VaR Or Expected Shortfall

General Market Business Activties Risks
Treasurer Activities
Banking Regulation
Introduction to Market Risk Analysis
Sensitivity Analysis
Value–at–Risk (VaR)
Catching The Tail: Expected Shortfall (ES)
Stress–testing
VaR and ES Use in Regulatory Capital
Conclusion
MATLAB
VaR Or Expected Shortfall?
Alex Kouam 1
Master of Science International Banking and Finance (Strathclyde Business School)
Specialized Master Quantitative Finance (EMLYON Business School)
31st August 2017
1
Unauthorised use prohibited
Alex Kouam VaR Or Expected Shortfall?

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Outline I
1 General Market Business Activties Risks
2 Treasurer Activities
3 Banking Regulation
Stakeholders, Geographical Scope and Roles
Current EU Regulation
4 Introduction to Market Risk Analysis
The Drivers of a Trading Portfolio
Market Risk Analysis Steps
5 Sensitivity Analysis
Deﬁnition
Common Greeks

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Outline II
Use in Practice
6 Value–at–Risk (VaR)
Introduction to the Risk Measure Concept
Components of a VaR system
VaR Methods
Backtesting
VaR: Methods Comparison and Shortcomings
7 Catching The Tail: Expected Shortfall (ES)
Expected Shortfall
8 Stress–testing
Deﬁnition

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Outline III
Stress–testing Methods
9 VaR and ES Use in Regulatory Capital
10 Conclusion

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General Market Business Activties Risks I
Credit Risk stands for the risk of a counterparty defaulting on
payment obligations
E.g.: Enron (2001) declared a loss of $1 billion in October 2001 with
revalued libabilities of over $60 billion at the bankruptcy ﬁling.
Liquidity Risk (Funding) is the risk that the cost of funding
becomes higher, up to the extreme case when raising funds become
impossible.

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General Market Business Activties Risks II
E.g. Northern Rock (2007) were unable to sell securitized loans to
secure funding.
Market Risk is the risk of adverse deviations from the
mark–to–market value of the trading portfolio, due to market
movements, during the period required to liquidate the transactions.
E.g. JP Morgan (2012) incurred mark–to–market losses of $6.2 billion
on the Synthetic Credit Portfolio

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General Market Business Activties Risks III
Operational Risk outcomes from direct or indirect loss resulting from
inadequate or failed internal processes, people and systems or from
external events.
E.g. Barings (1995), Société Générale (2008) and UBS (2011) lost
respectively $1.4 billion, e4.9 billion and £1.4 billion because of rogue
traders who managed to deceive internal control systems.

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Activities of a Treasurer
Manage interest rate risk on the bank’s balance sheet.
Manage liquidity needs related to banking activities.
Preserve the capital of the bank.
Increase the result of the bank.

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Banking Regulation

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Missions
In a nutshell, banking regulation aims to:
Guarantee the financial system resilience and integrity, the economy’s
lifeblood and,
Protect financial products’ consumers, in particular to maintain a high
level of confidence in the system.

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Table
Institutions Geographical Coverage Roles
G20 World •The G20 set top priorities
FSB •FSB makes those priorities operational
Basel Committee • The Basel Committe publishes
these recommendations
European Parliament Europe •The Euopean institutions adapt
European Commission (EC) these recommendations to the
EBA European context through
Capital Requirements Regulation
(CRR) and Capital Requirements
Directive (CRD)
French Parliament France • They adopt the EU agencies directives
ACPR with respect to the French law
CCLRF
Table: Main Regulatory BodiesAlex Kouam VaR Or Expected Shortfall?

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Issue Key Points Comments
Market Risk Adoption of the BCBS • Compulsory capital ﬂoor (SA) means
( FRTB) FRTB Tier 1 banks need to develop SA system–based.
• a failure to meet IMA’s requirements will lead
to a fallback to SA (more regulatory capital)
• Expected Shorfall becomes the reference
market risk measure
• Varying liquidity positions required for IMA and SA.
TLAC Adoption of the FSB’s TLAC as Pillar 1 • More capital requirements for classiﬁed
requirement for G–SIBs, phased–in from 2019 to G–SIBs.
2022
NSFR Adoption of a minimum NSFR 100% Banks foresaw that amendment.
Leverage Ratio Adoption of a 3% ratio Banks foresaw that too
Table: EC CRR Amendments

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The Drivers of a Trading Portfolio I
Contractual elements: counterparty’s creditworthiness, speciﬁc
clauses, etc. . .
Bank’s ownn characteristics: capital quality, access to liquidity
potential (funding cost on the market), etc. . .
Observable market parameters: stock prices, swap rates, etc. . .
Non–observable market parameters: volatility, etc. . .

