A 46-year-old man deposits a total of $2000 per period in an IRA until age 65. How much money will be in the account if the interest rate is 10% compounded semiannually with payments made at the end of each semiannual period?
To determine how much money will be in the account, we need to consider the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (the initial deposit)
r = the interest rate (in decimal form)
n = the number of times interest is compounded per period
t = the number of periods
In this case, the principal amount (P) is $2000, the interest rate (r) is 10% (0.10 in decimal form), the number of times interest is compounded per period (n) is 2 (semiannually), and the number of periods (t) is 65 - 46 = 19 (since the man will deposit each period until age 65).
Plugging in these values into the formula:
A = 2000(1 + 0.10/2)^(2*19)
Now, let's simplify the equation step-by-step:
A = 2000(1 + 0.05)^(38)
A = 2000(1.05)^(38)
Calculating the value inside the parentheses:
A = 2000(1.05^38)
A = 2000(5.42867544044)
Finally, we can find the future value of the investment:
A = $10,857.35
Therefore, there will be approximately $10,857.35 in the account at age 65 if the interest rate is 10% compounded semiannually with payments made at the end of each semiannual period.