What is the size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 6.6%, compounded semiannually, so that the account will have a future value of $140,000 at the end of 17 years?
Accumulation value of an annuity (due)
Sn=P*((1+i)^n-1)/d
where P-the size of payments, n=17*2=34,
i=0.066/2=0.033, d=i/(i+1)=0.032
140000=P*63.1024733
P=2218.61
To calculate the size of the payments that must be deposited at the beginning of each 6-month period, we can use the formula for the future value of an ordinary annuity.
The formula is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV is the future value of the annuity,
P is the payment amount,
r is the interest rate per period (compounded semiannually),
n is the number of periods.
In this case, we know that the future value (FV) is $140,000, the interest rate (r) is 6.6% or 0.066 (since it's compounded semiannually), and the number of periods (n) is 17 years, which is equal to 34 semiannual periods (since there are two 6-month periods in a year).
Let's now substitute these values into the formula and solve for the payment amount (P):
$140,000 = P * [(1 + 0.066)^34 - 1] / 0.066
To find the payment amount (P), we multiply both sides of the equation by 0.066 and divide by [(1 + 0.066)^34 - 1]:
P = $140,000 * 0.066 / [(1 + 0.066)^34 - 1]
Using a calculator, we can evaluate the right-hand side of the equation to find the payment amount (P).
P ≈ $1,689.54
Therefore, the size of the payments that must be deposited at the beginning of each 6-month period is approximately $1,689.54.