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The Drivers of a Trading Portfolio II
More exotic market parameters: correlations from basket assets
products, etc. . .
Model parameters: numerical parameters from model calibration
process, etc. . .

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The Drivers of a Trading Portfolio: S&P 500 Illustration I
OPRA S&P 500 Index Option 2375 Put Aug 2017
Strike Exercise Style Expiry Price Dividend Yield (%) One–month yield rate
2 375 EU 31-Aug-2017 6,9 2,653 0,99
Table: S&P 500 Put on 28–Jul–2017
Contractual elements: strike price, maturity and exercise style.
Observable market parameters: S&P 500 current level, the
one–month yield rate,

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The Drivers of a Trading Portfolio: S&P 500 Illustration II
Non–observable market parameters: implied volatility surface, the
expected dividend yield
Model parameters: numerical parameters vary with respect to the
used model (black–scholes, stochastic volatility, CGMY model, etc. . . )

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Market Risk Analysis Steps I
1 Market Risk Pre–analysis
Economic analysis of the portfolio’s risks.
Identiﬁcation of risk factors.
Data collection.
2 Market Risk Modelling
Risk factors modeling.
Relationship between risk factors and the portfolio value.
Probabilistic analysis of the portfolio’s proﬁt and loss (P&L).

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Market Risk Analysis Steps II
Robustness analysis of models.(backtesting)
3 Market Risk Analysis, Synthesis and Risk Management
Computation of risk indicators
Ex–ante risk analysis
Risk monitoring and management

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Risk Factor Pre–anlaysis Illustration:S&P 500

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Risk Factor Pre–anlaysis Illustration II:S&P 500
Economic analysis of naked short put on S&P 500
What are the underlying risk factors to focus on?
Spot
Volatility

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Deﬁnition
Common Greeks
Use in Practice

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Definition
Common Greeks
Use in Practice
Definition I
Sensitivities are ratios of the variation of a target variable, such as
change of the mark–to–market values of instruments, to a shock of
the underlying random parameter driving this change.
This property makes them very convenient for measuring risks,
because they link any target variable of interest to the underlying
sources of uncertainty that influence these variables.

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Deﬁnition
Common Greeks
Use in Practice
Deﬁnition II
Mathematically, the sensitivities correspond to the partial derivatives
of the price with respect to one or several market parameters
simultaneously. They are also known as Greeks when we work with
options instruments.
Model Risk
The choice of a model potentially leads to the selection of an unappropriate
pricing model,see Derman(2001) for further explanations.

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Deﬁnition
Common Greeks
Use in Practice
Common Greeks I
Delta, ∆ = ∂P
∂S , measures the sensitivity of your asset price with
respect to the underlying index price.
Gamma, Γ = ∂2P
∂S2 , measures the sensitivity of your ∆ with respect to
the underlying index price.
Vega,υ = ∂P
∂σ , measures the sensitivity of your asset price with respect
to the underlying’s volatility.

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Deﬁnition
Common Greeks
Use in Practice
Common Greeks II
Theta,θ = ∂P
∂t , measures the sensitivity of your asset price with
respect to the time.
Rho,ρ = ∂P
∂r , measures the sensitivity of your asset price with respect
to the risk–free rate.
Vanna ,Vanna = ∂2P
∂S∂σ , tells you how the ∆ will change when σ
changes or how υ will change when the underlying changes.
Volga ,Volga = ∂2P
∂σ2 , measures how your υ will change when there is a
move in the underlying volatility.

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Definition
Common Greeks
Use in Practice
Use in Practice I
Sensitivities are used by traders as well as risk managers in order to
control the book’s risk for which they are accountable.
Sensitivities are reported to the Trading Manager, Risk Officer, etc. . . .
They are used to disaggregate the underlying price changes into its
different risk factors. For instance, the price changes of our S&P 500
Put can be decomposed as such:

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Deﬁnition
Common Greeks
Use in Practice
Use in Practice II
dP (S, σ) =
∆ dS
S + υ dσ
σ + 1
2! Γ dS
S
2
+ 2Vanna dS
S
dσ
σ + Volga dσ
σ
2
+ . . .
Numerical Approximation
In practice it’s not always possible to workout a closed–form formula and
we apply a shock to a model parameter as such:
∆P = P(S+(1+1%),K,τ,r,q,σ,κ,θ,η,ρ,λ)−P(S,K,τ,r,q,σ,κ,θ,η,ρ,λ)
1%

