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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, βMPSCS") 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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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
β’ Using a multiperiodal structured model we are going to investigate
the rationale behind FVA, MVA and KVA
β’ The model represents a useful tool to understand the relations
between valuation adjustments, market parameters and regulatory
constraints
β’ Three main lessons can be learned from the model
β’ How to allocate capital to different business units
β’ How to manage funding strategies
β’ Hot to price banking products
4
FVA, MVA, KVA
β’ MVA & FVA measure the impact on Equity due to IM and VM bankβs
obligations after entering derivatives contract, using debt to finance
those obligations.
β’ 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
β’ 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 β Uniperiodal case
Assume:
- the risk meausure is the risk neutral one
- the bank will default if π΄(π) < πΏππ‘, where
- πΏππ‘is the amount of debt and interests to be paid and
- ππ‘ = 1 + ππ π‘,
- π π‘ is the funding spread set in t
- the risk free rate is zero.
The value of Equity in π‘ is
πΈπ‘ = πΌπ‘ πππ₯ π΄(π) β πΏππ‘ , 0
The value of the Liabilities in π‘ is
πΌπ‘ πππ π΄(π), πΏππ‘ = πΏππ‘ β πΌπ‘ πππ₯ πΏππ‘ β π΄(π), 0
6
The Model β Uniperiodal case
The spread π π‘ is set by the
creditor such that πΏ β€ πΏππ‘ β πΌπ‘ πππ₯ πΏππ‘ β π΄(π), 0
the spread must be sufficient to remunerate the
risks
In the following we will assume that the creditor is always Β«fairΒ», i.e
the minimum spread is applied:
πΏ = πΏππ‘ β πΌπ‘ πππ₯ πΏππ‘ β π΄(π), 0
N.B.
if π π‘ is fair πΈ(π‘) = π΄ π‘ β πΏ
Proof: πΈ(π‘) = π΄ π‘ β πΏππ‘ + πΌπ‘ πππ₯ πΏππ‘ β π΄(π), 0 = π΄ π‘ β πΏ
Put-Call Parity
7
The Model β Uniperiodal case
What is the impact of a new investment on the equity value of the bank?
Assume at π‘+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 π‘+
Assets Liabilities
πΈ(π‘+) = πΌπ‘+ max π΄(π) + πΆ β πΏππ‘ β βπΏππ‘+ , 0
πΆ = βπΏ = πΌπ‘ βπΏππ‘+π π΄ π +πΆ>πΏππ‘+βπΏππ‘++ π΄(π)
βπΏ
πΏ + βπΏπ π΄(π)+πΆ<πΏππ‘+βπΏππ‘+
In case of default the assets will be used for a partial
reimburse proportionally to the face value of the liabilities
C doesnβt depend
upon t
8
The Model β Uniperiodal case
π = π = 1
Note that:
β’ βπΏ = πΆ
β’ If π΄π‘ β« C β ππ‘+ β ππ‘
Hence, the variation in the equity value is
πΌπ‘ max π΄(π) + πΆ(π) β πΏππ‘ β βπΏππ‘+, 0 β πΌπ‘ max π΄(π) β πΏππ‘ , 0 =
β βπΆ β ππ π‘ β πΌπ‘ π π΄π>πΏπ‘ππ‘
This is the amount of money
shareholders requires in order to invest
borrowed money in a risk free asset
π‘+
Assets Liabilities
π΄(π‘+)+ πΆ(π‘+)
πΏ π‘+ + βπΏ
Equity
πΌπ‘ πππ₯ π΄(π) + πΆ β πΏππ‘
β βπΏππ‘+ , 0
Assets Liabilities
π΄π‘(π)+ πΆ
πΏππ‘ + βπΏππ‘+
Equity
πππ₯ π΄ π + πΆ β πΏππ‘
β βπΏππ‘+ , 0
9
The Model β Uniperiodal case
What if the asset is not risk free? There may be as well negative impacts
(Β«funding costsΒ») and positive ones (Β«funding benefitsΒ»), depending on
the volatility and correlation with the previous assets and its risk.
π΄ π‘ = 100
ππ΄ = 20% πΏ = 90
π π‘ = 6.60%
ΞπΏ = π΄1(π‘+) = 10
10
The Model β 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
π π‘ 2π 3π
πΏ πΏππ‘ πΏππ‘ππ πΏππ‘πππ2π
π π‘ 2π
πΏ πΏππ‘ πΏππ‘ππ
11
The Model β Multiperiodal case
π π‘ 2π
πΏ πΏππ‘ πΏππ‘ππ
We evaluate the equity by
means of the Β«tower properyΒ»
πΌ πΈ2π β±π πΈ(π‘) = πΌ πΌ πΈ2π β±π |β±π‘
Letβs look at the value of πΌ πΈ2π β±π in the following 2 cases
π΄ π β₯ πΏππ‘ π΄ π < πΏππ‘
The bank finance the debt +
interest at a new fair spread.
