Determine the annual percentage yield, or the effective interest rate, for invested at over 20 years compounded daily. Round your answer to the
nearest hundredth of a percent, if necessary.

Question

Determine the annual percentage yield, or the effective interest rate, for

invested at

over 20 years compounded daily. Round your answer to the
nearest hundredth of a percent, if necessary.

UpStudy AI · Accepted Answer

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%.
The effective interest rate is approximately 3.28%.

Determine the annual percentage yield, or the effective interest rate, for invested at over 20 years compounded daily. Round your answer to the
nearest hundredth of a percent, if necessary.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Pre Calculus Questions

Determine the annual percentage yield, or the effective interest rate, for invested at over 20 years compounded daily. Round your answer to the nearest hundredth of a percent, if necessary.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Pre Calculus Questions

Determine the annual percentage yield, or the effective interest rate, for invested at over 20 years compounded daily. Round your answer to the
nearest hundredth of a percent, if necessary.