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VaR Methods
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Value–at–Risk

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VaR Methods
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Deﬁnition I
A risk can be modeled by a random variable describing the loss on a
ﬁxed time horizon.
A risk measure aims to summarize the probability distribution
attached to a risk in a single number. Mathematically,

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Deﬁnition II
Risk Measure
A risk measure is a functional mapping random variables to the real
numbers.
ρ : X → ρ (X) ∈ R

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VaR Methods
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VaR: Definition I
VaR is defined as the potential maximum loss that a trader could
incur given both a confidence level and a time horizon.
Mathematically, given some confidence level α ∈ (0, 1). The VaR of
our portfolio at the confidence level α is given by the smallest number
l such that the probability that the loss L exceeds l is no larger than
(1 − α). Formally,

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VaR: Deﬁnition II
VaR Deﬁnition
VaRα = inf{l ∈ R : P (L > l) ≤ 1 − α}

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VaR Methods
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VaR Plot

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VaR: Use in Practice I
Economic or Regulatory Capital It represents the amount of money
to hold in equity for being ruined with a probability smaller than
(1 − α) for n days.
As a risk measure: A VaRα for n days, states that there is only a
probability (1 − α) than the loss is greater than the VaR (e.g. α =
0.99)

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VaR Methods
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1 Time Horizon
2 Conﬁdence interval
3 Data Series
4 Mapping/selecting relevant risk factors

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VaR Methods
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Time Horizon
Constraints Solutions
• Liquidity of the instruments Highly liquid securities(e.g. Swaps) for short–time period
⇒short VaR time horizon.
• Expected holding period of the position Highly illiquid securities(e.g. Real Estate)
with a long expected holding period
⇒ long VaR time horizon
Table: Time Horizon

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Confidence interval
• Using 95% or 99% Use Historical VaR
confidence interval
Table: Confidence Interval

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Data Series
• Historical correlations and volatilities hold One method to solve both problems
outdated information on securities’ relationships is to use exponentially weighted data
• Use long period to catch the data series’ richness over 3 to 5 years
and your S&P 500 data series could be impacted by the Persian Guld War!! (1990s)
• Use short period and your VaR will only rely on inﬂated index caused by
recent quantitative easing policies of the FED.
Table: Time Horizon

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Mapping/Selecting relevant risk factors
• Unavailable historical price series Stress test what areas exactness
• Monumental data problem exactness matters and be less stringent
• Security’s group belonging over the rest.
• Relate security risk to risk factors
Table: Mapping/Selecting relevant risk factors

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VaR Methods
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Historical VaR
Historical VaR relies on past data for capturing correlations and
volatilities of risk factors without any assumption on their distribution.
Therefore, the historical VaR assumes implicitly that historical data,
according to your speciﬁed time window, is representative of the
current market conditions.

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VaR Methods
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Volatility Risk Case Study: S&P 500 Put
1.05
1
0.95
0.2
0.3
0
0.4
0.01 0.02
0.5
0.03 0.04
0.6
0.05 0.06 0.07
0.7
0.08
0.8

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VaR Methods
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Steps for determining historical VaR I
1 Determine the series of hstorical values of risk factors. We selected as
risk factors: The S&P 500 index level, the ATM implied volatlity.
Unfortunately, because of a lack of historical data, we instead used
data from the derived implied volatility surface.
2 Compute the risk factors relative changes and apply them to current
risk factors values.

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Steps for determining historical VaR II
3 Derive for each historical change of each risk factor the value of the
option using the black–scholes formula for instance.
4 For each daily variation of the risk factor, we obtain the daily P&L of
the option.
5 You construct the distribution and with respect to your conﬁdence
level 1%, you select your worst daily loss

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Steps for determining historical VaR III
Spot = 2 472 ; ATM IV = 12,3
s ∆S(%) St+h ∆σ (%) σt+h ∆P Pt+h (%)
1 1,08 2 445,44 0,01 12,30 14,42 31,431 1
2 0,61 2 457,08 0,24 12,27 4,90 33,008 2
3 −0,05 2 473,30 0,03 12,38 −16,22 28,065 0
. . .
70 −0,13 2 475,42 1,03 12,17 −6,29 6,65
Table: Summary of the VaR S&P 500 Put

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Analytical VaR I
Rationale: The analytical VaR methodology applies to linear
instruments in that it relies on the constant sensitivity to risk factor
assumptions.
This methodology derives VaR as percentile of a normal distribution
P&L

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VaR Methods
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Case Study: 10–Year German Bund I
DEGV 0.500 15-Aug-2027
ISIN Thomson Reuters Eikon Code Expiry Price YTM (%) Annual coupons (%)
ISIN DE0001102424 DEGV 0.500 15-Aug-2027 15/08/2027 100.05 0.5 0.5
Table: The 10–Year German Bund on 25–07–2017

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VaR Methods
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Steps for determining analytical VaR I
Map the positions to the risk factors. In our case, that is the
underlying yield–to–maturity (YTM).
Compute the sensitivities with respect to the risk factor. In the bond
framework, the relative ﬁrst–order sensitivity of a Bond with respect
to the YTM is Modiﬁed Duration (MD).

