Master’s Thesis Econometrics, Operations Research and
Actuarial Studies
The Credit Value Adjustment of Inflation-Linked
Swaps
Ting An Phoa
s2381761
February 5, 2019
The Credit Value Adjustment of
Inflation-Linked Swaps
Ting An Phoa
Abstract
The introduction of IFRS 13 requires the inclusion of counterparty credit risk within fair value accounting standards. For inflation-linked deriva-tives, standard practice to price this risk is lacking. In this thesis we investigate the Credit Value Adjustment of Year-on-Year inflation-linked swaps. The goal is to find a methodology that prices the counterparty credit risk of this product consistent with the market prices of other products. To this end, both the CVA and the models for the pricing of inflation-linked derivative products are discussed. Then, these topics are combined and the set-up of a tool for the CVA calculation of inflation swaps is described and a simulation engine is implemented. The scenarios are generated by making use of the Jarrow-Yildirim model, which follows the macroeconomic relationship between the nominal and real interest rate and inflation. This framework has the advantage that it is very flexible. We obtain exposure profiles with characteristics that are expected from the more studied and simpler interest rate derivatives. The downside of the proposed model is that the real rate parameters are hard to calibrate. A sensitivity analysis was performed and the results show that care has to be taken when calculating the CVA. Especially an incorrect estimate of the real rate volatility could lead to significant mispricing of the counterparty credit risk.
1
Introduction
Before the financial crisis, participants in the financial markets considered credit risk on over-the-counter (OTC) derivative transactions mostly insignificant (Centrus, 2018). It was assumed that the default probability of large financial institutions was negligible, also due to the too big to fail mentality (Dash, 2009). When the financial crisis hit in 2007, liquidity tightened and the risk of bankruptcy became real. Market prices of OTC derivatives be-gan to diverge from counterparty to counterparty and substantial losses in mark-to-market (MtM) value were made on trading books. The calculation of these valuation adjustments from the risk-free price became standard market practice as part of trading among banks. This is relevant especially for OTC derivatives, since those are privately traded, formulated and negotiated between two parties. Exchange-traded derivatives in contrary, are traded in a common transparent market place, where the exchange guarantees the cash flows promised by the contract (Pykhtin and Zhu, 2006). As a consequence, the credit risk of these products is very low and depends on the survival of the exchange.
More recently, the introduction of IFRS 13 required the inclusion of this counterparty credit risk (CCR) within fair value accounting standards. Counterparty credit risk is defined by, among others, Pykhtin and Zhu (2006) as “the risk that the counterparty in a financial con-tract will default prior to the expiration of the concon-tract and will not make all the payments required by the contract”. CCR differs from more traditional credit risk in two ways: the exposure is uncertain and the risk is bilateral. The credit exposure at the time of default of a counterparty is the positive part of the contract’s market value and hence equals zero in case the contract value is negative. Only the current contract value is known and the value changes unpredictably over time with the market. Hence, the future exposure is uncertain. Since the value of the contract can change sign and either counterparty can default, CCR is bilateral. The pricing adjustment made to account for the credit risk of your counterparty to you is known as the credit value adjustment or CVA. The debt (or debit) value adjustment (DVA) is the CVA as seen from the perspective of your counterparty. The CVA and DVA depend on the future exposure. Since there is always a chance that the market changes and the exposure becomes positive, both CVA and DVA will be positive right until the maturity of the contract. Extensive work has been done regarding the CVA and DVA of standard interest rate swaps, as will be discussed in section 2.1. For the CVA and DVA calculation of inflation-linked swaps (ILS) standard practice is still lacking and this will be the focus of this thesis.
the market. An example illustrating how an ILS works is provided in Figure 1. An ILS can
Figure 1: Example of an inflation-linked swap, obtained from J.P. Morgan (2012) be used to transfer inflation risk. Pension funds and insurance companies who have inflation exposure can hedge this risk by entering an ILS where they receive inflation. Entities on the other side of the trade include utilities and infrastructure companies, which in many cases raise their prices by an amount linked to inflation (ISDA, 2014).
The main goal of this thesis is to propose a methodology for the calculation of the CVA for inflation-linked swaps consistent with market prices for other (inflation-linked) products. Since both CVA-DVA modelling and inflation-linked derivative pricing are very complex tasks, simplifying assumptions have to be made. Therefore, we will not claim that the methodology proposed in this thesis provides the “correct” CVA value, rather it should be seen as a first attempt and providing a framework for further development.
2
Literature
Since the inflation-linked swap market is still an emerging market and CVA calculation has only become relevant after the 2008 financial crisis, not much research into the topic has been done. The only study regarding the CVA of inflation products that was found was done by Petrov (2015). The focus of Petrov was at deriving an exact formula for the CVA of a basic zero-coupon inflation-indexed swap. For the year-on-year inflation-indexed swap, however, no related literature was found. In this section we will explain the concepts of CVA and DVA and give background information regarding inflation-linked derivative pricing models. It should be noted that the focus of this thesis is on the CVA for IFRS 13, which is a pricing issue, and not on the CVA for Basel III, which is a risk metric (Zarpellon, 2014). Therefore, risk-neutral measure and parameters for e.g. default probabilities and exposure should be used (Gregory, 2012).
2.1
CVA-DVA
According to Ahlberg (2013), the CVA and DVA are the market prices of CCR. CVA and DVA play an important role in the IFRS 13 “Fair Value Measurement” accounting standards for OTC derivatives. Solum Financial (2013) highlighted some key concepts for fair value accounting:
• Fair value: “The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date”, also referred to as an “exit price” (paragraph 9).
And more specifically related to counterparty credit risk:
• “The entity shall include the effect of the entity’s net exposure to the credit risk of that counterparty or the conterparty’s net exposure to the credit risk of the entity in the fair value measurement when market participants would take into account any existing arrangements that mitigate credit risk exposure in the event of default” (paragraph 56).
This seems to require both CVA and DVA adjustments to be made to the value of deriva-tives and to be reflected in the exit price. To explain the CVA and DVA calculation, we will first assume that the entity/institution itself is default free and only the counterparty can default before contract maturity. This is called the Unilateral CVA (UCVA) and simplifies the arguments, without losing the idea of the intuition behind CVA and DVA computation. Later it is discussed how the inclusion of a positive entity default probability influences the computation. In order to keep the explanation of the UCVA tractable, a single swap without collateral or netting sets is considered.
Gregory (2012) states that the UCVA is the difference between the risky and risk-free value of a financial transaction:
The UCVA component at time t = 0 for a contract with maturity T is calculated as follows:
UCVA(0, T ) = Z T
0
(1 − Rec(t))D(0, t)E[E(t)]dPD(t) (2)
and can be divided in the following components (Gregory, 2012):
• Loss given default (LGD(t)): If a counterparty defaults, generally, not the entire ex-posure is lost. The percentage of the claim that can be recovered for a default at time t is called the recovery rate (Rec(t)). The LGD(t) is given by 1 − Rec(t). Often the recovery rate is assumed to be constant over time and the LGD is simplified to LGD = 1 − Rec and can be taken out of the integral (Brigo, Morini, and Pallavicini, 2013). The holders of OTC derivative contracts are normally in the same class, in terms of potential claims on the defaulted company, as senior bondholders. Hence, recovery rates are generally reasonably high (Gregory, 2010).
• Discount factor (D(0, t)): Future losses should be discounted using the relevant dis-count factor. Hull and White (2013) state that it is most appropriate to disdis-count derivatives by using the overnight index swap (OIS) rate as proxy for the risk-free rate, even if they are uncollateralized.
