The VaR Estimation in Historical Simulation Approach Open Issues and Some Practical Proposals...

The VaR Estimation in Historical Simulation Approach Open Issues and Some Practical Proposals

Michele Bonollo [email protected] Rinaldi – Prometeia [email protected]

Conference on Numerical Methods in Finance, Paris 2009

mailto:[email protected]

mailto:[email protected]

2

Index

PART I Market Risk Mgt and Historical simulation approach The stylized context vs. the real world context

The challenges of the real world numbers

Historical Simulation approach. Review of the canonical steps

PART II Historical simulation: Open isues, some possible approaches Issue 1. Scenario P&L. multidimensional full evaluation vs. marginal full evaluation

Issue 2. VaR Estimation. Scenario weighting by and quantile etimation

Issue 3. Component VaR. Expected return approach vs hybrid parametric approach

PART III Just a practical view of the reporting system

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PART I

4

Introduction …

Once (3-4 years ago) a (world famous) financial mathematics researcher asked me “What about your work”; I said “I work on market risk, VaR

computation”. He said “Again VaR ? Is is just a quantile …”. While exiting from his University, my feeling was “why so hard to meet theoretical and applied

perspective?”.

I do not know the exact answer: in my experience the real world problems are always cross among several fields of knowledge: asset management, financial instruments, financial mathematics, statistics, computation science, regulatory contraints, reporting processes and so on. The theoretical research is (must be)

very deep on each task. In the next slides a (vey small) step to take in to account both them

5

Market Risk Management: stylized view vs. real world

In the usual book description, one has two keys concepts concerning the VaR

The single instrument/position j

The portfolio, i.e. the vector of weights w = (w1, …,wj, …,wN)

The implied underlying idea is that one has to compute the risk measures (VaR, ES, ..) for the whole portoflio, for the single instrument/position, and at most for a few number of subportoflios, following the asset class or other clustering variable

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Market Risk Management: stylized view vs. real world

In the actual risk managament process, the portolio is a complex multilevel tree, where the different levels refer to:

The banks of the groups, the types of strategies, the families of products, the risk factors, …..

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Historical Simulation approach. The canonical steps

Let

t=1..T the id of scenarios: T = 250, 500 daily

j = 1…N the number of position/instruments

m = (m1…mK) the vector of market paramers underlying

f( ) the pricing functions

The steps are

1. Collect time series for underlyings/market parameters mt

2. From data to shocks/returns st. Compound, or continuous, …

3. Evaluation of Scenario P&L: PLj,t = fj(mjt + sj

t) – fj(mj)

4. Aggregate scenario P&Ls for the required cluster P&LCt = …

5. Estimate the Quantile – VaR or any other risk measure (ES, CVaR, ..)

8

The challenges of the real world numbers

Some numbers (magnitudes) from our bank, the 4-th in Italy

100.000 elementary positions, the > 90% derivatives

portfolio tree with 1.000 nodes

100 billions € of Notional in derivatives

> 1.000 elementary risk factors (IR buckets, underlyings, …)

As concerns the number of variables for which to apply a possible clustering, we have > 10 variables, related to: portfolio/desk, risk factor, product family, issuer/counterparty

Each day, we deliver (at least) 892 “standard” VaR, by .txt file. Moreover the Risk Manager can browse the whole portoflio and to compute the VaR for each required cluster or risk factor (equity, interest, forex) class. The combinations (hypercube D = 10)

∞ 20 millions of pricing each day: (Instrument x Scenario x RFactor)

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Historical Simulation: the basic schema

From the single positions j P&L

… to the cluster P&L

PTF A = Deal1 + Deal2

Deal

(a possible) VaR estimation

PTF

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Risque Merlino

Pricing Sophis (3) Pricing Prometeia (3)MaPaC (1-2)

Staging Area

Repository (4)

Process 1

Process 2

Process 3

Process …

Estrattori

Calcolo P&L Calcolo HVaR Risk Contributions Browsing

QlikView (5)

Exp

or

t

Clu

sterin

g

Cle

an

ing

VaR

C

om

pu

tatio

n

Rep

ortin

g

Positio

ns

Fro

nt O

ffice

Pricin

g

En

gin

es

Bloomberg ReutersMoneyMate

Provider N

Historical Simulation: Canonical algorithm in the IT actual architecture

ShocksShocks

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PART II

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Issue 1: ScenarioP&L, Full evaluation and marginal Full evaluation Each day t the market parameters mt move togehter