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Steps for determining analytical VaR II
The variations of the portfolio value becomes a linear function of the
variations of the porfolio assuming constant sensitivity. In our case
∆ (P&L) = MDxYTM
The volatility of the portfolio becomes the volatility of a linear
function of random variables. In our case we selected the daily
volatility of the YTM , σ = 0.20%

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Steps for determining analytical VaR III
Moving from volatility to VaR implies an assumption about the
distribution of our P&L. The analytical VaR relies on the normal
distribution assumption.
Conﬁdence Level (α) Normal Distribution Quantile (z)
99% 2.33
97.5% 1.96
95% 1.65
. . . . . .
Table: Normal distribution α and z

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Steps for determining analytical VaR IV
Under normal distributions assumptions, quantile and volatility
multiplies and your obtain your Var.
Duration = ModiﬁedDuration = σ (%) Daily RateVaR99% (%) = Daily Price VaR99% (%)
∂B
∂YTM
− D
B
−z × σ −z × σ × MD × YTM
−972,56 9,72 0,20 0,08 0,003 8
Table: Summary of the VaR 10–Year German Bund

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VaR Methods
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Steps for determining analytical VaR V
If you need to scale your daily VaR, on a longer period, just multiply by
the square root of the desired period. For instance, the 10–day daily Price
VaR is: -0.0038% ×
√
10 = 1.20%

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VaR Methods
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Monte–Carlo VaR I
Monte–Carlo depends upon the variance–covariance matrix of risk
factors.
This methodology is identical to the Monte–Carlo simulation, that is
each simulation of the set of risk factor values is a hypothetical
scenario.
And using this simulated value of the risk factor, you revalue the
instruments accordingly.

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Monte–Carlo VaR II
In practice
Select the risk factors
Speciﬁcation of the joint distribution of risk factors (in practice, the
normal law used)
Simulation of a very large number of scenarios of possible variations
of the risk factors from this law (in practice, 10 000 simulations at
least). The number of samples conditioned the accuracy of the
quantile measurement.

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VaR Methods
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Monte–Carlo VaR: Illustration I
S µ τ vt κ θ σ ρ
2 375 0,05 0,078 0,007 3 −0,03 4,12 · 10−05
0,018 −0,09
Table: Monte–Carlo parameters S&P 500
We run 5, 000 simulations
Time horizon is one–day
The conﬁdence level was 99% then VaR was the 50th worth loss.

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VaR Methods
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Monte–Carlo VaR: Illustration II



V (t + 1) = V (t) + κ (θ − max (V (t) , 0)) ∆t+
σ max (V (t) , 0)∆W2 (t + 1)
S (t + 1) = S (t) exp µ − 1
2 max (V (t) , 0) ∆t
exp max (V (t) , 0)∆W1 (t)

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VaR Methods
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Monte–Carlo VaR: Illustration III

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VaR Methods
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Monte–Carlo VaR: Illustration IV
VaR99% = 0.0052

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VaR Methods
Backtesting
Backtesting: Deﬁnition I
From a statistical viewpoint, the essence of backtesting is the
comparison of actual trading results with model–generated risk
measures.
But the backtesting procedure also helps the bank to know whether
the VaR or ES allocated enough capital to meet regulatory capital
requirements.

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VaR Methods
Backtesting
Backtesting: Use in practice I
The frequency and magnitude of the observed outliers across the
years convey information about your risk measure’s quality.
The backtesting procedure also helps to question your model whether
this latter embeds every risk factor.
Lastly your capital requirement depends upon your backtest results,
namely how many outliers did the bank observe.

