Risk Management
Lecture 3
Jan Palczewski
University of Leeds
February 2, 2023
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 1 / 16
In lecture 3
1 Some additional comments on VaR and ES
2 Approaches for computing VaR and ES in practice
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 2 / 16
Some additional comments on VaR and ES
Use of VaR to compute regulatory capital
Regulatory capital calculation for trading book of a bank
RC(t) = max
(
VaRt,100.99; k
1
60
60
Â
j=1
VaRtj+1,100.99
)
+CSR,
where 3 k  4 (determined by the regulator) and CSR is a
component for specific risk.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 3 / 16
Some additional comments on VaR and ES
VaR and ES
ESa is an example of coherent risk measure.
VaRa is not coherent (as it does not satisfy convexity).
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 4 / 16
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Some additional comments on VaR and ES An example through a picture
An example through a picture
FigureJan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 5 / 16
Approaches for computing VaR and ES in practice
Approaches for computing VaR and ES
Here, we will give some "recipes" used by practitioners.
Variance-Covariance method (also known as normal method)
Historical estimation
Monte Carlo simulation
It is important that we understand the underlying assumptions, the main
steps, the advantages and disadvantages of each method.
We follow the presentation from the book by Embrechts, Frey, McNeil.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 6 / 16
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Approaches for computing VaR and ES in practice
Variance-Covariance method
The underlying assumptions of this method are:
Assumption
I X ⇠ N(m,⌃)
(That is, the random vector of risk factor changes follows a multivariate
Normal distribution with mean vector m 2 Rd and covariance matrix ⌃)
I The linearized loss Llin is a good approximation for the loss L.
The main steps of this method are:
Use historical data to estimate the mean m and the covariance
matrix ⌃.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 7 / 16
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Approaches for computing VaR and ES in practice
Variance-Covariance method
Consider the linearized loss Llin .
Llin has the structure (cf. Lecture 1 and 2):
Llin =

c+b0X

where b is a d-dimensional vector and c is a real constant.
As X follows a (multivariate) Normal distribution (by assumption),
and as Llin is written as a known linear function of X, we get
LLin ⇠ N(cb0m,b0⌃b).
Remark: Note that cb0m is a real number and that b0⌃b is a
(non-negative) real number. Hence, N(cb0m,b0⌃b) denotes the
(univariate) Normal with mean µ=cb0m and variance s2 = b0⌃b.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 8 / 16
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Approaches for computing VaR and ES in practice
Variance-Covariance method
Compute VaRa and ESa by using the explicit formulas established
in the Gaussian framework.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 9 / 16
Approaches for computing VaR and ES in practice
Variance-Covariance method
Advantages:
analytic method (no simulation), easy to implement.
Disadvantages (weaknesses):
the linearized loss may not offer good approximation for the loss;
normality of risk factors changes may not be a realistic assumption.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 10 / 16
Approaches for computing VaR and ES in practice
Historical Estimation
Idea: This method uses historical data for the risk factor changes and
an estimate for the probability distribution of the loss based on the
empirical distribution.
There are no assumptions on the probability distribution of X made: this
is a non parametric method.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 11 / 16
Approaches for computing VaR and ES in practice
Historical Estimation
We place ourselves at t .
The main steps of this method are:
Collect a historical dataset Xt(n+1), . . . ,Xt2,Xt,Xt .
Remark: Note that n corresponds to the size of the data set.
Each data point in the data set is a vector of dimension d .
Use the above data to "get" a dataset of losses:eLt = l[t](Xt),eLt = l[t](Xt), . . . ,eLt(n+1) = l[t](Xt(n+1)),
where l[t] is the loss operator at time t (cf. Lecture 1).
Remark: eLt corresponds to the portfolio loss that we would have if
the risk factor changes observed between t2 and t were to
recur.
Use this dataset to compute an empirical distribution estimate (e.g.
"histogram estimate") of the loss probability distribution.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 12 / 16
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Approaches for computing VaR and ES in practice
Historical Estimation
Compute an estimate for VaRa(Lt+) and ESa(Lt+).
An estimator for VaRa(Lt+) can be obtained as follows:
I Order the data points of the set from the smallest loss to the
greatest loss:eL(1)  eL(2)  . . . eL(n).
I An estimator for VaRa(Lt+) is given by:
eL([n(1a)]),
where [n(1a)] denotes the largest integer not exceeding
n(1a).
How can an estimator for ESa(Lt+) be obtained?
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 13 / 16
Approaches for computing VaR and ES in practice
Historical Estimation
Advantages:
Easy to implement.
No assumption about distribution and dependence structure of
risk-factors.
Disadvantages:
-In this approach, "the worst case is never worse than what has
happened in the past".
-Need of a large sample of relevant historical data to get reliable
estimates of VaRa and ESa; missing data can cause problems.
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 14 / 16
Approaches for computing VaR and ES in practice
Monte Carlo simulation
Monte Carlo simulation (in risk management) is a general name for any
approach which involves simulation from an explicit parametric model
for the risk factor changes.
Main steps of this approach:
Choose a d-dimensional parametric model for the vector of risk
factor changes X
Use historical data Xt(n+1), . . . ,Xt2,Xt,Xt of past risk
factor changes to calibrate the d-dimensional model.
Use the calibrated model to simulate M (possible) values of risk
factor changes for the next period (that is, the period from t to
t+): eX1t+,eX2t+, . . . ,eXMt+.
Use the simulated risk factors to simulate losses the period from t
to t+) using the loss operator l[t].
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 15 / 16
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Approaches for computing VaR and ES in practice
Monte Carlo simulation
Advantages: We can choose the number of simulations M
ourselves
(M can be chosen much bigger than n).
Disadvantages: simulation can be computationally intensive;
"results are as good as is the model used" (think of model risk).
Jan Palczewski (Univ. of Leeds) MATH5340 2022/23 February 2, 2023 16 / 16
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