《3 - Time Value of Money - Part 2 (Annuities》.ppt

Quarterly (from 2.16b): FV in one year = $1,000 x (1.12551)1 = $1,125.51 The annual rate (EAR) from 2.16b produces the same FV as the quarterly rate from 2.16a, so the annual rate from 2.16b does reflect the quarterly compounding that is taking place. Monthly (from 2.16d): FV 1 year (12 mos.) = $1,000 x (1 + EAR)1  = $1,000 x (1 + .12683)1 = $1,126.83 Example 2.16 (Cont.) (Compounding within the year) 2-31; CN29 Note: continuous compounding uses the natural antilog function, e, since this is the value obtained from  1 + 1 m as m ? infinity m We calculate a periodic rate by chopping an APR into pieces to reflect compounding throughout the year. We then create a future value factor for the year by taking one plus the periodic rate and raising that to the number of periods within one year. Under continuous compounding we would be dividing the APR into an infinite number of pieces and raising one plus that number to an infinite power, so we take the limit as the number of compounding periods (m) approaches infinity (see above). 2-32; CN30 Thus, an EAR under continuous compounding, for a given APR, can be found as follows: EARcontinuous = eAPR – 1 EAR under continuous compounding for a 5% APR =e.05 – 1 = .05127 = 5.13% EAR A future value factor for a given APR under continuous compounding is simply eAPR for one year, and [eAPR]t for “t” years. Remember, to raise a power to a power, multiply the exponents together, so this equals eAPR*t 2-33; CN30 Compute the EAR for a 10% APR compounded continuously, and compute the future value of $1,000 invested at that rate for one year, and for five years. 2.17a. EAR: e.10 – 1 = 10.52% 2.17b. FV1: 1,000 x e.1 = 1,105.17 2.17c. FV5: 1,000 x [e.1]5 = 1,000 x e.5 = 1,648.72 Example 2.17 (Continuous compounding, EAR, and future value) 2-34; CN30 Review example 2.10. It