Irr and equivalence of cash-flow streams, loans, and portfolios of bonds

We show, through a Linear Algebra approach, that a general deterministic cash-flow stream admits a given Internal Rate of Return (irr, either constant or time-varying) if, and only if, it can be replicated by a suitable portfolio of bonds, each with yield to maturity equal to that same irr. Five particular replicating portfolios are examined, including and generalizing other representations known from the the literature, which allow for a unified, irr-based, interpretation of apparently diverse objects. Considering the amortization of a loan as a particular case, further equivalences are found and lead to some original consideration.