**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.

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**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*= 2*T*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 ...Get *Interest Rate Swaps and Their Derivatives: A Practitioner's Guide* now with O’Reilly online learning.

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