πΌ πΈ2π π΄ π > πΏππ‘ = π΄ π β πΏππ‘
The bank try to finance the debt
+ interest at a new fair spread,
but no one is willing to lend
moneyβ¦
πΌ πΈ2π π΄ π β€ πΏππ‘ = 0
Proof in the following slide
12
The Model β Multiperiodal case
Why if π΄ π < πΏππ‘ no one is willing to lend money?
Letβs have a look at the fair value of the debt in the limit of an
infinite spread
limπ πββ
πΌπ min π΄(2π), πΏππ‘ππ = πΌπ π΄(2π) = π΄ π < πΏππ‘
The maximum fair value of the debt is always
lower than the amount to be financed!
πΌ πΈ2π β±π = max(A π β πΏππ‘ , 0) Combining the two cases we have that
Therefore the equity can be priced as
πΈ π‘ = πΌ max(A π β πΏππ‘, 0) β±π‘
Exactly the same as in the uniperiodal setting
13
The Model β Multiperiodal case
How is the FVA affected by the financing strategy of the bank?
Letβs consider the purchase at time π‘+ of a risk free asset (cash) whose
maturity is greater than π (the bond maturity), say 2π
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
πΈ(π‘+) = πΌπ‘+ max π΄(π) + πΆ β πΏππ‘ β βπΏππ‘+ , 0
The FVA is proportional to the financing Β«periodΒ», not
to the maturity of the asset, i.e. the following still
holds!
πΉππ΄ β βπΆ β ππ π‘ β πΌπ‘ π π΄π>πΏππ‘
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 β 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 bankβs
obbligations
- The maturity of the whole bank debt
equal to the derivativeβs one
- The IM is uncorrelated with the total
bank assets (πΌπ(π‘) βͺ π΄(π‘))
πππ΄π’ππ β βπΌπ‘ π π΄π>πΏππ‘ πΌπ π‘π π π‘(π‘π β π‘πβ1)
π:π‘πβ‘π
π
IM(t) β Expected Initial Margin
16
MVA β 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 bankβs
obbligations
- The maturity of the whole bank debt
equal to the derivativeβs one
- The IM is uncorrelated with the total
bank assets (πΌπ(π‘) βͺ π΄(π‘))
πππ΄π’ππ β βπΌπ‘ π π΄π>πΏππ‘ πΌπ π‘π π π‘(π‘π β π‘πβ1)
π:π‘πβ‘π
π
IM(t) β Expected Initial Margin
Spread never
resets
17
MVA β Multiperiodal case
πππ΄ππ’ππ‘ β βπΌπ‘ π(π΄ π1 > πΏ1 πΌπ π‘π π π‘(π‘π β π‘πβ1)
π:π‘πβ‘π1
π
+
β πΌπ‘ π(π΄ ππ > πΏπ) (πΌπ π‘π β πΌπ ππβπ )π ππβ1(π‘π β π‘πβ1)
π:π‘πβ‘ππ
π=1:π‘1β‘ππβ1
π:ππβ‘π
π=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 π1 ππ πππ
πππ πππ
πππ
18
MVA β Multiperiodal case
-1.000.000
-
1.000.000
2.000.000
3.000.000
0 1 2 3 4 5
π΄π½π¨πππππ < π΄π½π¨πππ
ππ πππ πππ
πππ πππ
πππ΄ππ’ππ‘ β βπΌπ‘ π(π΄ π1 > πΏ1 πΌπ π‘π π π‘(π‘π β π‘πβ1)
π:π‘πβ‘π1
π
+
β πΌπ‘ π(π΄ ππ > πΏπ) (πΌπ π‘π β πΌπ ππβπ )π ππβ1(π‘π β π‘πβ1)
π:π‘πβ‘ππ
π=1:π‘1β‘ππβ1
π:ππβ‘π
π=2
19
FVA for Collateral
πΉππ΄ππ’ππ‘π β β πΌπ‘ π(π΄ ππ > πΏπ) (πΈπΈ π‘π β πΈπΈ ππβ1 )π ππβ1(π‘π β π‘πβ1)
π:π‘πβ‘ππ
π=1:π‘1β‘ππβ1
π:ππβ‘π
π=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.
