Analytic CDO Tranche Price Formula

I present an analytical formula for the value of the jth tranche (of N tranches) that grants the holder ownership of the jth bond to default for the simplest case of a collateralized debt obligation (CDO) backed by N (typically 125 for many actual securities) zero-coupon bonds that all have the same spread and maturity; the same recovery; a single value for pairwise default-probability correlation, ρ, between any two bonds (i.e., a homogeneous pool); but also interest rates of zero, and then using the standard credit default swap market jump-to-default mathematical machinery (but not any stochastic process for credit). The critical theoretical step is to value the joint-survival probability of m bonds in an N bond homogeneous system with all pairwise default-probability correlations identical. Graphs of spread by correlation are presented for a selection of tranches in a variety of typical cases and demonstrate the power of this simple and fast CDO tranche pricing formula