In their model for credit returns, Rachev, Schwartz and Khindanova (2000) assumed a linear relationship between the returns of a risky credit instrument and the returns of a comparable risk-free credit instrument.
For such a credit instrumenti, the returns are described by
Ri=ai+biYi+Ui, (2)
where
• Ri are the log-returns of an assetithat is subject to credit risk;
• Yi are the log-returns of a risk-free asset;
• Ui is the disturbance. It represents the spread or the premium for the credit risk;
• ai andbi are constants which are obtained by ordinary least squares (OLS) estimation.
In this linear model, the returns of both the risky (Ri) and the risk-free (Yi) credit instru- ment are assumed to follow a strictly stable law. Moreover, the disturbance term (Ui) is a strictly stable random variable:
• Ui ∼Sα(σα, βα, àα), 1< α <2;
• Yi∼Sγ(σγ, βγ, àγ), 1< γ <2.
For credit instruments, the log-returnRi,t at timetis defined as Ri,t =log
Pi,t,T Pi,t−1,T−1
, (3)
wherePi,t,T is the price of an instrument i subject to credit risk with maturity date T evaluated at timet. The log-returns of the riskless assetYi,t are determined by
Yi,t =log
Bi,t,T Bi,t−1,T−1
, (4)
withBi,t,T as the price of the risk-free asset with maturity dateT evaluated at timet. This means that all prices used for the calculation of the returns are determined on the basis of constanttime-to-maturity. Therefore, the time series of log-returns (bothYi,t andRi,t) is calculated such that the time-to-maturity is the same for allt.
It must be noted that Yi,t and Ri,t are not directly observable for individual bonds whose market price movements are recorded on a daily basis. The pricesBi,t,T, Bi,t−1,T−1, Bi,t−2,T−2, . . . are calculated from the yield curve of riskless treasury bonds and Pi,t,T, Pi,t−1,T−1, Pi,t−2,T−2, . . . are derived from a yield curve generated from risky bonds representing a similar level of credit risk (e.g., having equal credit ratings). Such an approach enables us to deal with constant time-to-maturity. This is crucial, because for the prices of individual bonds, time-to-maturity decreases with increasing timet. However, a decreasing time to maturity does have an effect on the credit returns. Thus, the advantage of the approach in (3) and (4) is that we do not have to pay attention on the influence of a changing time-to-maturity.
Ch. 10: Stable Non-Gaussian Models 409
The effect of changing time-to-maturity on credit returns can be demonstrated by a small example with two riskless zero-bonds: the first one has a time-to-maturity of one year, the other one has a time to maturity of two years. Furthermore, the term structure is assumed to be flat, and therefore, both securities have equal yields. If the yield of both increases by the same percentage, then the price of the two-year bond reacts more sensitively, compared to the one-year bond.
However, the approach of modelling the returns as in (3) and (4) is very difficult to imple- ment in practice. Historical data of daily yield curves is available for treasury bonds, but it is practically impossible to observe a time seriesPt,T, Pt−1,T−1, Pt−2,T−2, . . .for an indi- vidual bond. We would have to define a number of different credit risk categories and assign individual bonds with different maturities to those.7We would use the prices of the bonds assigned to the same risk category in order to generate the corresponding yield curve.8
In order to avoid such difficulties, we look for a more practical way to define the credit returns. Obviously, a risk manager would prefer to deal with the observed real prices of a bond to fit a model, rather than deriving prices from yield curves that have to be gener- ated before. Moreover, each yield curve only represents an average credit risk level. Our approach proposed in the following paragraph determines the individual credit risk of the analyzed bond.
A new approach to define the returns Ri andYi. From the historical yield curve data of treasury bonds, we can construct daily prices for any riskless bond with given coupon, coupon dates, and maturity. Thus, we can generate a corresponding riskless bondi with identical specifications for each risky corporate bondi. We define the returnRi,t of a risky corporate bond as its actual (observable) daily price movement:
Ri,t=log Pi,t,T
Pi,t−1,T
. (5)
Here, time-to-maturity is no longer kept fixed. The returnRi,t is that of an individual bondiwith fixed maturity dateT. The riskless returnsYi,t are defined the same way:
Yi,t=log Bi,t,T
Bi,t−1,T
. (6)
This riskless bondihas the same specifications (maturity, coupon, coupon dates), as the risky bondi.
