Application of value at risk on Moroccan exchange rates

Introduction

Morocco is the 5th largest economy in Africa. Its currency is not freely convertible. However, the Moroccan course is established on the basis of a basket of currencies Euro and Dollard which are the main country partner's currencies. The basket consists of 80% Euro and 20% Dollar. As a result, all foreign-based companies are exposed to a currency risk related to these two currencies. For this reason, this article presents a modeling of the exchange rate of the two currencies in relation to the Moroccan currency according to the Value at Risk method.

Theoretical overview

Foreign exchange risk is one of the market risks, it is linked to "the uncertainty of the exchange rate of one currency compared to another in the short or medium term. Not knowing what will be the evolution of an exchange rate at three or six months causes difficulties. " [1] . It is therefore the risk related to the variation of exchange rate between the date of the commitment and the date of settlement.

Types and sources of foreign exchange risk: Foreign exchange risk may arise from commercial or financial transactions carried out by the company internationally. It can also depend on the international development of the company and the investments it has made abroad. Sources of foreign exchange risk can thus be categorized as following:  Risks arising from commercial operations;  Risks resulting from financial activities;  Economic risks sometimes called competitiveness risks or induced risks;  Risks related to cross-border investments.

Some reminders of probability and statistics

Before presenting the Value at Risk method, it is important to recall some probabilistic definitions.  Density and distribution function We consider a real stochastic variable or continuous real random variable (r.r.v), noted X, whose probability law, for an occurrence x, is defined by the density function𝑓 𝑋 (𝑥), assumed to be continuous, such that 𝑓 𝑋 (𝑥)≥ 0:

We denote by F X (. ) the distribution function or cumulative distribution function associated withX, such as:

 Some Moments: For a real random variable (r.r.v) X, the mean and the variance are two particular moments, but more generally it is possible, under certain assumptions, to define the population of moments and the population of centered moments in the following way:

-The moment of order 1: the mean The mean, denoted E (X), corresponds to the moment of order one (k = 1):

The moment of order 2: The variance The variance, denoted by V (X), is defined by the central moment of order two (k = 2):

The central moment of order 3: Skewness For the purpose of the VAR, it is important to define the 3rd order centered moment.

Skewness is a distribution asymmetry measure from which we can define a new measure, namely the Skewness coefficient:

𝜎 3 -The central moment of order 4: Kurtosis The 4th order centered moment is defined as follows:

The new measure to deduce from Kurtosis is the degree of Kurtosis excess:

Evaluation of currency risk by the VaR 'Value at Risk' method

By definition, VaR is the maximum loss that a portfolio manager may experience during a certain period of time with a given probability. Statistically, the Value at Risk can be defined as a quantile of the distribution of the theoretical P & L (Profit and Loss) of a portfolio, resulting from the possible movements of market risk factors, over a fixed time horizon. The P & L of a portfolio value P over a horizon of h days is given by:

P&L(t ; h)= ΔhPt=Pt + h -Pt

By definition, the VaR over h days with a confidence level (1-α) is the value R such as the probability of losing R at the most, beyond h days, is equal to (1-α). VARα; h = R such as the Probability P (ΔhPt<R) = 1-α. This is illustrated by the graph:

Graph of the density function P&L

The parametric method or variance-covariance approach

The parametric approach tries to define a formula describing the distribution of gains / losses. To calculate a day's VaR we use the formula𝑽𝒂𝑹 𝜶% = 𝝁 -𝒒 * 𝝈, with q a quantile taken from the normal law table (1.96 for α = 5% and 2, 33 for α = 1%), whereas for 10 days the formula becomes 𝑽𝒂𝑹 𝜶% = √𝟏𝟎(𝝁 -𝒒 * 𝝈). This method is based on several assumptions, mainly:  The variations of risk factors follow a normal law.  The relation between the variations of value's portfolio and the variations of market variables is linear.

