VaR Flashcards
What does the Risk-Metrics formula look like?
State a general parametric Conditional VaR model
Which distributions can be used for parametric conditional VaR?
Distributions from the ‘location-scale’ family, such as Student’s T and Normal
What are the steps to implement Filtered Historical Simulation?
1) Demean and devolatilze returns (e.g. compute standardized residuals) 2) Sort and chose alpha*Nth smallest observation 3) Rescale with volatility forecast and add conditional mean
Discuss Pros and Cons of Filtered Historical Simulation
Pros: Few distributional assumptions GARCH advantages Quantile converges to true quantile Cons: Only location scale families to incorporate conditional information Slow convergence
Describe Weighted Historical Simulation
What is a key benefit of Weighted Historical Simulation
No distributional assumptions while incorporating recent conditioning information
What are ways of calculating Unconditional VaR?
Parametric Estimation - Fully parametric mean model with constant variance. Estimate model with MLE Non-parametric (historical simulation) - Empirical distribution of raw returns - Can be smoothed Parametric Monte Carlo - Estimate model for conditional mean and variance - Simulate model for many periods (enough so you don’t need smoothing) - Use simulated distribution - Can give you fat tails because of time-varying volatility - Depends on parameter estimation (bad)
What are 3 ways of evaluating forecasts?
1) Standard Generalized Mincer-Zarnowitz (regress hit on past hits and VaR; HITS should be unforecastable and have mean 0 or alpha according to specification) 2) Likelihood based testing (exploiting Bernoulli characteristics) 3) Probit/Logit (alternative structure to 1) )
What complicates density forecasting?
2+ step ahead forecasts are complicated since variance is stochastic. Even if returns are conditionally normal, we get a variance-mixture of normals, which is not normal. We must ‘integrate out’ the uncertainty in the future variance to determine the future distribution
What is a fan plot?
A plot illustrating density forecasts X-axis number of steps a ahead Y-axis are variable values Darkness indicates density
What is a QQ plot?
Plots of observed values against distribution, matched on quantiles Uses demeaned and devolatilzed observations
Show the steps to perform an LR test on a VaR model
Estimate alpha as HITS/total observations Compute likelihood under estimated alpha and true alpha Test stat: 2*(diff in loglikelihood with estimated and true alpha) is distributed chi^2 with 1 df. Remember that in the test, the log likelihood uses all the observations (sums of individual log density)
What is the Berkowitz test?
It transforms data to uniform (using the assumed CDF) and then transform into std. normal (using inverse std normal) Tests for standard normality by running an AR(1) or higher regression, hypothesizing slope and intercept is zero and variance is one. Tests with maximum likelihood (chi^2 w 3 dfs)
Define Expected Shortfall
Et [r_(t+1) | r_(t+1) < -VaR_(t+1)]
What are the conditions for a coherent risk-measure?
1) Drift invariance 2) Homogeneity 3) Monotonicity 4) Subadditivity
Are VaR and ES coherent risk measures?
VaR is not, ES is. Sometimes violates subadditivity. Can construct to portfolios, the sum of which have larger risk than the sum of the parts
What does the GMZ for VaR look like?
What is the likelihood for a Bernoulli?
Show a LR test for a VaR model
What is a Kolmogrov-Smirnov test?
Show that an ARCH(1) is a AR(1)
Show that a GARCH(1,1) is an ARMA(1,1)
Define conditional VaR
