Your friend claims that he invested $5,000 seven years ago and that this investment is worth $38,700 today. For this to be true, what annual rate of return did he have to earn? Assume the interest compounds annually.
5000 * (1+x)^7 = 38700
Solve for x
x = 33.96
To determine the annual rate of return, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment ($38,700)
P = the initial investment ($5,000)
r = annual interest rate (unknown)
n = number of times interest is compounded per year (assume once, as it compounds annually)
t = number of years (7)
Plugging in the values, we have:
38,700 = 5,000(1 + r/1)^(1*7)
Divide both sides of the equation by 5,000:
38,700/5,000 = (1 + r/1)^(7)
Now, we can simplify further:
7.74 = (1 + r)^7
Taking the seventh root of both sides:
(1 + r) ≈ 1.1517
Now, subtract 1 from both sides:
r ≈ 1.1517 - 1 = 0.1517
The annual rate of return that your friend had to earn is approximately 15.17%.
To calculate the annual rate of return, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, $38,700)
P = the principal amount (in this case, $5,000)
r = the annual interest rate (what we're trying to find)
n = the number of times interest is compounded per year (in this case, once annually)
t = the number of years (in this case, 7 years)
Let's substitute the given values into the formula and solve for r:
38,700 = 5,000(1 + r/1)^(1 * 7)
Divide both sides of the equation by 5,000:
7.74 = (1 + r)^(7)
Take the seventh root of both sides of the equation:
(1 + r) = 7.74^(1/7)
Subtract 1 from both sides of the equation:
r = 7.74^(1/7) - 1
Using a calculator, we find that:
r ≈ 0.1631 (rounded to four decimal places)
So, your friend had to earn an annual interest rate of approximately 16.31% to turn a $5,000 investment into $38,700 over seven years, assuming the interest compounds annually.