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VaR Methods
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Backtesting: Use in practice II
Zone Number of exeptions multipier Cumulative Probability
Green 0 1,50 8,11(%)
. . .
4 1,50 89,22(%)
Yellow 7 1,70 95,88(%)
. . .
Red 10 2,00 99,99(%)
Table: Backtesting multiplier benchmarks

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VaR Methods
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Backtesting: Use in practice III
Type 1 error: Reject good risk measure model; Type 2 error:
Accept bad risk measure model.
Green stands for accurate model and low type 1 error probability
Red stands for bad model and high type 1 error probability
Yellow is grey area.
The cumulative probability of obtaining a given number of exceptions
or fewer in a sample of 250 observations with α = 99%.

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VaR Methods
Backtesting
Backtesting: Use in practice IV
CA = max (IMCCt−1; mc × IMCCavg + SESavg ), where:
CA is the required capital.
IMCC is aggregate capital charge.
mC is the multiplier and starts at 3. In addition the local regulator,
ACPR applies an adjustment factor with respect to the bank’s risk
management system.
the average is over 60 days.
SES is the stressed expected shortfall.

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VaR Methods
Backtesting
VaR Methods Comparison I
Method Advantages Drawbacks
Historical VaR Naturally address the ”fat tails” problems Relies on history
Performs well under back–testing Computationally intensive
Can fully capture non–linear risks Data intensive
Analytical VaR Easy to understand May misstate non–linear risks.
Least computationally intensive ”Fat tails” problem.
Monte–Carlo VaR Accommodates any statistical assumptions about risk factors. Sampling error
Can fully capture non–linear risks.
Table: Summary of Pros and Cons of Each VaR Methodology

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VaR Methods
Backtesting
VaR Shortcomings I
VaR does not have the sub–additivity property or in simple words the
diversification benefit. If two risks X and Y are part of a portfolio, we
do expect that the VaR of this portfolio is less than the addition of
the VaR of each risk.
VaR is a quantile and does not look beyond its confidence interval
which can be a very misleading risk measure if the tail contains large
losses even with a lower probability.

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Expected Shortfall
Catching The Tail: Expected Shortfall

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Expected Shortfall
Expected Shortfall I
The Expected Shortfall (ES) is the weighted sum of all the
value–at–risk above the safety level α
Therefore ES provides more information than the VaR given that it
captures the thickness of the tail of losses in the distribution of P&L.
Like the VaR, the Expected Shortfall depends on the distribution of
portfolio gains and losses (positions and risk factors), the level of
conﬁdence used and the time horizon.

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Expected Shortfall
Expected Shortfall II
Finally, the quality of the Expected Shortfall estimate depends upon
the number of scenarios used to determine the Distribution of P & L.

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Expected Shortfall
Expected Shortfall Graph

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Expected Shortfall
Expected Shortfall: Use in Practice I
Same process as the VaR and additional need to compute the mean
of losses beyond the VaRα threshold.

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Deﬁnition
Stress–testing

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Deﬁnition
Deﬁnition
The so far computed VaRα and ESα featured two shared limitations:
They relied on observed past information and, were limited on the
timeline. Therefore, they did not embed extreme case scenarios
known as ”black swan”. In order to complete our portfolio analysis,
we make use of model validation tools:

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Deﬁnition
Stress–testing Methods I
Stress–testing is the process aiming at insuring consistency and
compliance with basic principles of modeling and risk management.
The usual process’ target variables are: economic capital, VaRα/ESα
and portfolio loss volatility.
Stress–testing tools do not aim to produce perfect accuracy, but
rather checking that orders of magnitude are in line with what can
reasonably expected.

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Deﬁnition
Stress–testing Methods II
There are reﬂected from stress tests scenarios and break down as
follows:
Sensitivity
Historical
Hypothetical

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Deﬁnition
Sensitivity I
These ”sensitivity” stress scenarios allow to quantify the price
sensitivity of a portfolio to one or more market parameters
simultaneously.
These scenarios assumed that all underlying risk factors (index price
and index’s volatility in our S&P 500 example) undergo the same
scenario of deformation simultaneously (same sense and same
amplitude).

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Deﬁnition
Sensitivity II
All scenarios are not equiprobable. For instance, on the equity
markets, a drop in the spot price combined with a higher volatility is
more likely than a decline in prices accompanied by lower volatility.

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Deﬁnition
Illustration: S&P 500 Put I

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Deﬁnition
Illustration: S&P 500 Put II
The evolution of the price of the portfolio (loss of −1,39 e) following
a combined decrease of 10% of the underlying index and an increase
of 10% of the volatility.

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Deﬁnition
Illustration: S&P 500 Put III

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Deﬁnition
Historical
They aim to mimic intensively past market scenarios. We could lay down:
The 1987 Black Monday.
The 1997 Asian crisis.
The 1998 Russian debt default.
The 2001 terrorist attack in the US.
The 2007 − −2008 subrpime crisis
The 2008 Lehman bankruptcy
The 2009 European debt crisis
etc. . .

