and represents the optimal amounts of P(t, Ti) to include in the hedge portfolio when hedging P(t,T). This theorem is proved by differentiating equation 1 with respect to Δi and subsequently solving for Δi. The corollary is proved by substituting the result of theorem into equation 1.
which declines monotonically as N increases. the residual variance in corollary enables the effectiveness of the hedge portfolio to be evaluated. Therefore corollary is the basis for studying the impact of including different bonds in the hedged portfolio. for N = 1, the hedge parameter in the theorem reduces to
Equation 4 coincides with the ratio of the hedged parameters. When T1 = T, the hedge parameter equals minus one. Economically, this fact states that the best strategy to hedge a bond is to short a bond of the same maturity. This trivial approach reduces the residual variance in equation 1 to zero as ∆1 = −1 and P = P1 implies L1 = M11.