With this new approach, the original linear risk-return relation Ri =ai+biYi +Ui
remains, but its componentsRi,Yi, andUi now have a different meaning.Ri andYi are individual bond returns, and the disturbanceUiincorporates both credit spread and the risk of time-to-maturity.
7 For example, one can use the rating grades assigned by Standard & Poor’s or Moody’s to define a risk category.
8 For example, see McCulloch (1971, 1975).
410 B. Martin et al.
For all empirical examinations in this chapter, we used the model with the returns defined in (5) and (6). In the following, we present a brief summary of advantages and disadvan- tages of both approaches:
The model whose returns are defined by Equations (3) and (4), abandons the problem of changing time-to-maturity. This is its main advantage. The disadvantage of such an ap- proach is that yield curves have to be modelled for a number of different risk levels (e.g., corporate credit ratings), and for the risk free (treasury) bonds. After fitting the parameters aandbof Equation (2), we can simulate future scenarios for each yield curve integrating a model for the riskless returns. Such a framework would enable us to simulate future daily returns for each time-to-maturity. With the simulated yield curves, we would then be able to calculate the future returns of individual bonds.
With our model defined by the returns in (5) and (6) we neither need to construct yield curves for a number of risk levels (of risky bonds), nor do we have to simulate future representations of those yield curves by applying a complex term structure model. Thus, we can directly simulate future returns of individual bonds by generating representations ofYi andUi according to their fitted distributions.
The advantage of the chosen approach is that we can work with the actual historical price data and spread information of the individual bonds, instead of generating yield curves, each for a certain risk grade. Such yield curves only represent the average of a risk grade.
Studies found that in some cases a higher rated bond can even have a larger credit spread than bonds with a lower rating grade. This is due to the fact that the range of credit spreads within a given rating grade can be relatively wide and that the spread ranges of neighboring grades are usually overlapping. A reason for this effect could be that the market values the creditworthiness of an issuer differently than the rating agencies do. Sometimes the market can anticipate a change in the credit quality of an issuer before the rating agencies react.
The construction of a yield curve for a given credit grade usually requires data from a large number of bonds with various issuers. The yield curve of a single issuer is calculable even only for large corporations with many issued bonds.
2.1. Credit risk evaluation for single assets
In order to obtain the Credit Value-at-Risk (CVaR) for a bondiover a time horizon of one period, we perform the following steps:
• We create a corresponding risk-free treasury bond with equal maturity, coupon, and coupon dates.
• The estimates forai andbiare calculated with OLSE.
As in Rachev, Khindanova and Schwartz, the estimates are given by ˆ
ai= T
t=1Yit2T
t=1Rit−T
t=1YitT
t=1RitYit TT
t=1Yit2−(T
t=1Yit)2
, (7)
bˆi=TT
t=1RitYit−T
t=1Yit
T
t=1Rit
TT
t=1Yit2−(T
t=1Yit)2 , (8)
Ch. 10: Stable Non-Gaussian Models 411
wherei=1, . . . , N;t=1, . . . , T.
With the estimatesaˆi andbˆi, we obtain the residualsUi,
Ui=Ri− ˆai− ˆbiYi. (9)
• Finally, we perform a stable fit forUi andYi.
• In order to calculate the CVaR of assetifor one period, we simulate 1000 representations ofRi=ai+biYi+Ui.
2.2. A stable portfolio model with independent credit returns
Suppose there arendifferent credit instrumentsi(bonds) in a portfolio, and letvi be the weight of securityiwithin the portfolio.9The return of the portfolio is given by
Rp= n
i=1
viRi, (10)
with Rp=
n
i=1
vi(ai+biYi+Ui)= n
i=1
viai+ n
i=1
vibiYi+ n
i=1
viUi, (11)
and n
i=1
vi=1. (12)
Rpcan be expressed by
Rp= n
i=1
viai+Yp+Up, (13)
withYP andUpgiven by
Yp= n
i=1
vibiYi (14)
9 vican also be negative in case short-selling is allowed.
412 B. Martin et al.
and Up=
n
i=1
viUi. (15)
The constantapof the total portfolio is
ap= n
i=1
viai. (16)
As we assume theRi to be driven by independentα-stable distributions, this also means that both theUi and theYi,i=1, . . . , n, are independent of each other. We further assume that both theUi and theYi,i=1, . . . , n, are characterized by a common index of stability (αfor theUi,γfor theYi). A common stability index allows an easy analytical solution for the parameters of the distributions forUpandYp. For the properties of stable distributions, see Samorodnitsky and Taqqu (1994).
