Why do we use QQ plots
To inspect if an empirical dist matches a theoretical dist
Parametric approach def
e.g. delta-normal approach - explicitly assumes a dist for underlying observations
QQ plot
When on one axis - quantiles of hypothesized dist, on another - quantiles of your experimental distribution. Helps to confirm if your data is coming from hypothesized dist
Parametric VS non parametric diff
Non param - you don’t specify the underlying dist. It’s data driven, not assumption driven.
In parametric - you specify an underlying dist
Bootstrap method
- draw sample from data
- record the VaR
- “return” the sample (so sampling with replacement)
- Take the average VaR as a final estimate
Fisher-Tippett theorem
As sample size n gets large, the dist of extremes converges to GEV - Generalized Extreme Value distribution
Frechet, Gumberl, Weibull dist
eps>0 - Frechet dist
eps=0 - Gumbel dist
Eps<0 - Weibull dist (don’t often appear in fin models)
tail index
eps in the GEV distribution
Expected Shortfall is also known as …
CVaR (Conditional VaR)
Diff POT and GEV
POT - Peaks over Thresh
GEV - Generalized Extreme value
both are dists
- GEV required estimation of 1 more parameter than POT
- POT shares 1 parameter with GEV (tail (or shape) parameter ksi)
- GEV focus on dist of extremes, POT - focus on dist of values exceeding a set threshold. Threshold selection can introduce extra uncertainty
- Nature of the data may make one preferable to the other
exceptions / exceedances
instances where actual loss esceeded predicted VaR level
of actual observ in back testing, that fall outside a given VaR conf level. For a conf level of 95%, exceptions should occur <5% of the time
Under Basel rules, bank VaR models must use which conf level?
99%
type I and II errors in terms of backtesting
Type 1 - rejecting accurate model
Type 2 - failing to reject inaccurate model
What are the exceedance requirements for banks
99% conf level VaR on 250 days horizon -> only 2.5 exceptions within 250 day horizon.
Banks are penalized with higher capital requirements if >=5 exceptions observed
Why is it called “unconditional” coverage? Diff with conditional?
Backtesting the model with uncond cov -> timing of your exceptions is not considered. We’re not worried about independence of exceedances or when they happen - only in the total number of exceedances
Conditional - condiders “bunching” (clustering together) of exceptions, considers their independence. Reviews number and timing of exceptions for independence
LR_ind 95%CL critical value
3.84
3 methods of mapping for fixed income securities
principal mapping (simplest)
duration mapping
cash flow mapping (most precise)
Tracking error VaR def
measure of diff btw VaR of the target portfolio and benchmark portfolio
regulatory capital
Regulatory Capital is the minimum amount of capital a bank must hold as required by regulators (e.g. Basel III, central banks) to absorb UNEXPECTED losses and stay solvent.
economic capital
Economic Capital is the amount of capital a bank internally estimates it needs to cover all its risks (credit, market, operational, etc.) at a certain confidence level. It’s used for risk management and internal decision-making. EC is often larger than reg capital, but not always, depending on the bank’s internal models
Marginal VaR def
change in portfolio value due to small change in weight of a particular portfolio position
Benchmarking VaR model def
Comparing performance of the bank’s VaR model to another VaR model
Exceedance-based test
countes # of times actual losses exceed the VaR threshold
3 properties of exceedance-based testing
- Unconditional coverage - define across the entire observations set if the model is properly calibrated to estimate VaR
- Independence - check is there’s clustering of exceedances, e.g. proba of exc. on one day are related to proba of exc. on another day
- Conditional coverage - determing if the model jointly solves property 1 and 2