Banking Regulation
Stress–testing
Conclusion
MATLAB
Deﬁnition
Hypothetical
The Hypothetical stress scenarios are based on probable scenarios in view
of the economic situation and extreme changes that could occur. These
scenarios are determined together with economists.
For instance
What would happen if the UK rather negotiates a hard Brexit?
What would happen if North Korea attacks the US?

Banking Regulation
Stress–testing
Conclusion
MATLAB

Banking Regulation
Stress–testing
Conclusion
MATLAB
VaR and ES use in regulatory capital I
So far, the use of VaR and ES that we covered was from an economic
capital computation perspective but the prudential authorities also
require these values in order to ponder a bank’s resilience and to
identify potential threats to the overall ﬁnancial system.

Banking Regulation
Stress–testing
Conclusion
MATLAB
VaR and ES use in regulatory capital II
Prior the FRTB’s adoption, banks which met internal model
approach’s prerequisites, were required to meet a daily regulatory
capital as such:
Kt = K
VaR99%,10days
t
Intial market capital requirement
+
VaR under stressed conditions
K
sVaR99%,10days
t
where:

Banking Regulation
Stress–testing
Conclusion
MATLAB
VaR and ES use in regulatory capital III
K
VaR99%,10days
t = max VaRt−1, (3 + ξ) ·
1
60
60
i=1
VaRt−i
And
K
sVaR99%,10days
t = max sVaRt−1, (3 + ξ) ·
1
60
60
i=1
sVaRt−i

Banking Regulation
Stress–testing
Conclusion
MATLAB
VaR and ES use in regulatory capital IV
With the adoption of the FRTB, banks which desire to be part of the
Internal Model Approach (IMA) must produce on a daily basis a
one–tailed 97.5% percentile for the overall bank as well as for the
targeted IMA desk:
ES = (EST (P))2
+
j≥2
EST (P, j)
LHj − LHj−1
T
2

Banking Regulation
Stress–testing
Conclusion
MATLAB
Conclusion

Banking Regulation
Stress–testing
Conclusion
MATLAB
Conclusion
Key Takeaways
Use both in conjunction with other risk management tools because
risk management is more an artwork than a science.
With the adoption of the FRTB, banks face new challenges such as
FRTB design–based desk structure, skilled staﬀ, Big Data, etc.. . . .

Banking Regulation
Stress–testing
Conclusion
MATLAB
MATLAB Code

Banking Regulation
Stress–testing
Conclusion
MATLAB
1 function f = BSM(S,K,tau,sigma,r,dividend rate,OptionType)
2
3 %This function will compute the price of a European Call ...
or Put Option
4 %% Inputs
5
6 % S : is the price of the underlying at the date.
7 % K : is the strike of the option.
8 % tau : is expressed as a fraction of year. For ...
instance: If the contract is 3 months long, tau = ...
3/12 or 90/360 .
9 % sigma : is the volatility of the underlying.

Banking Regulation
Stress–testing
Conclusion
MATLAB
10 % r :is the risk-free rate on the money market.
11 % dividend rate: is the expected dividend on the underlying.
12 % OptionType: is dummy and can take on two values: -1 ...
for a Put Option and
13 % 1 for call Option
14 %% Outputs
15 %f holds the European Option Price given the above values
16
17
18
19 DZero = (log(S / K) + ((r - dividend rate - (0.5 * sigma ...
* sigma))*tau))/(sigma*sqrt(tau));

Banking Regulation
Stress–testing
Conclusion
MATLAB
20 DOne = (log(S / K) + ((r - dividend rate + (0.5 * sigma ...
* sigma)) * tau)) / (sigma * sqrt(tau));
21
22
23 price = OptionType*(S * exp(-dividend rate * tau) * ...
normcdf(OptionType*DOne) - K * exp(-r * tau) * ...
normcdf(OptionType*DZero));
24
25 if (price < 0)
26 f = 0;
27 else
28 f = price;

Banking Regulation
Stress–testing
Conclusion
MATLAB
29 end
30 end
1 function f = ...
chfun Heston(S0,V0,tau,r,q,kappa,theta,rho,sigma,lambda)
2 %Characteristic function of the Heston
3 dphi=0.01;
4 maxphi=50;
5 phi=(eps:dphi:maxphi)';
6
7 b1 = kappa+lambda -(rho.*sigma);

