Consider the following annuity scheme: regular payments of $200 are made every two months at the end of the month (in other words, there are six equally spaced payments over the year) into an account with a nominal rate of 6% compounded monthly. Note, this is not an ordinary annuity or an annuity due. What is the value of the annuity in 10 years? What is the present value of the annuity?
Find the amount (future value) of the ordinary annuity. (Round your answer to the nearest cent.)
$750/month for 18 years at 7%/year compounded monthly
To calculate the value of the annuity in 10 years and the present value of the annuity, we can follow these steps:
Step 1: Determine the periodic interest rate.
Since the nominal rate is given as 6% compounded monthly, we need to find the monthly interest rate. We can do this by dividing the annual nominal rate by the number of compounding periods in a year. In this case, the monthly interest rate is 6% / 12 = 0.5%
Step 2: Calculate the future value (FV) of the annuity.
To find the future value of the annuity in 10 years, we can use the formula for the future value of an ordinary annuity:
\[ FV = P \left( \left(1 + r\right)^n - 1\right) / r \]
Where:
FV = Future value of the annuity
P = Payment amount ($200)
r = Monthly interest rate (0.5%)
n = Total number of compounding periods (10 years x 12 months/year = 120 months)
Substituting the values into the formula, we get:
\[ FV = 200 \left( \left(1 + 0.005\right)^{120} - 1 \right) / 0.005 \]
Calculating this expression, we find that the future value of the annuity in 10 years is approximately $25,140.54.
Step 3: Calculate the present value (PV) of the annuity.
To calculate the present value of the annuity, we can use the formula for the present value of an ordinary annuity:
\[ PV = P \left(1 - \left(1 + r\right)^{-n}\right) / r \]
Where:
PV = Present value of the annuity
P = Payment amount ($200)
r = Monthly interest rate (0.5%)
n = Total number of compounding periods (10 years x 12 months/year = 120 months)
Substituting the values into the formula, we get:
\[ PV = 200 \left(1 - \left(1 + 0.005\right)^{-120}\right) / 0.005 \]
Calculating this expression, we find that the present value of the annuity is approximately $17,006.99.
Therefore, the value of the annuity in 10 years is approximately $25,140.54, and the present value of the annuity is approximately $17,006.99.