9.18 More on Price vs. Reinvestment Risk: Prelude

Theorem:

Duration is the point at which Price- and Reinvestment-Risks exactly offset one another. If one assumes an “instantaneous” change in yield, there will be no equal risk-offsets and the actual return an investor earns (i.e., the “Realized Compound Yield”) will be unequal to the expected yield (i.e., the “Yield to Maturity”) – unless the bond is sold at the Duration Point in time.

Proof:

This theorem will be proved in three consecutive steps. In the course of this discussion, we will assume that any change in Yield-to-Maturity will match up, percentagewise, with the new Reinvestment Rate.

By way of method, you will recall that initially – as a matter of mathematical definition and necessity – the YTM and the Reinvestment Rate must be the same. It is therefore reasonable to assume that the new YTM will equal the new Reinvestment Rate. This same assumption will be assumed in the following steps.

Step One:

If a simple Par bond is purchased and considered to have been instantaneously sold at the duration point, there will be perfect price- and reinvestment-rate offset. This is a necessary condition since we have not assumed any change in yields.

The offset can be observed by equating the RCY and YTM. It is however insufficient to prove our theorem because we will still observe perfect offset if the bond is sold at other than the Duration point.

Step Two:

Suppose we indeed assume an instantaneous change in yield. Now, we will have a non-Par bond. We will observe that if the bond is sold at the Duration point, there will be a perfect offset. RCY = YTM. This too is a necessary condition of our Theorem. It is however insufficient as we would also need to show – in the negative – that if this bond is sold at other than the Duration point there will be no precise offset and that the RCY will not equal the YTM.

Step Three:

We will assume both yields have changed and that the bond is sold at any point in time other than the Duration point. The RCY will not equal the YTM. This condition Is both necessary and sufficient. Case closed.

Read on intrepidly and return to this page when directed below for review and consolidation of thought and method.