Transcript The New Slides - Columbia University

Practical Problems with Building
Fixed-Income VAR Models
Rick Klotz
Managing Director
Global Head of Market Risk Management
Greenwich NatWest
Value At Risk: Definition
The value at risk (VAR) of a portfolio is the loss in value in the
portfolio that can be expected over a given period of time (e.g., 1-Day)
with a probability not exceeding a given number (e.g., 5%).
Probability (Portfolio Loss < - VAR) = K
K = Given Probability
Visualizing VAR: An Example
A one day VAR of $10mm using a probability of 5% means that there is
a 5% chance that the portfolio could lose more than $10mm in the next
trading day.
5%
1.645 Std Dev
-10MM
Possible Profit/Loss
VAR and Capital Requirements For Banks
Regulatory Capital = Market Risk Capital + Specific Risk Capital +
Counterparty Risk Capital
Market Risk Capital = Max [Ave. of 10-Day 99% VAR x Multiplier,
yesterday’s 10-Day 99% VAR]
Back-Testing A VAR Model



Calculate 1-Day 95% VAR for a (changing) portfolio each day for some
substantial period of time (e.g., 100 Days)
Compare the P/L on the succeeding trading day with the previous close of
business day’s VAR
Count the number of times the loss exceeds the VAR
25,000,000
20,000,000
15,000,000
10,000,000
5,000,000
P/L
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
(5,000,000)
(10,000,000)
(15,000,000)
95% 1 day VAR
(20,000,000)
(25,000,000)
49
VAR
The Need For VAR Model Accuracy

If the VAR is systematically “too low”, the model is
underestimating the risk and you tend to have too many
occasions where the loss in the portfolio exceeds the VAR. This
can lead to an increase in the “multiplier” for the capital
calculation.

If the VAR is systematically “too high”, the model is over
estimating the risk and your regulatory capital charge will be too
high
Building A VAR Model: Estimating the
Change in the Value of the Portfolio

Estimate the change in the value of the portfolio  P, as a function of
the change in the value of risk factors
, . ..,n
(e.g.,
, may be
 1
 1
the change in 1-year U.S. interest rates,
may be the change in 2 2
year U.S. interest rates, etc.).
Example:
 
 i i 
i 
Bi 

 i
2
 i
2
1
i (i ) 2

2
Building A VAR Model: Basic Methodologies
1) Variance/Covariance Method - Use historical variances and
i 

covariances
of risk factors,
,
to estimate how large 1.645 (for 5%) is for the
distribution of
.
Building A VAR Model: Basic Methodologies
2) Historical Simulation Method - Take an historical period, say the last
501 trading days, and calculate
2
1


   A       
j
i ij 2
i ij 
Where   Change in Risk Factor  ; from Day j to Day j  1.
ij
i
Order j  from highest to lowest and take the 475th as the VAR
Building A VAR Model: Basic Methodologies
3) Monte Carlo Simulation Method - Simulate a set of 500 (for
example) j  by choosing i ,  ij for risk factors ( i can be
historical or implied from options, ij are usually historical).
thej 
Order
from highest to lowest and take the 475th as the
VAR.
Model Specific Issues
Historical
Simulation
Most sensitive to bad data
Variance CoVariance
Generally some
kind of normal
distribution is assumed
Monte Carlo
Simulation
Hard to program
Approximation to full
revaluation can produce
significant errors
Options can present a
problem
Approximation to full
revaluation can produce
significant errors
General Challenges for VAR Models
• Obtaining Good Historical Data
• Finding a “complete” set of risk factors - fixed income VAR
models generally miss bond specific information (e.g., issuer
specific risk)
• How to weight historical data to accurately determine a 1-day
VAR.
Obtaining Good Historical Data



Poor Data
– Even actively traded markets can have “noisy” historical data
– Less actively traded markets can pose a significant challenge to
finding clean historical data
– Historical data can be misleading if a market is maturing over that
period
Missing Data
– It may be difficult to find historical data in relatively new (e.g., U.K. Asset
Backeds) or inactive markets (e.g., inverse I.O.s)
Asynchronous Data
– The data for risk factors that are traded against each other (e.g., Mortgages and
Treasuries, Futures and Cash Securities, etc.) must reflect simultaneous closes.
Finding a Complete Set of Risk Factors

Fixed Income VAR models generally miss bond (even market) specific
information
– Coupon to coupon trades
– Basis trades
– Issuer specific risk
– Some market specific risks (e.g., U.K. Asset Backeds)

Some risk factors are mapped to risk factors that have adequate
historical data but may not be good proxies
How Should Historical Data Be Weighted
To Calculate a 1-Day VAR?

Regulators require that you use at least one year of historical data

An option trader buying or selling a 1-Day option would give very
little weight to “old” data