Hull and White (2000)

3.1.1 A general analysis assuming defaults at discrete times We assume that we have chosen a set of N bonds that are either issued by the reference entity or issued by another corporation that is considered to have the same risk of default as the reference entity.² We assume that defaults can

2By the same risk of default we mean that the probability of default in any future time interval, as seen today, is the same.

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happen on any of the bond maturity dates. Later we generalize the analysis to allow defaults to occurs on any date. Suppose that the maturity of the ith bond is t_i with t₁ < t₂ < · · · < t_N. Here we follow Hull and White (2000) that we assume interest rates are constant, recovery rate are known, and claim amounts are known.³

Because interest rates are deterministic, the price at time t of the no-default value of the jth bond is F_j(t), where F_j(t) is the forward price of the jth bond for a forward contract maturing at time t assuming the bond is default-free (t < t_j). If there is a default at time t , the bondholder makes a recovery at rate ˆR⁴ on a claim of C_j(t). It follow that the present value of the lose, α_ij from a default on the jth bond at time t_i is

v(ti)[Fj(ti) − ˆRCj(ti)] (7)

where v(t_i) denotes present value of $1 received at time t_i with certainty.

There is a risk-neutral probability, p_i of default at time t_i which incurs the loss α_ij. The total present value of the loss on the jth bond is, therefore,

3It can be shown that, for either of these two assumptions, if default event, interest rates, and recovery rates are mutually independent, the following equations (7) and (8) are still true for stochastic interest rate, uncertain recovery rate , and uncertain default probability providing the recovery rate is set equal to its expected value in a risk-neutral world.

4The recovery rates can in theory vary according to the bond and the default time. We assume, for ease of exposition, that all the bonds have the same seniority in the event of default by reference obligation and that the expected recovery rate is independent of time.

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given by

where B_j denotes the price of the jth bond today and G_j denotes the price of the jth bond if there were no probability of default ( that is, the price of a treasury bond promising the same flows as the jth bond ). This equation allows the p’s to be determined inductively

p_j = G_j− B_j −Pj−1 i=1p_iα_ij

α_ij (9)

3.1.2 Extension to situation where defaults can happen at any time

The analysis used to derive equation (9) assumes that default can take place only on bond maturity dates. We now extend it to allow defaults at any time.

We assume that q(t) is constant and equal to q_i for t_i−1< t < t_i. Setting

β_ij = Z ti

ti−1

v(t)[F_j(t) − ˆRC_j(t)]dt (10)

a similar analysis to that used in deriving equation (9) gives

q_j = Gj − Bj −Pj−1 i=1qiβij

β_jj (11)

The parameters βij can be estimated using standard procedures, such as Simpson⁰s rule, for evaluating a definite integral.

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3.2 Duan (2010)

3.2.1 The hierarchical intensity model

For firm (i,j) at time t, which is the jth member of the ith group where i = 1, . . . , K and j = 1, . . . ni, we assume its default is following a process

dM_ijt= α_ijtdN_ct+ β_ijtdN_it+ dN_ijt (12)

where Nct, Nit, and Nijt are Poisson processes with intensities λct, λit, and λ_ijt, respectively. Moreover, λ_ct, λ_it, and λ_ijt are independent for all i’s and j’s. α_ijt and β_ijt are Bernoulli random variables taking value of 1 with probabilities p_ijt and q_ijt ( 0 with probabilities 1 − p_ijt and 1 − q_ijt ). We assume α_ijt and β_ijt are independent across different firms. The Poisson process N_ct is a common process shared by all firms, By the additively of independent Poisson processes, the equation (12) can be reduced to

dM_ijt= α^{d ∗}_ijtdN_ijt^∗ (13)

where = stands for distributional equivalence, N^d _ijt^∗ is a Poisson process with intensity λct+ λit+ λijt, and α^∗_ijt is a Bernoulli random variable taking value of 1 with a probability p^∗_ijt ( 0 with a probability 1 − p^∗_ijt ). It is clear that M_ijt is a Poisson process with intensity p^∗_ijt(λ_ct+ λ_it+ λ_ijt). If we look at a firm individually, the hierarchical intensity model is equivalent to the Duffie, et al (2007) model. Note that

p^∗_ijt = λ_ct

λ_ct+ λ_it+ λ_ijtpijt+ λ_it

λ_ct+ λ_it+ λ_ijtqijt+ λ_ijt

λ_ct+ λ_it+ λ_ijt (14) 7

Following Duan (2010), we let the Poisson intensities be the functions of some common state variables X_t, group-specific state variables Y_it and firm-specific factors Z_ijt. Thus we have

λ_ct= F (X_t−) (15)

λ_it= G(X_t−, Y_it−) (16)

λ_ijt= H(X_t−, Y_it−, Z_ijt−) (17) pijt= P (Xt−, Yit−, Zijt−) (18) q_ijt= Q(X_t−, Y_it−, Z_ijt−) (19)

where i = 1, . . . , K, j = 1, . . . , n_i, and t₋ denotes the left time. F, G, and H must be non-negative functions. P and Q must be bounded 0 and 1. In practice, one can only observe discretely sampled data, and t− means using the data available at time t − ∆t.

3.2.2 Maximum likelihood estimator

We just need to estimate default intensities, so our the log-likelihood function is a special case of Duan (2010) the log-likelihood function. Following Duan (2010), we also assume ϕ are the parameters governing F, G, H, P, and Q functions. Let D_T be the data set related to X_t, Y_it, and Z_ijt from time 1 to time T and I_t be a matrix with rows respecting different groups and the column dimension equals the maximum number of firms in groups. This matrix corresponding the status of all firms. Prior to default for a firm, its corresponding entry in I_t is assigned to 0 otherwise it switches to 1. In order

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to reflect the time at which different firms enter the sample, we also use V, a matrix matching the dimension of I_t, to capture these entry time. Thus our log-likelihood function is

C_ijt⁽⁴⁾ =1{V (i,j)>t−1}+ 1{V (i,j)≤t−1}{1_{{It−1(i,j)6=0}}

+ 1{It−1(i,j)=0}1{It(i,j)=0}(1 − q_ij(t−1))(1 − p_ij(t−1))e^{−λij(t−1)∆t} + 1{I_t−1(i,j)=0}1{I_t(i,j)=1}[p_ij(t−1)+ q_ij(t−1)+ (1 − e^{−λij(t−1)∆t})

− p_ij(t−1)(1 − e^{−λij(t−1)∆t}) − q_ij(t−1)(1 − e^{−λij(t−1)∆t}) + p_ij(t−1)q_ij(t−1)(1 − e^{−λij(t−1)∆t})]}

In order to implement the model, one must specify the intensity functions. In this paper, we let F (x₁, . . . , x_n) = e^{a0+a1x1+···+anxn}, since we know that from Duan (2010) it will make the log-likelihood function great than F (x1, . . . , xn) = ln(1+e^{a0+a1x1+···+anxn}). Similarly, the functions G and H are in the same form but allow for different coefficients. The default probability function corre-sponding to the common shock is the same as Duan (2010), p(x₁, . . . , x_n) =

1

1+e−b0−b1x1−···−bnxn. The default probability functions corresponding to the group-specific shock are similarly specified. Needless to say, the coefficients can be different.