$100 is deposited at the beginning of every week for five years in an account that pays 14%/a, compounded weekly.
a. What type of annuity is this?
b. Find the future value of the annuity using the formula.
b. Used online calculator.
Int. = $11,735.43
Future value = $37,835.43.
a. This is an example of an ordinary annuity because the deposits are made at the beginning of each week for five years.
b. To find the future value of the annuity, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r)^n - 1] / r
where:
FV = future value
P = periodic payment (in this case, $100 per week)
r = interest rate per compounding period (14% per year, compounded weekly, so r = 0.14/52)
n = number of compounding periods (5 years * 52 weeks per year)
Plugging in the values into the formula:
FV = $100 * [(1 + 0.14/52)^(5*52) - 1] / (0.14/52)
Calculating the exponent term first:
[(1 + 0.14/52)^(5*52)] = [(1 + 0.002692)^(260)] ≈ 1.95687
Now substituting the values back into the formula:
FV = $100 * (1.95687 - 1) / (0.14/52)
Simplifying further:
FV ≈ $100 * 0.95687 / (0.14/52)
FV ≈ $100 * 0.95687 * (52/0.14)
FV ≈ $36902.14
Therefore, the future value of the annuity is approximately $36902.14.