Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 6%/year compounded monthly. If the future value of the annuity after 10 yr is $55,000, what was the size of each payment?
From S(n) = R[(1+i)^n - 1]/i where R = the periodic payment, n = the number of periods, i = the periodic interest in decimal form and S(n) = the accumulated sum, you have S(n) = $55,000, n = 10(12) = 120 and i = 6/(100(12)) = .005.
Solve for R
To find the size of each payment in an ordinary annuity, we can use the formula for future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the future value of the annuity
P is the size of each payment
r is the interest rate per compounding period
n is the number of compounding periods
In this case, we are given:
FV = $55,000
n = 10 years
Since the interest rate is given as 6% per year compounded monthly, we need to convert it to the interest rate per compounding period.
The interest rate per compounding period can be found using the formula:
r_per_period = (1 + r_per_year)^(1/number_of_periods) - 1
In this case, the interest rate per period is given as 6% per year compounded monthly, which means there are 12 compounding periods per year.
r_per_period = (1 + 6%/12)^(1/12) - 1
Now we can substitute the values into the formula for future value:
$55,000 = P * ((1 + r_per_period)^n - 1) / r_per_period
Now we need to solve for P. We can rearrange the formula as follows:
P = FV * r_per_period / ((1 + r_per_period)^n - 1)
Substituting the values we calculated earlier:
P = $55,000 * r_per_period / ((1 + r_per_period)^n - 1)
P = $55,000 * (1 + 6%/12)^(1/12) - 1) / ((1 + 6%/12)^(10*12) - 1)
Now we can calculate P using a calculator:
P ≈ $554.56
Therefore, the size of each payment in the annuity is approximately $554.56.
To find the size of each payment, we can use the future value formula for an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value
P = Payment amount
r = Interest rate per period
n = Number of periods
In this case, the future value (FV) is given as $55,000, the interest rate (r) is 6% per year (or 0.06), compounded monthly, and the number of periods (n) is 10 years.
We need to convert the interest rate to a monthly rate. Since the interest is compounded monthly, the monthly interest rate (r_m) can be calculated as:
r_m = (1 + r)^(1/12) - 1
Substituting the given values:
r_m = (1 + 0.06)^(1/12) - 1
Now, we can calculate the size of each payment (P) using the future value formula:
$55,000 = P * ((1 + r_m)^(12 * 10) - 1) / r_m
Solving for P:
$55,000 = P * ((1 + r_m)^120 - 1) / r_m
Multiply both sides by r_m:
$55,000 * r_m = P * ((1 + r_m)^120 - 1)
Divide both sides by ((1 + r_m)^120 - 1):
P = ($55,000 * r_m) / ((1 + r_m)^120 - 1)
Now, we can substitute the value of r_m and calculate P:
P = ($55,000 * ((1 + 0.06)^(1/12) - 1)) / (((1 + 0.06)^(1/12))^120 - 1)
Calculating this expression will give us the size of each payment.