The common index of stability is calculated as an average from the stability indices of the distributions of the individualUi andYi, weighted according to formula (10):
α= n
i=1|vi|αi n
i=1|vi| (17)
and γ=
n
i=1|vi|γi
n
i=1|vi| . (18)
With the common stability index, the parametersβ,σ,àhave to be re-estimated for the individualUi andYi first.
The assumption of independent returns gives us an analytical solution for the portfolio’s UpandYp.
The parameters ofUpandYpare then determined by the following expressions:
σUp= n
i=1
|vi|σUiα
1/α
, (19)
βUp= n
i=1[sign(vn i)βUi(|vi|σUi)α]
i=1(|vi|σUi)α , (20)
àUp= n
i=1
viàUi, (21)
Ch. 10: Stable Non-Gaussian Models 413
σYp= n
i=1
|vibˆi|σYiγ
1/γ
, (22)
βYp= n
i=1[sign(vibˆi)βYi(|vibˆi|σYi)γ] n
i=1(|vibˆi|σYi)γ , (23)
àYp= n
i=1
viàYi. (24)
The portfolio’s returnsRpare given by (13).
2.3. A stable portfolio model with dependent credit returns
This section introduces a solution for modelling the dependence between credit returns on the one hand, and integrating the skewness-property of their distributions on the other hand.
Each variableUi andYi is split into a dependent symmetric and into an independent skewed component. Both components are independent of each other;
Ui=Ui(1)+Ui(2), (25)
Yi =Yi(1)+Yi(2). (26)
By the example ofUi, we show the derivation of the parameters for the two independent components. Both components are defined to have identical stability indices:
Ui(1)∼Sα(σ1, β1,0), (27)
Ui(2)∼Sα(σ2, β2,0). (28)
Because of the independence ofUi(1)andUi(2), the parameters’ values ofUi are calcu- lated as follows:
σ= σ1α+σ2α1/α
, (29)
β=β1σ1α+β2σ2α
σ1α+σ2α . (30)
Ui(1) is symmetric, thereforeβ1=0. We also set equal values for the scale parameters, σ1=σ2=σ∗.
Thus, the parameters ofUi are:
σ=21/ασ∗, (31)
414 B. Martin et al.
β=1
2β2. (32)
Summing up the results for the parameters, we have:σ1=σ2=σ∗=2−1/ασ,β2=2β (β2is for the skewed componentUi(2)), andβ1=0 (β1is for the symmetrical component Ui(1)),
Ui(1)∼Sα 2−1/ασ,0,0
, (33)
Ui(2)∼Sα 2−1/ασ,2β,0
. (34)
Analogously,Yi is split intoYi(1)+Yi(2), and their parameters are obtained the same way.
The return of the credit instrumentiis then given by Ri,t =a+b Yi,t(1)+Yi,t(2)
+ Ui,t(1)+Ui,t(2)
. (35)
The symmetric componentsYi,t(1)andUi,t(1)are used to incorporate the dependence among then assets. The dependence structure of the SαS10 vectors(U1(1), U2(1), . . . , Un(1))and (Y1(1), Y2(1), . . . , Yn(1)) is modelled by representing them as sub-Gaussian vectors. Thus, (U1(1), U2(1), . . . , Un(1))is represented as
U1(1), U2(1), . . . , Un(1)
∼ A1/2G1, A1/2G2, . . . , A1/2Gn
, (36)
whereAis a totally skewedα/2-stable random variable with
A∼Sα/2
cosπ α
4 2/α
,1,0
and G=(G1, G2, . . . , Gn) is an n-dimensional Gaussian zero mean random vector.
LetRij =EGiGj, i, j =1, . . . , n, be the covariances within the vector G=(G1, G2, . . . , Gn). Then(U1(1), U2(1), . . . , Un(1))is generated by simulating a representation of the Gaussian vectorGwith correlated elementsG1, G2, . . . , Gnand an independent represen- tation of theα/2-stable random variableA.11
The generation of vector(Y1(1), Y2(1), . . . , Yn(1))is performed analogously.
10ASαSvector is a symmetrically stable random vector.
11There are various ways to model the dependence. For example, see Rachev, Khindanova and Schwartz (2000).
Ch. 10: Stable Non-Gaussian Models 415