange rate risk in stock market», Journal of Financial and Quantitative Analysis », 26, (3), 363–376.

(8) Jorion, (2007). « Vers la crise du capitalisme américain », La découverte/M.A.U.S.S, Paris 13^ème.

(9) Jordan J.V. & Mackay R.J. (1997), « Assessing value at risk for equity portfolios:

implementing alternative techniques ». Derivatives Handbook, Robertpp.265-309 (10) Heckman James, (1983). «Are Unemployment and Out of the Labor Force

Behaviorally Distinct Labor Force States? » , Journal of Labor Economics, 1983, vol.1, issue 1, pp 28-42.

(11) Hull, (2007). « Risk Management and Financial Instituions », Pearson International Edition.

(12) Hurlin, et al., (2007). «Un test de validité de la Value-at-Risk », halshs- 00257309,HAL

(13) Kadouri H. & al. (2016). «Evaluation du risque de change selon les directives de Bâle II», Revue : Finance et Finance Internationale, N°4 Juillet 2016.

(14) Saber Mouna (2018), « Risk assessment : contribution of the VAR compared to the stress test », Revue du Contrôle et de la Comptabilité et de l’Audit, N°6, Septembre.

(15) Schipperijn (2018). « Gestion des risques : diversification, couverture et assurance », Haute École de Gestion de Genève (HEG-GE).

(16) Sofia A. (2002). « Estimation de la valeur à risque en finance ». Université du Québec à Trois-Rivières. Page 67.

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(17) Szegö, (2002). « Measures of risk », Journal of Banking & Finance. Volume 26, Issue 7, pages 1253-1272.

(18) Shokoofeh B. & Navidi S. (2017). «Portfolio performance evaluation in Mean-CVAR framework : a comparison with non-parametric methods value at risk in Mean-VAR analysis», Operations Research Perspectives, volume 4, pages 21-28.

(19) Y.Bakkar (2018). « Apports de la Théorie des Valeurs Extrêmes au calcul de la Value-at-Risk », School of Business and Governance,Tallinn University of Technology–TalTech