• Probability of default (PD(t)): The cumulative probability of default gives the
prob-ability of default until time t. Clearly this starts at PD(0) = 0 and tends towards
limt→∞PD(t) = 1, since eventually all counterparties default (Ahlberg, 2013). The
probability of default between two specified future time points (t1, t2 : t1 ≤ t2) is given
by q(t1, t2) = PD(t2) − PD(t1). Hence, PD(t) must be monotonically increasing. In
order to calculate this term structure of default probabilities a differentiation between real-world and risk-neutral default probabilities can be made. Real-world probabilities are used for risk management purposes and risk-neutral probabilities reflect the mar-ket price of risk. Historical data and credit rating can be used to estimate real-world probabilities. Risk-neutral default probabilities can be estimated using market data and often credit default swaps (CDSs) are used. For companies where CDS data is not available, the default probability curve will have to be estimated differently. Bond prices, bond spreads or default probabilities of comparable companies might be used. • Exposure (E(t)): The amount that would be lost in case the counterparty defaults on
the contract at time t. A key characteristic of exposure is that it depends asymmet-rically on the value of the derivative contract. If the contract is an asset and hence has a positive present value, the exposure corresponds to the claim on the defaulted counterparty. However, if the contract is a liability the institution still has to pay the (creditors of the) defaulted counterparty. The exposure can be seen as the replacement costs of a contract. Note that in order to calculate the UCVA we need both the current and future exposure. The exposure at time t is given by
where V (t, T ) represents the value of a contract at time t with maturity T . The relation between value and exposure for a fixed point in time is illustrated in Figure 2. It can be seen that the exposure has a similar profile as the pay-off of a call option. From this characteristic we can conclude that the volatility of the value of the contract plays a key role in determining the future exposure and CVA. Where the current exposure is known, the future exposure has to be defined probabilistically regarding uncertain market movements. This is illustrated in Figure 3. It should be noted that the expected future exposure can be positive, also if the expected future value is negative. A typical exposure profile of a swap initially trading at par is provided in Figure 4. The exposure is zero at initiation and increases over time as the underlying risk factors can diverge further from current values. The exposure decreases towards zero again since less remaining coupon payments are due to be paid.
Figure 2: Exposure as a function of value, for a fixed point in time
In practice, the UCVA calculation in equation (2) is often approximated by discretizing time into buckets and we obtain the discretized CVA function of Gregory (2012):
UCVA = (1 − R)
m
X
i=1
D(0, t0i)E[E[t0i]]PD(ti−1, ti), (4)
where tm = T , t0i denotes the midpoint of the interval (ti−1, ti] and PD(ti−1, ti) is the
proba-bility of default between time ti−1 and ti.
2.1.1 BCVA and DVA
Figure 3: The distribution of the future value (represented by scenarios), where the grey area represents the future exposure. Figure obtained from Gregory (2012)
Figure 4: The exposure profile of an interest rate swap (Pykhtin and Zhu, 2006)
((U)DVA). IFRS 13 states on the DVA: “The fair value of a liability reflects the effect of non-performance risk. Non-performance risk includes, but may not be limited to, an entity’s own credit risk” (Solum Financial, 2013). The value adjustment assuming that both the entity and counterparty can default is called the bilateral credit value adjustment (BCVA). It should be noted that the BCVA is not simply the sum of the UCVA and UDVA, since also the order of default is relevant. Assuming that the institution and the counterparty cannot default simultaneously and that defaults are independent, Gregory (2012) defined the discretized BCVA as
BCV A = (1 − RecC) m
X
i=1
E[DEI(ti)][1 − PD,I(0, ti−1)]PD,C(ti−1, ti)
+ (1 − RecI) m
X
i=1
E[DEC(ti)][1 − PD,C(0, ti−1)]PD,I(ti−1, ti), (5)
where Rec(·), DE(·) and PD,(·) denote the recovery rate, discounted exposure and probability
term of equation (5) is close to the UCVA, but with an additional factor for the institution’s survival probability. Inclusion of the own default probability and DVA is necessary in order to get an equal view on the derivative value from the perspective of the institution, counterparty and other market participants and hence arrive at a single market price, as is also done for bonds (EY, 2014). However, DVA is a controversial concept and could have worrying implications (Gregory, 2009). First, the BCVA may be negative and hence increase the value of the derivative. So, a risky derivative can have a higher value than a risk-free derivative. Second, if the derivative contracts trade at the mid-market price, the sum of the BCVAs and hence the overall amount of CCR will be zero. A practical objection is that in order to hedge the DVA, a firm has to sell protection on itself. Another counter-intuitive characteristic of DVA is that the value of a firms equity increases as its credit quality declines and vice versa. Furthermore, the DVA is not easily realizable, just as an individual cannot realize gains on their own life insurance policy (Gregory, 2012). Gregory (2009) argues that a full reduction of CCR with DVA is inappropriate. However, an institution should be able to reduce CCR in line with the systematic component of their default risk, which is hedgeable.
2.1.2 Wrong-way risk
The (U)CVA, (U)DVA and BCVA are determined by both market risk factors and credit risk factors. The underlying market factors affect the no-default value of the contracts and hence the exposure. The movements in credit spreads influence the value adjustments via the prob-ability of default. A key simplifying assumption that is usually made is the independence of these factors (Hull and White (2012), Gregory (2012)). Wrong-way risk is the term used for a situation where there is a positive dependence between the two and right-way risk is the situation where this dependence is negative. Wrong-way risk is considered unfavourable since the exposure is more likely to be high when the probability of default is high, so the CVA will increase. Right-way risk will reduce counterparty risk. Estimating the amount of wrong-way or right-way risk is a difficult task and requires a good knowledge of the counter-party’s business (Hull and White, 2012). Essentially this corresponds to integrating credit risk and market risk, which is very complex (Gregory, 2012).
For an extensive overview of CCR and CVA we further refer to Gregory (2012) and Brigo et al. (2013).
2.2
Inflation-linked derivative pricing
2.2.1 Inflation and inflation-linked swaps
The Oxford dictionary describes inflation as “a general increase in prices and fall in the purchasing value of money”. To measure this fall (or rise, in case of deflation) in purchasing value of money baskets of goods are used. Price indices refer to different baskets of goods and are calculated by various national agencies for different currencies (Dodgson and Kainth, 2006). Inflation products are linked to these indices. For the EU area, Eurostat publishes the HICPxT (Harmonised Indices of Consumer Prices excluding Tobacco). For the United Kingdom, the ONS (Office for National Statistics) measures the RPI (Retail Price Index) and the CPI (Consumer Price Index), where the CPI excludes housing-related expenses. In the USA, the CPI-U (Consumer Price Index for All Urban Consumers) is published by the BLS (Bureau of Labor Statistics). The history of the yearly percentage change of the RPI is depicted in Figure 5. One can observe that different inflation environments have existed. The highest yearly inflation was measured in August 1975 at 26.9%. Over the last years the inflation has been considerably lower and in 2009 even deflation occurred. These different environments influence the purchasing value of money highly. If the nominal income, i.e. the income in terms of money, increases, the real income, i.e. the income in terms of a composite of goods, can be decreasing. Fisher (1930) describes this important macroeconomic phenomenon in terms of interest. He describes that there are two economies, a real and a nominal economy, and that inflation links the two. This is captured in the Fisher equation:
rr(t, T ) = rn(t, T ) − E[i(t, T )], (6)
where rr(t, T ) denotes the cumulative real interest rate from time t up to time T , rn(t, T )
the nominal cumulative rate and E[i(t, T )] denotes the expected overall inflation from t to T . The relation between the inflation rate i(·, ·) and the inflation index I(·) is given by
i(t, T ) = I(T ) − I(t)
I(t) . (7)
For continuous interest rates, the Fisher equation can be rewritten to e(rn−rr)(T −t) = E[I(T )
I(t)], (8)
(Kruse, 2011).
