E6-10
Assume that Serena Williams desires to accumulate $1 million in 15 years using her market fund balance of $182,696. At what interest rate must Serena's investment compound annually?
I came up with 12,179.733 per year; but know this is incorrect.
For compounded annually
R = rate, n = number of years
FV = orig investment * ((1 + R)^n)
1,000,000 = 182,696 * (1 + R)^15
5 = (1 + R)^15
(15th root(5)) = 1 + R
1.11 = 1 + R
.11 = R
please check my math
To find the interest rate at which Serena's investment must compound annually, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Future value of the investment ($1 million)
P = Present value of the investment ($182,696)
r = Annual interest rate
n = Number of times interest is compounded per year
t = Number of years
We know that A = $1,000,000, P = $182,696, and t = 15.
The formula can be rearranged to solve for r:
r = ( (A/P)^(1/(n*t)) ) - 1
Plugging in the values, we get:
r = ( ($1,000,000 / $182,696)^(1/(1*15)) ) - 1
r = ( 5.4695 ) - 1
r ≈ 4.4695
So, the interest rate at which Serena's investment must compound annually is approximately 4.4695, or 446.95% per year.
Note: The interest rate seems high because the investment needs to grow from $182,696 to $1 million in 15 years, which requires significant growth.