Single and Dual Duration and Convexity Formulae: A Reference for Perpetuities, Annuities, Coupon Bonds, and Zero Coupon Bonds, and for Both Growing and Nominal Cash Flows, in Multiple Forms

Abstract

In this reference note, intended for easy reference use by finance generalists, we derive the formulae for a) the prices and dual and single durations and convexities for b) growing and nominal instruments, including c) coupon bonds, zero coupon bonds, annuities, and perpetuities, across d) the seven most useful and relevant combinations of coupon, discounting, and other cash flow details (forms)—all in a relatively compact presentation. These forms include 7 variations of:

• Continuous or discrete coupons

• Payment timing, i.e., payments made at the end of the period (most common) or payments made at the beginning of the period (annuity due)

• Growth beginning right away during the first period, or growth only beginning during the
second period (the conventional form for most growing perpetuities and annuities)

• Continuous or discrete compounding.