Figure 5: Time series of monthly RPI percentage change over 12 months (1948-2018) (Office for National Statistics)
The inflation-linked derivative market is an OTC market and products can be tailor-made to match the requirements of the parties involved. Hughston (1998) provides an overview of inflation-linked derivative products. By far the most liquid interest linked derivative is the inflation-linked swap and we will focus on this product. Other derivative products are e.g. caps, floors and swaptions. Comprehensive data on the market activity of ILS is not available since it is an OTC market. However, Fleming and Sporn (2013) state that the U.S. ILS market has grown quickly to a daily trading volume around 190 to 350 million U.S. dollars, based on data from a leading broker and survey data. J.P. Morgan (2012) and Hurd and Relleen (2006) confirm this trend for the U.S. and European market respectively. Clarus Financial Technology (2015) states that the January 2015 average daily trading volume from U.S. Swap Data Repository is 780, 600 and 465 million U.S. dollar for USD, EUR and GBP notional respectively.
Two types of swaps are available: zero-coupon inflation-indexed swaps (ZCIIS) and year-on-year inflation-indexed swaps (YYIIS) (Mercurio, 2005). A ZCIIS is a bilateral contract where at maturity T (and M =T − t0 years, where t0 denotes the initiation date) a fixed
amount is swapped for an amount linked to an index. There is a wide range of maturities. The fixed amount paid is
N [(1 + K)M − 1] (9)
and the floating amount received is N [I(T )
I(t0)
− 1], (10)
For the YYIIS, payments are made at each time Ti. The fixed payment is
N φiK. (11)
The floating payments depend on the change in index compared to the previous time period. So for the floating payment we have
N [ψi
I(Ti)
I(Ti−1)
− 1] (12)
where ψiand φidenote floating and fixed year fractions respectively for the interval [Ti−1, Ti].
2.2.2 Inflation-linked derivative pricing models
In this thesis we consider the IFRS 13 “Fair Value Measurement” CVA, which is a pricing issue. Therefore, we need arbitrage free scenarios and hence cannot use the standard econo-metric approaches for modelling inflation based on a time series of data under the historical probability measure. We need to resort to pricing models under the risk-neutral probability measure that use no arbitrage assumptions to define prices as in the framework of Harrison and Kreps (1979) and Harrison and Pliska (1981). There are two main modelling approaches for the no arbitrage pricing of inflation-linked derivatives: A foreign currency analogy and an approach based on the lognormal forward market model (Kooistra, 2017). Both can be seen as extensions of interest rate derivative pricing models. Therefore, we will first introduce some important interest rate models.
2.2.3 Short rate models
The development of interest rate models started with short rate models. The short rate (r(t)) is the instantaneous time dependent interest rate at time t. It can be seen as the growth rate of the bank account or money market account over infinitesimal time. The bank account (B(t) with B(0) = 1) can be seen as the return on a risk-less investment and develops according to
dB(t) = r(t)B(t)dt ⇐⇒ B(t) = eR0tr(s)ds, B(0) = 1. (13)
The instantaneous forward rate (f (t, T )) is the time t rate for the short rate at time T (T ≥ t). The relationships between the short rate, the yield to maturity curve (R(t, T )), the forward rates and bond prices (P (t, T ), T ≥ t) are given by
r(t) = R(t, t) = limT →tR(t, T ), (14) R(t, T ) = − 1 T − tlnP (t, T ) = 1 T − t Z T t f (t, s)ds. (15)
of the past. The price of a bond is assumed to be determined by the time t assessment of the segment {r(s), t ≤ s ≤ T }. Finally, Vasicek (1977) assumes that the market is efficient. Then under the risk-neutral measure Q, the arbitrage free price (Xt) at time t is given by
Xt = EQt {D(t, T )XT}, (16)
where EQt denotes the expected value under the risk-neutral measure Q with filtration Ft
and D(t, T ) is the discount factor, which determines the time t amount of money that is equivalent to an amount of one at time T :
D(t, T ) = B(t) B(T ) = e
−RT
t r(s)ds. (17)
To illustrate this general model, Vasicek (1977) provides a specific case where the market price of risk is assumed to be constant and the short rate follows an Ornstein-Uhlenbeck process (see Appendix A) under the risk-neutral measure Q:
dr(t) = a[b − r(t)]dt + σdWQ(t), (18)
where b > 0 is the long term mean, σ2 the variance, a > 0 is the mean reversion speed of the process and dWQ(t) is a standard Wiener process. The Vasicek model provides the convenient possibility of characterizing the entire yield curve process in terms of the short rate process and two deterministic functions of time (Mamon, 2004).
This model has some important drawbacks. The first drawback is that since only one factor is used to model the short rate, there is perfect correlation between interest rates for all ma-turities at every time t. This implies that a shock to the short rate results in a parallel shift of the entire yield curve. Yield curves are known to change shape (Nelson and Siegel, 1987) and the model should reflect this behaviour . Multi-factor short rate models could be used to represent curve evolution to different shapes. A second drawback is that the parameters a, b and σ are assumed to be constant over time. Therefore, the model implied initial term structure generally cannot perfectly fit the current term structure as observed from market prices of zero-coupon bonds. Hull and White (1990) extended the Vasicek model by making the long term mean time dependent to perfectly match the market implied term structure at calibration. The Hull-White extended Vasicek model dynamics under Q are given by
dr(t) = [θ(t) − ar(t)]dt + σdWQ(t), (19)
2.2.4 Models for inflation-linked derivative pricing
Hughston (1998) outlined a general theory for the pricing of inflation-linked derivatives in a complete market setting and ties this with the HJM framework (Heath, Jarrow, and Mor-ton, 1992) for interest rate derivatives. An analogy with the foreign exchange methodology is proposed, where the price index acts as exchange rate between the nominal and real market. Let {n, r} be the subscripts used for nominal and real rates and prices respectively and let I(t) denote the value of the CPI at time (t). As usual in HJM frameworks, uncertainty in the future is modelled using a multidimensional Brownian motion and assuming no arbi-trage. Then under the risk-neutral probability measure Q, Hughston (1998) writes the price processes of the bonds and the CPI as
dPn(t, T ) Pn(t, T ) = rn(t)dt + σn(t, T )dWQ(t), (20) dPr(t, T ) Pr(t, T ) = [rr(t) + (λr(t) − λn(t))σr(t, T )]dt + σr(t, T )dWQ(t), (21) dI(t) I(t) = [rn(t) − rr(t)]dt + (λn(t) − λr(t))dW Q (t), (22)
where λn(t) and λr(t) denote the nominal and real risk premium vector respectively. Note
that equation (22) resembles the Fisher equation as given in equation (6).
2.2.5 Jarrow and Yildirim model
Jarrow and Yildirim (2003) (JY) make use of this foreign exchange analogy and propose a pricing model where a 3-factor HJM model is used to consistently price inflation-linked bonds, nominal bonds and related derivatives. They first introduce the processes for the nominal and real instantaneous forward rates and inflation index with correlated Brownian motions WnP(t), WrP(t) and WIP(t) under the real world probability measure P, where n denotes the nominal rate, r denotes the real rate and I denotes the inflation index:
dfn(t, T ) = αn(t, T )dt + σn(t, T )dWnP(t), (23) dfr(t, T ) = αr(t, T )dt + σr(t, T )dWrP(t), (24) dI(t) I(t) = µI(t)dt + σI(t)dW P I (t), (25)
where αk (k ∈ {n, r}) and µI are random and σk(t, T ) and σI(t) are deterministic functions
of time that are subject to some technical smoothness and boundedness conditions, for which we refer to Jarrow and Yildirim (2003). The correlations are simply given by
dWnP(t)dWrP(t) = ρn,rdt, (26)
dWnP(t)dWIP(t) = ρn,Idt, (27)
For the pricing of derivatives, the model needs to be arbitrage free. Jarrow and Yildirim use the theory of Girsanov (1960) and compute the Q-Brownian motions for the three processes by rewriting the Brownian motions to
WlQ(t) = WlP(t) − Z t
0
λl(s)ds, (29)
where λl(t) is the market price of risk, for l ∈ {n, r, I}. Note that the correlations do not
change.
