  The minimum (worst case) value of a long-only portfolio of bonds, over a
convex set of yield curves and spreads, can be estimated by its sensitivities
to the points on the yield curve. We show that sensitivity based estimates are
conservative, \ie, underestimate the worst case value, and that the exact worst
case value can be found by solving a tractable convex optimization problem. We
then show how to construct a long-only bond portfolio that includes the worst
case value in its objective or as a constraint, using convex-concave saddle
point optimization.
