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

Define Drift Invariance

If adding a constant to the return, the risk measure must decline by this sum

Define monotonicity

If p1 first-order stochastically dominates P2, the risk of p1 must be lower than that of p2

Define first-order stochastic dominance

If p1 is higher than p2 in all possible states of the world, p1 FOSD p2

Define value at risk?

Value at risk satisfies: Pr ( R_t < - VaR_t ) = alpha

What is the difference between risk-metrics and any general GARCH-based parametric VaR model?

In RiskMetrics, the vol. intercept is zero, alpha and beta sum to 1 and returns are assumed to be conditionally normally distributed. Also mean is assumed to be zero

What are some ways of evaluating density forecasts?

Informally: Fan Plots QQ plots Formally: Kolmogrov-Smirnov test Berkowitz test

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?

How do you perform a probit/logit test on a VaR model

Regressed modified HITs (VAR excedences) on a set of variables transformed through a 'squash' function.

The 'squash' funciton can be logstic function or Normal CDF. The variables are intercept, VaR and past HITS. Only intercept should be non-zero if model is correcrly specified. Intercept should be inverse squash function of alpha.

What is variance of a Bernoulli?

p(1-p)

What is the general formula for a t-test?

(Estimate - null_value) / standard error