Then, by using Ito’s lemma, Jarrow and Yildirim obtain the following processes: dPn(t, T ) Pn(t, T ) = rn(t)dt − Z T t σn(t, s)dWnQ(t), (30) dPr(t, T ) Pr(t, T ) = rr(t) − σI(t)ρr,I Z T t σr(t, s)ds dt − Z T t σr(t, s) dWrQ(t). (31) dI(t) I(t) = [rn(t) − rr(t)] dt + σI(t)dW Q I (t) (32)
Jarrow and Yildirim pre-specified a one factor volatility function for both the nominal and real interest rates. The volatility functions are given by
σk(t, T ) = σke−ak(T −t) ∀k ∈ {n, r}, (33)
where ak and σk are constants.
This is the volatility of the (extended Hull-White) Vasicek model, as was shown by e.g. Mamon (2004). The nominal and real zero-coupon bond price are then given by
Pk(t, T ) = Ak(t, T )e−Bk(t,T )rk(t), (34) Bk(t, T ) = 1 ak [1 − e−ak(T −t)], (35) Ak(t, T ) = Pk∗(0, T ) Pk∗(0, t)e Bk(t,T )fk∗(o,t)− σ2k
4ak(1−e−2akt)B 2
k(t,T ) ∀k ∈ {n, r}, (36)
where Pk∗(0, ·) and fk∗(o, t) denote the market observed bond price and forward rate (Hull and White, 1990).
In these formulas the bond price is a function of the corresponding short rate and not of the instantaneous forward rate, for which Jarrow and Yildirim (2003) originally specified there model. So, it is necessary to rewrite the dynamics of the instantaneous forward rates to the dynamics of the nominal and real short rate under the Q measure. The derivation for the nominal short rate can be found in Appendix C.1 and the derivation of the real rate follows analogously. The inflation dynamics under Q were given in equation (32) and are included here for completion. We have
drr(t) = [θr(t) − ρr,IσIσr− arrr(t)]dt + σrdWrQ(t) (38)
dI(t) = [rn(t) − rr(t)]I(t)dt + σII(t)dWIQ(t), (39)
where, θk(t) = δfk(0, t) δT + akfk(0, t) + σ2 k 2ak (1 − e−2akt). (40)
The solutions of these linear stochastic differential equations are (for t > s) rn(t) = rn(s)e−an(t−s)+ Z t s e−an(u−t)θ n(u)du + Z t s e−an(u−t)σ ndWnQ(u) (41) rr(t) = rr(s)e−ar(t−s)+ Z t s e−ar(u−t)(θ
r(u) − ρr,IσrσI)du +
Z t
s
e−ar(u−t)σ
rdWrQ(u)
(42) I(t) = I(s)eRst(rn(u)−rr(u))du−
1 2σ 2 I(t−s)+σI(WIQ(t)−W Q I (s)), (43)
and are found in e.g. Petrov (2015).
The foreign currency analogy and the model of Jarrow and Yildirim (2003) have the advan-tage that they are very intuitive and reflect the macroeconomic phenomenon as described by Fisher (1930). Moreover, these Gaussian models are analytically tractable and allow for closed form formulas for various derivatives (Mercurio, 2005). However, the Jarrow and Yildirim model has also some limitations. The main challenge is the difficulty of estimating the real rate parameters, as the real rate is not directly observable in the market and is constructed using the nominal term structure and inflation-linked swaps (Mercurio, 2005).
2.2.6 Mercurio’s market models for inflation derivatives
The second main type of models for the pricing of inflation derivatives are market models, which were simultaneously developed by Mercurio (2005) and Belgrade, Benhamou, and Koehler (2004). Mercurio states that the advantage of this approach is that the model parameters are better understood and more accurately calibrated to market data. In Mer-curio’s first market model it is assumed that the nominal and real forward rates follow a lognormal Libor market model and that the forward CPI is a martingale under the corre-sponding forward measure and follows a driftless geometric Brownian motion. The dynamics of the model are provided in Appendix B and for further details we refer to the original pa-per. The YYIIS value depends on the volatilities of nominal and real forward rates and their correlations for each payment time Ti and is more complex in both input parameters
and calculations than the model of Jarrow and Yildirim (2003) (Mercurio, 2005). Because of the precise relation between two consecutive forward CPIs and the nominal and real forward rates, Mercurio states that it cannot be assumed that the volatilities are positive constants for all payment times Ti. In order to get approximately constant volatilities σI,i,
just as the Jarrow and Yildirim model, depends on the real rate volatility, which may be hard to calibrate (Mercurio, 2005). In order to overcome this drawback, Mercurio proposed a second market model. The dynamics the forward inflation (Ii−1(t)) under the Ti-forward
measure QTi are given in Appendix B and depend on the nominal forward rate, making
the calculation of the expected inflation rather involved. Hence, Mercurio (2005) chose to freeze the drift at its time t value, which leads to the introduction of a correction function (Hi(t)) (see Appendix B), which makes Ii−1(Ti−1)|Ft lognormally distributed under QTi.
The YYIIS pricing function can now be given by a fully-analytical formula and depends on the (instantaneous) volatilities of the nominal forward rates, the (instantaneous) volatilities of the forward inflation index and their correlations. A drawback of this model is that this approximation, the freezing of the drift of the forward inflation dynamics, may be rough for long maturities (Mercurio, 2005). This especially holds for nonzero correlation between nominal forward rates and forward inflation indices (ρI,n,i) (Malvaez, 2005). A comparison
between his models and the Jarrow-Yildirim model was presented by Mercurio, from which he concluded that the models are equivalent for YYIIS, however could imply different prices for away-from-the money derivatives. Kruse (2011) stated that another limitation of these lognormal forward CPI models is that the macroeconomic concept of Fisher, relating the interest rates and inflation, is not reflected.
3
CVA calculation by simulation
3.1
Default probability and loss given default
Hull and White (2012) state that default probabilities (qi) for defaults between time ti−1and
ti, are usually calculated from CDS spreads observed in the market. A good estimate of the
risk-neutral default probability between time 0 and ti is 1 − exp[−sLGDiti] where si denotes the
credit spread for maturity ti (Hull and White, 2012). It follows that
qi = exp[
−si−1ti−1
LGD ] − exp[
−siti
LGD], (44)
where it should be noted that the qis are unconditional default probabilities. Ahlberg (2013)
states that a differentiation can be made between dynamic modelling of the default proba-bilities and a static default curve. The above presented methodology is static. According to Brigo et al. (2013), for the dynamic model, the default intensity γ(t) = s(0,t)LGD can be modelled using e.g. (positive) short rate models. Then, qi = exp[−
Rti
ti−1γ(t)dt]. For a static default
curve, these default intensities are a deterministic function of time. The static default curve is much simpler, since no short rate model for γi has to be specified for which e.g. the
volatility of CDS spreads is a necessary input. Modelling the default intensities dynamically has the advantage that it provides the possibility to include wrong-way risk e.g. by the inclusion of market risk factors. In this thesis it is chosen not to incorporate wrong-way or right-way risk as will be explained in section 3.2, and a static default term structure is chosen. In order to estimate the risk-neutral default probabilities from the CDS spreads, also the LGD has to be known. It is practice to assume a constant LGD since the true recovery rate is known only a considerable time after the default occurs and since this is in line with how CDSs are priced (Ahlberg, 2013). In this thesis we will assume a recovery rate of R=40%, following the standard ISDA (International Swaps & Derivatives Association) conventions (ISDA and Markit, 2009). This results in a LGD of 60%. It should be noted that the LGD scales the CVA value linearly and hence plays an important role. If no CDS data is available, other indicators of creditworthiness, like traded debt, could be considered or an approximation based on comparable companies could be employed.
3.2
Independence of components
In section 2.1.2 it was explained that the CVA depends on both credit and market risk factors and that often the simplifying assumption of independence between these components is made. In our case of inflation swaps, this yields the independence between the default probability and inflation and interest rates. This is a strong assumption and could lead to right-way or wrong-way risk. However, modelling the dependence requires in-dept knowledge of the business of both parties. Therefore, the independence assumption is followed in this thesis.
