Case

Amount of

initial deposit

Nominal

annual​ rate, r

Compounding​ frequency, m

​(times/year)

Deposit period

​(years)

A

​$2 comma 600

8

​%

2

7

B

​$49 comma 000

12

​%

6

5

C

​$1 comma 100

6

​%

1

12

D

​$19 comma 000

17

​%

4

7

Compounding​ frequency, time​ value, and effective annual rates   For each of the cases in the following​ table,

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​:

a.  Calculate the future value at the end of the specified deposit period.

b.  Determine the effective annual​ rate, EAR.

c.  Compare the nominal annual ​rate,​ r, to the effective annual​ rate,

EAR. 

What relationship exists between compounding frequency and the nominal and effective annual​ rates?

Question content area bottom

Part 1

a.  The future value of case A at the end of year

7

is

​$enter your response here

. 

​(Round to the nearest​ cent.)

To solve this problem, we'll calculate the future value (FV) and the effective annual rate (EAR) for each of the cases provided. Here's the step-by-step approach:

### Formula for Future Value (FV)

The formula to compute the future value (FV) when interest is compounded periodically is:

\[
FV = P \left(1 + \frac{r}{m}\right)^{mt}
\]

where:
- $ P $ = Initial deposit (principal)
- $ r $ = Nominal annual interest rate (as a decimal)
- $ m $ = Compounding frequency (times per year)
- $ t $ = Deposit period (in years)

### Formula for Effective Annual Rate (EAR)

The effective annual rate (EAR) is calculated using the formula:

\[
EAR = \left(1 + \frac{r}{m}\right)^m - 1
\]

where:
- $ r $ and $ m $ are as defined above.

Let's compute these values for **Case A**:

### Case A Details:
- Initial Deposit, $ P = 2600 $
- Nominal Annual Rate, $ r = 8\% = 0.08 $
- Compounding Frequency, $ m = 2 $ (Semiannually)
- Deposit Period, $ t = 7 $ years

#### Step 1: Compute the Future Value (FV) for Case A

\[
FV = 2600 \left(1 + \frac{0.08}{2}\right)^{2 \times 7}
\]

#### Step 2: Compute the Effective Annual Rate (EAR) for Case A

\[
EAR = \left(1 + \frac{0.08}{2}\right)^2 - 1
\]

Let's calculate these values.### Results for Case A:

1. **Future Value (FV) at the end of 7 years**:  
   $ \text{FV} \approx $ **$4502.36**

2. **Effective Annual Rate (EAR)**:  
   $ \text{EAR} \approx $ **8.16%**

### Comparison of Nominal Annual Rate and EAR

- The nominal annual rate $ r = 8\% $ is less than the effective annual rate (EAR = 8.16%).  
- This relationship shows that with semiannual compounding (twice a year), the effective annual rate is slightly higher than the nominal rate due to the effect of compounding more frequently.

Would you like to proceed with calculating the future value and EAR for the remaining cases (B, C, and D)? 

### Related Questions:
1. How would the results change if the compounding frequency was increased?
2. What impact does the deposit period have on the future value?
3. How is the EAR influenced by different nominal rates?
4. What happens if the nominal rate remains constant, but the compounding frequency varies?
5. Can you derive the formula for EAR from the formula for future value?

### Tip:
Increasing the compounding frequency generally increases the effective annual rate, which can lead to a higher future value over the same period.

For each of the cases in the following table, calculate the future value at the end of the specified deposit period, determine the effective annual rate (EAR), and compare the nominal annual rate to the effective annual rate. What relationship exists between compounding frequency and the nominal and effective annual rates?

Learn how to calculate the future value and effective annual rate (EAR) for compounded interest scenarios. This detailed guide covers different cases, including various initial deposits, compounding frequencies, and nominal annual rates. Understand the relationship between compounding frequency and effective annual rates. Suitable for Grades 10-12 students.

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