Final answer:
To reach $5000 in 10 years with a semiannual compound interest rate of 6%, one must deposit $2768.38 today. This calculation is made using the formula for compound interest with inputs respective to the given conditions.
Step-by-step explanation:
To find out how much money should be deposited today to accumulate to $5000 in 10 years at a 6% interest rate compounded semiannually, we need to use the formula for compound interest:
P = A / (1 + r/n)^(nt)
Where:
- P = principal amount (initial investment)
- A = amount of money accumulated after n years, including interest.
- r = annual interest rate (decimal)
- n = number of times that interest is compounded per year
- t = number of years the money is invested for
Given that A = $5000, r = 6% or 0.06, n = 2 (since interest is compounded semiannually), and t = 10, we can plug in the values:
P = 5000 / (1 + 0.06/2)^(2*10)
P = 5000 / (1.03)^20
P = 5000 / 1.80611
P = $2768.38
So, an amount of $2768.38 should be deposited today to have $5000 in 10 years with the interest compounded semiannually at a rate of 6%.