SunGard Adaptiv CVA volatility

1. RISK MANAGEMENT
CVA Volatility - Minimizing the Aftershocks
BART PIRON
principal risk consultant, SunGard
Anticipate.

2. Contents.
1 Introduction
2 CVA Accuracy
2 CVA Volatility
3 Dynamic Date Grid
4 Daily CVA
5 Conclusion
Some CVA calculation
methodologies can
produce undesired artefacts
overstating the CVA PL
volatility. This paper explains
how this can happen and
how it can be simply and
efficiently be rectified.

3. CVA Volatility - Minimizing the Aftershocks 01
Where does the problem lie?
Suppose for a moment that Figure 1 accurately describes the
exposure profile for a given counterparty. This counterparty
is assumed to have a master netting agreement, but not a
collateral agreement. The graph may look a bit awkward at
first glance. The spike in the exposure is caused by
transactions being short or long a given risk factor, but with
different maturities. As transactions mature of before others
they reduce the overall netting effect and hence there is a
sudden jumpin the exposure. This example has been chosen
on purpose, not every portfolio will show such high spikes,
but almost every portfolio will show peaks and troughs in the
exposure profile when the latter is accurately calculated.
Now let us look at how a typical CVA calculation deals with an
exposure profile like this. Calculating the exposure at every
single business day in the future is prohibitively expensive so
the exposure is usually only calculated for a number of ‘grid’
days, for example: 1 day, 2 days, 1 week, 2 weeks, 1 month,
3 months, 6 months and every 6 months beyond that. Figure 2
below shows what this looks like for our example exposure
profile. The dots on the curve represent the grid dates.
As only a limited number of grid dates are used (in this case
39), some of the detail which was in Figure 1 went missing,
but at least the essential shape is still there. Even
the shape of the spike can be seen clearly, but notice it is
much lower than in the first graph. There happened to be
no grid point coinciding with the peak of the spike. Therein
lies the problem. Too much detail is missed for the CVA to
be calculated accurately.
0
500,000
1,200,000
1,500,000
2,000,000
2,500,000
3,000,000
3,500,000
4,000,000
4,500,000
0 1,000 2,000 3,000 4,000 5,000 6,000 7,000
Figure 1 Expected Exposure Profile
INTRODUCTION
The purpose of CVA is, amongst others, to correct the
Mark-to-Market of the derivatives book for the effects of
credit risk. Hence any volatility of the CVA measure also
impacts the valuation of the derivatives on the balance
sheet and the PL. PL is influenced by a large number
of factors and these should be made as transparent as
possible. Unexplainable PL volatility is not generally
welcomed. Some CVA calculation methodologies can
produce undesired artefacts overstating the CVA PL
volatility. This paper explains how this can happen and
how it can be simply and efficiently be rectified.

4. 02 CVA Volatility - Minimizing the Aftershocks
CVA Accuracy
The exposure profile in Figure 1 was generated by calculating
the exposure for every single day for the next 15 years under
every scenario and then averaging the values for each day.
This is the most accurate calculation which is possible. Based
on this profile the CVA1
turns out to be 27,884.
When calculating with a Fixed Date Grid as shown in Figure 2,
the CVA comes out as 20,763. This is more than 25% less than
the accurate calculation, a significant underestimate. But this
lack of accuracy is not the only problem with the Fixed Date
Grid. It also leads to more subtle problems.
CVA Volatility
We will look first at how CVA affects the quarterly PL results
and compare a Daily Date Grid to a Fixed Date Grid.
In Figure 3 below the portfolio is held constant to simulate a
counterparty where no trading activity is taking place. To
minimize the effects of other factors, we hold the market data2
constant and recalculate CVA at a number of points in time.
The only effect on the CVA should be due to the portfolio
aging and roll-off, and some small effects due to market data
being held constant (rather than risk neutral drift). In theory,
the CVA should change very slowly over time.
The see-saw pattern in the Fixed Date Grid line is caused by
the grid calculation dates coinciding or not with particular
peaks and troughs in the exposure profile. This volatility is
hence entirely artificial.
Whereas the values in Daily Date Grid calculation are relatively
smooth and decrease over time, the CVA under the Fixed
Date Grid is far more volatile. Expressed as a percentage
of the initial value, the CVA volatility of the Daily Date Grid
column is 10% and for the Fixed Date Grid it is 31%.
As the Bilateral CVA feeds into the accounting PL it will
produce artificial PL changes. This will make any attribution
analysis of the PL to its underlying causes particularly
troublesome and difficult to explain to management.
1
BILATERAL CVA IS BEING USED IN THIS EXAMPLE. BUT THE SAME REASONING
APPLIES TO UNILATERAL CVA, DVA, FVA OR ANY OTHER XVA.
2
WE COULD ALSO EVOLVE THE MARKET DATA RISK NEUTRALLY OVER TIME,
BUT THAT DIFFERENCE WOULD BE VERY SMALL.
Figure 2 Exposure Profile with a Fixed Grid
0
200,000
400,000
800,000
600,000
1,000,000
1,200,000
0 1,000 2,000 3,000 4,000 5,000 6,000
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
0 3 6 9 12 15 18 21 24 27 30 33 36
Daily Date Grid
Fixed Date Grid
Figure 3 CVA Comparison with a Fixed Grid as a graph

