To determine how much more you would earn in the first investment than in the second investment, we need to calculate the future value of both investments using the appropriate formulas.
Investment 1: $48,000 at 12% compounded annually for 30 years
The formula for future value (FV) when compounded annually is:
[
FV = P(1 + r)^n
]
Where:
- ( P ) = principal amount (initial investment)
- ( r ) = annual interest rate (as a decimal)
- ( n ) = number of years
For the first investment:
- ( P = 48,000 )
- ( r = 0.12 )
- ( n = 30 )
Calculating the future value:
[
FV_1 = 48,000(1 + 0.12)^{30}
]
[
FV_1 = 48,000(1.12)^{30}
]
Calculating ( (1.12)^{30} ):
[
(1.12)^{30} \approx 29.959
]
Now, calculate ( FV_1 ):
[
FV_1 \approx 48,000 \times 29.959 \approx 1,438,992
]
Investment 2: $48,000 at 6% compounded monthly for 30 years
The formula for future value when compounded monthly is:
[
FV = P(1 + \frac{r}{m})^{mn}
]
Where:
- ( m ) = number of compounding periods per year (12 for monthly)
For the second investment:
- ( P = 48,000 )
- ( r = 0.06 )
- ( n = 30 )
- ( m = 12 )
Calculating the future value:
[
FV_2 = 48,000\left(1 + \frac{0.06}{12}\right)^{12 \times 30}
]
Calculating ( \frac{0.06}{12} = 0.005 ):
[
FV_2 = 48,000(1 + 0.005)^{360}
]
Calculating ( (1.005)^{360} ):
[
(1.005)^{360} \approx 6.022575
]
Now, calculate ( FV_2 ):
[
FV_2 \approx 48,000 \times 6.022575 \approx 289,082
]
Difference in Earnings
Now, we can find the difference between the two future values:
[
\text{Difference} = FV_1 - FV_2
]
[
\text{Difference} \approx 1,438,992 - 289,082 \approx 1,149,910
]
Conclusion
You would earn approximately $1,149,910 more on the first investment than on the second investment.
Final Answer: $1,149,910 (rounded to the nearest