Fair Value and FVA – A Quant’s Perspective
Shahram Alavian*†
Royal Bank of Scotland
Shahram.Alavian@rbs.com
7th Annual Credit Risk in Banking Summit, Berlin
February 6, 2014
*The opinions expressed are those of the author’s and do not necessarily reflect the views of the author's employer, or any
member of its staff. †The author is grateful to Quantitative Analytics Group’s support at RBS. All errors are author’s.
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2
Motivation
Debt Value Adjustment (DVA) has been around for some time now
“The
twelve months ended December 31, 2013 and December 31, 2012 include
positive (negative) revenue of $(681) million and $(4,402) million, respectively,
related to the movement in DVA.“ , [Notes, Morgan Stanley, Q4 (2013)].
On the other hand, the fair value guidelines from financial accounting standards for
funding cost are still unclear if not absent. Nevertheless, Funding cost is starting to be
included in fair value adjustments
“For the first time this quarter, we were able to clearly observe the existence of
funding costs in market clearing levels … As a result, the Firm recorded a $1.5B loss
this quarter … FVA – which represents a spread over LIBOR – has the effect of
“present valuing” market funding costs into the value of derivatives today, rather
than accruing the cost over the life of the derivatives..”, [JPMorgan Q4 (2013)] .
Question
Under fair value accounting, what would be the relationship between FVA
and DVA? and how would it impact the P&L in the trading book?
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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In a Nutshell
Statement
Under fair value accounting of DVA and FVA, a trading book’s funding cost offsets
the DVA [due to changes in its bond prices]. Bond price movements should no longer
impact the trading book’s P&L.
Approach
Define “Fair Value Adjustment” as an access over the default free price
Define a simple economy of a firm, its counterparty and a liquid funding market
Assign two total adjustments to the firm; one facing the market, and another, its
counterparty. Each adjustment is a net of both Credit and Funding adjustments.
Show that in the market facing adjustment, the funding cost offsets the market
facing DVA. This would be our main result.
Main Assumption
A firm’s fair value adjustment should obey the conservation of value. This means
that its cost should be its counterpart’s benefit and vice versa. If not, the firm can
create fictitious loss or profit.
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Formulating “Fair Value Adjustment”
1. Fair valuation guidelines (such as FAS157, IFRS13) aim to bring balance sheet transparency and
standardization to the investors and the firm’s management. Fair value adjustment’s objective is
to include the impact of the credit (default risk) on the price of the asset (at the point of exit
from the market)
2. Define “Fair Value Adjustment” as the difference between the fair value and the default free
price.
∆
≡
V (t ) − V (t )
(1)
Adjustment
Fair Value
Default Free Price
4. [Assumption] Under fair value accounting, a firm should follow the “Conservation of Value”:
One Counterparty’s benefit(cost) is its counterparty’s cost(benefit).
Adjustment = Cost + Benefit
( 2)
Credit = CC + CB *
Funding [FVA] = FC + FB
(3)
( 4)
CC = Credit Cost [CVA]
CB = Credit Benefit [DVA]
FC = Funding Cost
FB = Funding Benefit
*To remain consistent with the rest of the presentation we continue to use CB and CC to refer to DVA and CVA,
respectively.
Remark: For conservation of value to hold, an adjustment should “face” an entity that holds
the offsetting position, only. If not, fictitious loss or profit is created.
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Funding Value Adjustment (FVA)
1. We define FVA as the credit adjustment to the value of a default risk free cash account, wholly
allocated to fund positions of a single portfolio.
2. The adjustment can either be positive (funding benefit) or negative (funding cost)
3. The adjustment faces the entity that pays(receives) interest into(from) the cash account, only.
4. Therefore, a counterparty faces a firm’s FVA if it pays or receives interest, not just coupon.
Some Examples of Who a Firm Could Face for its FVA
Counterparty
Firm receives a spread over OIS in a
back-to-back
trade
between
two
collateralized counterparties. It has a
funding benefit facing each counterparty.
Market
Firm receives OIS
from the
counterparty and pays above OIS to the
market: funding cost to the market, only.
