Answer
The effective interest rate is approximately 3.28%.
Solution
To determine the annual percentage yield (APY) or the effective interest rate for $400 invested at 3.23% over 20 years compounded daily, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (in decimal form).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the time the money is invested for in years.
Given:
- \( P = \$400 \)
- \( r = 3.23\% = 0.0323 \) (in decimal form)
- \( n = 365 \) (compounded daily)
- \( t = 20 \) years
Substitute the given values into the formula:
\[ A = 400 \left(1 + \frac{0.0323}{365}\right)^{365 \times 20} \]
Now, we can calculate the amount of money accumulated after 20 years, including interest, using the formula above.
Calculate the value by following steps:
- step0: Calculate:
\(400\left(1+\frac{0.0323}{365}\right)^{365}\times 20\)
- step1: Divide the terms:
\(400\left(1+\frac{323}{3650000}\right)^{365}\times 20\)
- step2: Add the numbers:
\(400\left(\frac{3650323}{3650000}\right)^{365}\times 20\)
- step3: Multiply the terms:
\(8000\left(\frac{3650323}{3650000}\right)^{365}\)
- step4: Simplify:
\(8000\times \frac{3650323^{365}}{3650000^{365}}\)
- step5: Rewrite the expression:
\(2000\times 4\times \frac{3650323^{365}}{3650000^{365}}\)
- step6: Rewrite the expression:
\(2000\times 4\times \frac{3650323^{365}}{2000^{365}\times 1825^{365}}\)
- step7: Reduce the numbers:
\(4\times \frac{3650323^{365}}{2000^{364}\times 1825^{365}}\)
- step8: Rewrite the expression:
\(4\times \frac{3650323^{365}}{16^{364}\times 125^{364}\times 1825^{365}}\)
- step9: Rewrite the expression:
\(4\times \frac{3650323^{365}}{4^{728}\times 125^{364}\times 1825^{365}}\)
- step10: Reduce the numbers:
\(1\times \frac{3650323^{365}}{4^{727}\times 125^{364}\times 1825^{365}}\)
- step11: Multiply the fractions:
\(\frac{3650323^{365}}{4^{727}\times 125^{364}\times 1825^{365}}\)
The amount of money accumulated after 20 years, including interest, is approximately $8262.61.
To find the annual percentage yield (APY) or the effective interest rate, we can use the formula:
\[ APY = \left(1 + \frac{r}{n}\right)^n - 1 \]
Substitute the given values into the formula:
\[ APY = \left(1 + \frac{0.0323}{365}\right)^{365} - 1 \]
Now, we can calculate the APY or the effective interest rate.
Calculate the value by following steps:
- step0: Calculate:
\(\left(1+\frac{0.0323}{365}\right)^{365}-1\)
- step1: Divide the terms:
\(\left(1+\frac{323}{3650000}\right)^{365}-1\)
- step2: Add the numbers:
\(\left(\frac{3650323}{3650000}\right)^{365}-1\)
- step3: Rewrite the expression:
\(\frac{3650323^{365}}{3650000^{365}}-1\)
- step4: Reduce fractions to a common denominator:
\(\frac{3650323^{365}}{3650000^{365}}-\frac{3650000^{365}}{3650000^{365}}\)
- step5: Transform the expression:
\(\frac{3650323^{365}-3650000^{365}}{3650000^{365}}\)
The annual percentage yield (APY) or the effective interest rate for $400 invested at 3.23% over 20 years compounded daily is approximately 3.28%.
Therefore, the effective interest rate for this investment is 3.28%.
Reviewed and approved by the UpStudy tutoring team
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