If a savings fund pays interest at a rate of 10% per year compounded semiannually, how much money invested now will amount to $5000 after 1 year?
Please round the answer to the nearest cent.
To find out how much money you need to invest now, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value
P = the principal (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the future value (A) is $5000, the annual interest rate (r) is 10% or 0.10, the interest is compounded semiannually, so n=2, and t=1 year.
Now, we can plug in these values and solve for P:
5000 = P(1 + 0.10/2)^(2*1)
Simplify the equation:
5000 = P(1 + 0.05)^2
Square the value inside the parentheses:
5000 = P(1.05)^2
0.05 represents 5% (10% divided by 2), so (1.05)^2 is equivalent to multiplying the principal by (1 + 5%) twice.
Let's calculate (1.05)^2:
(1.05)^2 = 1.1025
Now, substitute this value back into the equation:
5000 = P(1.1025)
To solve for P, divide both sides of the equation by 1.1025:
P = 5000 / 1.1025
P ≈ $4,526.73 (rounded to the nearest cent)
So, you would need to invest approximately $4,526.73 now to have $5000 after 1 year, with an interest rate of 10% compounded semiannually.