3.3
Exposure profile
price the ZCIIS and YYIIS. His methodology is outlined here.
For the ZCIIS, the value at time t, 0 ≤ t ≤ T is (for t and T in years) ZCIIS(t, T, I(0), N ) = N EQ e−RtTrn(s)ds I(T ) I(0) − (1 + K) T |Ft . (45) Since I(0) is known and I(t)Pr(t, T ) = EQ
h
e−RtTrn(s)dsI(T )|Ft
i
, the no arbitrage price at time t = 0 is
ZCIIS(0, T, I(0), N ) = N I(0)
I(0)Pr(0, T ) − Pn(0, T )(1 + K)
T
= NPr(0, T ) − Pn(0, T )(1 + K)T . (46)
For the YYIIS, the time t value of the fixed and floating legs with payoffs as given in equations (11) and (12) are given by Mercurio (2005) as
Y Y IISf ixed(t, T , K, Φ, N ) = M X i=m E[e− RTi t rn(s)dsN φ iK|Ft] = N K M X i=m φiPn(t, Ti), (47) where m = min{i : t < Ti}, M = max{j : Tj ≤ T }, Φ = {φ1, ..., φM} and T = {T1, .., TM},
and Y Y IISf loating(t, T , Ψ, N ) = N M X i=m ψiE[e− RTi t rn(s)ds I(Ti) I(Ti−1) − 1 |Ft], (48) where Ψ = {ψ1, ..., ψM}.
So, the time t value for each payoff at time Ti is given by
Y Y IISf loating(t, Ti−1, Ti, ψi, N ) = N ψiE[e− RTi−1
t rn(s)dsP
r(Ti−1, Ti)|Ft]−N ψiPn(t, Ti),
(49) What remains is the expectation E[e−RtTi−1rn(s)dsP
r(Ti−1, Ti)|Ft], which represents the time t
nominal value of a payoff equal to Pr(Ti−1, Ti) (in nominal units) at time Ti−1. If real rates
were deterministic this would not impose problems since then E[e−
RTi−1
t rn(s)dsP
r(Ti−1, Ti)|Ft] = Pn(t, Ti−1)Pr(Ti−1, Ti), (50)
The Jarrow and Yildirim model as outlined in section 2.2.5 with (extended Hull-White) Vasicek volatility, see equation (33), is chosen in this thesis to model the rates. This model is preferred over the lognormal forward CPI market models because of the severe conse-quences of the limitations of these models when used in simulation. The freezing of the drift as in Mercurio’s second market model could lead to problems for swaps with long maturities. Moreover, the connection between inflation and nominal and real interest rates as described by Fisher (1930) is essential for simulation (Kooistra, 2017) and is not reflected in the market models. As was stated in section 2.2.5, the main drawback of the Jarrow and Yildirim (2003) model is the calibration of the real rate parameters. We will analyze the consequences of this problem by testing the sensitivity of the exposure and CVA to these parameters. In order to get the discounted value of the forward price of the real bond in equation (49), Mercurio (2005) underlined that a further adjustment has to be made, since forwards are only martingales under their respective forward measure (Jamshidian, 1997). For example, Pr(Ti−2, Ti−1) and Pr(Ti−1, Ti) cannot be martingales under the same probability measure.
The floating payment I(Ti)
I(Ti−1) in equation (48) resets at time Ti−1and the numeraire is changed
from the money market account to the Ti−1nominal bond (Pn(t, Ti−1)) and hence the nominal
risk-neutral measure Q is changed to the Ti−1-forward measure QTi−1, so that e− RTi−1
t rn(s)ds
can be taken out of the expectation:
Y Y IISf loating(t, Ti−1, Ti, ψi, N ) = N ψiPn(t, Ti−1)EQ
Ti−1
[Pr(Ti−1, Ti)|Ft] (51)
−N ψiPn(t, Ti).
The forward prices of the real bonds with different maturities and measures can be corrected by a “convexity adjustment” (Belgrade et al., 2004), depending on the nominal and real interest rate volatility and their correlation to account for the change of measure. Mercurio (2005) states that this convexity adjustment is obtained by changing the dynamics for the real rate (equation (38)) to the Ti-forward measure:
drr(t) = [θr(t) − arrr(t) − ρr,IσrσI− ρn,rσnσrBn(t, Ti)]dt + σrdWQ
Ti
r (t). (52)
Then, the value of the floating payment of the YYIIS at time Ti is
Y Y IISf loating(t, Ti−1, Ti, ψi, N ) = N ψiPn(t, Ti−1)
Pr(t, Ti) Pr(t, Ti−1) eC(t,Ti−1,Ti)−N ψ iPn(t, Ti), (53) where,
C(t, Ti−1, Ti) = σrBr(Ti−1, Ti)[Br(t, Ti−1)(ρr,IσI−
1 2σrBr(t, Ti−1) (54) + ρn,rσn an+ ar (1 + arBn(t, Ti−1))) − ρn,rσn an+ ar Bn(t, Ti−1)]
So, E[e−RtTi−1rn(s)dsP
r(Ti−1, Ti)|Ft], the expected present value of a real zero-coupon bond
is equal to the forward price of the bond times a correction factor. The correction factor depends on the nominal rate, real rate and inflation volatilities and correlations and vanishes in case of deterministic real rates (since σr = 0).
The time t value of the swap, for the holder paying fixed, is obtained by summing all the payments: Y Y IIS(t, T , Φ, Ψ, N ) = N M X i=m ψiPn(t, Ti−1) Pr(t, Ti) Pr(t, Ti−1) eC(t,Ti−1,Ti) − N M X i=m ψiPn(t, Ti) − N K M X i=m φiPn(t, Ti) (55) (56) where for the first payments we can use the value of a ZCIIS:
Y Y IIS(t, T , Φ, Ψ, N ) = N ψm I(t) I(Tm−1) Pr(t, Tm) + N M X i=m+1 ψiPn(t, Ti−1) Pr(t, Ti) Pr(t, Ti−1) eC(t,Ti−1,Ti) (57) − N M X i=m (ψi+ φiK)Pn(t, Ti).
It can be seen that the price of the swap depends on the nominal and real bond price Pk(t, Ti) ∀k ∈ {n, r}, which depend on the corresponding interest rates and two
deter-ministic functions of time. Then, for each scenario, the exposure at time t is computed by
E(t, T ; Φ, Ψ, N ) = max(Y Y IIS(t, T ; Φ, Ψ, N ), 0). (58)
So, for a given contract, this framework provides the convenient characteristic that the exposure profile can be computed by simulating the nominal and real interest rates. Petrov (2015) states that the year fractions φi and ψi are usually around one, corresponding to a
yearly exchange of cash flows. This assumption will be followed in this thesis and we set φi = ψi = 1 for all i. So, the exposure at time t for the party paying fixed is
E(t, T , N ) = N ∗ max I(t) I(Tm−1) Pr(t, Tm) + M X i=m+1 Pn(t, Ti−1) Pr(t, Ti) Pr(t, Ti−1) eC(t,Ti−1,Ti) (59) − M X i=m (1 + K)Pn(t, Ti), 0 ,
3.4
Discount factor
Since the exposure in the future depends on the interest rates, special care should be taken when discounting. It is not possible to simply separate the discount factor and the future exposure since we have to account for the convexity effect (Gregory, 2012). This convexity effect arises since the nominal rate and the exposure are correlated. When the nominal rate and the exposure are high, a larger discount factor should be used. In order to do this, we discount per scenario, before calculating the average exposure of the scenarios. Jamshidian (1997) calls this “discounting along the path” and states that this is a characteristic of sim-ulation under the risk-neutral measure. Discounting the average exposure with the expected nominal interest rate would lead to overestimation of the CVA for the floating inflation re-ceiver.
