Nancy Bellow promised her son she would pay him $600 a quarterly for four years.If Nancy can invest her money in a 6% ordinary annuity, how much would she need to invest today
i = .06/4 = .015
n = 4x4 = 16
Present Value = 600( 1 - 1.015^-16)/.015
= 8478.88
To determine how much Nancy would need to invest today, we can use the formula for the present value of an ordinary annuity.
The formula is: PV = PMT x [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present Value (the amount Nancy needs to invest today)
PMT = Payment per period ($600)
r = Interest rate per period (6% or 0.06)
n = Total number of periods (4 years or 4 quarters)
Using the given values, we can substitute them into the formula:
PV = $600 x [(1 - (1 + 0.06)^(-4)) / 0.06]
Let's calculate it step by step:
1. Calculate the value inside the brackets: (1 - (1 + 0.06)^(-4))
This equals 1 - (1.06)^(-4) ≈ 1 - 0.79209 ≈ 0.20791
2. Divide the result from step 1 by the interest rate: 0.20791 / 0.06 ≈ 3.46517
3. Multiply the result from step 2 by the payment per period to get the present value:
$600 x 3.46517 ≈ $2,079.10
Therefore, Nancy would need to invest approximately $2,079.10 today in order to fulfill her promise to pay her son $600 quarterly for four years, assuming a 6% ordinary annuity.
To determine how much Nancy would need to invest today, we can use the formula for the present value of an ordinary annuity.
The formula for the present value of an ordinary annuity is:
PV = PMT * [1 - (1 + r)^(-n)] / r
Where:
PV = Present Value (the amount Nancy would need to invest today)
PMT = Payment amount per period ($600 in this case)
r = Interest rate per period (6% or 0.06 as a decimal)
n = Total number of periods (4 years, which is equivalent to 4 quarters)
Plugging in the values into the formula, we get:
PV = $600 * [1 - (1 + 0.06)^(-4)] / 0.06
Now, let's calculate this using a calculator or a spreadsheet:
PV = $600 * [1 - (1 + 0.06)^(-4)] / 0.06
= $600 * [1 - (1.06)^(-4)] / 0.06
≈ $600 * [1 - 0.82228] / 0.06
≈ $600 * 0.17772 / 0.06
≈ $3,543.60
Therefore, Nancy would need to invest approximately $3,543.60 today in order to fulfill her promise to her son.