CVA - VaR
Shahram Alavian
Royal Bank of Scotland
Etienne Koehler
Barclays Capital
Shahram_Alavian@yahoo.com Etienne.Koehler@univ-paris1.fr
January 2012
In Nutshell
What we say …
How we go about it ….
 Statement
 Preliminaries
 Calculate VaR on CVA using a
single batch run instead of multiple
runs; once for each sensitivity.
 Benefit
 Faster execution of VaR
 Approach
 Insert a 1-day time horizon in the
simulation timeline.
 Model the distribution of the
change in value, at 1-day horizon,
as a function of the underlying
factors.
 Use this function to reprice the
CVA changes in the same way a
trade’s pricing formula is used to
calculate its VaR.
 Credit Value Adjustment (CVA )
 Value at Risk (VaR)
 VaR for CVA
 The Proposed Approach
 VaR properties
 VaR and Monte Carlo simulation
 VaR for CVA
 Examples
 A non-linear payoff
 Application to Regulatory VaR
 Concluding Remarks
 Limits of the approach
 Challenges
2
Preliminaries
 Credit Value Adjustment (CVA )
 Value at Risk (VaR)
 VaR for CVA
3
Preliminaries - CVA
 Definition
 It is a Fair Value Adjustment to an
OTC Trade.
 It is equivalent to the risk that a
dealer
has
taken
on
its
counterparty’s credit by entering
into an OTC trade
Dealer
Dealer
 Management
 Accounting
–
Managing
its
volatility in the balance sheet.
 Risk Mitigation – Selling protection
to
other
Proposed
business
functions (desks)
 Regulatory Capital – Minimizing
the cost of regulatory capital
Counterparty
=
 Objective
 Accounting, risk mitigation, and
regulatory
 Unilateral versus bilateral
Defaultable Cash flow
Default Free Cash flow
Counterparty
+
Dealer
Option to Default
Counterparty
4
Preliminaries – CVA Continued …
 Main Components of CVA
 Market component
 Credit component (including default)
 Cross gamma component
 Hedging CVA
 Inconsistencies in hedging from different objectives
 Hedging for Default?
 Hedging for Capital
 Hedging for volatility
 The hedged CVA book, П is
  CVA CVAH
5
Preliminaries – VaR
 Value at Risk (VaR) Is A
 measure of unexpected loss due to market move.
 tool for calculating capital
 qualitative representation of the volatility of the balance-sheet.
 risk-limit measure for trading desks
 Sensitivity Based
 Sensitivities are sent to market risk department. VaR and its relevant break-down
numbers are then returned.
 Not all sensitivities are included
 Historical Based
 Recent (3 year) daily time series
 “stressed period “
6
Preliminaries – VaR for CVA
 CVA Volatility leads to:
 CVA Capital
 CVA as a fair value adjustment brings volatility to the balance sheet
 For firms using unilateral CVA (no DVA), the unexpected loss due to a rise in
credit spread of the market can be very large.
 Newly introduced regulatory CVA capital (BASEL III)
 Limit on CVA-VaR
 Level - how much?
 Bucket – Region, industry, currency, …
7
Proposed Approach
 VaR properties
 VaR and Monte Carlo simulation
 VaR for CVA
8
Proposed Approach – VaR Properties
1. Instantaneous:
It excludes the duration of the portfolio, omits any cash flow
during the time horizon and limits the change of value to the instantaneous change in
the underlying risk factors, only.
2. Conditional: It is conditional on its initial value at current time.
3. Functional:
Similar to a pricing function, it is a function of the risk factors driving
its value. This feature provides the means for generating VaR from any arbitrary
distribution.
9
Proposed Approach – VaR and Monte Carlo
 Monte Carlo Approach
 N risk factors are simulated under a joint process. For each path j and time t,
we have (dependence on t is implied)

RFj  RF j , RF j ,, RF j
(1)
( 2)
(N)

j
 Valuation of an asset π conditional on each time t and path j will result
 j t    RF j , t 
 Therefore, we have a distribution of π, as a function of its underlying risk
factors, for each time t.
 Defining a 1-day Implied VaR.
 Calculate the conditional values of the asset at δ = 1-day time horizon for each path j.
 j  
In future slides, we replace π with expected exposure (EE) and CVA itself.
10
Proposed Approach – VaR and Monte Carlo
1.
Create an Instantaneous Change
Calculate the change in values of the asset at 1-day time horizon for each path j.
 j ,0   j     0
This is not an instantaneous change. Therefore, we pick up another path j’.
 j ',0   j '     0
and create an instantaneous change by
 j , j '   j ,0    j ',0  
  j    j '  
2.
<= This change is from any j’ to any j
Create a Conditional Change
Having simulated a large number of paths, one can find a path for which
 j '     0
So we have a conditional change from j’ which given the above assumption, we a have
conditional on spot.
11
Proposed Approach – VaR and Monte Carlo
3.
Create a Functional Change
Assuming there exists a model, M, representing the conditional changes as a function of the
changes in the risk factors
 j , j '    M  RF j , j ' 
with
RF j , j '  RF j '  RF j
Using a linear regression model, for example,


