Calculate the Doubling Time for Different Investment Scenarios
What is the doubling time for the following investments?
a) $5000 invested at 8%, simple interest. (Round to the nearest year.) Doubling Time = [Select ]
b) $5000 invested at 8%, compounded quarterly. (Round to the nearest quarter.) Doubling Time = [Select ]
c) $5000 invested at 8%, compounded continuously. (Round to two decimal places.) Doubling Time = [Select ]
Answer:
The doubling time for the three investments are as follows:
a) $5000 invested at 8%, simple interest: Doubling Time = 9 years.
b) $5000 invested at 8%, compounded quarterly: Doubling Time ≈ 9.01 quarters (rounded to the nearest quarter).
c) $5000 invested at 8%, compounded continuously: Doubling Time ≈ 8.66 years (rounded to two decimal places).
Explanation:
To calculate the doubling time for each investment, we will use the appropriate formula based on the type of interest.
a) Simple Interest:
For simple interest, we use the formula Doubling Time = 72 / r, where r is the interest rate.
Given that $5000 is invested at 8% simple interest, we can calculate the doubling time as follows:
Doubling Time = 72 / 8 = 9 years.
b) Compounded Quarterly:
For compounded interest, we use the formula Doubling Time = ln(2) / (n * ln(1 + r/n)), where n is the number of compounding periods per year.
Given that $5000 is invested at 8% compounded quarterly, we can calculate the doubling time as follows:
Doubling Time = ln(2) / (4 * ln(1 + 0.08/4)) ≈ 9.01 quarters (rounded to the nearest quarter).
c) Compounded Continuously:
For compounded continuously, we use the formula Doubling Time = ln(2) / (r * ln(1 + r)), where r is the interest rate.
Given that $5000 is invested at 8% compounded continuously, we can calculate the doubling time as follows:
Doubling Time = ln(2) / (0.08 * ln(1 + 0.08)) ≈ 8.66 years (rounded to two decimal places).