13th Fixed Income Conference Florence Presentation

13th Fixed Income Conference
Florence, Italy, 19th - 20th October 2017
Understanding balance-sheet dynamics
impact on FVA, MVA, KVA
Andrea Gigli Tommaso Gabbriellini
Head of Interest Rate, Inflation & XVA Head of Quants
MPS Capital Services MPS Capital Services
andrea.gigli@mpscs.it tommaso.gabbriellini@mpscs.it

____________________________________________________________________________________________________________________
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
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and/or employees thereof (jointly, “MPSCS") is responsible for any errors or omissions, or for the results obtained from the use of this
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results obtained from the use of this information, and without warranty of any kind, express or implied, including, but not limited to
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3Andrea Gigli, 13th WBS Fixed Income Conference, Florence, 19th - 20th October 2017
Goal of the talk
• To investigate the rationale behind FVA, MVA and KVA using a
structured model under a multi-period setting.
• To understand the relationships between valuation adjustments,
model parameters and encoded regulatory constraints using Monte
Carlo simulation method.
• To show
• How to allocate capital on different business units
• How to manage funding strategies
• How to price banking products
after having specified the bank’s utility function, funding policy and
regulatory constraints.

FVA, MVA, KVA
• MVA & FVA measure the impact on the bank’s Equity value of the IM
and VM obligations after entering derivatives contract, due to the
choice of using debt to finance those obligations.

FVA, MVA, KVA
• Regulatory requirements impose that the leverage of the balance
sheet remains below a predefined threshold, requiring additional
equity if not. KVA measures the impact on the Equity due to

FVA, MVA, KVA
• Regulatory requirements impose that the leverage of the balance
sheet remains below a predefined threshold, requiring additional
equity if not. KVA measures the impact on the Equity due to
• In order to evaluate the cost-opportunity of entering a derivative
contract and to compensate shareholders for negative variations in
the equity value, a charge equal to MVA, FVA, and KVA might be
applied on derivatives pricing.

The Model – One Period case
Assume:
- the probability meausure is the risk neutral one,
- the risk free rate is zero,
- the bank will default if , where
- is the total asset value in T
- is the amount of debt and interests to be paid in T
- 1 , where is the funding spread the market set in t.

Assume:
The value of the Equity in is
, 0
The value of the Liabilities in is
, , 0
Equity
Assets
Debt
Assets

Assume:
The value of the Equity in is
, 0
The value of the Liabilities in is
, , 0
Equity
Assets
Debt
Assets
It is the Asset side of the
balance-sheet which
determines the «stochasticity»
of the Equity value

Given that the credit risk determinants are defined by the dynamics
of the Assets mix at time t, one can derive the credit spread from
assuming that the creditors will apply the minimum spread sufficient
to remunerate risk.
, 0
Using the put-call parity it can be shown that the Equity value in t is
Discounted value in t of the Liabilities amount in T

The Model – Uniperiod case – Risk-free Asset
How does a new investment impact the Equity value of a bank?
Let C be the value of a risk free asset. Assume (1) the bank issues new
debt in for funding the risk free asset purchase and (2) the maturity of
the new asset is the same of the debt. The Equity value will change to
max ∆ "
, 0

Fair spread in
Assets Liabilities
max ∆ "
, 0

Fair spread in
Assets Liabilities
max ∆ "
, 0
1
Assets Liabilities
∆
Equity
# $
∆ ", 0%&
Assets Liabilities
A
∆
Equity
$
∆ "
, 0%

1
If ≫ C ∆ → *
then the variation in the equity value is
max ∆ , 0 max , 0
* ∙ ∙ , -./0121
This is the amount of money shareholders
requires in order to invest borrowed
money in a risk free asset
Assets Liabilities
∆
Equity
# $
∆ "
, 0%&
Assets Liabilities
A
∆
Equity
$
∆ "
, 0%

The Model – Uniperiod case – Risky Asset
What if the new asset is not a risk free one?

The Model – Uniperiod case – Risky Asset
What if the new asset is not a risk free one?
The Equity valuation adjustment will depend on the volatility of the new
asset and the correlation between the original asset portfolio and the
new asset.

The Model – Multiperiod case
In a multi-period setting we assume that the bank rolls its debt repeatedly
2 3
> > ?>
1

In a multi-period setting we assume that the bank rolls its debt repeatedly
For the sake of simplicity, we analyze the case where the bank rolls its
debt just once
2 3
> > ?>
2
>
1

2
>
We evaluate the equity by
means of the «tower propery»
> @>> @> |@
The value of @> depends on the moneyness of the option on the
Asset portfolio

2
>
We evaluate the equity by
means of the «tower propery»
> @>> @> |@
The value of @> depends on the moneyness of the option on the
Asset portfolio
B
The bank finance debt and
accrued interests at the new
fair spread.
> C
No one will be willing to lend
money to the bank and equity
fall to zero
> D 0
@> max A , 0 → max A , 0 @

How is the Equity value affected by the financing strategy of the bank?

