Person A opens an IRA at age 25, contributes $2000 per year for 10 years, but makes no additional contributions
thereafter. Person B waits until age 35 to open an IRA and contributes $2000 per years for 30 years. There is
no initial investment in either case.
d. Determine the interest rate for which the two IRA's have equal value at age 65.
PLEASE HELP ME. i need to compare my answer with u.^^
To determine the interest rate for which the two IRA accounts will have equal value at age 65, we can use present value and future value formulas.
Let's start with Person A's IRA account, where they contribute $2,000 per year for 10 years, starting at age 25. We can calculate the future value of their contributions using the formula:
FV = P * [(1 + r)^n - 1] / r
Here, FV represents the future value, P is the annual contribution, r is the interest rate, and n is the number of years of contributions.
In Person A's case, P = $2,000, n = 10, and the initial investment is $0. Therefore, the future value of Person A's IRA account at age 65 is:
FV_A = $2,000 * [(1 + r)^10 - 1] / r
Now let's move on to Person B's IRA account, where they contribute $2,000 per year for 30 years, starting at age 35. Again, we can use the same formula to calculate the future value:
FV = P * [(1 + r)^n - 1] / r
In Person B's case, P = $2,000, n = 30, and the initial investment is $0. Therefore, the future value of Person B's IRA account at age 65 is:
FV_B = $2,000 * [(1 + r)^30 - 1] / r
To find the interest rate (r) for which the two IRA accounts have equal value at age 65, we can set FV_A equal to FV_B:
$2,000 * [(1 + r)^10 - 1] / r = $2,000 * [(1 + r)^30 - 1] / r
Now we can solve this equation for r. Since it involves a bit of algebraic manipulation, I'll provide you with the simplified result:
(1 + r)^10 = (1 + r)^30 / (1 + r)^20
(1 + r)^10 = (1 + r)^10 * (1 + r)^10 / (1 + r)^20
Canceling out the (1 + r)^10 term on both sides, we get:
1 = (1 + r)^10
Taking the 10th root of both sides, we find:
1^(1/10) = (1 + r)^(10/10)
1 = (1 + r)
r = 0
This means that in order for the two IRA accounts to have equal value at age 65, the interest rate would have to be 0%.
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