Nonlinear Duration

Treasury Bonds as a Mutual Fund

By Steven J. Grisafi, PhD.

Imagine the United States Treasury bond market as a mutual fund. If one knew the original issue of all extant treasury bonds one could compute the quantity known as duration for this market. Only the thirty year debenture is actually a bond; all other treasury securities are called notes. Knowing the duration for treasury bonds one would know the extent to which the dirty bond price would drop with a rise in its interest rate. So then what do we understand as the function that duration is performing? The duration is a linear proportionality constant! A mutual fund is a closed system. Although its capitalization can fluctuate, its manager knows exactly how much money flows into, and out of it, and which securities are responsible for these changes. Knowing the money flows creates a non-equilibrium stationary state. The distinction between equilibrium and non-equilibrium is irrelevant because our mathematical procedures solve the system equations for stationary points regardless of whether or not there exist non-zero flows into the system. Hence, the derivation of the mathematical description for the inverse relationship between a dirty bond price and its corresponding interest rate would be possible if the market is effectively closed.

To some extent, one could neglect losses to the market from individuals who sell their securities so as to stuff the proceeds of the sale under their mattresses. Consequently, one would only need to know the actions taken by large market share owners of the security. But is this linear proportionality constant we know as the duration a satisfactory measure of the inverse relationship between a dirty bond price and its interest rate? We ought to do better. If we had the capability to monitor the income stream of all extant debentures in a market then we know the ratio of the total payout to the total cost over the total lifetime of the market. This, in essence, is our mathematical description for the inverse relationship between dirty bond price and interest rate. To approximate it with a linear proportionality constant means that each and every time there are large scale capitalization changes to the market a modification to the proportionality constant is required. As a means of keeping mutual fund investors aware of changes to their portfolio, this is satisfactory. But the United States Treasury has the capability to do much better than this. To whom do they make this information available?