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Question 1170650: Calculate the present value of the following annuity streams:
a. $5,000 received each year for 5 years on the last day of each year if your investments pay 6 percent compounded annually.
b. $5,000 received each quarter for 5 years on the last day of each quarter if your investments pay 6 percent compounded quarterly.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's calculate the present value of these annuity streams.
**a) $5,000 received each year for 5 years at 6% compounded annually.**
This is an ordinary annuity, where payments are made at the end of each period.
* Payment (PMT) = $5,000
* Number of periods (n) = 5 years
* Interest rate per period (r) = 6% or 0.06
The formula for the present value of an ordinary annuity is:
PV = PMT * [1 - (1 + r)^-n] / r
Plugging in the values:
PV = 5000 * [1 - (1 + 0.06)^-5] / 0.06
PV = 5000 * [1 - (1.06)^-5] / 0.06
PV = 5000 * [1 - 0.747258] / 0.06
PV = 5000 * [0.252742] / 0.06
PV = 5000 * 4.21236666667
PV ≈ $21,061.83
**b) $5,000 received each quarter for 5 years at 6% compounded quarterly.**
This is also an ordinary annuity, but with quarterly payments.
* Payment (PMT) = $5,000
* Number of periods (n) = 5 years * 4 quarters/year = 20 quarters
* Interest rate per period (r) = 6% / 4 = 1.5% or 0.015
The formula for the present value of an ordinary annuity is the same:
PV = PMT * [1 - (1 + r)^-n] / r
Plugging in the values:
PV = 5000 * [1 - (1 + 0.015)^-20] / 0.015
PV = 5000 * [1 - (1.015)^-20] / 0.015
PV = 5000 * [1 - 0.742470] / 0.015
PV = 5000 * [0.25753] / 0.015
PV = 5000 * 17.1686666667
PV ≈ $85,843.33
**Answers:**
a. The present value of the annual annuity is approximately $21,061.83.
b. The present value of the quarterly annuity is approximately $85,843.33.
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