But, for compliance and strategic views, one often want to decomposte the different sources of risk in a single instrument/portfolio: equity, interest, forex

So the straigh way is to aply marginal shocks stk and then to evaluate the marginal P&Lj,t,k

But P&Lj,t k P&Lj,t,k

If we suppose that the pricing functions are smooth and follow a taylor representation, it

happens because k P&Lj,t,k consistes of the sum of all the “pure” derivatives, up to ∞, in the

taylor expansion, and the interaction terms are lost. Very often the first two terms are enough in the expansion, so the difference is mainly due to the terms ou fo the diagonal in the Hessian matrix of the pricing function

How to “reconciliate” the two measures? We remind the “data oriented” HS schema requires to use a unique large table (millions of rows) and we can not row by row deal with different cases

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PLAIN VANILLA OPTION

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0,0%

1,0%

Delta PV (Marginal Full evaluation)

% Difference - Full / Marginal Full

The graph below is an example of the difference for a plain vanilla option (SPMIB call) using two different approaches: Full Evaluation and Marginal Full Evaluation. % is small

Issue 1: ScenarioP&L, Full evaluation and marginal Full evaluation

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Here we have a large, higly exotic portfolio (napoleon, altiplano, rainbow, ..). We poit out that the difference still are quite small

Aletti Bank

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Full Evaluation

% Difference


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So what do we do? This interesting hard problem ha fortunately a small impact on computations. So depending from the different situations, we

have some different strategies: For linear or quasi linear portfolio (bond, equity) we can put by definition

P&L SUM of marginal P&L

In other cases we take the residual and we split it to the different sources in any way. Consider that this task is very frequent. For example an equity

in $ has each day a new value Vt that is Vt = Vt-1 x (1+Rt)(1+FXt), where R

anf FX represent the share and the $ return. The interaction effect (R x FX) has to be splitted. The issue is well konwn also in asset management, as “performace contribution – attribution”, see Brinson, Carino, ….

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Here we have to consider a trade off between some different goals:

The quantile estimation, e.g. the simplest empirical quantile (the 5-th worst scenario if T =500, = 99%), has a high variance behaviour. This is a well known problem in order statistics theory (see David, Huber, ..), but is forgotten from practitioners.

From a risk management perspective, one has to optimize the back testing statistics. In other words, the out of sample P&L that exxcee the ex-ante VaR must be close to the expected frequency. In a 1-year VaR estimation = 99%, I would expect that only 2.5 times the daily P&L is below the VaR prediction. The accuracy of back testing implies a different capital requiremet by the central bank. If good, the capital requirement is (approximately) 3 times the 10-days 99% VaR

The risk changes over time but it remains unobservable. We can use (see Boudoukh 1996, Finger 2008) also in the non parametric historical simulation approach a weighting technique à la RiskMetrics. In this case, we weight the probability of scenario before estimating the quantile

Issue 2: The quantile estimation and scenario weighting by

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VaR 99%

VaR 99% lambda 0,99

VaR 99% lambda 0,98

VaR 99% lambda 0,97• Holding period: 250

day• Confidence level: 99%

The graph below is an example of VaR calculation using different lambda parameters for a large exotic portfolio “Structured Product”. With is (now) more conservative


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The VaR of the portfolio has been calculated using different parameters Lambda and is on a portfolio composed of the following indexes:

• Nikkey 225• S&P 500• EUROSTOXX 50E

The three indexes have thesame weight in the portfolio composition

The table shows expected and Effective outliers data Portfolio’s Profit & Loss compared with VaR calculated with different Lambda

PORTFOLIO EXPCTED OUTLIERS EFFECTIVE OUTLIERS

VAR 99% 2,5 7

VAR 99% ( 0,99) 2,5 5

VAR 99% ( 0,98) 2,5 5

VAR 99% ( 0,97) 2,5 6

Backtesting Analysis

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P&L VAR 99%(Lambda 0,98)

VAR 99% VAR 99%(Lambda 0,97)

VAR 99%(Lambda 0,99)

• Holding period: 250 day

• Confidence level: 99%


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The second exercise of Lambda Weighting was calculated for a portfolio with the followes shares:

• IT0001976403• IT0003856405• IT0001063210• IT0001334587• IT0000072725• DE0005557508• DE0005752000• DE0005140008• FR0000121261

