Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
Closed-Form Expression of A CVA on A Bond
Axel Delamarre, Bertrand Guillon, Abdelkader Kadaoui, Jérémy Lisbona and Félix
Pariente*
Financial products valorization, vanilla or exotic, has become more time-consuming since
2007 financial crisis. Financial actors became conscious of risks which did occur between
the different stakeholders, leading regulators to change with Basel directives some of
most fundamental rules in the financial landscape. But counterparty risk became more
important and regulators imposed Credit Valuation Adjustment (CVA), a risk measure
corresponding to the capital charge that banks must keep in order to cover counterparty
failure in portfolios. In this paper two closed form expressions which evaluate a CVA on a
coupon-bearing bond are given. Using short rate stochastic models to represent evolution
of interest rates, implementation of CVA formula is provided in R programming language.
Keywords: CVA, Capital Charges, Basel III, Expected Exposure, Bond Valuation, HullWhite Model, Vasicek Model, Nelson-Siegel-Swensson Model
JEL Codes: F34, G21 and G24
1. Introduction
After 2007 crisis and the default of major financial institutions like Lehmann Brothers, it became
important for financial actors to valuate counterparty risk for their contracts. To this aim, Basel
committee for International Banking introduced for banks in 2009 a new capital charge called CVA [1]
[2] [3]. This charge corresponds to adjustment in the valuation of a contract when taking into account
its exposure to default. Owing to time and power involvement needed by today calculation of CVA, it
seemed important to reduce the burden.
One way proposed here is to find an analytical mathematical formula allowing to approximate CVA
charge without using heavy calculation methods.
The upsides of closed-form expression for such a calculation are the possibility of light
implementation. As a first step in risk management [4] and CVA calculation, the CVA on a couponbearing bond is evaluated in the paper. Interest rate models like Hull-White model and its successors
are also explored. This model is used to determine zero-coupon prices needed in evaluation of bond
value at any time in the future. In the sequel, CVA will be first presented with definition and original
formula. Bond valuation method playing a key role in valuation of expected exposure will be
discussed next, and different short-rate models used to express zero-coupon prices will be analyzed.
Finally implementation of the formulae will be showcased and analyzed before a conclusion with the
results.
2. Credit Valuation Adjustment
Counterparty risk is the risk when a counterparty does not respect a financial contract: for example,
the risk that someone owing money will never pay it back, or the risk that a company having issued
bonds goes bankrupt. For a long time, this risk was not taken into consideration due to supposed
solidity of major financial actors, since then called “too big to fail”. After 2007 crisis, financial
institutions were forced by regulators to consider counterparty risk and to evaluate risk change of a
*
Financial Engineering Students from ECE Paris Graduate School of Engineering, France. Email: fparient@ece.fr
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Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
contract by varying a certain amount of money put aside: the Credit Valuation Adjustment. The CVA
can be seen as unilateral or bilateral. From either end of the contract, the counterparty risk is not the
same. In the case of a swap, both parties can possibly default, so the CVA calculated by first party
must consider its own default risk so the CVA is called bilateral. Here the CVA for a single couponbearing bond is considered. A bond is a contract where only issuer default can affect the contract life,
so for a bond a unilateral CVA calculation is used. Usually, CVA calculation is performed by banks by
doing Monte-Carlo simulations. In Basel formula for CVA, the Monte-Carlo technique is used to
approximate the expected exposure to the counterparty by doing a bond valuation with a dynamic
short-rate. CVA Basel formula is used in this first part, and approximation of expected exposure
calculation will be studied, as this part of the formula is not written in a closed-form.
Starting from general CVA formula [5] [6], a unilateral CVA under risk-neutral expectation of
discounted loss writes:
∫
[
]
(1)
where
is the present instant,
is the maturity of the instrument,
the recovery rate of the
counterparty,
is the discount factor,
is the exposure to the counterparty,
is the time of
default and
is the counterparty default probability between and . Integration variable is
, and when focusing on bonds specifically, (1) simplifies to
∫
so
(2)
where the discount factor is substituted for the zero-coupon price
, and the exposure for the
bond future (random) value at time ,
.
represents counterparty probability of survival
using a reduced form approach:
with
a piecewise constant function [7] [8].
∫
Finally, integral calculation is discretized over interval subdivisions between present and maturity:
∑
(3)
More details about the time instants, the zero-coupon value and the bond value will be detailed in the
rest of this paper.
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Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
3. Bond Valuation
As seen on CVA formula (3), explicit evaluation of the term:
is required, where
is the (random) future bond value. Some notations are introduced in order to describe the
different time instants:

First, the set
is defined. It contains N dates corresponding to
subdivisions of the time interval between the present and the bond maturity. A larger N
corresponds to more instants where the bond’s value has to be calculated.

Then, the set
is defined. It contains the M dates when coupons
are paid. The last coupon is paid and the notional is reimbursed at
. At the term of this
bond, all coupons have been paid so
.

