Value at Risk Intro

8/4/2019 Value at Risk Intro

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Value-at-Risk Intro

March 2007

Juhani Huopainen

2007-2011 Juhani Huopainen (huopainen on gmail)


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VALUE-AT-RISK

1. What is VaRand what its not

2. VaR-methods

3. VaR-model: creation

4. VaR-metrics and their use

5. Advanced models

6. Stress tests and scenario analysis

7. Future challenges

Schedule



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Assume the trading position iscomposed of one long Q2 forwardcontract

Price of the contract is 27 euros,

underlying value 58.968 euros is thisthe total amount at risk?

Is the risk the collateral requirement ofthe position?

You have a sell stop-order in themarkets at 26 euros. Is the risk 27-26 =

2.184 euros?What if you are addionally short a Q3forward? is the total risk 123.883euros? Or zero?

Defining The Risk


ENOQ2-07, 1.3.2005 - 26.1.2007

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1 29 57 85 113 141 169 197 225 253 281 309 337 365 393 421 449 477 505 533 561 589 617 645 673 701 729 757 785 813 841 869 897 925 953 981

Risk is notequal to

position size

margin/collateral requirement

stop-loss

sum of individual risks


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VaR measures the largest expectedloss over a certain period of timeunder normal market conditions ata certain confidence level.

A company can say that its dailytrading VaR equals1 million with99% confidence level

This means that under normal

market conditions a loss larger than1 million would on average happen

once every hundred days.

Defining VaR

2007-2011 Juhani Huopainen(huopainen on gmail)


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Well keep the presentation short

and quickly calculate the correctVaR for one bought forwardcontract.

We need the historical loss that isseen only once in hundred days

Solution: we take 1000 daily forwardreturns, put them in ascendingorder and pick the tenth largest lossday (-4.65%)

We do the same for 5% VaR andpick the 50th largest loss, -1.96%

ENOQ2-07, 1.3.2005 - 26.1.2007

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1 29 57 85 113 141 169 197 225 253 281 309 337 365 393 421 449 477 505 533 561 589 617 645 673 701 729 757 785 813 841 869 897 925 953 981

example Q2-07, 1st Mar 2005 26th Jan 2007 (1000)


4 -6.98%

5 -6.44%

6 -6.41%

7 -6.08%

8 -5.61%

9 -5.48%

10 -4.65%

11 -4.62%

12 -4.21%

13 -3.93%

14 -3.72%

15 -3.67%


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example Q2-07, 1st Mar 2005 26th Jan 2007 (1000)


Historical method what wouldhave happened

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-7.00% -6.00% -5.00% -4.00% -3.00% -2.00% -1.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00

p1%: -4,65

p5%: -1,96%

Variance method:

Mean zero, standard deviation 1.28%so 1% worst day would be

-2.32635 * 1.28% = -2.98%

..and comparable 5% worst day

-1.64485 x 1.28% = -2.1%Difference? skew -1.37, kurtosis 12.7


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The bad and the good:

Assumption: returns normally distributed

Assumption: the correlations between different instruments is assumed to bestable. (we only had one instrument in our example, and still borked)

Generally, tomorrow is assumed to be like yesterday

Nonlinear instruments (e.g. options) do not fit into the model

There is no single, official, correct VaR-method. This makes comparing themodels or using the model outputs difficult

VaR that deals with these weaknesses is bound to become complicated, heavyand still rely on other assumptions.

Already in trouble and only the 7th slide...



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VaR is better than nothing

VaR is better than the alternatives, especially when it is combined with limitsand scenario analysis

Portfolio theory is comfortable with only normally distributed world. More

advanced VaR models do not suffer from this limitation Portfolio theory would think that playing lottery is very risky, as the potential

payoff makes the results volatile. VaR thinks the risk is the possibility oflosses, not profits

VaR on is a simple dashboard figure that investors and regulators want andeven management can understand. Like it or not, VaR has to measured

...but its not that bad



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2. VaR-methods

Sharpe, Markowitz et al experimented with rudimentary VaR calculationsalready in 1950s, but lack of computational power made any practical

applications unrealistic

The end of Bretton Woods in 1971, oil crisis, inflation, volatility of the interest

rates, government debt created new markets derivatives and their pricingwere discovered: leverage became possible