For discounting along the path from time T back to time t0, the discount factor is
com-puted for every scenario j ∈ {J } by the continuously compounded nominal short rate as Dj(t0, T ) = e
−RT
t0rn,j(s)ds. (60)
3.5
Discretization
The evolutions as given in equations (41), (42) and (43) will be used to generate scenarios. However, before they can be applied in simulation, the first integral in the dynamics of the nominal and real rate is rewritten following Petrov (2015). Moreover, the Euler discretization method is applied, since it is not possible to simulate over continuous time. The derivation for the real rate can be found in Appendix C.3 and the nominal rate is analogous. Then for discrete time intervals the nominal rate, real rate and inflation rate dynamics are finally given by Petrov (2015) as
rn(t) = rn(s)e−an(t−s)+ Θn(t) − Θn(s)e−an(t−s)+
p
var(rn(t)|Fs) · Xn (61)
rr(t) = rr(s)e−ar(t−s)+ Θr(t) − Θr(s)e−ar(t−s)− (1 − e−ar(t−s))
ρr,IσrσI ar (62) +pvar(rr(t)|Fs) · Xr I(t) = I(s)e Rt s(rn(u)−rr(u))du− 1 2σ 2 I(t−s)+σI √ t−s·XI (63) where, Θk(t) = fk(0, t) + σ2 k 2a2 k (1 − e−akt)2 ∀k ∈ {n, r} (64) and X = Xn, Xr, XI 0
are the normally distributed random variables that play the role of the correlated Brownian motions, with correlation matrix Σ =
The variances of the processes are given by V ar(rk(t)|Fs) = σ2 k 2ak (1 − e−2ak(t−s)) ∀k ∈ {n, r}, (65) V ar(ln(I(t))|Fs) = σI2(t − s). (66)
The discrete approximation of the discount factor as defined in equation (60) is given by Dj(t0, T ) =
Y
u:tu∈{t0,T }
e−rn,j(tu)(tu−tu−1). (67)
We finally obtain the UCVA by summing over time and over all scenarios:
U CV A(t0, T , N ) = LGD ∗ X i:ti∈{t0,T } EQ[D(t0, ti)E(ti, T , N )|t0] ∗ qi (68) = 0.6 ∗ X j:j∈{J } X i:ti∈{t0,T } Ej(ti, T , N ) ∗ qi∗ Dj(t0, ti), (69)
where we filled in the value of the loss given default: LGD = 0.6.
4
Data
The inflation-linked derivative market is an over-the-counter market and consequently data availability is limited. Moreover, except for the inflation swap market, the derivative market is not liquid, leading to large bid-ask spreads and unreliable market prices. In this thesis, euro nominal swap data and bond data is used to compute the term structure of nominal interest. The HICPxT, published by Eurostat, is used as inflation index. Furthermore, zero-coupon inflation-linked swaps are used to compute the term structure of real interest. The data is obtained from Datastream.
4.1
Inflation data
For the Euro area, the HICPxT is used as the underlying for the inflation swap market. The history of the HICPxT with reference year 1996 is depicted in Figure 6. Especially in the post crisis years some seasonality effects can be observed. The corresponding history of the 12-month inflation rate is shown in Figure 7. It can be observed that the inflation rate has been relatively low over the entire period starting in 1996 upto 2018 and even deflation occurred following the 2008 financial crisis and the eurocrisis.
4.2
Nominal term structure
Figure 6: History of the HICPxT, with reference year 1996=100
Figure 7: History of the HICPxT 12 month inflation rate
approximate the nominal risk-free rate, Euro overnight index swap (OIS) rates are used. This swap is currently seen as the instrument closest to risk-free in the market (Hull and White, 2013). However, since the OIS quotes are available only for a range of maturities from one week to ten years, it is necessary to extend the yield curve to longer maturities. It was chosen to use the German government nominal zero-coupon Euro swap curve for this purpose, since the German Bunds are considered as very reliable. The maturities range from 1 year to 50 years. The OIS curve and the German bond swap curve have overlapping maturities from 1 year to 10 years. Over this time period the average difference between the rates, i.e. the additional risk premium of the German bond over the OIS, is taken. Then for the longer maturities, the OIS curve is approximated by decreasing the German bond curve by this additional risk premium. In Figure 8 the OIS curve, the German bond swap curve and the extended OIS curve are shown. The bond price is then given by
Pn(t, T ) = e−Rn(t,T )(T −t) (70)
and shown in Figure 13.
In section 5, the calibration of the parameters to the data will be explained. The parameters will be estimated using historical data. For this goal it is important that a representative sample is used. In Figure 9 the historical values of the one month OIS rate are shown. It can be seen that during and after the 2008 financial crisis this rate dropped significantly. Therefore, only data from the first of January 2010 and onward is used for the calibration of the model. For consistency, only the OIS data is used for calibration, hence only the rates upto 10 years.
4.3
Real term structure
Figure 8: Extending the yield curve based on market data the 29th of June 2018
Figure 9: History of the 1 month OIS rate Figure 10: History of the 1 year real rate
structure and the market ZCIIS quotes, which are denoted by K = K∗(T ) and ensure the ZCIIS trades at par at time t = 0 (Mercurio, 2005). From equation (46), Mercurio (2005) derived that
ZCIIS(0, T, I(0), N ) = 0 = NPr(0, T ) − Pn(0, T )(1 + K∗(T ))T
⇐⇒ Pr(0, T ) = Pn(0, T )(1 + K∗(T ))T (71)
Figure 11: Market quotes of the ZCIIS rate on the 29th of June 2018
Figure 12: The real yield curve based on mar-ket data on the 29th of June 2018
Figure 13: Nominal bond prices on the 29th of June 2018
Figure 14: Real bond prices on the 29th of June 2018
4.4
Default data
The default probabilities are provided by Zanders1 and obtained following standard ISDA
methodology with a recovery rate of 40%. Three term structures of monthly default proba-bilities are shown in Figure 15: the Dutch government default probability curve, the default probability curve of a Dutch bank and the default probability curve of a German bank. It can be seen that both the level and the shape of the curves are significantly different. The Dutch government has the lowest default probability. Also, the curve is relatively flat with a jump in 2023 and a smaller jump in 2026. For the Dutch bank, the default probability is low initially and increasing until 2029. The German bank is less creditworthy initially and the default probability is increasing the first years. Afterwards, the default probability is decreasing and at the end of the time horizon equal to the Dutch bank.
Figure 15: The default probability curves based on CDS market data on the 29th of June 2018
5
Calibration
The model that was introduced in section 2.2.5 with volatility function as specified in equa-tion (33) includes eight parameters: an, ar, σn, σr, σI, ρn,r, ρn,I, ρr,I. Ideally, these
parame-ters are calibrated on market prices, for example on inflation-linked caps or floors. However, since we could not obtain this data, the parameters will be calibrated using time series of the HICPxT and the nominal and real historical term structures constructed as described above, following the methodology as proposed by Jarrow and Yildirim (2003).
From the dynamics of the nominal and real bond prices under the risk-neutral measure Q in equations (30) and (31) it is derived that the bond returns evolve according to a normal distribution ∆Pn(t, T ) Pn(t, T ) − rn∆t ∼ N 0, Z T t σn(t, s)ds 2! , (72) ∆Pr(t, T ) Pr(t, T ) − rr(t) + ρr,IσI(t) Z T t σr(t, s)ds ∆t ∼ N 0, Z T t σr(t, s)ds 2! . (73)
If daily observations are used, ∆t = 2601 , it is possible to neglect the expected return on the bond in the estimation since it is small compared to the standard deviation (Jarrow and Yildirim, 2003). This simplifies the estimation of especially the real parameters since it is no longer necessary to estimate σI and ρr,I first. So, now we have
var ∆Pk(t + ∆, T ) Pk(t, T ) = σ 2 k(e−ak(T −t)− 1)2∆t a2 k ∀k ∈ {n, r}. (74)
series of the nominal or real zero-coupon bond prices and is shown by the dots in Figures 16 and 17. Then, the parameters (an, ar, σn, σr) can be estimated by non-linear regression
across the estimates of the variances of the nominal and real bond prices of different matu-rities. Note that an, ar, σn and σr should be positive constants.