Let’s consider the purchase at time of a risk free asset whose
maturity is 2 (greater than the bond maturity ).
In our setting, the equity can be computed as if the maturity of the
purchased asset is the same as of the debt
"
max ∆ "
, 0

Let’s consider the purchase at time of a risk free asset whose
maturity is 2 (greater than the bond maturity ).
In our setting, the equity can be computed as if the maturity of the
purchased asset is the same as of the debt
"
max ∆ "
, 0
;< * ∙ ∙ , -E/021
The FVA is proportional to the financing «period», not
to the maturity of the asset, i.e. the following still
holds!
;< * ∙ ∙ , -E/021

Which is the impact on the Equity value due to the funding of the
collateral (both IM and VM) in the multiperiodal case?
Suppose the bank enters in a derivative with a collateralized counterpary
and hedge it with a risk free counterparty.
An application for FVA/MVA

An application for FVA/MVA
RiskFree CTPRiskFree CTP
BankBank
Collateralized
CTP
Collateralized
CTP
Initial MarginInitial Margin
Collateral
account
Collateral
account
Which is the impact on the Equity value due to the funding of the
collateral (both IM and VM) in the multiperiodal case?
Suppose the bank enters in a derivative with a collateralized counterpary
and hedge it with a risk free counterparty.

KVA - Regulatory obligations
What is the impact of the regulatory obbligations on Equity values changes in
the previous cases (KVA)?
Regulator requires that the balance-sheet 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.

KVA - Regulatory obligations
What is the impact of the regulatory obbligations on Equity values changes in
the previous cases (KVA)?
Regulator requires that the balance-sheet 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.
For the sake of simplicity, let the regulatory constraint be defined as
`a b
∑ dM e MM
B %
where % is a (simplified) regulatory ratio.

A case for FVA/KVA
In our case we assume:
• regulatory capital is the equity value given by the structural
model,
• any capital raise will be invested into the same assets portfolio
observed at the time of capital raising,
• the minimum amount of equity will be raised in order to respect
the regulatory ratio: fghijk
∑ limnnojii
B p%
fghijk
∑ limnnojii
p%
• creditors have perfect knoweldge of the bank’s balasheet and the
capital dynamics due to the regulatory constraints.

A case for FVA/KVA
d
t
d 1 r t d: :
%
Δ s ∙ ∙ , It NuvwxUV vu
y0
0 y0
1 r : , uvwxUV vu
To measure the impact of adding a new asset to the balance-sheet, the
previous hypotheses require to solve the following equations problem

A case for FVA/KVA
To measure the impact of adding a new asset to the balance-sheet, the
previous hypotheses require to solve the following equations problem
d
t
d 1 r t d: :
%
Δ s ∙ ∙ , It NuvwxUV vu
y0
0 y0
1 r : , uvwxUV vu
• t max : 1 r Δ "
, 0
• r is the amount of cash at bank’s disposal after the capital
increase, which is reinvested in the existing asset mix
• "
in 1 "
is the fair spread on the debt issued to purchase
the new risky asset.
• "
, r are the unknown variables which can be found by means of a
root find numerical algorithm at each step of the Monte Carlo.

where FVA and KVA are tightly bounded.
A case for FVA/KVA
The impact on the Equity value at time t is
FVA&KVA rFVA&KVA r
~ max 1 r •, 0 ≃ Δ• ⋅ r
KVA ~ r
≃ Δ• ⋅ r
As an example, a capital increase to invest in the
same asset portfolio has the following impact on
existing shareholders

A case for FVA/KVA – Numerical results
ƒ „ 10%
100
3- 0.2
90
6.60%
Δ : 10
d 1
d: 0.4
;< & •<

ƒ „ 10%
100
3- 20%
90
6.60%
d 1
† 0.5
3: 30%
;< & •<

ƒ „ 10%
100
3- 20%
90
6.60%
Δ : 10
d 1
† 0.5
;< & •<

ƒ „ 10%
100
3- 20%
90
6.60%
Δ : 10
d 1
3: 30%
;< & •<

Conclusions
• We showed that the impact of FVA and MVA on the Equity value
depends on the rolling frequency of the debt.

Conclusions
• Once regulatory constraints are introduced it is not possible to
separate KVA and FVA components easily but you can see the
interaction of their effects through MC simulation.

Conclusions
• The way both regulatory constraints and bank’s utility function are
incorporated into the model determines the optimal ALM, pricing and
transfer price policies.

Conclusions
transfer price policies. In our simple case,
• the optimal ALM strategy requires to reduce the duration of banks
liabilities in periods of distressed conditions and increase the duration of
liabilities in period of flourishing conditions;

Conclusions
• the optimal Pricing policy requires to price assets independently from
the bank’s funding cost, although funding is a function of the whole asset
portfolio risk;

Conclusions
• the optimal Pricing policy requires to price assets independently from
the bank’s funding cost, although funding is a function of the whole asset
portfolio risk;
• the optimal Transfer Price Policy requires the bank to fund any business
unit accordingly to the marginal contribution to the whole asset portfolio
risk.

Thanks
Questions?
Andrea Gigli Tommaso Gabbriellini
Head of Interest Rate, Inflation & XVA Head of Quants
MPS Capital Services MPS Capital Services
andrea.gigli@mpscs.it tommaso.gabbriellini@mpscs.it

13th Fixed Income Conference Florence Presentation

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