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

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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
004
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 ...

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