The estimate of the volatility of the inflation rate is computed as follows
ˆ σI = 1 ∆tvar ∆I(t) I(t) 12 . (75)
Here, ∆t = 121, since the HICPxT data is available only monthly. Finally, for the estimates of the correlations we have
ˆ ρn,r = corr ∆rn(t), ∆rr(t) , ˆρn,I = corr ∆rn(t), ∆I(t) I(t) , ˆρr,I = corr ∆rr(t), ∆I(t) I(t) (76) where again ∆t = 121. The resulting parameter estimates are
Estimated parameter Value Estimated parameter Value
ˆ an 0.049856 σˆI 0.0015908 ˆ ar 0.028533 ρˆn,r 0.18722 ˆ σn 0.0073579 ρˆn,I 0.11264 ˆ σr 0.010970 ρˆr,I -0.19067
By filling in these estimates for the parameters into the right hand side of equation (74), the fit can be shown and is depicted in Figures 16 and 17.
6
Results
In this section the results of the proposed methodology for the CVA calculation of YYIIS are presented. First, the scenarios that were generated for the nominal, real and inflation rate are provided. Then, the values, exposure profiles and CVA values for swaps with different characteristics regarding time to maturity, fixed rate and real rate parameters are given.
6.1
Generated scenarios
Using the discretized form of the model as specified in section 3.3, with parameter estimates as provided in section 5, J = 1000 scenarios were generated. Increasing the number of scenarios to 2000 hardly affected the results and it was concluded that 1000 scenarios is enough, at least to analyze the applicability of the model. To illustrate this, a comparison of the exposure profiles for J = 1000 and J = 2000 is given in Figure 39 in Appendix D. As an approximation for the short rate, the 1 year rate was taken, since for the ZCIIS and hence the real rate, no data was available for shorter maturities. Monthly time increments are chosen in this thesis. In general, a finer time grid could make the simulation more accurate, but is more time-consuming. Since we are mainly interested in swaps with long maturities and the floating rate is the yearly inflation, we consider a monthly time grid as fine enough. In Figures 18, 19 and 20, samples of 50 randomly chosen realizations of the nominal rate, the real rate and the inflation rate are shown for a time horizon of 300 months. The mean of the 1000 scenarios is depicted by the thick black line and the dashed black lines represent the 95% confidence interval. The nominal rate scenarios are slightly increasing over time. The mean rate seems to follow the nominal yield curve. It can be observed that the spread between the scenarios increases first and later stabilizes. For the real rate, the same pattern can be observed. However, the range is wider, which can be explained by the lower mean reversion and higher volatility parameter. The inflation rate scenarios continue to diverge over a longer period. The mean inflation is stable and low, which is conform the current low inflation environment. The confidence intervals and the median of the rates can be found in tables 2, 3 and 4 in Appendix D. The rates are normally distributed if viewed on a specific timepoint, since the nominal and real rate and the inflation dynamics follow (geometric) Brownian motions. This is illustrated in Figures 40, 41 and 42 in Appendix D. Since the rates are normally distributed and hence symmetrical and given the current low interest and inflation environment, a considerable part of the scenarios displays negative rates.
6.2
Exposure profile
Figure 18 Figure 19
Figure 20
The simulated nominal rate (Figure 18), real rate (Figure 18) and inflation rate (Figure 20) for 50 randomly chosen scenarios obtained by the model and parameters as defined in sections 3 and 5. The mean rate of the 1000 scenarios is depicted by the thick black line and the dashed black lines represent the 95% confidence interval. Time is depicted in months
In order to investigate if the model functions as expected, the behaviour is tested for different maturities T and for different fixed rates K. The profiles of the discounted exposure of the inflation receiver and payer for different maturities that trade at par are provided in Figures 27 and 28. The exposure increases with maturity for both payers, which is conform expectations. It is observed that the peak exposure of the swap with a maturity of 300 months is roughly three times as high as the peak exposure of the swap with a maturity of 120 months. This stresses the importance of CVA calculations for swaps with longer maturities. For values of K 6= 0.0132, the value of the swap at initiation is not equal to zero. For lower K, the initial value is positive and for higher rates the value is negative. This is illustrated in the profiles of the discounted exposure of the inflation receiver and inflation payer for different fixed rates K in Figures 29 and 30. The maturity is set to 300 months for all considered K. For the inflation receiver, it can be observed that for K = 0 the exposure is already considerable at initiation. For rates K above the par rate, the value is negative and hence the initial exposure is zero. For the inflation payer this is vice versa. Then, the exposure is considerable for the relatively high K and the exposure for K = 0 is low. As was explained before, the main shortcoming of the Jarrow and Yildirim model is the difficulty of the estimation of the real rate parameters. In order to analyze the sensitivity of the model towards these parameters, the model was implemented for different values of ar and σr independently, for maturity T = 300 and with fixed rate K set to 0.01. The
parameters ar and σr are set to 0.5, 0.75, 1, 1.5 and 2 times the estimated values.
Figures 31 and 32 show the exposure profiles for varying values of the real rate mean re-version parameter ar. For higher ar, the exposure is lower than for lower mean reversion.
A higher mean reversion parameter causes a lower volatility of the real rate and hence also a lower volatility of both inflation and the value of the YYIIS. Since the exposure has an “option like” relationship with the value of the swap, it is lower for less volatile swap values. Another observation that can be made is that the decrease in exposure is stronger for the party paying the floating rate. An explanation is that a higher arcauses a lower real rate and
hence a higher inflation rate (because of the macroeconomic concept as outlined by Fisher (1930)). Therefore, the value of the YYIIS increases for the inflation receiver. This increase in value and exposure offsets the decrease in exposure because of the lower volatility partially. From Figures 33 and 34 it can be concluded that the exposure of both parties is sensi-tive towards changes of the real rate volatility σr, where the effect is much stronger for the
inflation payer. For the party receiving the floating inflation, the overall exposure decreases slightly as σr increases. This can be explained since for a higher real rate volatility parameter
σr, the real rate will be higher, causing a lower inflation rate and hence a lower value and
exposure for the inflation receiver. However, this effect is offset by the higher volatility of the real rate and hence also of the inflation rate and value of the swap. For the floating payer, these two effects work in the same direction. Here, a higher real rate volatility parameter σr
and hence a lower inflation causes an increase in value. Together with the higher volatility of the value this results in a far higher exposure. For a lower σr, the effect works in opposite
direction. For a doubling of σr the maximum exposure increases from 0.126 to 0.451 and
Figure 21 Figure 22
Figure 23 Figure 24
Figure 25 Figure 26
100 Randomly chosen scenarios of the sim-ulated value over time for the holder of a swap, receiving floating inflation and paying fixed rate K = 0.0132, with maturity of 25 years (Figure 21) and the corresponding ex-posure for the inflation receiver (Figure 23) and payer (Figure 25). Time is depicted in months
The value profile of a swap (Figure 22), exposure profile for the inflation receiver (Figure 24) and inflation payer (Figure 26) paying fixed rate K = 0.0132 and with maturity of 25 years, calculated by taking
the mean of the 1000 scenarios. Time is
Figure 27 Figure 28
Figure 29 Figure 30
Figure 31 Figure 32
Figure 33 Figure 34
Above: The exposure profiles for different values of ar for the party
re-ceiving inflation (Figure 31) and paying inflation (Figure 32). Below: The exposure profiles for different values of σr for the party receiving inflation
6.3
CVA value
With the nominal rate, serving as discount factor, and the exposure as presented above, using the LGD as defined in section 3.1 and the default probabilities as presented in sec-tion 4.4 the UCVA can be computed. The value is calculated by summing over time and over all scenarios using equation (69). The resulting UCVA values are provided in Table 1. The base case values are computed for notional N = 1, maturity T = 300, trading at par (K = 0.0132), real rate mean reversion ar = 0.028533 and real rate volatility σr = 0.010970.
