14.You need to have $15,000 in five years to pay off a home equity loan. You can invest in an account that pays 5.75 percent compounded quarterly. How much will you have to invest today to attain your target in five years?
5.75% compounded quarterly ---> i = .014375
principal( 1.014375^20) = 15000
principal = 11275.10
10,000
To calculate the amount you need to invest today to attain your target amount in five years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future value or target amount ($15,000 in this case)
P = the principal amount (the initial investment we're trying to determine)
r = the annual interest rate (5.75% or 0.0575 as a decimal)
n = the number of times interest is compounded per year (quarterly compounding, so 4 times)
t = the number of years (in this case, 5 years)
Plugging in the values, the formula becomes:
15,000 = P(1 + 0.0575/4)^(4*5)
Now, let's solve for P:
Divide both sides by (1 + 0.0575/4)^(4*5):
P = 15,000 / (1 + 0.0575/4)^(4*5)
Calculating the right-hand side of the equation:
P = 15,000 / (1 + 0.014375)^(20)
P = 15,000 / (1.014375)^20
Using a calculator or spreadsheet:
P ≈ 12,539.39
Therefore, you would need to invest approximately $12,539.39 today to attain your target amount of $15,000 in five years, considering the given interest rate and compounding frequency.