If a reference portfolio of 3 investment grade names with the following 1-yr CDS rates:c(1) = 56bps, c(2)=80bps, c(3)=137bps, The recovery rate is the same for all names at R=25%Average spread = 91bps (56+80+137)/3The notional amount invested in every CDO tranche is $1. Consider the questions:(a) what are the corresponding default probabilities? 91bps/(1-25%)=0.01213, correct?(b) How would we use this information in predicting defaults? don't understand(c) Suppose the defaults are uncorrelated, what is distirbution of number of defaults during 1 year? symmetric?(d) How much would the 0-33% tranche loss under these conditions?(e) Suppose there are 3 tranches: 0-33%33-66%66-100%How much would each tranche pay over a year?(f) Suppose the default correlation goes up to 50%, answer questions (c)-(e) again.

a) no s/(1-R) = hazard_rate = lSo Proba_default = 1 - exp(-hazard_rate *T)l = 0.01213, p = Proba_default = 0.0588631c) P(N(T) = k) = (n,k) * p^(k) * (1-p)^(n-k) (N number of defaults)d)EL[0-33] = 33 * P(L > 33) + sum( k = 0 , round(33/(1-R)), k * (1 - R) * P(N(T) = k))etc .....