πΉππ΄π’ππ β βπΌπ‘ π π΄π>πΏππ‘ πΈπΈ π‘π π π‘(π‘π β π‘πβ1)
π:π‘πβ‘π
π
ππ½π¨πππππ < ππ½π¨πππ
(*)
(*) 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
πΈππ’ππ‘π¦
π€ππ΄π π ππ‘ππ
β₯ π₯%
where x% is the regulatory ratio.
21
A case for FVA/KVA
In our model we assume:
β’ regulatory capital is the equity value given by the structural model
β’ bank operates on the regulatory threshold
β’ new capital will be invested proportionally into existing assets
β’ creditors have perfect knoweldge of the bankβs 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
πΈ(π‘)
π€π΄(π‘)=
πΈ(t+)
π€ 1 + πΌ π΄(t+) + π€1π΄1(π‘+)= π₯%
π΄1 = ΞπΏ = πΌπ‘+ π₯πΏππ‘+π πππ‘βπππππ’ππ‘ππ +π₯πΏ
πΏ+π₯πΏ1 + πΌ π΄ π + π΄1 π πππππ’ππ‘ππ
πΌπ‘+ max (π΄1 + 1 + πΌ π΄ π β πΏππ‘ β ΞπΏππ‘+ , 0)
23
A case for FVA/KVA
This leads to the following equations problem
πΈ(π‘)
π€π΄(π‘)=
πΈ(t+)
π€ 1 + πΌ π΄(t+) + π€1π΄1(π‘+)= π₯%
π΄1 = ΞπΏ = πΌπ‘+ π₯πΏππ‘+π πππ‘βπππππ’ππ‘ππ +π₯πΏ
πΏ+π₯πΏ1 + πΌ π΄ π + π΄1 π πππππ’ππ‘ππ
- πΌπ΄ π‘+ is the amount of cash raised in the capital increase and reinvested in the existing asset
- π π‘+ in ππ‘ = 1 + ππ π‘+ is the fair spread on the debt issued to purchase the new risky asset.
- π π‘+, πΌ are the unknown variables which can be found by means of a root find numerical algorithm.
πΌπ‘+ max (π΄1 + 1 + πΌ π΄ π β πΏππ‘ β ΞπΏππ‘+ , 0)
24
FVA and KVA are tightly bounded and represents two sides of the same
coinβ¦
A case for FVA/KVA
The impact on shareholders who were long equity at t is FVA&KVA
FVA&KVA = πΈ π‘+ β πΈ π‘ + πΌπ΄ π‘+
πΈ π‘+ = πΌπ‘+ max ( 1 + πΌ π΄ π β πΎ, 0) β πΈπ‘ + Ξπ΅π β πΌπ΄(π‘+)
KVA = πΈ π‘+ β πΈ π‘ + πΌπ΄ π‘+ β β(1 β Ξπ΅π) β πΌπ΄(π‘+)
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 β Numerical results
πππ‘ππ = 10% π΄ π‘ = 100
ππ΄ = 20% πΏ = 90
π π‘ = 6.60%
ΞπΏ = π΄1 π‘+ = 10
π€ = 1
π€1 = 0.4
26
A case for FVA/KVA β Numerical results
πππ‘ππ = 10% π΄ π‘ = 100
ππ΄ = 20%
πΏ = 90
π π‘ = 6.60%
π€ = 1
π = π. π
π1 = 30%
27
A case for FVA/KVA β Numerical results
πππ‘ππ = 10% π΄ π‘ = 100
ππ΄ = 20%
πΏ = 90
π π‘ = 6.60%
π€ = 1
π = βπ. π
π1 = 30%
28
A case for FVA/KVA β Numerical results
πππ‘ππ = 10% π΄ π‘ = 100
ππ΄ = 20%
πΏ = 90
π π‘ = 6.60%
ΞπΏ = π΄1 π‘+ = 10
π€ = 1
π = 0.5
29
A case for FVA/KVA β Numerical results
πππ‘ππ = 10% π΄ π‘ = 100
ππ΄ = 20%
πΏ = 90
π π‘ = 6.60%
ΞπΏ = π΄1 π‘+ = 10
π€ = 1
π1 = 30%
30
Conclusions
β’ 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.
β’ Once regulatory constraints are introduced it not possible to separate KVA and FVA components easily.
β’ 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.
β’ The model defines the Pricing Policy: even if assets fair values do not depend on bankβs funding cost, a pricing policy should also take into account of potential losses on equity value due to funding level.
β’ 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