ABSTRACT
This paper attempts to analyze bond portfolios through polynomial algebra. Viewing such a portfolio as a series of future payments, we argue that it is completely determined by its price-yield function. In a discrete-time model with smallest time unit δ, the price-yield function is a polynomial with real coefficients. It is a well-known algebraic result that such polynomials factor as a finite product of irreducible ones. Each factor represents a short duration portfolio, and these must be somehow related to the original portfolio. With this in mind, we define an instrument which has the effect of multiplying one bond portfolio by another. These “bond products” can be created using forward contracts. In effect, they amount to investing first in one portfolio and at maturity reinvesting the proceeds in the second. This concept gives meaning to polynomial factorizations of bond portfolios. Any such factorization will have the same price-yield function as the portfolio it factors. A consequence of this is that the duration and convexity of a portfolio can be computed in terms of the duration and convexity of the factors. In particular, the duration of a bond product is the sum of the durations of its factors. This raises the possibility of using factorizations to estimate sensitivity to changes in the shape of the yield curve, but limited numerical experimentation has so far proved inconclusive.