Bank risk profiling was oldfashioned +500 crude oil futures and -200 porkbelly futures cannot be added by delta method

By 1993 several banks had developed a proprietary VaR-method. Because oflarge derivative losses by many corporations and banks, J.P.Morgan published

its own RiskMetrics-VaR RM VaR documentation and factor correlation data were free. JPMs idea was

not to become a service provider, instead it wanted to trade derivatives withothers

History



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2. VaR-methods

Variance-covariance method (VCV i.e. delta-normal)

Historical simulation

Monte Carlo simulation

Three main types



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2. VaR-methods

VCV-method: calculate historical returns and variances and covariances for allthe instruments in the portfolio.

Instead of full covariance matrix calculation, the problem can be solved byusing only few variables (factors). In classical portfolio theory only the beta

or codependence of a stocks price between a benchmark index is calculated,instead of all the covariances between all the individual stocks.

These factors are identified and selected with the help of cluster analysis andPCA (principal component analysis)

Assumption risk factors (and instrument price returns) are multivariatenormal and the price of the portfolio is linearly dependent on these factors

Assumption 2 the historical numbers give a good approximation of the future

Parametric variance-covariance method



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2. VaR-methods

Expected return can be assumed to be an average of historical returns (?)

Expected returns are usually not an issue, since they are a relatively smallsource of risk compared to variance of returns

Variance can be estimated in many ways:

long term average

moving average (accounts for heteroscedasticity)

GARCH (also accounts for mean reversion)

EGARCH (also accounts for asymmetrical variance response)

FIGARCH (also accounts for long-term memory effects)

etc., etc. Variance clusters and is predictable to a certain extent




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Q3-07 ja Q2 07-forwardsprice correlation 0.98 andreturn correlation 0.88

Great, the example positionon slide 3 (long/short) has avery small risk!

Traditional VaR agrees withour intuition and we get asmall number


-15%

-10%

-5%

0%

5%

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15%

-15% -10% -5% 0% 5% 10% 15%

2. VaR-methods



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Q3-07 ja Q2 07-forwards

price correlation 0.98 andreturn correlation 0.88

Even a high correlationdoes not explaineverything:

R2=0.882=0.77

Short-term correlation isvery volatile: 10-day corrvaries between 0.1 and 1.

Naive VCV-thinking givesus too too low VaR-number


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10day

90day180day


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2. VaR-methods

1. Well-behaving correlation

2. Nonlinear function

3. Perfect correlation & outlier

4. Zero correlation & outlier

All four have the samemean, standard deviationand correlation.



Source: http://en.wikipedia.org/wiki/Correlation_and_dependence


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2. VaR-methods

Open questions

Length of time period for the VCV calculation? If the time span is short, resultsare volatile, if long, changes are not picked up

Period of time chosen? Is the period representative? Will it be different now?

Factors or individual instruments? By using factors, one could pay more attention in making sure they are well-

behaved and representative of the issues but this could also backfire

Classic case GBP/DEM in 1992, before devaluation should you use forward orspot rates in risk calculations, does history have any value empirically?

VCV can be calculated in real time




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2. VaR-methods

Well take four years of daily market data, get approximately 4 x 52 x 5 = 1000

data points for the value of the position

Transparent like no other, least amount of assumptions except that tomorrowwill be like yesterday

No assumptions of normal distribution or stabil VCV-matrix Still one has to decide how long and what period to use

Does not solve the problem of instruments that have not been (i.e. new futuresand option series)

With modern computing results in real time.

Non-parametric: Historical simulation



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2. VaR-methods

1. Decide how many times paths are iterated (N)

2. Create market models for all instruments (or factors) and simulate an imaginarydaily price changes

3. Calculate change in portfolios value, given the simulated prices

4. Repeat N times

End result is N number of portfolio values, where one can easily locate the 1%VaR figure.

MC is only as good as the market model If VCV cannot be used because of lackof historical data or because of heavy non-normality or existence of options inthe portfolio, MC is the only way to go.

MC is like the historical simulation, but with made-up data

Computationally the heaviest, in practice not available in realtime

Non-parametric: Monte Carlo



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Common to all


Whatever method you use, the same grunt workhas to be done:

Identification and classification of historical pricechanges or factors, estimating the parameters oftheir joint distribution

Defining the position to be measured and pricing it

Getting the VaR metrics out by combining the two


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Position mapping


Position mapping is critical but often thought to be a secondary issue

Should one include whole firm, risk management or only tradingpositions? Should they be calculated separately or combined?