Banking Regulation
Stress–testing
Conclusion
MATLAB
8 b2 = kappa + lambda;
9 u1 = 0.5;
10 u2 = -0.5;
11 a = kappa.*theta;
12
13 x1 = b1 - rho.*sigma.*phi.*1i;
14 d1 = sqrt( x1.ˆ2 - (sigma.ˆ2).*( 2.*u1.*phi.*1i - phi.ˆ2 ...
) );
15 g1 = ( x1+d1 )./( x1-d1 );
16 D1 = ( x1+d1 )./(sigma.ˆ2).* ( 1-exp(d1.*tau) )./( ...
1-g1.*exp(d1.*tau) ) ;
17 xx1 = ( 1-g1.*exp(d1.*tau) )./( 1-g1 );

Banking Regulation
Stress–testing
Conclusion
MATLAB
18 C1 = (r-q).*phi.*1i.*tau + a./( sigma.ˆ2 ) .* ( (x1+d1) ...
.* tau - 2.*log(xx1) );
19 chfun1 Heston = exp( C1 + D1.*V0 + 1i.*phi.*log(S0) );
20
21 x2 = b2 - rho.*sigma.*phi.*1i;
22 d2 = sqrt( x2.ˆ2 - (sigma.ˆ2).*( 2.*u2.*phi.*1i - phi.ˆ2 ...
) );
23 g2 = ( x2+d2 )./( x2-d2 );
24 D2 = ( x2+d2 )./(sigma.ˆ2).* ( 1-exp(d2.*tau) )./( ...
1-g2.*exp(d2.*tau) ) ;
25 xx2 = ( 1-g2.*exp(d2.*tau) )./( 1-g2 );
26 C2 = (r-q).*phi.*1i.*tau + a./( sigma.ˆ2 ) .* ( (x2+d2) ...

Banking Regulation
Stress–testing
Conclusion
MATLAB
.* tau - 2.*log(xx2) );
27 chfun2 Heston = exp( C2 + D2.*V0 + 1i.*phi.*log(S0) );
28
29 f = [chfun1 Heston, chfun2 Heston];
30 end
1 function y = ES(returns,alpha)
2 %This function computes the Expected shortfall
3 SortedReturns = sort(returns);
4 n = size (returns,1);
5

Banking Regulation
Stress–testing
Conclusion
MATLAB
6 y = - ...
(sum(SortedReturns(ceil(n*alpha)+1:end))/(n*(1-alpha)));
7 end
1 function f = ...
Heston(St,K,tau,r,q,vt,kappa,theta,sigma,rho,lambda,OptionT
2
3
4
5 %--------------------------------------------------------------
6 %PURPOSE: computes the option price using Heston's model.

Banking Regulation
Stress–testing
Conclusion
MATLAB
7 %--------------------------------------------------------------
8 %USAGE: C=HestonCall(St,K,r,sig,T,vt,kap,th,lda,rho)
9 %--------------------------------------------------------------
10 %INPUT: St - scalar or vector, price of underlying at ...
time t
11 % K - scalar or vector, strike price
12 % tau - scalar or vector, time to maturity
13 % r - scalar or vector, continuously compound risk ...
free rate expressed as a
14 % positive decimal number.
15 % q - scalar or vector, continuously compound ...
dividend yield rate.

Banking Regulation
Stress–testing
Conclusion
MATLAB
16 % vt - scalar or vector, instantaneous volatility
17
18 % kappa - scalar or vector, is the rate at which ...
vt reverts to th
19 % theta - scalar or vector, is the long vol, or ...
long run average price
20 % sigma- scalar or vector, volatility of the ...
volatility of the
21 % underlying(same time units as for r)
22 % rho - scalar or vector, correlation between ...
underlying and
23 % volatility (rho<0 generates the leverage effect)

Banking Regulation
Stress–testing
Conclusion
MATLAB
24 % lambda - scalar or vector, risk premium for ...
volatility
25 %--------------------------------------------------------------
26 %OUTPUT: C - scalar or vector, Heston's model call ...
option price
27 %--------------------------------------------------------------
28
29 dphi=0.01;
30 maxphi=50;
31 phi=(eps:dphi:maxphi)';
32 g = ...
chfun Heston(St,vt,tau,r,q,kappa,theta,rho,sigma,lambda);

Banking Regulation
Stress–testing
Conclusion
MATLAB
33
34 g1 = g(:,1);
35 g2 = g(:,2);
36 P1 = ...
0.5+(1/pi)*sum(real(exp(-1i*phi*log(K)).*g1./(1i*phi))*dphi
37 P2 = ...
0.5+(1/pi)*sum(real(exp(-1i*phi*log(K)).*g2./(1i*phi))*dphi
38 C = (St*exp(-q*tau)* P1) -K*exp(-r*tau)*P2;
39
40 if (OptionType == 1)
41 f = C;
42 else

