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Practical CVA and KVA Forum London, 24th - 26th April 2017
Reasons behind FVA, MVA, KVA Tommaso Gabbriellini Andrea Gigli Head of Quants Head of Fixed Income and XVA
MPS Capital Services MPS Capital Services
Disclaimer _______________________________________________________________________________________________________
These are presentation slides only. The information contained herein is for general guidance on matters of interest only and
does not constitute definitive advice nor is intended to be comprehensive.
All information and opinions included in this presentation are made as of the date of this presentation.
While every attempt has been made to ensure the accuracy of the information contained herein and such information has been
obtained from sources deemed to be reliable, neither MPS Capital Services, related entities or the directors, officers
and/or employees thereof (jointly, โMPSCS") is responsible for any errors or omissions, or for the results obtained from the use
of this information. All information in this presentation is provided "as is", with no guarantee of completeness, accuracy,
timeliness or of the results obtained from the use of this information, and without warranty of any kind, express or implied,
including, but not limited to warranties of fitness for a particular purpose. MPSCS does not assume any obligation whatsoever to
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any decision made or action taken in reliance on the information in this presentation or for any consequential, special or similar
damages, even if advised of the possibility of such damages.
This document represents the views of the authors alone, and not the views of MPSCS. You can use it at your own risk.
3
Goals of the talk
โข Using a multiperiodal structured model we are going to investigate
the rationale behind FVA, MVA and KVA
โข The model represents a useful tool to understand the relations
between valuation adjustments, market parameters and regulatory
constraints
โข Three main lessons can be learned from the model
โข How to allocate capital to different business units
โข How to manage funding strategies
โข Hot to price banking products
4
FVA, MVA, KVA
โข MVA & FVA measure the impact on Equity due to IM and VM bankโs
obligations after entering derivatives contract, using debt to finance
those obligations.
โข Regulatory requirements impose that the leverage of the balance
sheet remains below a predefined threshold KVA measures the
impact on the Equity as the bank fulfils the regulatory constraints
โข In order to compensate shareholders for negative variations in the
Equity value a charge equal to MVA, FVA, KVA might be needed.
5
The Model โ Uniperiodal case
Assume:
- the risk meausure is the risk neutral one
- the bank will default if ๐ด(๐) < ๐ฟ๐๐ก, where
- ๐ฟ๐๐กis the amount of debt and interests to be paid and
- ๐๐ก = 1 + ๐๐ ๐ก,
- ๐ ๐ก is the funding spread set in t
- the risk free rate is zero.
The value of Equity in ๐ก is
๐ธ๐ก = ๐ผ๐ก ๐๐๐ฅ ๐ด(๐) โ ๐ฟ๐๐ก , 0
The value of the Liabilities in ๐ก is
๐ผ๐ก ๐๐๐ ๐ด(๐), ๐ฟ๐๐ก = ๐ฟ๐๐ก โ ๐ผ๐ก ๐๐๐ฅ ๐ฟ๐๐ก โ ๐ด(๐), 0
6
The Model โ Uniperiodal case
The spread ๐ ๐ก is set by the
creditor such that ๐ฟ โค ๐ฟ๐๐ก โ ๐ผ๐ก ๐๐๐ฅ ๐ฟ๐๐ก โ ๐ด(๐), 0
the spread must be sufficient to remunerate the
risks
In the following we will assume that the creditor is always ยซfairยป, i.e
the minimum spread is applied:
๐ฟ = ๐ฟ๐๐ก โ ๐ผ๐ก ๐๐๐ฅ ๐ฟ๐๐ก โ ๐ด(๐), 0
N.B.
if ๐ ๐ก is fair ๐ธ(๐ก) = ๐ด ๐ก โ ๐ฟ
Proof: ๐ธ(๐ก) = ๐ด ๐ก โ ๐ฟ๐๐ก + ๐ผ๐ก ๐๐๐ฅ ๐ฟ๐๐ก โ ๐ด(๐), 0 = ๐ด ๐ก โ ๐ฟ
Put-Call Parity
7
The Model โ Uniperiodal case
What is the impact of a new investment on the equity value of the bank?
Assume at ๐ก+the bank issues new debt for funding a risk free asset whose
maturity is the same of the debt.