Finally, the set
is defined. It simply contains the coupon dates that
will be paid after a certain instant .
A bond value today is nothing more than the actualized sum of its future cash-flows. This is the
standard approach [9]
( ∑
)
(4)
where
is the bond’s notional amount,
its coupon rate and
the zero-coupon price or
discount factor. The CVA formula requires to consider bond value at future date
such that
, so it becomes a random variable. By linearity of the expected value, one gets
(
∑
In this expression, the only source of uncertainty comes from the
explained in the next section.
)
(5)
terms, which are
4. Interest Rate Models
Short-rate models are used to describe interest rates dynamics like EURIBOR. Subsequently, these
models are also useful to express a formula for the price of a zero-coupon, which is simply an amount
of money to be received in the future.
First, the theoretical foundations of more general Hull-White model using stochastic calculus are
introduced. Then, analytical formulas are provided in two special cases corresponding to Vasicek and
Nelson-Siegel-Svensson models.
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Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
1. The Hull-White Model
In one factor Hull-White model, interest rate dynamics are described by the following stochastic
differential equation
(6)
a deterministic map. Using Ito’s lemma to
where
is a standard brownian motion and
differentiate the function
one gets
(7)
In the rest of this section will be considered as the zero-coupon time-to-maturity. By integrating
between present moment and
one can find the solution of the model
∫
∫
(8)
Pricing theory shows that the value of zero-coupon is given by the formula
*
( ∫
)
+
(9)
where
is the expected value under the risk-neutral measure
conditioned by all the known
information
until date . The integral inside the exponential here denoted
is
∫
used to rewrite equation (9) as
(
(
)
∫
(10)
)
However, if this price is considered at a future time instant
, its expression will use the future
interest rate
which is unknown today, so it has to be considered as a random variable.
Substituting (8) into (10) gives
(
where
√
)
(11)
is a standard gaussian random variable and
∫
∫
(12)
(13)
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Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
With the right specification for
with
, the value of a zero-coupon in the future can be calculated
(
)
(14)
2. The Vasicek Model
The Vasicek model can be considered as a special case of Hull-White model where
is the constant:
(15)
Here
can be considered as a long-term equilibrium value of the interest rate and is the same
than in the Hull-White model. The future zero-coupon price is the following closed-form expression
(
)
(16)
where now, simplifying equation (12), one obtains
(
)
(17)
3. The Nelson-Siegel-Svensson Model
The Nelson-Siegel-Svensson model proposes a 6-parameter description of the yield curve, out of
which an expression for
, derived in [10] can be used in the Hull-White framework
The future zero-coupon price is as in (16) where
(18)
with
as above and
as detailed in the appendix.
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Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
5. Implementation
As an illustration, computations are performed on the very simple example of the following bond:
Notional amount: 100$, maturity: 1 year, coupons payments: after 6 month and at the term, coupon
rate: 3.5%, risk-free rate: 2.5%.
Figure 1. Bond Cash Flows (arrows) and Time Discretization.
By increasing the number of time steps, one can observe how CVA value gets more accurate. As
expected, the curve obtained using the Nelson-Siegel-Swensson requires more capital because of
the larger number of parameters to determine.
Figure 2. CVA Value vs Number of Time Steps with Vasicek Model (down), and Nelson-Siegel-Swensson Model
(up)
6. Conclusion
2007 crisis has shown the limit of existing financial regulations. As a consequence, prudential rules
have been upgraded in order to come up with a new set of rules containing the CVA measure. The
first step in present work has been to rewrite CVA formula specifically for bonds and without
continuity condition. The second step has been to explain and to use the single factor interest rate
models to find the zero-coupon bond price. Finally the expression of a coupon-bearing bond value is
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Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
used in CVA formula.
7. Acknowledgments
The authors are very much indebted to ECE Paris School of Engineering for having provided the
necessary environment for the development of the work, to Dr Yves Rakotondratsimba for advices
during the study and to Pr Michel Cotsaftis for help in the preparation of the manuscript.
References
[1] Basel Committee on Banking Supervision, December 2010 (review June 2011),
Basel III: A global regulatory framework for more resilient banks and banking
systems.
[2] A. Sokol, May 2012, A Practical Guide to Fair Value and Regulatory CVA, Proc.
PRMIA Global Risk Conf., New York.
[3] H.J. Stein, L.K. Pong, 2011, Counterparty Valuation Adjustments, in Credit Risk
Frontiers, Subprime Crisis, Pricing and Hedging, CVA, MBC, Ratings and Liquidity, T.R.
Bielicki, D.Brigo, F. Patras, eds., Chap.12, J. Wiley and Sons, New York.
[4] Z.G. He, W. Xiong, 2012, Rollover Risk and Credit Risk, J. Finance, Vol.67,
pp.391-430.
[5] Pykhtin M. and Zhu S., July-August 2007, A Guide to Modeling Counterparty
Credit Risk, GARP Risk Review, Related SSRN Research Paper.
[6] M. Realdon, 2013, Credit Risk, Valuation and Fundamental Analysis, Intern. Rev.
Financial Analysis, Vol.27, pp.77-90.
[7] Dominic Kane and Stuart Turnbull, April 2003, Valuation of Credit Default Swaps,
Lehman Brothers Quantitative Credit Research.
[8] Ahlberg J., September 2013, Credit Value Adjustment, Zurcher Kantonalbank and
LUNDS Universitet.
[9] Burgess N., August 2014, Bond Option Pricing using the Vasicek Short Rate
Model, Related SSRN Research Paper.
[10] Peter Lochte Jorgensen, March 20, 2013, On the Reconciliation of the Extended
Nelson-Siegel and the Extended Vasicek Models, SSRN Research Paper.
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Proceedings of 7th Annual American Business Research Conference
23 - 24 July 2015, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978-1-922069-79-5
Appendix A: Expression of Delta function
*
√
(
(
(
(
(
(
)
(
)
)
(
)
(
)
(
)
)
)
(
)
(
)
)
)
+
8