Final answer:
To double a $10,000 investment in 20 years with daily compounding interest, the necessary annual interest rate is approximately 3.5%.
Step-by-step explanation:
Estevan wants to know what interest rate is necessary for his $10,000 investment to double to $20,000 in 20 years with the interest being compounded daily. To find this, we can use the compound interest formula: A = P(1 + r/n)^(nt), where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
In this case, to double the investment, A would be $20,000, P is $10,000, n is 365 (since the interest is compounded daily), and t is 20 years. We need to find the annual interest rate r that makes this equation true.
Here is the equation restated with Estevan's numbers:
20,000 = 10,000(1 + r/365)^(365*20)
To isolate r, we first divide both sides by 10,000:
2 = (1 + r/365)^(365*20)
Next, we take the 365th root of both sides to remove the exponent:
(2)^(1/(365*20)) = 1 + r/365
Subtract 1 from both sides:
(2)^(1/(365*20)) - 1 = r/365
Finally, multiply both sides by 365 to solve for r:
365 * ((2)^(1/(365*20)) - 1) = r
Using a calculator, we find that the required interest rate r is approximately 0.035 or 3.5%.