The fair spead on the new debt must be such that:
Fair spread
in ๐ก+
Assets Liabilities
๐ธ(๐ก+) = ๐ผ๐ก+ max ๐ด(๐) + ๐ถ โ ๐ฟ๐๐ก โ โ๐ฟ๐๐ก+ , 0
๐ถ = โ๐ฟ = ๐ผ๐ก โ๐ฟ๐๐ก+๐ ๐ด ๐ +๐ถ>๐ฟ๐๐ก+โ๐ฟ๐๐ก++ ๐ด(๐)
โ๐ฟ
๐ฟ + โ๐ฟ๐ ๐ด(๐)+๐ถ<๐ฟ๐๐ก+โ๐ฟ๐๐ก+
In case of default the assets will be used for a partial
reimburse proportionally to the face value of the liabilities
C doesnโt depend
upon t
8
The Model โ Uniperiodal case
๐ = ๐ = 1
Note that:
โข โ๐ฟ = ๐ถ
โข If ๐ด๐ก โซ C โ ๐๐ก+ โ ๐๐ก
Hence, the variation in the equity value is
๐ผ๐ก max ๐ด(๐) + ๐ถ(๐) โ ๐ฟ๐๐ก โ โ๐ฟ๐๐ก+, 0 โ ๐ผ๐ก max ๐ด(๐) โ ๐ฟ๐๐ก , 0 =
โ โ๐ถ โ ๐๐ ๐ก โ ๐ผ๐ก ๐ ๐ด๐>๐ฟ๐ก๐๐ก
This is the amount of money
shareholders requires in order to invest
borrowed money in a risk free asset
๐ก+
Assets Liabilities
๐ด(๐ก+)+ ๐ถ(๐ก+)
๐ฟ ๐ก+ + โ๐ฟ
Equity
๐ผ๐ก ๐๐๐ฅ ๐ด(๐) + ๐ถ โ ๐ฟ๐๐ก
โ โ๐ฟ๐๐ก+ , 0
Assets Liabilities
๐ด๐ก(๐)+ ๐ถ
๐ฟ๐๐ก + โ๐ฟ๐๐ก+
Equity
๐๐๐ฅ ๐ด ๐ + ๐ถ โ ๐ฟ๐๐ก
โ โ๐ฟ๐๐ก+ , 0
9
The Model โ Uniperiodal case
What if the asset is not risk free? There may be as well negative impacts
(ยซfunding costsยป) and positive ones (ยซfunding benefitsยป), depending on
the volatility and correlation with the previous assets and its risk.
๐ด ๐ก = 100
๐๐ด = 20% ๐ฟ = 90
๐ ๐ก = 6.60%
ฮ๐ฟ = ๐ด1(๐ก+) = 10
10
The Model โ Multiperiodal case
In our multiperiodal settings we assume that the bank rolls its debt at its
maturity.
For the sake of simplicity, we analyze the case where the bank rolls its
debt just once
๐ ๐ก 2๐ 3๐
๐ฟ ๐ฟ๐๐ก ๐ฟ๐๐ก๐๐ ๐ฟ๐๐ก๐๐๐2๐
๐ ๐ก 2๐
๐ฟ ๐ฟ๐๐ก ๐ฟ๐๐ก๐๐
11
The Model โ Multiperiodal case
๐ ๐ก 2๐
๐ฟ ๐ฟ๐๐ก ๐ฟ๐๐ก๐๐
We evaluate the equity by
means of the ยซtower properyยป
๐ผ ๐ธ2๐ โฑ๐ ๐ธ(๐ก) = ๐ผ ๐ผ ๐ธ2๐ โฑ๐ |โฑ๐ก
Letโs look at the value of ๐ผ ๐ธ2๐ โฑ๐ in the following 2 cases
๐ด ๐ โฅ ๐ฟ๐๐ก ๐ด ๐ < ๐ฟ๐๐ก
The bank finance the debt +
interest at a new fair spread.
๐ผ ๐ธ2๐ ๐ด ๐ > ๐ฟ๐๐ก = ๐ด ๐ โ ๐ฟ๐๐ก
The bank try to finance the debt
+ interest at a new fair spread,
but no one is willing to lend
moneyโฆ
๐ผ ๐ธ2๐ ๐ด ๐ โค ๐ฟ๐๐ก = 0
Proof in the following slide
12
The Model โ Multiperiodal case
Why if ๐ด ๐ < ๐ฟ๐๐ก no one is willing to lend money?