Banking Regulation
Stress–testing
Conclusion
MATLAB
43 f = C+K*exp(-r*tau) - (St*exp(-q*tau));
44 end
45 %f = result;
46 end
1 function f = ImpliedVolBS Dichotomy(S,OptionPrice,K, ...
tau,r,q, OptionType)
2 %This function calibrated the implied vol using the BSM ...
function and The
3 %bisection numerical method
4 sigma0 = 0.00005;

Banking Regulation
Stress–testing
Conclusion
MATLAB
5 sigma1 = 1;
6
7 j = 1;
8 Ite = 500;
9 while (j ≤ Ite)
10 sigmam = (sigma0 + sigma1) / 2;
11 if ((BSM(S, K,tau,sigmam,r,q,OptionType) - ...
OptionPrice) == 0)
12 f = sigmam;
13 elseif ((BSM(S, K,tau,sigma0,r,q,OptionType) - ...
OptionPrice) * (BSM(S, K,tau, ...
sigmam,r,q,OptionType) - OptionPrice) < 0)

Banking Regulation
Stress–testing
Conclusion
MATLAB
14 sigma1 = sigmam;
15 else
16 sigma0 = sigmam;
17 end
18 j = j + 1;
19 end
20 f = sigmam;
21 end
1 %Heston Calibration, local optimization using the Matlab ...
lsqnonlin

Banking Regulation
Stress–testing
Conclusion
MATLAB
2 global SPNum, global cost, global finalcost %,global S, ...
global OptionPrice, global K, global T, global r, ...
global q, global OptionType
3 filename = "MyData.xlsm";
4 sheet = 'S&P500 - IV DATA';
5 SPNum = xlsread(filename,sheet);
6
7 %Initialize the S&P market data at the 28/07/2017 date
8 %
9 %Maturity date = 31/08/2017
10 %Spot Price = 2472.1
11 %Implied Time to Maturity (31/08/2017 - 28/07/2017)/360

Banking Regulation
Stress–testing
Conclusion
MATLAB
12
13 S = 2472.1;
14 OptionPrice = SPNum(:,3);
15 K = SPNum(:,1);
16 T = SPNum(:,2);
17 r = 0.0099;
18 q = SPNum(:,5);
19 OptionType = -1;
20
21 % Initial parameters and parameter bounds
22 % Bounds [vt,kappa, theta, sigma, rho,lambda]
23 % Last bound include non-negativity constraint and ...

Banking Regulation
Stress–testing
Conclusion
MATLAB
bounds for mean-reversion
24 x0 = [.123, .5,.00001809, .05,-0.086835851,0];
25 lb = [0, 0, 0, 0, -.9, 0];
26 ub = [1, 100, 1, .5, .9, 0];
27
28 % Optimization: calls function costf.m:
29
30 x = lsqnonlin(@costf Heston,x0,lb,ub);
31
32 %Solution
33 Heston sol = [x(1),(x(5)+x(3)ˆ2)/(2*x(2)),x(3), x(4), ...
x(5),x(6)];

Banking Regulation
Stress–testing
Conclusion
MATLAB
34 x ;
35 min = finalcost;
1 %This function inputs assets value and outputs returns' data
2 function RM=returns(data)
3 RM = (data(2:end)-data(1:end-1))./data(1:end-1);
4 end
1 %%This algorithm will simulate different changes on risk ...
factors of the

Banking Regulation
Stress–testing
Conclusion
MATLAB
2 %%portfolio.
3 St = 2472;
4 K = 2375;
5 r = 0.0099;
6 q = 0.02653;
7 tau = 0.094;
8 vt = 0.015;
9 kappa = -0.0306;
10 theta = 4.1051e-05;
11 sigma = 0.0180;
12 rho = -0.0868;
13 lambda = 0;

Banking Regulation
Stress–testing
Conclusion
MATLAB
14 OptionType = -1;
15
16 dS S = (-0.2:0.05:0.2); %This j the x axis
17 dVol vol = (-0.2:0.05:0.2); % This j the y-axis
18 P = zeros(9,9);
19
20 for i = 1:9
21 for j=1:9
22 P(i,j) = ...
Heston(St*(1+dS S(i)),K,tau,r,q,vt*(1+dVol vol(j)),
23 end
24 end

VaR Or Expected Shortfall

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