1. Value at Risk
June 7, 2013
Amir Khwaja
www.clarusft.com

2. Agenda
 The Need for VaR
 Definition of VaR
 Uses of VaR
 VaR Methods
 VaR - Historical Simulation
 Changes since the Financial Crises of 2008
 Strengths and Weakness
 Summary

3. The Need for VaR
 Different Asset Classes use their own measures
 Fixed Income – Duration
 Interest Rates – DV01
 FX – Currency position
 Commodity – Number of contracts
 Equities – Number of shares
 Using these to compare risk in these portfolios is like
comparing apples and oranges
 An investor or owner needs a simple measure that can be
used in a consistent way to compare risk between these
portfolios
 And with the ability to aggregate risk appropriately
 Value at Risk is that risk measure

4. Definition of VaR
 How much could we lose over a specified holding period
with a defined probability
 So if a portfolio has a VaR of $20 million
 We need to know the confidence level used to calculate
 We need to know the holding period (time horizon)
 Say confidence level is 99% and holding period is 5 days
 This then means “We would expect to lose $20 million
or more over a 5 day period, in 1 out of 100 business
days”
 Note it does not tell us whether on that 1 day we could
lose $21 million or $200 million!
 So if we are relying only on VaR for the answer to that
then we are going to be in trouble
 For that we need Tail Measures or Stress Testing

5. Uses of VaR
 Made public by JP Morgan in 1994 with RiskMetrics
 Widely adopted in the industry very quickly after that
 Particularly for Derivatives where measures such as gross
notional or position in contracts units, are not that
insightful
 Basel II Capital Accord for Market Risk – 1995
 Internal Model Capital is VaR times a multiplier set for each
bank by its regulator as between 3 & 4
 Banks report VaR in Annual Financial Statements – 1997
 Internal 1d VaR and Regulatory 10d VaR
 Clearing Houses for Initial Margin – 1999 (LCH SwapClear)
 Margin from defaulting member used to cover the market risk
loss for the period it takes to close-out the portfolio

6. VaR Methods
 Three main methods
 Parametric (aka Variance-Covariance or Delta-Gamma)
 Historical Simulation
 Monte-Carlo Simulation
 Different assumptions, calculation steps, compute
efficiency but similar numbers for standard portfolios
 The most common is Historical Simulation
 As easiest to understand
 Simple assumptions on distributions of returns
 So if for our $20 million VaR portfolio, we also said that we
had used 5 Years of history as well as 99% and 5d
 We would say that “given how the market has performed
in the past 5 years” our VaR estimate is $20 million

7. VaR - Historical Simulation
 It relies on choosing
 A historical period, e.g. 4 Years
 A holding period e.g. 5 days
 Generating daily holding period returns in this period
 Calculating the P&L impact on a portfolio by applying
these returns to today
 Ordering the P&L outcomes by decreasing loss
 Interpolating for a chosen confidence level e.g. 99%

8. VaR - Historical Simulation
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
80.00
USD 5Y Swap Rate
5d returns (bps)
Sep08 to Sep12
20Nov08 > -60bps
13Dec10 > 20bps

9. VaR - Historical Simulation
 Assume our portfolio has a PV01 of $1million
 Assume for simplicity that USD 5Y Swap is the only risk factor
 For a 1 bps rise in the 5Y Swap rate, our Profit will be $1m
 For a 1 bps fall in the 5Y Swap rate, our Loss will be $1m
 We can calculate the PL Series for our portfolio by multiplying the
bps returns on each day by $1 million, which is shown below
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
80.00
Profit Loss
Sep08 to Sep12
20Nov08 > $60m