5. CVA Volatility - Minimizing the Aftershocks 03
Figure 4 Expected Exposure with Dynamic Dates
500,000
1,200,000
1,200,000
1,200,000
1,200,000
1,200,000
1,200,000
1,200,000
1,200,000
0
2
14
90
274
548
913
1,278
1,643
1,853
2,005
2,009
2,067
2,250
2,374
5,756
2,432
2,617
2,739
2,797
2,981
3,104
3,164
3,345
3,470
3,528
3,835
4,200
4,565
4,931
5,296
5,620
5,661
5,756
0
Dynamic Date Grid
So how do we get rid of both the artificial volatility and the
complications in PL attribution analysis? One way would
be to always calculate CVA using a Daily Date Grid. This is
guaranteed to produce accurate and smooth results, but,
as will be shown below, this can be prohibitively expensive.
Fortunately, there is a better solution. For most trades, credit
exposure changes only slowly over time, except when there
is a significant effect on the trade, such as a cash flow or an
option exercise date. By dynamically inserting additional
valuation dates per trade into the Fixed Date Grid, these
shocks in the exposure are made explicit. This requires 2
additional valuation dates for each significant event, one
before and one after the event, in order to see the shock.
For some portfolios adding another 2 valuation dates for
every significant event would rapidly lead to valuing the
portfolio at every business day, especially on the short end,
resulting again to a prohibitively expensive calculation. But it
is not necessary to value all trades at these additional event
dates, if only the trades which have a significant event at that
point are valued then the other ones can have their value
interpolated between their adjacent dates, as they will be
valued at their own significant event dates as well.
Figure 4 below shows the exposure from our example
portfolio using a Dynamic Date Grid. Each dot corresponds to
one date on the Fixed Date Grid or to a generated date. Note
how closely it resembles the graph from Figure 1 which was
done by simulating for every day for the next 15 years. Not
only does it get the shape right, it also has almost exactly the
same values for each date.
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
0 3 6 9 12 15 18 21 24 27 30 33 36
Daily Date Grid
Dynamic Date Grid
Figure 5 CVA Comparison with a Dynamic Date Grid

6. 04 CVA Volatility - Minimizing the Aftershocks
Let us now look at the CVA calculation with a Dynamic Date
Grid. In the table below you can see the CVA calculated for the
end of every quarter, again with all portfolio and market data
held constant as described in CVA Volatility. The comparison
between the simulation for the Daily Date Grid and the
Dynamic Date Grid results in differences which are nowhere
larger than 1%.
On Figure 5 above it becomes very hard to see the difference
between the CVA calculated by both methods, and that is
exactly the point.
The accuracy achieved through calculating with Dynamic
Date Grid comes at a comparatively low price. The number
of valuations necessary for this portfolio for the various
cases is shown in the table below.
The increase in the number of valuations (and hence the
resource consumption) from the Fixed Date Grid to Dynamic
Date Grid is only 16%. Another increase of two orders of
magnitude is needed to get to the accuracy of Daily Date Grid,
which is barely different from the result reached by Dynamic
Date Grid.
Daily CVA
Many banks have CVA desks managing the daily CVA PL.
When calculating daily CVA the effects are similar. On Figure 7
below you can see how daily CVA changes for a fixed portfolio
using the various methods over a one month period. The
sudden change in the Fixed Date Grid profile comes from a
fixed date all of a sudden picking up a spike in the exposure.
Note that again it is hard to spot the difference between the
Daily Date Grid and the Dynamic Data Grid whereas the Fixed
Date Grid method produces an unwanted jump in CVA.
Fixed Date Grid 1,400,005
Dynamic Date Grid 1,630,005
Daily Date Grid 187,650,005
Figure 6 Number of valuations required
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
1 2 3 4 5 6 7 8 9 10 11 12 15 16 17 18 19 22 24 25
Fixed Date Grid
26
Daily Date Grid
Dynamic Date Grid
Figure 7 Daily CVA over one Month

7. CVA Volatility - Minimizing the Aftershocks 05
Conclusion
Calculating CVA with a fixed grid of simulation dates, which is
still being done by a large number of banks, leads to problems
with the accuracy of the CVA result, but more importantly
increases the volatility of the CVA artificially and makes PL
attribution to market factors difficult or even impossible.
Similar effects occur for other xVA measures based on
simulation through time.
This can be remedied by simulating the exposure on a Daily
Date Grid, instead of just a Fixed Date Grid, but this method
is computationally very expensive. Using a Dynamic Date Grid
is fast and accurate giving most of the benefits of a Daily Date
Grid in terms of PL management with the performance of a
Fixed Date Grid.

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