For all UCVA values the number of scenarios (J ) is set to 1000, except for the row J = 2000. For the different maturities (T ), the UCVA values are given for the swaps trading at par (note that K differs for each maturity) and ar and σr the same as in the base case. For the
different fixed rates (K), the maturity is set to 300 months and arand σrare set equal to the
base case. The UCVA values for the different real rate parameters ar and σr are provided
for T = 300 and K = 0.01.
From Table 1 and Figure 35, it can be observed that the UCVA increases strongly with time to maturity. For example, for the Dutch government as counterparty, if the maturity doubles from T = 120 to T = 240 months, the UCVA value is multiplied by roughly six. Also for the other counterparties and maturities, the UCVA increase is steeper than the increase in maturity. This is explained by the strong increase in discounted exposure for longer maturity swaps. The default probability also plays a role. The effect depends on the shape of the default probability. The influence of the different default probability struc-tures of the different counterparties is illustrated in Figure 36. Here, the CVA value for a Dutch bank and a German bank are compared with the CVA where the Dutch Government is the counterparty. Especially for the German bank the influence of the different default probabilities is clear. For T = 120 the CVA is around eight times as high as for the Dutch government, whereas for T = 300 this is only five times.
The UCVA for different values of the real rate mean reversion ar and volatility σr is
il-lustrated in Figures 37 and 38, where K = 0.01 and T = 300. For an increasing ar, the CVA
is slightly decreasing. This effect is stronger for the floating payer, following the exposure profile. Note that for the inflation receiver in swaps with a shorter maturity this might not be the case, since the exposure over the first payments is higher as ar increases. When
varying the real rate volatility, we find that the UCVA for the party paying fixed decreases as σr increases. For the party that pays the floating inflation however, the CVA increases
drastically as the real rate volatility increases. If σr is twice as high as estimated, the UCVA
Table 1: UCVA values for different settings of J , T , K and the real rate parameters ar and σr for the three different entities
NL Dutch bank German Bank
Paying Fixed Floating Fixed Floating Fixed Floating
Base case 0.0087 0.0091 0.0230 0.0235 0.0398 0.0395 J = 2000 0.0086 0.0090 0.0226 0.0234 0.0392 0.0391 T = 120 0.0009 0.0009 0.0029 0.0028 0.0074 0.0070 T = 180 0.0028 0.0027 0.0082 0.0078 0.0171 0.0160 T = 240 0.0056 0.0055 0.0153 0.0150 0.0284 0.0272 K = 0.00 0.0151 0.0047 0.0412 0.0116 0.0767 0.0179 K = 0.010 0.0100 0.0078 0.0267 0.0201 0.0473 0.0330 K = 0.020 0.0063 0.0122 0.0161 0.0324 0.0268 0.0568 K = 0.025 0.0048 0.0149 0.0123 0.0400 0.0198 0.0720 ar = 0.0143 0.0107 0.0094 0.0281 0.0242 0.0488 0.0401 ar = 0.0214 0.0102 0.0085 0.0270 0.0219 0.0471 0.0362 ar = 0.0428 0.0094 0.0063 0.0250 0.0161 0.0439 0.0266 ar = 0.0571 0.0092 0.0052 0.0244 0.0133 0.0431 0.0217 σr= 0.00548 0.0102 0.0025 0.0274 0.0063 0.0493 0.0100 σr= 0.00823 0.0101 0.0046 0.0268 0.0118 0.0476 0.0191 σr= 0.0165 0.0092 0.0163 0.0241 0.0426 0.0411 0.0724 σr= 0.0219 0.0082 0.0282 0.0212 0.0743 0.0356 0.1287
Figure 35: The CVA for the three counter-parties and different maturities
Figure 37: The CVA for the three counter-parties and different ar
Figure 38: The CVA for the three counter-parties and different σr
7
Conclusion and discussion
In this thesis we researched the credit value adjustment of year-on-year inflation-indexed swaps from a pricing or IFRS 13 “Fair Value Measurement” accounting perspective. We think we developed a promising methodology where realistic results are obtained. However, both CVA calculation and the arbitrage free pricing of inflation-linked derivatives are very complicated topics. Moreover, since the inflation derivative market is still an emerging and over-the-counter market, data availability is limited. Therefore some strong assumptions are made in this thesis and we acknowledge that our work can best be seen as a first attempt rather than a definitive “best practice” approach.
Since the goal of this thesis was to find a methodology that calculates CVA values for the YYIIS consistent with the market prices of related products, the risk-neutral pricing approach was assessed as the most suitable. This market price based approach already has strong assumptions itself. The default probabilities are based on CDS spreads and to cal-culate them the recovery rate is assumed to be constant and 40%. This simple approach is questionable, especially in financially difficult times. The model used for the calculation of the exposure and the discount factor is based on short rate models. In our approach the short rate follows an Ornstein–Uhlenbeck process and hence is assumed to be mean reverting and normally distributed. This has the consequence that, also given the current low inter-est and low inflation environment, negative nominal, real and inflation rates are generated. Moreover, the usually made assumption of independence between the default probability and market value and hence exposure is followed. It was necessary to make this strong assumption since integrating market and credit risk is be very complicated and requires in dept knowledge of the counterparty.
computed over time by simulating the nominal and real short rate and the CPI for each scenario. With the values for the scenarios at different timepoints, the exposure profile is computed. The model is based on the foreign exchange analogy, where the nominal rate can be seen as the domestic rate, the real rate as a foreign rate and the CPI acts as the exchange rate and follows the import macroeconomic concept outlined by Fisher. The resulting CVA values and exposure profiles were analyzed for different maturities and fixed rates and are promising in the sense that they follow the stylized concepts as found for regular interest rate swaps. As was expected, CVA calculation is especially relevant for swaps with longer maturities and that have significant positive value.
The main disadvantage of the Jarrow-Yildirim model is that the real rate parameters are hard to estimate. Our analysis shows that the CVA value is sensitive to these parameters. We found that the change in CVA value for the party paying the floating inflation rate is bigger when these parameters change than for the inflation receiver. This is because for the inflation payer a decrease in the real rate mean reversion or an increase in volatility both make the swap value more volatile and higher, whereas for the inflation receiver these effects offset each other partially. If the real rate volatility turns out to be higher than estimated, this could lead to serious mispricing of the counterparty credit risk. If the volatility doubles, the CVA could be four times higher than estimated.
All in all we think that our proposed methodology serves the goal of market price con-sistent CVA calculation for inflation-linked swaps well and that this thesis provides a good basis for further development. The model we proposed has shown to be able to follow the desirable properties and will also provide possibilities to consider portfolios of swaps, netting and collateral. Care has to be taken however with the estimation of the real rate parame-ters. We showed that the CVA value reacts strongly to different values of these parameparame-ters. Especially an underestimation of the real rate volatility can have drastic effects. Since the parameters were calibrated historically, only using post-crisis data, where both interest rates and inflation have been (artificially) low and stable, it could be that the real rate volatility turns out to be larger than was estimated, leading to misspecification of the CVA. This provides a good starting point for further research. Ideally the parameters are calibrated on traded assets. It would be interesting to research how calibration on e.g. inflation-linked options affects the estimation of the parameters and the resulting CVA value. Also, we have made some important assumptions when we calculated the exposure and these assump-tions provide good possibilities for further research. We chose the Jarrow-Yildirim model to simulate the rates. It would be interesting to compare the results of this model with the CVA calculated using the market model type of models. Finally, we assumed independence between the default probability and exposure. Research into this relationship can largely contribute to the correct pricing of counterparty credit risk for inflation-linked swaps.
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