Interest Rate Swaps and Their Derivatives: A Practitioner's Guide

CHAPTER 1

Bonds: It′s All About Discounting

Before we delve into all the good stuff (swaps and options), let us review some fixed income basics.

TIME VALUE OF MONEY: FUTURE VALUE, PRESENT VALUE

Following the classical fixed income gospels, we remember that the Future Value, FV, on a horizon date of an investment PV at an annual interest rate of r , compounded m times a year, for N whole compounding periods is

FV = PV(1+r/m)^N

For example, if m = 1, we have annual compounding FV = PV(1 + r )^N, and N is the number of years until the future horizon date. If m = 2, we have semiannual compounding (standard for U.S. Treasury securities) FV = PV(1 + r /2)^N, and N = 2T is the number of whole semiannual periods until the horizon date (T years from now).

The above formula can be easily generalized to incorporate horizon dates that are not a whole number of compounding periods away. We compute T as the number of years between the investment date and the horizon date, according to some day count basis, and come up with:

FV = PV(1 + r/m)^Tm

From college math courses, we recall that as you increase the compounding frequency, the above, in the limit, becomes

and r is then referred to as the continuous compounding rate.

An alternative to using compounded rates is to use simple or noncompounding interest rates:

FV = PV(1 + r T)

where T is the number of years (can be fractional) to the horizon ...