What is position? Long-term financing costs, credit risks, operative risks?

How to include nonlinear instruments and other exoticity?

When to calculate VaR? A daily figure calculated at the end of the day?How about intraday positions and intraday risks for longer positions?


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Historical inference


Factors or instruments?

Length and choice of data period, possible weighting scheme to givegreater weight to recent data over older data

VCV-matrix calculated from historical, implied or econometric models?

How to accommodate the weak predictability of covariance estimatesconfidence intervals, statistical significance, seasonal models?

Can the system handle non-normal distributions or nonlinearity and

should this be accommodated for when planning the data collection?


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Testing


Naturally VaR-model should betested before implementation

The only practical way of testing ishistorical simulation

VaR could be teached with older

data, then checked how it works without-of-sample period (e.g. how oftenthe 1% threshold is trespassed)

Basle-standard 250 days:

Zone Violations Scaling factor

Green 0 4 3.00

Yellow 5 3.40

Yellow 6 3.50

Yellow 7 3.65

Yellow 8 3.75

Yellow 9 3.85Red 10+ 4.00


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Var-metrics


VaR: max loss usually

Conditional VaR (ExpectedTail Loss, Expected Shortfall):when losses go beyond VaR-limit, how much they are onaverage

Minimizing CVaR alsominimizes VaR


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Var-metrics


Profit/VaR (vs. Sharpe Ratios

Proft/Standard Deviation)

Marginal VaR: if you add1 to oneportfolio component, how much yourVaR changes

Incremental VaR: The change inVaR from adding a position to theportfolio

Component VaR: The change inVaR from removing a position fromthe portfolio

PaR Profit-at-risk

Relative VaR

Cash Flow at Risk

EBITDA at Risk

Long Term VaR

Short Term VaR

Trading VaR/trading

Stop loss x VaR

Counterparty VaR


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Use of VaR-metrics


In practice a well-modeled VaR figure at 99% level means that in a normalyear losses larger than the VaR figure are met on average 2.5 times.

The higher the probability figure, the smaller the tail section that is underexamination. The further one goes down the tail, the less experience (anddata) there is, and one should be less confident in the resulting lossestimates.

Rule of thumb: the VaR-period (e.g. 1 or 10 days) should be selected sothat usually there are no major changes in the portfolio during that time.

There is no meaningful way of calculating a daily VaR for a high-frequencyoperation

With a selection of a longer VaR-period, one ends up with less data


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5. Advanced models

Cornish-Fisher


On slide 4 we noticed that the assumption of normality leads to too lowVaR estimates

z(cf) = z(c) +1/6 * {z(c)^2 - 1}*S + 1/24 * {z(c)^3 - 3*z(c)}*K - 1/36 * {2*z(c)^3 - 5*z(c)) * S^2

where z(cf) is Cornish-Fisher critical value

z(c) is the critical value for the probability 1-a assuming normality (-2.3399%)

S is skewness and K is kurtosis eli hntien paksuus

Now for the 99% case we get a higher z-value (-5.6, -2.33), and VaRestimate would be instead of 2.33 x 1.28% (=2.98%) this: 5.6 x 1.28%(=7.18%).


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5. Advanced models

Cornish Fisher and the old example


Historical method what would have happened

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-7.00% -6.00% -5.00% -4.00% -3.00% -2.00% -1.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00

p1%: -4,65

p5%: -1,96%

Daily VaR CF 99% = -7,18%

Daily VaR CF 95% = -2,23%

Variance method:

Mean zero, standard deviation 1.28%

so 1% worst day would be

-2.32635 * 1.28% = -2.98%

..and comparable 5% worst day

-1.64485 x 1.28% = -2.1%Difference? skew -1.37, kurtosis 12.7


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5. Advanced models



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5. Advanced models

After Cornish-Fisher


Cornish-Fisher allows working with non-normal distributions as long asthere are no other hidden surprises besides non-normal kurtosis andskewness.

How to include options and other non-linear instruments?