Market + Counterparty
Firm receives spread from counterparty
and pays spread to the market. Firm has a
funding benefit facing the counterparty
and funding cost facing the market
OIS+s
Collateralized
Counterparty1
Figure 1
OIS
Firm
OIS
OIS+s
cash
Money Market
cash
Firm
OIS+s
OIS
cash
Money Market
cash
Firm
OIS+s1
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
OIS+s2
Collateralized
Counterparty2
One-way CSA
Counterparty
MTM>0
One-way CSA
Counterparty
MTM>0
6
Defining Market and the Firm
Money Market (“Market”)
1. Definition: By “Market” we mean an entity that borrows or lends money through its
participants. It is a conceptual entity incorporating all its active participants. When
we make a market facing adjustment, we do so facing this entity.
2. In this presentation, we denote ∆M to refer to the total fair value adjustment of a
firm facing the market.
3. Market borrows when a firm invests its access cash and lends when a firm borrows
for its funding needs.
4. [Assumption] Market does not default.
5. Market is transparent to all its participants. It has no adjustments.
Firm’s Trading Book
1. Under a bilateral trade, each counterparty has a financial commitment towards the
other.
2. The variation of these credit hybrid contingent commitments around the default free
ones generates either a benefit (positive) or a cost (negative) adjustment.
3. In this presentation, we denote ∆ to refer to the total fair value adjustment of a firm
facing the counterparty.
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A Simple Economy
1. Our economy is made up of a firm, its
counterparty and the money market.
2. In this economy, the money market is the
source of the funding for both the firm and
its counterparty.
3. At any given time, either the firm or the
counterparty is in need of funding.
4. With no loss of generality, we assume it is
the firm.
Figure 2
∆
Counterparty
Firm
∆M
Money Market
5. The firm faces two sources: The money market and its counterparty. Its total
adjustment (∆Firm) is composed of the market facing adjustment (∆M) and the
counterparty facing adjustment (∆): ∆Firm = ∆ + ∆M ( 2)
7. The money market has no adjustments. Due to the firm’s conservation of value,
∆M = 0. (3)
In a trading book, the fair value adjustments
are those facing the counterparty, only.
∆Firm = ∆
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
( 4)
8
Fair Value Adjustment – Formulation
1. To each adjustment we assign a Credit and a Funding
∆ = Credit + Funding
∆ M = Credit M + Funding M
Figure 3
(5)
( 6)
Counterparty
2. Market has no adjustments ( ∆M = 0 ).
∆ M = CC M + CBM + FC M + FB M = 0
(7)
∆
3. Market never defaults (CCM = 0). Therefore, it only borrows at
risk free rate (FBM = 0). Our final expression is
CBM + FC M = 0
(8)
Firm
Remarks*
The above relationship says that the market facing DVA(CBM ) is always offset
by the market facing funding cost(FCM). Practically speaking, we should still expect
some residual due to the mismatch between the liability and the funding needs’
maturities.
Calculating FCM allows us to carve out CBM from the aggregate liability benefit.
The phrase “market facing” is important as the counterparty facing Funding
Adjustment follows the same logic as the counterparty Credit Adjustment.
∆M
Money Market
*In the Appendix, we will derive the expression for the funding cost.
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Example*: Fair Value This!
A default free counterparty sells a one year, uncollateralized, European
option to the firm (1 year PD = PFirm ) for which it receives the full
premium, $1, up-front. Calculate the firm’s initial adjustments (assume 0
interest rate and loss given default = LGD).
Funding = 0
Counterparty Facing (CVA)
The counterparty does not default
The counterparty has no exposure to the firm
(9)
CC = 0
CB = 0
(11)
∆=0
(12)
(10)
*For illustration purposes only
Premium
CB M = + LGD PFirm
∆ M = CB M + FC M = 0
Spread
The firm borrows for one year to fund the premium
The liability bond is priced below risk free by,
Firm
Bond
Market Facing Total Adjustment
CFM = − LGD PFirm
Premium
The counterparty pays or receives no
interest
Default Free
Counterparty
Option
Counterparty Facing (FVA)
Figure 4
(13)
(14)
Money Market
(15)
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Example
Let’s work out a couple of questions on this example
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Example: Questions
Question
Surely, we should have been able to apply the funding
cost through discounting the position, right?