Letโs have a look at the fair value of the debt in the limit of an
infinite spread
lim๐ ๐โโ
๐ผ๐ min ๐ด(2๐), ๐ฟ๐๐ก๐๐ = ๐ผ๐ ๐ด(2๐) = ๐ด ๐ < ๐ฟ๐๐ก
The maximum fair value of the debt is always
lower than the amount to be financed!
๐ผ ๐ธ2๐ โฑ๐ = max(A ๐ โ ๐ฟ๐๐ก , 0) Combining the two cases we have that
Therefore the equity can be priced as
๐ธ ๐ก = ๐ผ max(A ๐ โ ๐ฟ๐๐ก, 0) โฑ๐ก
Exactly the same as in the uniperiodal setting
13
The Model โ Multiperiodal case
How is the FVA affected by the financing strategy of the bank?
Letโs consider the purchase at time ๐ก+ of a risk free asset (cash) whose
maturity is greater than ๐ (the bond maturity), say 2๐
Applying the same reasoning as before, the equity can be computed as
if the maturity of the newly purchased asset is the same as of the debt
๐ธ(๐ก+) = ๐ผ๐ก+ max ๐ด(๐) + ๐ถ โ ๐ฟ๐๐ก โ โ๐ฟ๐๐ก+ , 0
The FVA is proportional to the financing ยซperiodยป, not
to the maturity of the asset, i.e. the following still
holds!
๐น๐๐ด โ โ๐ถ โ ๐๐ ๐ก โ ๐ผ๐ก ๐ ๐ด๐>๐ฟ๐๐ก
14
An application for FVA/MVA
Suppose the bank enters in a back to back derivitave, one collateralized
and one not. Which is the impact on equity due to the funding of the
collateral (Initial Margin and Variation Margin) in the multiperiodal case?
RiskFree CTP
Bank
Collateralized
CTP
Initial Margin
Collateral
account
15
MVA โ Uniperiodal case
In this case we can treat the initial margin as a cash account whose exposure
varies (stochastically) through time.
-1.000.000
-
1.000.000
2.000.000
3.000.000
0 1 2 3 4 5
- we assume that the fraction of cash
coming back from the IM account is
used to buy back the bankโs
obbligations
- The maturity of the whole bank debt
equal to the derivativeโs one
- The IM is uncorrelated with the total
bank assets (๐ผ๐(๐ก) โช ๐ด(๐ก))
๐๐๐ด๐ข๐๐ โ โ๐ผ๐ก ๐ ๐ด๐>๐ฟ๐๐ก ๐ผ๐ ๐ก๐ ๐ ๐ก(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐
๐
IM(t) โ Expected Initial Margin
16
MVA โ Uniperiodal case
In this case we can treat the initial margin as a cash account whose exposure
varies (stochastically) through time.
-1.000.000
-
1.000.000
2.000.000
3.000.000
0 1 2 3 4 5
- we assume that the fraction of cash
coming back from the IM account is
used to buy back the bankโs
obbligations
- The maturity of the whole bank debt
equal to the derivativeโs one
- The IM is uncorrelated with the total
bank assets (๐ผ๐(๐ก) โช ๐ด(๐ก))
๐๐๐ด๐ข๐๐ โ โ๐ผ๐ก ๐ ๐ด๐>๐ฟ๐๐ก ๐ผ๐ ๐ก๐ ๐ ๐ก(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐
๐
IM(t) โ Expected Initial Margin
Spread never
resets
17
MVA โ Multiperiodal case
๐๐๐ด๐๐ข๐๐ก โ โ๐ผ๐ก ๐(๐ด ๐1 > ๐ฟ1 ๐ผ๐ ๐ก๐ ๐ ๐ก(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐1
๐
+
โ ๐ผ๐ก ๐(๐ด ๐๐ > ๐ฟ๐) (๐ผ๐ ๐ก๐ โ ๐ผ๐ ๐๐โ๐ )๐ ๐๐โ1(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐๐
๐=1:๐ก1โก๐๐โ1
๐:๐๐โก๐
๐=2
-1.000.000
-
1.000.000
2.000.000
3.000.000
0 1 2 3 4 5
Spread resets at each
refinancing date
Term similar to uniperiodal, but up to ๐1 ๐๐ ๐๐๐
๐๐๐ ๐๐๐
๐๐๐
18
MVA โ Multiperiodal case
-1.000.000
-
1.000.000
2.000.000
3.000.000
0 1 2 3 4 5
๐ด๐ฝ๐จ๐๐๐๐๐ < ๐ด๐ฝ๐จ๐๐๐
๐๐ ๐๐๐ ๐๐๐
๐๐๐ ๐๐๐
๐๐๐ด๐๐ข๐๐ก โ โ๐ผ๐ก ๐(๐ด ๐1 > ๐ฟ1 ๐ผ๐ ๐ก๐ ๐ ๐ก(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐1
๐
+
โ ๐ผ๐ก ๐(๐ด ๐๐ > ๐ฟ๐) (๐ผ๐ ๐ก๐ โ ๐ผ๐ ๐๐โ๐ )๐ ๐๐โ1(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐๐
๐=1:๐ก1โก๐๐โ1
๐:๐๐โก๐
๐=2
19
FVA for Collateral
๐น๐๐ด๐๐ข๐๐ก๐ โ โ ๐ผ๐ก ๐(๐ด ๐๐ > ๐ฟ๐) (๐ธ๐ธ ๐ก๐ โ ๐ธ๐ธ ๐๐โ1 )๐ ๐๐โ1(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐๐
๐=1:๐ก1โก๐๐โ1
๐:๐๐โก๐
๐=1
Collateral
account
As for MVA, under the same assumptions, we treat the future exposure on the collateral account as non stochastic and take instead the expected exposure.
๐น๐๐ด๐ข๐๐ โ โ๐ผ๐ก ๐ ๐ด๐>๐ฟ๐๐ก ๐ธ๐ธ ๐ก๐ ๐ ๐ก(๐ก๐ โ ๐ก๐โ1)
๐:๐ก๐โก๐
๐
๐ญ๐ฝ๐จ๐๐๐๐๐ < ๐ญ๐ฝ๐จ๐๐๐
(*)
(*) These are proxy formulas valid in the case of a derivative traded with payment in upfront.
20
KVA - Regulatory obligations
Regulator requires that the balancesheet of any banks be respectful of predetermined leverage ratios. Those constraints have an impact on the Equity dynamics over time, on the ROE of a bank, hence on the funding spread a bank can negotiate at the end of each funding period.
What is the impact of the regulatory obbligations on the ALM strategy of the
bank? How does this affect the equity value (KVA)?
For the sake of simplicity, let the regulatory constraint be defined as
๐ธ๐๐ข๐๐ก๐ฆ
๐ค๐๐ด๐ ๐ ๐๐ก๐๐
โฅ ๐ฅ%
where x% is the regulatory ratio.
21
A case for FVA/KVA
In our model we assume:
โข regulatory capital is the equity value given by the structural model
โข bank operates on the regulatory threshold
โข new capital will be invested proportionally into existing assets
โข creditors have perfect knoweldge of the bankโs balance sheet and
the dynamics due to the regulatory obligations (i.e. capital raising)
22
A case for FVA/KVA
This leads to the following equations problem
๐ธ(๐ก)
๐ค๐ด(๐ก)=
๐ธ(t+)
๐ค 1 + ๐ผ ๐ด(t+) + ๐ค1๐ด1(๐ก+)= ๐ฅ%
๐ด1 = ฮ๐ฟ = ๐ผ๐ก+ ๐ฅ๐ฟ๐๐ก+๐ ๐๐๐กโ๐๐๐๐๐ข๐๐ก๐๐ +๐ฅ๐ฟ
๐ฟ+๐ฅ๐ฟ1 + ๐ผ ๐ด ๐ + ๐ด1 ๐ ๐๐๐๐๐ข๐๐ก๐๐
๐ผ๐ก+ max (๐ด1 + 1 + ๐ผ ๐ด ๐ โ ๐ฟ๐๐ก โ ฮ๐ฟ๐๐ก+ , 0)
23
A case for FVA/KVA
This leads to the following equations problem
๐ธ(๐ก)
๐ค๐ด(๐ก)=
๐ธ(t+)
๐ค 1 + ๐ผ ๐ด(t+) + ๐ค1๐ด1(๐ก+)= ๐ฅ%
๐ด1 = ฮ๐ฟ = ๐ผ๐ก+ ๐ฅ๐ฟ๐๐ก+๐ ๐๐๐กโ๐๐๐๐๐ข๐๐ก๐๐ +๐ฅ๐ฟ
๐ฟ+๐ฅ๐ฟ1 + ๐ผ ๐ด ๐ + ๐ด1 ๐ ๐๐๐๐๐ข๐๐ก๐๐
- ๐ผ๐ด ๐ก+ is the amount of cash raised in the capital increase and reinvested in the existing asset
- ๐ ๐ก+ in ๐๐ก = 1 + ๐๐ ๐ก+ is the fair spread on the debt issued to purchase the new risky asset.