Quadratic (or Delta-Gamma) VaR

Beyond the scope of this presentation


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Scenario analysis


Weaknesses of VaR are well-known, also by the regularors. Even a weakVaR-model can measure risks reasonably enough in a normalenvironment, but what if something strange happens?

Scenario analysis is a close relative to historical analysis. One selects abad historical event and sees how the portfolio and the VaR measurewould have worked.

1987 stock market crash

1992- ERM crisis1997 Asian crisis

1998 Russian crisis, LTCM

2000 tech bubble

2006 Amaranth implosion2006 carbon credit collapse

2008 Lehman moment

2010 Euro crisis


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Stress tests


Stress test is a self-made scenario analysis

Anything, not only price changes, can be included in the stress test (e.g.liquidity desert, making opening or closing of positions impossible or very

expensive)

Correlations moving to -1 or +1

Increases in volatility

Changes in forward curves

Most famous stress tests recently have been the infamous Europeanbank tests (that did no even include the possibility of a sovereign failure)


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7. Finally

Development develops

Recognize &Avoid

Measure (e.g. marketVaR,creditVaR, operative VaR) andcontrol

Trading limits

Risk analysis

Allocate reserves andcapital

RAROC, Risk-AdjustedReturn on Capital

Active portfoliomanagement

Testing (stress and scenario)


Pricing

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7. Finally

In 2006 energy industry used $4.4 billion on portfolio- and risk managementsystems

In 2007 an estimated $5.25 billion were used (thank you, Amaranth!)(Carbon360 survey)

One third of hedge funds make their risk management work on Excel perhapsbecause they know that fancier stuff isnt more effective, or because they want

to show what they show to investors

Nordic SPAN uses VaR-based calculations for margin requirements

All the big participants use VaR, for internal and regulatory purposes

Large players (hedge funds) demand and get VaR-based margin practices from

their prime brokers

Lots of resources at play



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7. Finally

1. Conglomerate or departmental risk management

bottom-up or top-down

2. Official vs. internal-only

Interpreting the greeks, using and calculating volatility- and distribution forecasts

3. Market risk Length of sample, estimating the variance-covariance matrix, non-normality

4. Credit risk

using credit derivatives

5. Liquidity risk

Still badly known, usually integrated to market risk

Greatest challenges



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7. Finally

6. Operative risk- law, pricing- and model risks, roque dealers and risk managers. No standard

practice, view or certainty7. Nonlinear instruments

- Delta-method, delta-gamma-method, full revaluation (Monte Carlo)

8. Estimating volatilities Volatility of volatility? Historical, implied, econometric? Volatility curve and

smile?9. Estimating correlations

Same problems as with volatility estimations. Correlation derivatives could help.Time synchronization issues in products traded in different time zones

Greatest challenges



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7. Finally

Managing director wants to hire someone who can answer the question how muchis 2 + 2.

Engineers uses a slide ruler and states it is between 3.98 ja 4.02.Mathematician guarantees that she can prove it is 4 after two hours of non-trivial

calculations.

Physicist, by means of deduction, decides the magnitude of the answer is 1x101

.

Logician, after thinking for hours tells that the problem is solvable.The social welfate professional apologizes his lack of knowledge, but wants to tell it

is good that a topic of that importance was brought forward.The lawyer remembers a previous case where it was 4.Trader wants to know before answering are you looking to buy or sell.Risk professional gets up from the chair, checks the aisle so that nobody can hear

and whispers to the managers ear: what do you want it to be.

Mandatory Joke



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7. Finally

www.gloriamundi.org Largest source for VaR

www.rhoworks.com VaR-program for testing

www.riskmetrics.com

The oldest VaR-producer, the Risk Metrics official manuals, online courses etcare valuable stuff for learners

www.riskglossary.com

On the net



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7. Finally

Jorion, Philippe, Value at Risk: The New Benchmark for Managing Financial Risk2001

the benchmark book

Holton, Glyn A., Value-at-Risk: Theory and Practice2003

some think this is the best out there (www.value-at-risk.net) Dowd, Kevin, Beyond Value at Risk The New Science of Risk Management2003

Lots of material, but not that technical. Good for beginners who want to have anoverview on the topic

Javanainen, Timo, Analytical Delta-Gamma VaR Methods for Portfolios of ElectricityDerivatives2004

Readings



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Thank You!

2007 2011 J h i H i (h i il)

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