Factor models for CDOs

A CDO is a special transaction between investors and debt issuers through the inter-vention of a special purposed vehicle (SPV). The CDO structure serves as an efficient tool for banks to transfer and control their risks as well as decrease the regulatory capital. On the other hand, this product provides a flexible choice for investors who are interested in risky assets but have constrained information on each individual firm.

A CDO contract consists of a reference portfolio of defaultable assets such as bonds (CBO), loans (CLO) or CDS (synthetic CDO). A CDO contract deals in general with large portfolios of 50 to 500 firms. Instead of the individual assets of the portfolio, one can invest in the specially designed notes based on the portfolio according to his own risk preference. These notes are called tranches. There exist in general the eq-uity tranche, themezzanine tranchesand the senior tranche, and we suppose that the nominal amounts of tranches are denoted by N_E, N_M and N_S. The cash-flow of a tranche consists of interest and principal repayments which obey prioritization policy:

the senior CDO tranche which carries the least interest rate is paid first, then follow the lower subordinated mezzanine tranches and at last the equity tranche which carries excess interest rate. More precisely, the repayment depends on the cumulative loss of

the underlying portfolio which is given by Lt =Pn

i=1Ni(1−Ri)11_{τi≤t} where Ni is the nominal value of each firm. If there is no default, all tranches are fully repayed.

The first defaults only affect the equity tranche until the cumulative loss has arrived the total nominal amount of the equity tranche and the loss on the tranche is given by L^E_t =L_t11_[0,NE](L_t) +N_E11_]NE,+∞[(L_t) (3.1) Notice thatL^E_t =L_t−(L_t−N_E)⁺, which is the difference of two “call” functions with strike values equal 0 and N_E. The following defaults will continue to hit the other tranches along their subordination orders. The loss on the mezzanine tranche and the senior tranche is calculated similarly as for the equity tranche. Hence, for the pricing of a CDO tranche, the call function plays an important role.

On the other hand, the market is experiencing several new trends recently, such as the creation of diversified credit indexes like Trac-X, iBoxx etc. A Trac-X index consists in general of 100 geographically grouped entreprises while each one is equally weighted in the reference pool. All these credits are among the most liquid ones on the market, so the index itself reflects flexibility and liquidity. The derivative products based on this index are rapidly developed. The CDOs of Trac-X are of the same characteristics of the classical ones with standard tranches as [3%,6%], [6%,9%], [9%,12%], [12%,1], (or [12%,22%], [22%,1]) and the transaction of single tranche is possible.

We now present the factor model, which has become the standard model on the market for CDOs. The one-factor normal model has been first proposed by Andersen, Sidenius and Basu [1] and Gregory and Laurent [43]. It is a copula model which we introduce in Subsection 2.3.1.

We consider a static context and we neglect the filtration. The time horizon is fixed to beT. The default timesτ₁,· · ·, τ_nare defined as a special case in the Sch¨onbucher and Schubert’s model with the filtrationFbeing trivial. To be more precise, the default time is defined as the first time thatq_i(t) reaches a uniformly distributed thresholdU_i on [0,1], i.e.

τ_i= inf{t:q_i(t)≤U_i},

where q_i(t) is the expected survival probability up to time t. Clearly, q_i(0) = 1 and q_i(t) is decreasing. Moreover, q_i(t) can be calibrated from the market data for each credit.

The characteristic of the factor model lies in the correlation specification of the thresholdsU_i. Let Y₁,· · ·, Y_n andY be independent random variables whereY repre-sents a common factor characterizing the macro-economic impact on all firms and Y_i are idiosyncratic factors reflecting the financial situation of individual credits. Let

X_i =√ρ_iY +p

1−ρ_iY_i

be the linear combination ofY and Y_i. The coefficientρ_i is the weight on the common factor and thus the linear correlation betweenX_i andX_j is√ρ_iρ_j. The default

thresh-olds are defined by Ui = 1−Fi(Xi) where Fi is the cumulative distribution function of X_i. Then

τ_i = inf{t:√ρ_iY +p

1−ρ_iY_i ≤F_i⁻¹(p_i(t))}

wherep_i(t) = 1−q_i(t) is the expected default probability of creditibefore the timet.

It is obvious that conditioned on the common factorY, the defaults are independent.

The survival probability is P(τ_i > t) =P(√

ρ_iY +p

1−ρ_iY_i≥F_i⁻¹(p_i(t))) =p_i(t).

Conditioned on the common factor Y, we have P(τ_i ≤ t|Y) = F_i^{Y Fi−1(p}√^{i(t))−√ρiY} 1−ρi

where F_i^Y is the distribution function of Y_i. In particular, in the normal factor case where Y and Yi are standard normal random variables, Xi is also a standard normal random variable, then we have

p_i(t|Y) =P(τ_i ≤t|Y) =N

N⁻¹(p_i(t))−√ρ_iY

√1−ρi

. (3.2)

We note that although the Gaussian factor model is very popular among the practi-tioners, it can be extended without much difficulty to models containing several factors which can follow any distribution.

Each default before the maturity T brings a loss to the portfolio. Then the total loss on the portfolio at maturity is given by

L_T = Xn

i=1

N_i(1−R_i)11_{τi≤T}, (3.3) where N_i is the notional value of each credit i and R_i is the recovery rate. In the following, we suppose R_i is constant. Conditional on the common factor Y, we can rewrite

L_T = Xn

i=1

N_i(1−R_i)11

Yi≤^Fi−1(pi√^{(T))−√ρiY}

1−ρi

.

Hence, the conditional total loss L_T on the factor Y can be written as the sum of independent Bernoulli random variables, each with probability p_i(Y) = p_i(T|Y). In particular, for an homogenous portfolio where N_i, R_i and p_i(Y) are equal, L_T is a binomial random variable.

The common factor Y follows certain distribution. Denote by F(y) = P(Y ≤ y) the distribution function of Y. Then for any function h, if we denote by H(Y) = E[h(L_T)|Y], we have

E[h(L_T)] = Z

R

H(y)dF(y).

That is to say, we can study E[h(L_T)] in two successive steps. First, we consider the conditional expectationE[h(L_T)|Y] and second, we study the role played by the factor Y.

We recall that for a CDO tranche with the lower and upper barriers of the tranche A andB, the evaluation is determined by the loss on the tranche

L_T(A, B) = (L_T −A)⁺−(L_T −B)⁺. (3.4) Notice thatL_T(A, B) is the call spread, i.e. the difference between two European call functions. Hence, we are interested in calculating the expectation of the call function h(x) = (x−k)⁺.