Let

be the value at time zero of a bond portfolio with payments due at times and let

be the Macaulay duration of the portfolio. Here

and are the forward rates. Let be the future value of the portfolio at time ,

Put

Suppose the forward rates change instantaneously to new values . If K is an upper bound for the change in the slope of the forward rate curve,

then the change in the portfolio value is bound from below by

(page 198)