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3.2.3 Interest rates process and other parametres Interest rates

Moody's interest rate model is based on projecting a daily evolution of 3-month and 10-year term rates and linearly interpolating between them for rates of other tenors. Rates with tenors shorter than 3 months are assumed to be equal to the 3-month rate. 3-month and 10-year term rates follow a two-dimensional correlated Cox-Ingersoll-Ross (CIR) process, where R^s and R^l denote respectively 3-month and 10-year term rate processes:

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dR^s_t = ás(âs -- R^s_t)dt + ópR^s tdW s dR^l_t = ál(âl -- R^l_t)dt+ó JR^l_tdW_t^l

d (W^s,W^l)_t = ñdt

(Rs0, R^l ₀) = (r_s, rl)

In order to make sure that Euler's discretized sheme does not generate negative values for interest rates, the natural discretized CIR process is given below:

(R^s₀, Rl0) = (r_s, rl)
?k ? {1,..,T/Ät},

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(3.2)

R^s(kÄt) = |R^s((k -- 1)Ät) + ás(âs -- R^s((k -- 1)Ät))Ät + .. + ÄtR^s((k -- 1)Ät)Z1|

R^l(kÄt) = |R^l((k -- 1)Ät) + ál(âl -- R^l((k -- 1)Ät))Ät + .. + ó₀/ÄtR^l ((k -- 1)Ät)(ñZ1 + ñ²Z2)|

Recovery Rates

Default recovery rates for our N obligors are assumed to be random and follow marginal Beta distributions correlated through a one factor gaussian copula model. Given that the recovery rate RR_n of each obligor n follows a Beta distribution, it is characterized by its mean li_n and standard deviation ó_n. li_n and ó_n depend on the obligor's location, its type (corporate, sovereign,..) and the seniority of the CDS underlying reference obligation. The parameters á_n and O_n of the Beta(á_n, O_n) distribution are given below:

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?n ? {1, .., N}, { á_n = li_n ó2 ¹⁴¹1^-un 1 \

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O_n = (1 -- lin)(lin ón 2 1

Let RRG Law = N(0,1) denote the global recovery factor. The standard normal variable X_n describing the recovery rate of obligor n is given by:

X_n = V pGRR^G + V1 -- pGe_n

where p^G is a correlation parametre common to all obligors and en Law = N(0,1) the idiosyncratic recovery factor independent from the common factor RR^G and from all other idiosyncratic ones. The following proposition allows us to simulate a N-vector of recovery rates drawn from marginal Beta distributions correlated through a one factor gaussian copula:

Proposition 7. Recovery Rates Simulation

Let F;₇7¹ denote the invert of the cumulative distribution function of Beta(á_n, O_n)

law. Assume (RR^G, e1, .., €N) Law = N (0, _IN+1). Then the distributions of individual recovery rates are given by:

?n ? {1, .., N}, RRn ^Law= Fn ¹ (Ö(V pG RR^G + V1 -- pGe_n))