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Example: Questions
Aggregate entity
Default Free
Counterparty
Premium
Option
Q1: What is wrong with applying the funding cost through
discounting the position? (that old chestnut, again!)?
A1: This essentially means that we are applying the funding cost
facing the uncollaterallized counterparty while we should have
faced the market. To see the consequence of that, consider an
investor who owns both the firm and the counterparty. The
aggregate firm would now end up with no funding cost (due to
offsetting impact of the mtm) and positive credit benefit (CBM >
0), generating fictitious value for the aggregate entity. This would
invalidate the conservation of the value we started with.
Firm
Premium
Spread
Bond
Money Market
Figure 5
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Example: Questions
Question
Could we have generated a funding benefit?
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Example: Questions
Default Free
Counterparty
Premium
Firm
Premium
Spread
Bond
A2 (Different view): Imagine a second identical counterparty
entering in the new reduced-notional trade. According to the
previous slide, we would still have a funding cost and a CB(DVA)
(albeit, reduced). So the change from the full notional to the
reduced notional trade should bring us to the reduced funding
cost and a reduced CB(DVA)-> no funding benefit!
Aggregate entity
Option
Q2: We reduce the firm’s exposure. We surely create a funding
benefit, don’t we?
A2: No. We have only reduced the funding cost. A reduction of
funding cost does not produce a funding benefit in the same way
that a reduction of CC(CVA) does not produce a CB(DVA). The
funding cost will still offsets the reduced market facing CB(DVA)
since the difference in MTM will be used to pay down the debt.
Remark: While “funding benefit” is only a name, one can never initiate
an uncollateralized position to generate a funding benefit without taking
on risk; defeating the spirit of positive adjustment.
Money Market
Figure 5
*For illustration purposes only
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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What We Talked About
We Started With
1. [Scope] for us, fair value adjustments covered only Credit and Funding
Adjustments.
2. Market has no fair value adjustments facing participants (firms).
3. [Assumption] market does not default.
4. [Assumption] In fair value accounting, firms should obey conservation
of value.
We Concluded That
1. Total fair value adjustment facing the market is 0.
2. Under fair value accounting market facing funding cost and DVA (due
to bond price movement) offset each other.
Thank you!
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Appendix – General Expression for Funding Cost
We defined FVA to be the credit adjustment to the present value of the cash account allocated to fund a default
free position. Conditional on the firm surviving to t-δ , risky cash account , CȦ(t), will grow similar to the risk free
account, CA(t), if the firm continues to survive and pays 1-LGD if it defaults (assuming instantaneous payout on
default)
+
−
( A.1)
C A& (t ) = CA (t ) + CA (t ) (1 τ > t + (1 − LGD ) ⋅ 1 τ ≤ t ) ⋅ 1 τ > t − δ
= CA (t ) + CA (t ) (1 − 1 τ ≤ t + (1 − LGD
( A.2)
= CA (t ) + CA (t ) (1 − 1 τ ≤ t + 1 τ ≤ t −
( A.3)
+
−
+
−
)⋅ 1τ ≤t )⋅ 1τ >t −δ
LGD ⋅ 1 τ ≤ t ) ⋅ 1 τ > t − δ
= CA (t ) + CA (t ) (1 − LGD ⋅ 1 τ ≤ t ) ⋅ 1 τ > t − δ
( A.4)
= CA (t ) + CA (t ) ⋅ 1 τ > t − δ − CA (t ) ⋅ LGD ⋅ 1 t ≥ τ > t − δ
( A.5)
+
−
+
−
−
with CA(0) = -V(0). If we define FC (t1 , t2 ) [FC (0, 0) = 0] as the accrued funding cost between t1 and t2 ,
dFC (t,t) would be the instantaneous funding cost defined by
{
}
dFC (t , t ) = CA(t ) − CA& (t ) 1τ ,τ c >t −δ
Expanding the above,
{
( A.6)
}
dFC (t , t ) = CA(t ) − CA& (t ) 1τ ,τ c >t −δ
{
= {CA(t )
( A.7)
}
= CA(t ) + CA(t ) − CA(t ) − CA(t ) ⋅ 1τ >t −δ + CA(t ) ⋅ LGD ⋅ 1t ≥τ >t −δ 1τ ,τ c >t −δ
+
−
−
+
−
−
}
− CA(t ) ⋅ 1τ >t −δ + CA(t ) ⋅ LGD ⋅ 1t ≥τ >t −δ 1τ ,τ c >t −δ
−
−
( A.8)
( A.9)
= CA(t ) ⋅ 1τ ,τ c >t −δ − CA(t ) ⋅ 1τ >t −δ ⋅ 1τ ,τ c >t −δ + CA(t ) ⋅ LGD ⋅ 1t ≥τ >t −δ ⋅ 1τ ,τ c >t −δ
( A.10)
= CA(t ) ⋅ 1τ ,τ c >t −δ − CA(t ) ⋅ 1τ ,τ c >t −δ + CA(t ) ⋅ LGD ⋅ 1t ≥τ >t −δ ⋅ 1τ c >t −δ
( A.11)
= CA(t ) ⋅ LGD ⋅ 1τ c >t −δ ⋅ 1t ≥τ >t −δ
( A.12)
−
−
−
−
−
−
−
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Appendix – General Expression for Funding Cost
The total expected funding cost from t = 0 to t = T (T is the
longest maturity in the portfolio), is given by
FC = B (0, T ) E T [FC (0, T )]
( A.13)
T