 j , j '    M RFj , j ' ; β   k0  1 RF1, 0    2 RF1, 0      x RF2, 0   
k
k
 Putting It All Together

2
k

VaR 1day,99% [M  RF j ]
with

RF j  RF t j
(1)
 RF t j 1 ,, RF t j
(1)
(N)
 RF t j 1
(N)

Note that RF j represents historical and RFj represents simulated.
12
Proposed Approach – VaR for CVA
 Change of Value of a Hedged CVA Book
   CVA  CVAH
Re-writing the above in the same notation as previous section
  j 0  CVAj 0  CVAHj 0
Following our prescription, we need to calculate
CVA j  
which we can do using a rollback method like Least Squared MC, under two different
approaches.
13
Proposed Approach – Method I
 Joint Simulation of the Credit and Market

CVA j , j0    M CVA RF j , j0 ; β
 Step-by-Step

CVA  
1. Simulate the credit and market risk factors under a joint simulation
2. Calculate the conditiona l CVA j   at t   for every path j. This requires a rollback method.
3. Go through all paths and find the path j 0 : j  j 0 when CVA0  - CV A j   is the smallest.
4. Generate a set of N-1 changes (RF j,j0 ) from path j 0 for each of the market implied
risk factors [ to simplify notation RF j  RF j,j0 ]

5. Include all relevant terms in polynomial s of the regression model M CVA RF j ; β

6. Use CVA j   and RF j to calculate β in order to fix M CVA RF j ; β


7. Obtain all relevant sensitivit ies of CVA H in order to represent CVA H in a functional
form of CVA H (RF j )
8. Calculate the historical (daily) changes,  RF j .
9. VaR would be the 99 - percentile of  j .
14
Proposed Approach – Method II
 Simulation of the market, only
 CVA (bilateral case)
CVA   x k ,d EE k ,   x
k ,c
EE k , 
k 1
EE  
x
k ,c
x
k ,d

 LGD P

 LGD c P c t k   P c t k 1  Q d t k 
d
d
t k   P d t k 1 Q c t k 
Q  1 P
EE  From simulation and discounted
x  Not Simulated
15
Proposed Approach – Method II
 Simulation of the market, only
 Change in CVA (bilateral case)
CVA j   x k ,d EE kj ,   x k ,c EE kj , 
 Market
k 1
 EE k ,  x j
 x j
k ,d
k ,d
 EE k ,  x j
k ,c
 Credit
EE kj ,   x j EE kj ,   Cross Terms
k ,c
 CVA j change of CVA from its current va lue
 x k ,d is the spot
 x j
k ,d
is from daily historical change
 EE k ,  is the spot

 EE kj ,   M kEE,   RF j ; β

  RF j is from daily historical change
 EE k ,    requires roll - back of EE k ,  t k  from t  t k to t  
16
Proposed Approach – Method II
 Step-by-Step
1. Simulate all market risk factors
2. For every time horizon k :
(a) Calculate EE k ,    using a roll - back alogrithm
(b) Go through all paths and find the path j 0k  : j  j 0k  when EE k ,  0   EE k ,   
is the smallest
(c) Calculate RF j , j k  [ to simplify notation RF j  RF j , j k  ]
0
k

(d) Calculate β in order to fix EE k ,   M kEE RF j ; β

0
3. Calculate all relevant sensitivit ies of CVA H in order to represent CVA Hj in a
functional form of CVA H RF j 
4. Obtain the historical time series for RF j , x kj ,d and x kj ,d .

