Doctoral School of Finance and Banking Academy of Economic Studies Bucharest. Testing and ComparingValue at Risk Models – an Approach to Measuring Foreign Exchange Exposure -dissertation paper-. MSc student: Lapusneanu Corina Supervisor: Professor Moisa Altar. Bucharest 2001.

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Doctoral School of Finance and Banking Academy of Economic Studies Bucharest

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Doctoral School of Finance and Banking • Academy of Economic Studies Bucharest • Testing and ComparingValue at Risk Models – an Approach to Measuring Foreign Exchange Exposure • -dissertation paper- • MSc student: Lapusneanu Corina • Supervisor: Professor Moisa Altar Bucharest 2001

Corina Lăpuşneanu - Introduction • Introduction VaR is a method of assessing risk which measures the worst expected loss over a given time interval under normal market conditions at a given confidence level. Basic Parameters of a VaR Model Advantages of VAR Limitations of VaR

Corina Lăpuşneanu - Introduction Basic Parameters of a VaR Model • For internal purposes the appropriate holding period corresponds to the optimal hedging or liquidation period. • These can be determined from traders knowledge or an economic model • The choice of significance level should reflect the manager’s degree of risk aversion.

Corina Lăpuşneanu - Introduction Advantages of VAR • VaR can be used to compare the market risks of all types of activities in the firm, • it provides a single measure that is easily understood by senior management, • it can be extended to other types of risk, notably credit risk and operational risk, • it takes into account the correlations and cross-hedging between various asset categories or risk factors.

Corina Lăpuşneanu - Introduction Limitations of VaR: • it only captures short-term risks in normal market circumstances, • VaR measures may be very imprecise, because they depend on many assumption about model parameters that may be very difficult to support, • it assumes that the portfolio is not managed over the holding period, • the almost all VaR estimates are based on historical data and to the extent that the past may not be a good predictor of the future, VaR measure may underpredict or overpredict risk.

Corina Lăpuşneanu - Data and simulation methodology Data and simulation methodology • Statistical analysis of the financial series of exchange rates against ROL (first differences in logs): Testing the normality assumption Homoskedasticity assumption Stationarity assumption Serial independence assumption

Corina Lăpuşneanu - Value at Risk models and estimation results Value at Risk models and estimation results “Variance-covariance” approach Historical Simulation “GARCH models Kernel Estimation Structured Monte Carlo Extreme value method

Corina Lăpuşneanu - Value at Risk models and estimation results “Variance-covariance” approach • where Z() is the 100th percentile of the standard normal distribution Equally Weighted Moving Average Approach Exponentially Weighted Moving Average Approach

Corina Lăpuşneanu - Value at Risk models and estimation results Equally Weighted Moving Average Approach • where • represents the estimated standard deviation, • represents the estimated covariance, • T is the observation period, • rt is the return of an asset on day t, • is the mean return of that asset.

Corina Lăpuşneanu - Value at Risk models and estimation results GARCH models In the linear ARCH(q) model, the conditional variance is postulated to be a linear function of the past q squared innovations: • GARCH(p,q) model:

Corina Lăpuşneanu - Value at Risk models and estimation results • The constant correlation GARCH model • estimates each diagonal element of the variance- • covariance matrix using a univariate GARCH (1,1) • and the risk factor correlation is time invariant:

Corina Lăpuşneanu - Value at Risk models and estimation results Orthogonal GARCH • X = data matrix • X’X = correlation matrix • W = matrix of eigenvectors of X’X • The mth principal component of the system can be • written: • Principal component representation can be write: where

Corina Lăpuşneanu - Value at Risk models and estimation results The time-varying covariance matrix (Vt) is approximated by: • where • is the matrix of normalised factor weights • is the diagonal matrix of variances of principal components • The diagonal matrix Dt of variances of principal components is estimated using a GARCH model.

Corina Lăpuşneanu - Value at Risk models and estimation results • But, generally, variables are correlated. To account or this correlation, we start with a set of independent variables , which are then transformed into the , using Cholesky decomposition. In a two-variable setting, we construct: where is the correlation coefficient between the variables .