= B (0, T ) E  ∫ dFC (t , t )
δ

( A.14)
 dFC (t , t ) 
= ∫ EQ 

 η (0, t ) 
δ
( A.15)
T

η (s, t ) ≡ exp ∫ ru du 
S

( A.16)
T
T
where
Replacing dFC(t, t), with its equivalent from previous
slide gives the general expression for funding cost in
terms of the cash account
(
)
−


CA(t ) ⋅ 1τ c >t −δ
FC = ∫ E  LGD
⋅ 1 t ≥τ >t −δ 
η (0, t )


δ
T
Q
( A.17)
[
(
)
FC = ∫ B (0, t ) E t LGD CA(t ) ⋅ 1τ c >t −δ ⋅ 1t ≥τ >t −δ
−
]
1) Looking at the general expression for the funding cost,
the funding amount to be borrowed is a simple expression
of
−
( A.19)
N (t ,τ c ) ≡ CA(t ) ⋅ 1τ c >t −δ
The indicator function ensures that we do not borrow if
there is no position (due to counterparty’s default) and the
negative sign ensures that we borrow the shortfall only. All
this continues conditional on the firm’s survival (second
indicator function in the integrand.)
2) The expression obtained for the funding cost in this slide
is applicable to all situations, whether the firm faces the
counterparty, through a collateral arrangement, or the
market, since it was based on the cash account. What goes
into the cash account and who pays in (or out of) the
account makes the funding cost to face different entities.
To be complete, a risk free cash account should have three
sources of inflows(outflows): remaining cash that grows by
the risk free rate, the collateral amount , Col(t), which
follows from the collateral arrangement specifications, and
the current cash flow, CF(t),
CA(t ) = e rδ CA(t − δ ) + Col (t ) + CF (t )
( A.20)
In the next slides we will consider a market facing funding
cost of an uncollateralized position by just removing the
collateral component from the above expression.
Changing to the more intuitive t-Forward measure
T
Remarks:
( A.18)
δ
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Appendix – Funding Cost (Uncollateralized Position)
In this section we focus on deriving the funding cost due to an uncollateralized
position. We do this in two ways: an economic and an analytical approach.
+
]
( A.21)
Cash
[
FC = ∫ B (0, t ) E t LGD V (t ) ⋅ 1τ c >t −δ ⋅ 1 t ≥τ >t −δ
V(t)+
UnCollateralized
Counterparty
FC
Firm
Cash
A simple approach to relate the cash account to the mtm of an uncollateralized
position is to consider a case where an uncollaterlized trade is back-to-backed with a
fully collateralized hedging counterparty. When the mtm facing the uncollaterized
name is positive, V(t)+, the firm’s mtm would be negative to the hedging counterparty
who is entitled to the collateral. This cash needs to be funded unsecured from the
market to which the funding spread has to be paid. However, if the mtm facing the
counterparty is negative, V(t)-, the mtm facing the hedging counterparty would be
positive and they need to post the cash which can only be used to invest in a default
free account. Therefore, the funding cost can only be incurred when the mtm facing
the counterparty is positive. [We pay and receive OIS on collateral.]
Cash
Method 1- An Economic Approach
T
Figure 6
Collateralized
Counterparty
Money Market
δ
Method 2- An Analytical Approach
Our starting point for this approach is the definition of the risk free cash account. A risk free cash account that is wholly used to fund
an uncollateralized position starts with CA(0) = - V(0). As time goes by, the cash account from t – δ to t accrues with the risk free
rate. The accrual is applied to the remaining cash at time t – δ and then nets any cash transfer in (or out of) the account for any cash
flows , CF(t), that occurred at time t. Putting it in a more formal way,
CA(t ) = e rδ CA(t − δ ) + CF (t )
( A.22)
Note that there is no CF(0) and r is the short instantaneous rate. The above iterative equation gives
t j ≤t
CA(t ) = CA(0 ) η (0, t ) + ∑ CF (t j )η (t j , t )
( A.23)
t j >0
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Appendix – Funding Cost (Uncollateralized Position)
Selecting the risk neutral measure Q requires the calculation of CA(t) in units of η(0,t) as previously
defined
t ≤t
( )
j
CF t j
CA(t )
= − V (0 ) + ∑
η (0, t )
t j > 0 η (0, t j )
( A.24)
The above quantity is adaptable to the filtration Ft. We now write an identity that will come
handy for later-on.
t j ≤T CF (t j )
 t j ≤t CF (t j )
t j ≤T CF (t j )

