Find APYs (expressed as a percentage, correct to three decimal places). Then compare them to find the best investment option for 1 year. 4 banks offer CD. The first bank offers 4.96% compounded monthly. The second bank offers 4.95% compounded daily. The third bank offers 4.97% compounded quarterly. The fourth bank offers 4.94% compounded continuously.
a. Either the first or the second bank
b. The first bank
c. Either the first or the third bank
d. The fourth bank
e. The third bank
f. Either the third or the fourth bank
g. The second bank

To solve this problem, we need to calculate the Annual Percentage Yield (APY) for each bank. The APY is a way to express the interest rate as an annualized percentage rate, accounting for the effects of compounding. We can use the following formulas for each compounding method:

1. **Compounded Monthly:**
   \[
   \text{APY} = \left(1 + \frac{r}{n}\right)^n - 1
   \]
   where \( r = 0.0496 \) (4.96%) and \( n = 12 \) (monthly compounding).

2. **Compounded Daily:**
   \[
   \text{APY} = \left(1 + \frac{r}{n}\right)^n - 1
   \]
   where \( r = 0.0495 \) (4.95%) and \( n = 365 \) (daily compounding).

3. **Compounded Quarterly:**
   \[
   \text{APY} = \left(1 + \frac{r}{n}\right)^n - 1
   \]
   where \( r = 0.0497 \) (4.97%) and \( n = 4 \) (quarterly compounding).

4. **Compounded Continuously:**
   \[
   \text{APY} = e^r - 1
   \]
   where \( r = 0.0494 \) (4.94%).

Let's calculate each APY and compare them.

### 1. First Bank (4.96% compounded monthly):
\[
\text{APY} = \left(1 + \frac{0.0496}{12}\right)^{12} - 1
\]

### 2. Second Bank (4.95% compounded daily):
\[
\text{APY} = \left(1 + \frac{0.0495}{365}\right)^{365} - 1
\]

### 3. Third Bank (4.97% compounded quarterly):
\[
\text{APY} = \left(1 + \frac{0.0497}{4}\right)^{4} - 1
\]

### 4. Fourth Bank (4.94% compounded continuously):
\[
\text{APY} = e^{0.0494} - 1
\]

I'll calculate these now.Here are the calculated APYs for each bank:

1. **First Bank (4.96% compounded monthly):** 5.074%
2. **Second Bank (4.95% compounded daily):** 5.074%
3. **Third Bank (4.97% compounded quarterly):** 5.063%
4. **Fourth Bank (4.94% compounded continuously):** 5.064%

### Comparison:
- The **First** and **Second Banks** both offer the highest APY of 5.074%.
  
So, the best investment options are **either the First or the Second Bank**, making option **a. Either the first or the second bank** the correct choice.

---

Would you like more details or have any other questions?

### Related Questions:
1. How is APY different from the nominal interest rate?
2. Why does compounding frequency affect the APY?
3. How does continuous compounding compare to other compounding methods over different periods?
4. What other factors should be considered when choosing a CD apart from APY?
5. Can APY decrease if interest rates are compounded less frequently?

### Tip:
APY is particularly useful for comparing financial products because it accounts for the effects of compounding, making it easier to see the true return on investment.

Find APYs (expressed as a percentage, correct to three decimal places). Then compare them to find the best investment option for 1 year. 4 banks offer CD. The first bank offers 4.96% compounded monthly. The second bank offers 4.95% compounded daily. The third bank offers 4.97% compounded quarterly. The fourth bank offers 4.94% compounded continuously.

Learn how to calculate the Annual Percentage Yield (APY) for different compounding methods offered by four banks. This problem involves understanding key concepts such as compound interest and exponential growth. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation