If $86,000 is invested in an annuity that earns 5.2%, compounded quarterly, what payments will it provide at the end of each quarter for the next 3½ years?
Principal, P = $86000
Compounding period : quarterly
interest, i = 5.2%/4 = 1.3% = 0.013
Number of periods, n = 3.5*4 = 14
R = payment per quarter
P = R(1-(1+i)^(-n))/i, or
R = Pi/(1-(1+i)^(-n))
=$86000*0.013/(1-1/(1.013)^14)
=$86000*(0.0785876)
=$6758.54
This answer makes no sense. the "1-1" part of the fraction would make this DNE. You cannot divide by zero.
This is how the equation should look:
86,000[0.013/([1-(1+.013)^(-14)])]
To find the payments provided at the end of each quarter, we need to use the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value
P = Payment
r = Interest Rate per period
n = Number of periods
In this case, the principal (initial investment) is $86,000, the interest rate is 5.2% (or 0.052) compounded quarterly, and the number of periods is 3.5 years * 4 (since the interest is compounded quarterly).
Substituting the values into the formula:
FV = $86,000 * ((1 + 0.052/4)^(3.5 * 4) - 1) / (0.052/4)
Let's calculate this using a calculator or a spreadsheet:
FV = $86,000 * ((1 + 0.013)^14 - 1) / 0.013
FV = $86,000 * (1.013^14 - 1) / 0.013
FV ≈ $100,405.43
So, the future value of the $86,000 investment after 3.5 years is approximately $100,405.43.
To find the payments provided at the end of each quarter, we can rearrange the formula:
P = FV * (r / ((1 + r)^n - 1))
Substituting the values:
P = $100,405.43 * (0.052/4 / ((1 + 0.052/4)^(3.5 * 4) - 1))
P = $100,405.43 * (0.013 / (1.013^14 - 1))
P ≈ $2,793.34
Therefore, the annuity will provide approximately $2,793.34 at the end of each quarter for the next 3.5 years.