Financial Risk Management and Governance Other VaR methods. Prof. Hugues Pirotte

Transcription

1 Financial Risk Management and Governance Other VaR methods Prof. ugues Pirotte

2 Idea of historical simulations Why rely on statistics and hypothetical distribution?» Use the effective past distribution for all variables» Let s compare actual and normal distributions of returns x c Pr X x f( x) dx F ( x ) c c X c 00.00% 95.00% 90.00% 85.00% 80.00% 75.00% 70.00% 65.00% 60.00% 55.00% 50.00% 45.00% 40.00% 35.00% 30.00% 5.00% 0.00% 5.00% 0.00% 5.00% 0.00%,65 Dist. normale Dist. effective

3 3 Methodology Method. Compute returns and changes for all time-series of all risk sources Leave them in the same order!. Compute the new values of the positions held in the portfolio a) Apply the returns to latest underlying prices to generate new price series vi v b) Reprice all positions based on those new prices 3. Aggregate them» Aggregation can be an issue... n vi X... X... X Xn... m Risk sources m 4. Rank portfolio values in descending order 5. Choose the desired quantile leaving c% of the values above ij X X nm n observations

4 4 Ins & Outs

5 Accuracy 5 Kendall & Stuart (97): confidence interval for the quantile of a probability distribution estimated from sample data x q quantile of the distribution SE f ( x) q( q) n where n # obs f ( x) pdf at x

6 6 Extensions Weighting of observations» Standard weights: /n» EWMA idea: Boudoukh et al. (998) Weight given to change between day n i and day n i + i n Same as standard weighting scheme when We sum up weights until we reach the desired quantile The best value of can be tested using backtesting The effective sample size is reduced, unless we increase substantially n.

7 Extensions () Incorporating volatility updating (ull & White (998))» Use of: where v n v i v i v v i i n n current estimate of the volatility since i it applies to period between today and tomorrow. 7

8 Extensions (3) 8 Bootstrap. We resample from the same dataset of changes to recreate many new similar datasets.. The VaR is then calculated for each dataset. 3. The confidence on the VaR is given by the range taken on the distribution of VaRs.

9 9 Monte Carlo simulations > principle Why rely on a single scenario? Simulate many» ow? Use statistics to generate those distributional samples!» What is the advantage? We can reprice everything and...

10 0 The idea of random generation Random returns generation random?» The Wiener process ΔW t ~N 0, Δt z t ~N 0, ΔW t = z Δt» The generalized Wiener process X t W t t» The Ito process t,, X X t t X t W t The Geometric Brownian motion» Returns distribution S S t z t t i» Limit of the model ds S i d ln S ln ST ln St i t iz t

11 Ito s calculus

12 The Choleski decomposition Generating correlated randoms / / 0 V Var ) (,, / / 0 0 ' V ' ' ' ' ' ' ' ) ( et ) ( I Var Var

13 3 Binomial methods (tree methods)

14 4 Accuracy of simulations The effect of sampling variability» the empirical distribution of ST is only an approximation, unless the number of simulations (k) is extremely large» Monte Carlo implied independent draws k The standard error of statistics is inversely related to Methods to speed up convergence» Antithetic variable technique» Control variate technique» Quasi-random sequences (= QMC)

15 5 Comparison of models Delta-Normal (or var-covar) istorical Simulation Valuation Linear (Local) Full Full MonteCarlo Simulation Distribution Shape Normal Actual General Extreme events Low probability In recent data Possible Implementation Ease of computation Yes Intermediate No Communicability Easy Easy Difficult VaR precision Major pitfalls Excellent Non-linearities, fat tails Poor with short window Time variation in risk, unusual events Good with many iterations Model risk Inspired from Jorion, Financial Risk Manager andbook

16 6 Prof.. Pirotte References Some excerpts from:» ull (007), Risk management and Financial Institutions» The RiskMetrics technical document» Jorion (008), Financial Risk Manager andbook Others:» Jorion (000), «Risk Management Lessons from LTCM».