Market Risk and Var Models 2: Simulation Approaches

Question 1

What are the problems of the parametric VaR approach?

Answer 1

Non-normal distribution of market factors’ returns: higher kurtosis + skewness
Serial correlation of market factors’ returns
Non linear positions

Question 2

What are the four phases of the historical simulations approaches to VaR?

Answer 2

1. Selection of a historical sample of market factors’ returns
2. Revaluation of the portfolio for each of the historical values of the market factor
3. Reconstruction of the empirical frequency distribution of the portfolio market values
4. Identification of the desired distribution percentile, corresponding to the desired confidence level

Question 3

What are the advantages of historical simulations for VaR?

Answer 3

1. Easy to understand and communicate
2. No explicit underlying assumption concerning the functional form of the returns distribution
3. No need to estimate the variance-covariance matrix
4. Allows to capture the risk profile of portfolios with non linear and non monotonic sensitivity to market factors returns

Question 4

What are the disadvantages of the historical simulation approach for VaR?

Answer 4

1. Assumption of stability of the distribution of market factors’ returns
2. Computationally hard because of full valuation
3. Size of the historical sample, particularly when time horizon > 1 day
4. Bad definition of the distribution tails
5. Risk of overweighting or underweighting the extreme events in the historical sample
6. Increasing the size of the historical sample brings the risk of deviating from the distribution stationarity assumption

Question 5

What is the Hybrid Approach to VaR?

Answer 5

It is an attempt to combine the advantages of the parametric approach (decreasing weights through EWMA) and those of historical simulations (no normal distribution assumption)
Long historical series but more weight to recent data
Weight attributed to each individual historical return:
P_t-i=λⁱ/(Σⁿ_i=1λⁱ)

Question 6

What is the difference between a prudent and linear interpolation VaR in the hybrid approach?

Answer 6

For the prudent VaR the cutoff is the closes cumulative value to 1% which is lower than 1%, while for the realistic approach we do linear interpolation, by taking the weighted average of the variable, which is the value of the variable at the value higher and lower weighted by the distance of the cumulative weight from 1% divided by the distance between the cumulative weights.

Question 7

What problem of the historical and hybrid approach does the Monte Carlo Simulation approach solve?

Answer 7

It is most useful when there is a lack of data, as we just generate new data.

Question 8

What are the 5 steps for Monter Carlo Simulations in Risk Management?

Answer 8

1. Identify the distribution - f(x) - that best proxy the actual market factor returns distribution
2. Simulate the market factor evolution n times
3. Calculate the position market value in each scenario
4. Build the empirical probability distribution of the changes of the position’s market value
5. Cut the empirical distribution at the desired confidence level

Question 9

What are the 4 substeps to calculating the market value of the position in the Monte Carlo approach?

Answer 9

1. Extract a value U from a uniform distribution [0,1]
2. Calculate the value x of the function f(x) corresponding to the extracted U value
3. Determine the invesrse of the cumulative function of the original sample distribution
4. Repeat the previous steps a large number of times

Question 10

What are the 5 steps for monte carlo simulations with correlationn?

Answer 10

1. Estrimate variance-covariance matrix.
2. Decompose the original matrix into two symmetric matrices, A and A^T
3. Generate scenarios for the different market factors multiplying matrix A^T, which reflects the historical correlations of market factors returns, for a vector z of random numbers
4. Calculate the market value change corresponding to each of the simulated scenarios
5. Calculate VaR cutting the empirical probability distribution at the desired confidence level

Question 11

What are the advantages of Monte Carlo simulations?

Answer 11

1. Full valuation: no problems with non linear or non monotonic portfolios
2. Flexibility: possibility to use any probability distribution functional form
3. Simulating not only final values but also path: possibility to evaluate the risk profile of path dependent options

Question 12

What are the limits of Monte Carlo simulations?

Answer 12

1. Need to estimate market factors’ returns correlations (stability problem)
2. Computationally intensive
3. Need to make an explicit assumption about the functional form of the distributions of market factors’ returns