- ๐ ๐ก+, ๐ผ are the unknown variables which can be found by means of a root find numerical algorithm.
๐ผ๐ก+ max (๐ด1 + 1 + ๐ผ ๐ด ๐ โ ๐ฟ๐๐ก โ ฮ๐ฟ๐๐ก+ , 0)
24
FVA and KVA are tightly bounded and represents two sides of the same
coinโฆ
A case for FVA/KVA
The impact on shareholders who were long equity at t is FVA&KVA
FVA&KVA = ๐ธ ๐ก+ โ ๐ธ ๐ก + ๐ผ๐ด ๐ก+
๐ธ ๐ก+ = ๐ผ๐ก+ max ( 1 + ๐ผ ๐ด ๐ โ ๐พ, 0) โ ๐ธ๐ก + ฮ๐ต๐ โ ๐ผ๐ด(๐ก+)
KVA = ๐ธ ๐ก+ โ ๐ธ ๐ก + ๐ผ๐ด ๐ก+ โ โ(1 โ ฮ๐ต๐) โ ๐ผ๐ด(๐ก+)
To better understand the following numerical results it can be noted that
a capital increase has always a negative impact on existing shareholders
In fact
HINT
25
A case for FVA/KVA โ Numerical results
๐๐๐ก๐๐ = 10% ๐ด ๐ก = 100
๐๐ด = 20% ๐ฟ = 90
๐ ๐ก = 6.60%
ฮ๐ฟ = ๐ด1 ๐ก+ = 10
๐ค = 1
๐ค1 = 0.4
26
A case for FVA/KVA โ Numerical results
๐๐๐ก๐๐ = 10% ๐ด ๐ก = 100
๐๐ด = 20%
๐ฟ = 90
๐ ๐ก = 6.60%
๐ค = 1
๐ = ๐. ๐
๐1 = 30%
27
A case for FVA/KVA โ Numerical results
๐๐๐ก๐๐ = 10% ๐ด ๐ก = 100
๐๐ด = 20%
๐ฟ = 90
๐ ๐ก = 6.60%
๐ค = 1
๐ = โ๐. ๐
๐1 = 30%
28
A case for FVA/KVA โ Numerical results
๐๐๐ก๐๐ = 10% ๐ด ๐ก = 100
๐๐ด = 20%
๐ฟ = 90
๐ ๐ก = 6.60%
ฮ๐ฟ = ๐ด1 ๐ก+ = 10
๐ค = 1
๐ = 0.5
29
A case for FVA/KVA โ Numerical results
๐๐๐ก๐๐ = 10% ๐ด ๐ก = 100
๐๐ด = 20%
๐ฟ = 90
๐ ๐ก = 6.60%
ฮ๐ฟ = ๐ด1 ๐ก+ = 10
๐ค = 1
๐1 = 30%
30
Conclusions
โข We showed that the FVA and MVA impact on Equity depends on the rolling frequency of the debt and the ability of the market to price properly the funding spread at the time the debt is rolled.
โข Once regulatory constraints are introduced it not possible to separate KVA and FVA components easily.
โข The model defines an ALM Strategy: reduce the duration of liabilities in periods of distressed conditions and increase the duration of liabilities in period of flourishing conditions.
โข The model defines the Pricing Policy: even if assets fair values do not depend on bankโs funding cost, a pricing policy should also take into account of potential losses on equity value due to funding level.
โข The model defines a Transfer Price Policy: fund any business unit accordingly to the marginal contribution to the total risk of the Assets in the balance-sheet.
Questions?
Tommaso Gabbriellini Andrea Gigli Head of Quants Head of Fixed Income and XVAs
MPS Capital Services MPS Capital Services