To save money for his daughter's college tuition, Shen invests $258 every quarter in an annuity that pays 6% interest, compounded quarterly. Payments will be made at the end of each quarter. Find the total value of the annuity in 21 years.
We can solve this problem using the formula for the future value of an annuity:
\[A = P \times \left( \dfrac{(1 + r)^n - 1}{r} \right)\]
Where:
A = Total value of the annuity
P = Amount invested at the end of each payment period
r = Interest rate per period
n = Number of payment periods
In this case,
P = $258
r = 6% per year = 6/100 = 0.06 per quarter
n = 21 years * 4 quarters = 84 quarters
Let's substitute these values into the formula and calculate the total value of the annuity:
\[ A = 258 \times \left( \dfrac{(1 + 0.06)^{84} - 1}{0.06} \right) \]
Using a calculator, we find that
\[ A \approx \$27,855.76 \]
Therefore, the total value of the annuity in 21 years will be approximately $27,855.76.
To find the total value of the annuity in 21 years, we can use the formula for the future value of an annuity. The formula is:
FV = P * ((1 + r/n)^(n*t)-1) / (r/n)
Where:
FV = Future value of the annuity
P = Payment per period (in this case, $258 per quarter)
r = Interest rate per period (6%, or 0.06)
n = Number of compounding periods per year (4, since the interest is compounded quarterly)
t = Number of years (21)
Let's substitute the given values into the formula and solve for FV:
FV = 258 * ((1 + 0.06/4)^(4*21)-1) / (0.06/4)
First, let's calculate the value inside the parentheses:
(1 + 0.06/4)^(4*21) ≈ (1 + 0.015)^84 ≈ 1.015^84 ≈ 2.082980
Now, substitute the calculated value into the formula:
FV = 258 * (2.082980 - 1) / (0.015)
FV ≈ 258 * 1.082980 / 0.015 ≈ 18234.27
Therefore, the total value of the annuity in 21 years would be approximately $18,234.27.
To find the total value of the annuity in 21 years, we can use the formula for the future value of an annuity:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = future value of the annuity
P = periodic payment (in this case, $258)
r = interest rate per compounding period (in this case, 6% or 0.06), compounded quarterly
n = number of compounding periods (in this case, 21 years * 4 quarters per year = 84 quarters)
Substituting the given values into the formula, we have:
FV = $258 * [(1 + 0.06/4)^(21*4) - 1] / (0.06/4)
Calculating within the parentheses first:
FV = $258 * [(1 + 0.015)^(84) - 1] / 0.015
Now, let's calculate the value within the square brackets:
FV = $258 * [(1.015)^(84) - 1] / 0.015
We can calculate (1.015)^(84) using a calculator, which equals approximately 3.09568. Substituting this value into the formula:
FV = $258 * [3.09568 - 1] / 0.015
Simplifying further:
FV = $258 * 2.09568 / 0.015
FV ≈ $258 * 139.712
Finally, multiplying the periodic payment by the resulting factor:
FV ≈ $36,016.30
Therefore, the total value of the annuity in 21 years would be approximately $36,016.30.