You are 25 years old and inherit $65,000 from your grandmother. If you wish to purchase a $100,000 yatch to celebrate your 30th birthday, what compound annual rate of return must you earn?
To calculate the compound annual rate of return needed, we can use the future value formula:
FV = PV * (1 + r)^n
Where:
FV = Future value (the cost of the yacht)
PV = Present value (the inherited amount)
r = Annual rate of return (what we want to find)
n = Number of years
In this case, we have:
PV = $65,000
FV = $100,000
n = 5 years (from age 25 to age 30)
Rearranging the equation to solve for r, we get:
(1 + r)^n = FV / PV
Substituting the given values:
(1 + r)^5 = $100,000 / $65,000
Now, to isolate (1 + r), we take the fifth root of both sides:
1 + r = (100000 / 65000)^(1/5)
Now, subtract 1 from both sides to find r:
r = (100000 / 65000)^(1/5) - 1
Using a calculator, we can find:
r ≈ 0.0725
So, you would need to earn a compound annual rate of return of approximately 7.25% to afford the $100,000 yacht by your 30th birthday.
To determine the compound annual rate of return you must earn in order to purchase the yacht by your 30th birthday, we can use the future value formula for compound interest:
Future Value (FV) = Present Value (PV) * (1 + r)^n
Where:
FV = $100,000 (desired future value)
PV = $65,000 (current value/inherited amount)
r = compound annual rate of return
n = number of years (5 years from now - your 30th birthday minus your current age of 25)
Now let's solve for r:
$100,000 = $65,000 * (1 + r)^5
Divide both sides of the equation by $65,000:
$100,000 / $65,000 = (1 + r)^5
1.53846 ≈ (1 + r)^5
Now, we need to take the fifth root of both sides to isolate (1 + r):
(1.53846)^(1/5) = 1 + r
1.10204 ≈ 1 + r
Subtract 1 from both sides:
1.10204 - 1 ≈ r
0.10204 ≈ r
So, to purchase the $100,000 yacht by your 30th birthday, you must earn a compound annual rate of return of approximately 10.20%.