Q
E ∑
Ft  = ∑
Ft 
+ E ∑
 t j > 0 η (0, t j )
 t j > 0 η (0, t j )
 t j >t η (0, t j )

Q
( A.25)
With some rearrangements,
t j ≤t
∑
t j >0
CF (t j )
t j ≤T CF (t j ) 
= V (0) − E  ∑
Ft 
η (0, t j )
 t j >t η (0, t j ) 
= V (0 ) − V (t )
Q
( A.26)
( A.27 )
Calculating the conditional expectation of the uncollatralized cash account gives
 t j ≤t CF (t j )

 CA(t )

Q
E 
Ft  = − V (0 ) + E  ∑
Ft 
t j >0 η (0, t j )

η (0, t )

= − V (0 ) + V (0 ) − V (t )
= − V (t )
Q
( A.28)
( A.29)
( A.30)
We can now calculate the market facing funding cost for an uncollateralized position
Shahram Alavian - 7th Annual Credit Risk in Banking Summit, Berlin, 2014
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Appendix – Funding Cost (Uncollateralized Position)
−


 CA(t ) 
⋅
FC M (0, T ) = ∫ E  LGD 
1
1
 τ c >t −δ t ≥τ >t −δ 
η
0
,
t
(
)




δ

−
 
T



Q Q
Q  CA(t )

= ∫ E  E LGD E 
Ft  ⋅ 1τ c >t −δ ⋅ 1t ≥τ >t −δ H t ∨ Ft  

η (0, t ) 
δ
 

T
Q
T
{ [
= ∫ E Q E Q LGD[− V (t )] ⋅ 1τ c >t −δ ⋅ 1t ≥τ >t −δ H t ∨ Ft
−
]}
( A.31)
( A.32)
( A.33)
δ
Where Ht is the filtration generated with the discrete events of default τ and τk,. With
[− V (t )]− = [V (t )]+
( A.34)
and after changing measure to t-forward, we finally obtain the more intuitive expression
T
[
FC = ∫ B (0, t ) E t LGD V (t ) ⋅ 1τ c >t −δ ⋅ 1 t ≥τ >t −δ
+
]
( A.35)
δ
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