The Cornish-Fisher VAR

Under the assumption of normality of returns, asymmetry and kurtosis are zero. In reality, returns generally have negative asymmetries and high kurtosis. Using a normal quantile's limited development, we can take asymmetry and kurtosis into consideration; this is the Cornish-Fisher approximation:

z is the normal quantile at α: N (z) = α, S is asymmetry and K is kurtosis. In the case of zero asymmetry, the preceding expression is reduced to:

The Cornish Fisher VaR with a confidence level of 1-α (for example 99%) and a horizon T (for example 10 days) is then: 𝑉𝑎𝑅(𝛼, 𝑇) ≈ 𝜇𝑇 + 𝑧 𝑪𝒐𝒓𝒏𝒊𝒔𝒉-𝑭𝒊𝒔𝒉𝒆𝒓 * 𝜎 * √𝑇

The historical method

The historical simulation approach does not make assumptions about the distribution of portfolio returns. However, it is based on the hypothesis of stationarity of returns, in other words the distribution of the portfolio value variations for the horizon on which we calculate the VaR is well estimated by observations of these variations from historical data.

The Monte Carlo simulation

The method using Monte Carlo simulation is similar to the historical method as it is a total valuation method based on different scenarios. Like the historical method, the Monte Carlo method does not assume that market variables follow a normal distribution.

In the historical method, the scenarios have already occurred in the past. With the Monte Carlo method, however, the scenarios are generated at random. They follow a form similar to those that took place in the past but they are not limited by history.

Each risk factor has a dynamic that can be modeled with a term of uncertainty. By taking into account the dynamics of each risk factor as well as their correlation, it is possible to generate probable scenarios of collective evolution of these risk factors.

7. VAR using the Boostrap technique

The Boostrap technique is a statistical inference method dating from the late 1970s. This method is based on simulations (we are talking about resampling). Its principle is to generate on the basis of an initial series several other randomly reordered series, with or without replacement, so as to decrypt the distribution of the initial series.

Database and Applications

The exchange rate covered by this article is the exchange rate of the Moroccan currency "MAD" compared to the US Dollar and the Euro. Our series contains the exchange rates of the two currencies for the daily period from 01/01/2013 to 31/10/2016. To quantify the volatility of exchange rates, we have used the daily return of the two currencies as an indicator of exchange rate risk:

Return USD-MAD from 01/01/2013 to 31/10/2016

Return EUR-MAD from 01/01/2013 to 31/10/2016

The trajectory of the two portfolios is very volatile around an approximately zero value. The normality tests on our data show the rejection of normality, which is the case almost for all financial series. However, the incompatibility with the normal law is quasi-systematically decided on large size, even if the differences of distributions are weak. As a result, empirical approaches, especially graphical ones, remain important.

Return USD-MAD Return EUR-MAD

-Graphic illustration: On the two graphs below, we note that the distribution of our series is close to the normal distribution.

Return USD-MAD

Parametric VaR and VaR according to the Cornish-Fisher approximation a) Parametric VaR

The parametric VaR calculation is based on the normality hypothesis. But our two series do not adjust to a normal distribution. Nevertheless we will calculate the parametric VaR and then we will correct it with the Cornish-Fisher approximation.

USD/MAD EUR/MAD

References

1. Dupuy M, Cardebat Jm, Jercourel Y. Finance Internationale, Edition Dunod, PARIS, 2006.
2. Jean-Marcel Dalbarade, Mathématiques des marchés financiers, 2 ème édition ESKA, 1995.
3. François-Éric Racicot, Raymond Theoret, «La Value atRisk, Modèles de la VaR, simulations en Visual Basic. et autres mesures récentes du risque de marché », cahier de recherche 04-2006.
4. Thierry Roncalli. La Gestion des Risques Financiers, Collection Gestion, édition Economica, 2004.
5. Ben Vogelvang. Econometrics theory and application with EViews, Pearson Education Limited, 2005.