Post on 07-Apr-2018
8/4/2019 Value at Risk Intro
1/39
1
Value-at-Risk Intro
March 2007
Juhani Huopainen
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
2/39
2
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
3/39
3
1. What is VaRand what its not
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
2007-2011 Juhani Huopainen (huopainen on gmail)
ENOQ2-07, 1.3.2005 - 26.1.2007
20.00
25.00
30.00
35.00
40.00
45.00
50.00
55.00
60.00
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
8/4/2019 Value at Risk Intro
4/39
4
1. What is VaRand what its not
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)
8/4/2019 Value at Risk Intro
5/39
5
1. What is VaRand what its not
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
20.00
25.00
30.00
35.00
40.00
45.00
50.00
55.00
60.00
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)
2007-2011 Juhani Huopainen (huopainen on gmail)
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%
8/4/2019 Value at Risk Intro
6/39
6
1. What is VaRand what its not
example Q2-07, 1st Mar 2005 26th Jan 2007 (1000)
2007-2011 Juhani Huopainen (huopainen on gmail)
Historical method what wouldhave happened
0
50
100
150
200
250
300
350
400
450
500
-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
8/4/2019 Value at Risk Intro
7/39
7
1. What is VaRand what its not
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...
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
8/39
8
1. What is VaRand what its not
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
9/39
9
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
10/39
10
2. VaR-methods
Variance-covariance method (VCV i.e. delta-normal)
Historical simulation
Monte Carlo simulation
Three main types
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
11/39
11
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
12/39
12
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
Parametric variance-covariance method
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
13/39
13
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
2007-2011 Juhani Huopainen (huopainen on gmail)
-15%
-10%
-5%
0%
5%
10%
15%
-15% -10% -5% 0% 5% 10% 15%
2. VaR-methods
Parametric variance-covariance method
8/4/2019 Value at Risk Intro
14/39
14
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
2007-2011 Juhani Huopainen (huopainen on gmail)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1/3/05
3/3/05
5/3/05
7/3/05
9/3/05
11/3/05
1/3/06
3/3/06
5/3/06
7/3/06
9/3/06
11/3/06
1/3/07
10day
90day180day
8/4/2019 Value at Risk Intro
15/39
15
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.
Parametric variance-covariance method
2007-2011 Juhani Huopainen (huopainen on gmail)
Source: http://en.wikipedia.org/wiki/Correlation_and_dependence
8/4/2019 Value at Risk Intro
16/39
16
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
Parametric variance-covariance method
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
17/39
17
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
18/39
18
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
19/39
19
3. VaR-model: creation
Common to all
2007-2011 Juhani Huopainen (huopainen on gmail)
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
8/4/2019 Value at Risk Intro
20/39
20
3. VaR-model: creation
Position mapping
2007-2011 Juhani Huopainen (huopainen on gmail)
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?
8/4/2019 Value at Risk Intro
21/39
21
3. VaR-model: creation
Historical inference
2007-2011 Juhani Huopainen (huopainen on gmail)
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?
8/4/2019 Value at Risk Intro
22/39
22
3. VaR-model: creation
Testing
2007-2011 Juhani Huopainen (huopainen on gmail)
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
8/4/2019 Value at Risk Intro
23/39
23
4. VaR-metrics and their use
Var-metrics
2007-2011 Juhani Huopainen (huopainen on gmail)
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
8/4/2019 Value at Risk Intro
24/39
24
4. VaR-metrics and their use
Var-metrics
2007-2011 Juhani Huopainen (huopainen on gmail)
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
8/4/2019 Value at Risk Intro
25/39
25
4. VaR-metrics and their use
Use of VaR-metrics
2007-2011 Juhani Huopainen (huopainen on gmail)
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
8/4/2019 Value at Risk Intro
26/39
26
5. Advanced models
Cornish-Fisher
2007-2011 Juhani Huopainen (huopainen on gmail)
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%).
8/4/2019 Value at Risk Intro
27/39
27
5. Advanced models
Cornish Fisher and the old example
2007-2011 Juhani Huopainen (huopainen on gmail)
Historical method what would have happened
0
50
100
150
200
250
300
350
400
450
500
-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
8/4/2019 Value at Risk Intro
28/39
28
5. Advanced models
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
29/39
29
5. Advanced models
After Cornish-Fisher
2007-2011 Juhani Huopainen (huopainen on gmail)
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
8/4/2019 Value at Risk Intro
30/39
30
6. Stress tests and scenario analysis
Scenario analysis
2007-2011 Juhani Huopainen (huopainen on gmail)
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
8/4/2019 Value at Risk Intro
31/39
31
6. Stress tests and scenario analysis
Stress tests
2007-2011 Juhani Huopainen (huopainen on gmail)
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)
8/4/2019 Value at Risk Intro
32/39
32
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)
2007-2011 Juhani Huopainen (huopainen on gmail)
Pricing
}
}}
8/4/2019 Value at Risk Intro
33/39
33
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
34/39
34
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
35/39
35
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
36/39
36
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
37/39
37
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
38/39
38
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
2007-2011 Juhani Huopainen (huopainen on gmail)
8/4/2019 Value at Risk Intro
39/39
39
Thank You!
2007 2011 J h i H i (h i il)