Semiannually deposits of $400 are made for 10 years into an annuity that pays 7% compounded semiannually
Oh, the joy of compound interest! Let's calculate how much money you'll have at the end of those 10 years, shall we?
Now, since the deposits are made semiannually and the interest is also compounded semiannually, we first need to figure out the number of periods. We have 10 years with two deposits per year, which means a total of 20 deposits.
The interest rate is a delightful 7%, compounded semiannually. We know that the interest rate for each period is 7% divided by 2, which equals 3.5%.
Let's crunch the numbers and see what we get. Drumroll, please...
After 10 years, with $400 deposited semiannually at an interest rate of 7% compounded semiannually, you'll have approximately *cue the confetti* $11,345.18!
Now, that's a number worth celebrating! Just remember, don't spend it all on squirting flower lapel pins.
To calculate the value of an annuity with semiannual deposits over a given period, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r/n)^(nt) - 1) / (r/n)
Where:
FV = future value of the annuity
P = periodic payment (deposit)
r = annual interest rate
n = number of compounding periods per year
t = number of years
In this case, we have:
P = $400 (semiannual deposit)
r = 7% (annual interest rate)
n = 2 (compounded semiannually)
t = 10 years
Plugging in the values, we can calculate:
FV = 400 * ((1 + 0.07/2)^(2*10) - 1) / (0.07/2)
To simplify, let's perform the calculations step by step:
Step 1: Calculate the periodic interest rate (r/n):
Interest rate per period = 7% / 2 = 0.07/2 = 0.035
Step 2: Calculate the number of compounding periods (nt):
Number of compounding periods = 2 * 10 = 20
Step 3: Calculate the future value (FV):
FV = 400 * ((1 + 0.035)^(20) - 1) / (0.035)
Now let's evaluate it:
FV = 400 * ((1.035)^(20) - 1) / (0.035)
≈ 400 * (1.80717 - 1) / 0.035
≈ 400 * 0.80717 / 0.035
≈ 322.868 / 0.035
≈ 9225.942
Therefore, the value of the annuity after 10 years with semiannual deposits of $400 and an interest rate of 7% compounded semiannually is approximately $9225.942.