Find the present value of an annuity due that pays $4000 at the beginning of each quarter for the next 9 years.

Question

Find the present value of an annuity due that pays $4000 at the beginning of each quarter for the next 9 years. Assume that money is worth 6.6%, compounded quarterly. (Round your answer to the nearest cent.) 

Answer 1

Let's set the discount rate v.

This is the present value of 1 at a rate of 6.6% compounded quarterly for one quarter. In other words how much do I need presently to produce 1 after one quarter:

v = 1/ (1 + .066/4)

An annuity can be thought of as a geometric series:

PV = 4000 * v⁰ + 4000 * v¹ + 4000 * v² + ... + 4000 * v³⁵

(if pmt is at the beginning of each quarter then the first pmt is 4000 and the last pmt is discounted 35 quarters)

PV = 4000( v⁰ + v¹ + v² + ... + v³⁵ )

Use the formula for the sum of a geometric sequence:

PV = 109,708.78