5. For every j in the historical time series, and for each k , calculate M kEE,   RF j ; β

6. For every j in the historical time series, and for each k , calculate CVAkj
in CVA j  k CVAkj
7. VaR would be the 99 - percentile of  .
17
Examples
 A Power Option
 An Interest Rate Swap and Its Application to Regulatory VaR
18
Examples – A Power Option
 Motivation For This Example
 To show the effectiveness of linear regression when modelling a non-linear trade
 Motivation For This Trade
 The power option is a highly non-linear trade with an analytical pricing formula.
 Setup

payoff  max K  ST2 ,0

dS
 r dt  0.25 dw S S (0)  100
S
dH
 dt  0.3 dw H H (0)  0.02
H
E dw S dw H  

 Assumption
 Rates and Vols are
not stochastic.
 Counterparty has sold
us a put on its stock

r  0.05 CVAH  0
LGD c  1
Historical volatilit ies for both S and H  0.5.
Number of Paths  5,000
Number of historical elements  750
19
Examples – A Power Option [ ρ = 0 ]
For every strike, with ρ = 0, both S and H
were simulated. First, the VaR using
benchmark(BM) was produced by pricing
the CVA with current market data and then
pricing the CVA with each of the 750
scenarios. The VaR of this distribution, for
this strike, produces one point under BM
label. Separately, Method-I, and Method-II
were used to calculate their corresponding
VaR numbers. Each VaR makes a data
point under the Method-I and Method-II
labels, respectively. This process was then
repeated for strikes ranging from 4,000 to
40,000. The longest part of the exercise
was the generation of the VaR using BM,
since it had to reprice the CVA 751 times
for each strike value.
Method - I


M CVA , H , S ; β   k0   1 S    2 S      5 S    6 H    7 H    8 H S 
Method - II


k
k
2
k
5
k
k
2
k
M EE k  S ; β   k0   1 S    2 S      5 S 
k
k
2
k
5
20
Examples – A Power Option [ ρ = -0.95 ]
The same set of simulations was
repeated, for the same range of strikes,
using ρ =-0.95, incorporating a large
WWR. In this case, however, only
Method-I was calculated. This figure
illustrates the effect of WWR to VaR
using Method-I.
Method - I


M CVA , H , S ; β  k0  1 S    2 S      5 S    6 H    7 H    8 H S 
k
k
2
k
5
k
k
2
k
21
Examples – An Interest Rate Swap
 Motivation For This Example
 Sensitivity of the approach to various volatility values
 To compare the BASEL –III CVA regulatory capital using method-I.
 Motivation For This Trade
 Trade’s value can be negative creating the exposure non-linear and making
the rollback process challenging.
 Setup
H (0)  0.02 r (0)  0.05
Implied volatilit y of r and H  .2


E dw S dw H  
LGD c  1
 Assumption
 Current period is also
the “stress Period”
 There are no hedging
instruments.
Historical volatilit ies of H ,  H , varies from 0 to 1.1.
 varies from 0 to 1.
Number of Paths  8,000
Number of historical elements  750
22
Examples – Effect of Volatility on Approach
Since the regulatory VaR does not take
any contribution from the un-hedged
exposure variations, the first test would be
to compare the credit component of the
VaR for various historical spread
volatilities. This is done by generating a
time series for H while keeping the
historical volatility of the rates to zero. The
results are then compared with the
regulatory VaR. Using the regulatory VaR
as the benchmark, one can also observe
that the VaR methodology proposed here
can generate, from a single implied
volatility of 20%, the correct VaR for
different historical volatilities generated
from various time series.
Method - I


M CVA , H , r; β  k0  1 r    2 r      5 r    6 H    7 H    8 H r 
k
k
2
k
5
k
k
2
k
23
Examples – Regulatory VaR and WWR
The next objective is to compare the
regulatory and the proposed VaR for a
given historical volatility of market (both,
hazard rate and the rates) and for
various correlation. This is done by
performing a Monte Carlo simulation for
each level of ρ, generating the time
series matching the corresponding
correlation and calculating the VaR as
prescribed in method-I. Since the
regulatory VaR depends only on the
historical volatility of the hazard spread,
it produces a flat line. Note that for ρ=0,
method-I includes variations in the market
component and regulatory VaR does not.
Method - I


M CVA , H , r; β  k0  1 r    2 r      5 r    6 H    7 H    8 H r 
k
k
2
k
5
k
k
2
k
24
Concluding Remarks
 Limits of the Approach
 Implied Volatility – In cases where rollback methods were used to obtain
the conditional prices of option, there will be no volatility to regress against.
 Challenges
 A Robust rollback algorithm – Obviously, there is no analytical formula to
calculate the conditional CVA or exposures. This means we need a robust
rollback method in order to obtain convergent conditional values .
 Nonlinearity of the exposures – Even when using a rollback one still
needs a substantial number of paths for the exposures and CVA values to
converge as they are highly nonlinear.
 A different VaR platform – Almost in all cases, all desks use the same VaR
platform. This method now requires a different one.
25
Questions and Comments
Thank you
26