This shares have thesame weight in the portfolio composition

The table shows Expected and Effective outliers data Portfolio’s Profit & Loss compared with VaR calculated with different Lambda

PORTFOLIO EXP. OUTLIERS EFFECTIVE OUTLIERS

VAR 99% 4.01 7

VAR 99% ( 0,99) 4.01 5

VAR 99% ( 0,98) 4.01 6

VAR 99% ( 0,97) 4.01 8

• Holding period: 401 day


Backtesting Analysis

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P&L VaR 99% (Lambda 0,98)

VaR 99% VaR 99% (Lambda 0,97)

VaR 99% (Lambda 0,99)


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So what do we do? At now (the project is in progress and fine tuning):

As concerns the high variance of the estimator, the system allows to smooth the system by simple L-estimator, e.g. we do not take the 5-th worst case, but we average in a neighbor. We have not yet applied more sophisticated technique, such as Harrel-Davis estimator (the weight of the scneario is given by its frequency in a bootsrap sampling), Cornish-Fischer and so on. Here, for auditing reasosn and reporting cotraints, converge to simplicity …

As concerns l, from april the “official VaR” is computed with l = 0.98., but each day we compute as a check/warning also the the “plain vanilla” VaR, the is the simplest quantile estimation. Consider that the computation effort is very hard. So we compute 2 x 892 VaR (Portoflio, clustering, …). Each one of them requires: sum of 10.000 100.000 P&L, sort them, estimate VaR. With a 48 (!!) GB RAM server, 20-30 minutes.


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The additive decomposition of risk is veru useful. In the actual risk management process:

The risk limits are given as VaR limits or greeks limits (delta equivalent, basis point value, vega for 1% shift, ..). The strategic analysis of risk need to split the effect of the different desk /

porfolios on the risk: VaR = …

A good measure is the ComponentVaR (see Garman, Mausser, ..). Using for simplicity the % return notation, not the € P&L notation, if we have a partition of the portoflio indexed by i, the portfolio return is RP, the VaR is R*P, then

CVARi E(Ri | VaRP) = E (Ri | RP = R*P)

In the gaussian context, this reduces (see Garman, 96) to

CVaRi = R*P x i x c, between porfolio and subporfolio

If we apply the definition for the Historical case, if t* is the VaR scenario, we take the P&L for the

t* for the subportfolio as CVaRi (see Hallerbach). Nevertheless this simple tecnique may not be

used, becaues of high variance, low reliability.

Issue 3: Component VaR

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Here we see the weakness of the described “pure” non parametric estimation, based of the expectation definition. The portfolio model is of european blue chips, higly correlated (avg 0.7)


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A simple robust idea is to “plug-in” the beta-Garman formula in the non parametric approach. We compute over the T P&L scenario and then fixed the t* VaR-scenario, apply the formula to decompose it. Below the same portfolio, the same compute date. More reliable!


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The portfolio is composed by the followes shares:

• IT0001976403• IT0003856405• IT0001063210• IT0001334587• IT0000072725• DE0005557508• DE0005752000• DE0005140008• FR0000121261

• Holding period: 401 days


The chart below shows VaR Contribution (Component VaR) of each share in portfolio to Total VaR, by appling the hybrid Beta technique over 400 days

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IT0001976403 IT0003856405IT0001063210 IT0001334587IT0000072725 DE0005557508DE0005752000 DE0005140008FR0000121261 CVaR


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So what do we do? At now (the project is in progress and fine tuning):

Differently from , the CVaR is not yet published in the daily reporting, it is computed in order to make software test.

Consider that we could deliver a “number of CVaR” very high (> 1000) depending of all interesting clustering and partitioning of portfolios.

I think we will use hybrid approach or some way very cloe to it. To measure risk is importante, but even more important is that the top management “believes” the the risk meausers. An irregular or “strange” risk measure over time makes is useless


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PART III

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Distribution of the Scenario P&L of a large exotic portfolio

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The Tableau de Bord

Bank filter

Portfolio / Desk

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In relazione a una serie di anomalie “fisiologiche” o determinate da errori del batch Sophis sono èstato messo a punto un sistema di controllo, denominato Outliers, che permette di navigare e visualizzare tali casi: fair value nulli, P&L rilevanti; Le soglie di rilevazione delle anomalie sono parametriche, modificabili dall’utente

Reporting Historical VaR

The VaR Estimation in Historical Simulation Approach Open Issues and Some Practical Proposals...

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