Jim wants to deposit money in an account to save for a new stereo system in two years. He wants to have $4,000 available at the time. The following rates are available to him:
6.2% simple interest
6.1% compounded annually
5.58% compounded semiannually
5.75% compounded quarterly
Which account(s) should he choose if he wants to invest the smallest amount of money now? I was thinking the 5.57% compunded quarterly or 5.58% compounded semiannually
How much money must he invest to accumulate $4,000 in two years'time?
Imagine you start with $1 (or $1,000 if you prefer).
How much would you have at the end of two years, under each of the schemes?
For example, under the second, starting with $1, you would have
$1 * 1.06 = $1.061 at the end of the first year
$1.061 * 1.06 = $1.12466 at the end of the second year
So to end with $4000, you'd have to start with $4000 / 1.12466.
Do the same for the others, and you have your answer.
(And yes, it's one of those two.)
To determine which account Jim should choose in order to invest the smallest amount of money now, we need to compare the different interest rates and compounding options.
Let's calculate the future value using each option:
1) 6.2% simple interest: In this case, the future value can be calculated using the formula: FV = P(1 + rt), where FV is the future value, P is the principal (initial investment), r is the annual interest rate (as a decimal), and t is the time in years. Substituting the given values, we get FV = P(1 + 0.062 * 2).
2) 6.1% compounded annually: In this case, the future value can be calculated using the formula: FV = P(1 + r)^t, where FV is the future value, P is the principal (initial investment), r is the annual interest rate (as a decimal), and t is the time in years. Substituting the given values, we get FV = P(1 + 0.061)^2.
3) 5.58% compounded semiannually: In this case, the future value can be calculated using the formula: FV = P(1 + r/n)^(n*t), where FV is the future value, P is the principal (initial investment), r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the time in years. Substituting the given values, we get FV = P(1 + 0.0558/2)^(2*2).
4) 5.75% compounded quarterly: In this case, the future value can be calculated using the same formula as in case 3, but with different values for r and n: FV = P(1 + 0.0575/4)^(4*2).
Now, let's calculate the amount Jim needs to invest in each case to accumulate $4,000:
1) For the simple interest case, we need to solve the equation FV = P(1 + 0.062 * 2) = $4,000 for P.
2) For the compounded annually case, we need to solve the equation FV = P(1 + 0.061)^2 = $4,000 for P.
3) For the compounded semiannually case, we need to solve the equation FV = P(1 + 0.0558/2)^(2*2) = $4,000 for P.
4) For the compounded quarterly case, we need to solve the equation FV = P(1 + 0.0575/4)^(4*2) = $4,000 for P.
By comparing the results, we can determine which account(s) Jim should choose to invest the smallest amount of money now.