10. VaR - Historical Simulation
 This PL Series
 Has a PL value for each business day from 5 Sep 08 to 4 Sep 12
 A total count of 1043 values
 Each corresponds to a specific scenario date, starting on 5 Sep 08
 The first element represents the PL outcome of applying the 5-day
return shift between 1 Sep 08 and 5 Sep 08 to todays market data
and todays portfolio
 We call this the PL vector of the portfolio
 The first few elements of which are shown below
-26.51
-5.97
-12.75
-6.47
-3.29
-15.83
-14.76
-37.08
-9.98
-16.55
22.76
71.55
09/05/2008
09/08/2008
09/09/2008
09/10/2008
09/11/2008
09/12/2008
09/15/2008
09/16/2008
09/17/2008
09/18/2008
09/19/2008
09/22/2008
For these dates

11. VaR - Historical Simulation
 The PL vector can then be re-ordered by decreasing loss
 Keeping a note of the scenario date and PL of each
 The first part of this is shown below
1
2
3
4
5
6
7
8
9
10
11
11/20/2008
12/17/2008
10/21/2008
10/22/2008
11/21/2008
06/17/2009
08/14/2009
12/18/2008
10/06/2008
10/07/2008
09/16/2008
-62.58
-56.16
-47.79
-46.24
-44.95
-42.47
-41.29
-39.73
-39.62
-37.45
-37.08
For these datesRe-order by PL
 Now we can determine the VaR
 Which we will define as 99% or the loss of the 11th worst PL
 (We could define as 10th worst or interpolate between 10th and 11th)
 So VaR is $37.08m
 Occurs on the scenario date of 16-Sep-08, we call this the VaR Date
 This is the week of Lehman’s bankruptcy filing
VaR Date

12. VaR - Historical Simulation
0
50
100
150
200
250
-70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Frequency
5d PLs
Sep08 to Sep12
Mean -1.21
Standard Deviation 13.03
Kurtosis 2.63
Skewness 0.17
Range 134.13
Minimum -62.58
Maximum 71.55
VaR 99% -37.08
Count 1043
 A Histogram is a good way to view the PL vector
 Allocate each PL to a bin range
 Frequency is high for small PLs, giving the distribution below

13. VaR - Historical Simulation
 Zooming in to the largest losses
Mean -1.21
Standard Deviation 13.03
Kurtosis 2.63
Skewness 0.17
Range 134.13
Minimum -62.58
Maximum 71.55
VaR 99% -37.08
Count 1043
11th largest loss
Largest loss
Expected Shortfall

14. Changes since the Financial Crises of 2008
 Basell Capital Accord, introduced Stressed VaR
 So Trading Book Capital is the higher of Firm’s Multiplier *
VaR or Multiplier * Stressed VaR
 Where Stressed VaR covers a period of Market Stress
 The Financial Crises of 2008 qualifies as a period of stress
for most major markets
 Wider use of Tail measures
 Expected Shortfall (ES) or Worst Case Loss (WCL)
 Renewed focus that Stress testing must be performed and
the definition and results of these discussed within the firm
and with regulators
 Hence US & European Regulatory Stress Tests

15. Strengths and Weaknesses
 Strengths
 Reduces risk to a single $ amount
 Mark to market based measure (not original notional)
 Compare risk of different asset class portfolios
 Aggregate risk across portfolios
 Widely used since 1994
 Weaknesses
 Reduces risk to a single $ amount
 Assumptions may be complex
 Depends on market prices being observable and similar
behaviour to that observed in the past
 Long-tailed properties of financial markets
 Portfolio diversification is not there in a Crisis
 So correlation goes to 1

16. Summary
 It is crucial to understand any assumptions
 For VaR, these are
 Method e.g. Historical Simulation
 Confidence level e.g. 99.7%
 Holding Period e.g. 5 days
 Historical Period Used e.g. 5 Years
 And not use just a single VaR measure
 Or indeed discard VaR (and replace with what?)
 So, in addition to VaR
 Use Tail Measures e.g. ES or WCL
 Stress Tests – Historical and Hypothetical
 Independent price verification
 Gross measures

17. Contact Details
Our LinkedIn Page: Clarus Financial Technology
Our Website: www.clarusft.com
My contact details: amir@clarusft.com