Interest and TVM An account has a nominal rate of 4.49 . Find the effective annual yield, rounded to the nearest hundredth of a percent, with quarterly compounding, monthly compounding, and dally compounding. How does changing the compounding period affect the effective annual yield? (1) Click the icon to view some finance formulas.

Question

UpStudy AI · Accepted Answer

To find the effective annual yield (EAY) given a nominal interest rate and different compounding periods, we can use the formula:

$$
EAY = \left(1 + \frac{r}{n}\right)^n - 1
$$

where:
- $$ r $$ is the nominal interest rate (as a decimal),
- $$ n $$ is the number of compounding periods per year.

Given:
- Nominal rate $$ r = 4.49\% = 0.0449 $$

We will calculate the EAY for:
1. Quarterly compounding ($$ n = 4 $$)
2. Monthly compounding ($$ n = 12 $$)
3. Daily compounding ($$ n = 365 $$)

Let's perform these calculations step by step.

### Step 1: Calculate EAY for Quarterly Compounding
For quarterly compounding:
$$
n = 4
$$
$$
EAY_{quarterly} = \left(1 + \frac{0.0449}{4}\right)^4 - 1
$$

### Step 2: Calculate EAY for Monthly Compounding
For monthly compounding:
$$
n = 12
$$
$$
EAY_{monthly} = \left(1 + \frac{0.0449}{12}\right)^{12} - 1
$$

### Step 3: Calculate EAY for Daily Compounding
For daily compounding:
$$
n = 365
$$
$$
EAY_{daily} = \left(1 + \frac{0.0449}{365}\right)^{365} - 1
$$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+\frac{0.0449}{4}\right)^{4}-1$$
- step1: Divide the terms:
  $$\left(1+\frac{449}{40000}\right)^{4}-1$$
- step2: Add the numbers:
  $$\left(\frac{40449}{40000}\right)^{4}-1$$
- step3: Rewrite the expression:
  $$\frac{40449^{4}}{40000^{4}}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{40449^{4}}{40000^{4}}-\frac{40000^{4}}{40000^{4}}$$
- step5: Transform the expression:
  $$\frac{40449^{4}-40000^{4}}{40000^{4}}$$
Calculate or simplify the expression $$ (1 + 0.0449/365)^365 - 1 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+\frac{0.0449}{365}\right)^{365}-1$$
- step1: Divide the terms:
  $$\left(1+\frac{449}{3650000}\right)^{365}-1$$
- step2: Add the numbers:
  $$\left(\frac{3650449}{3650000}\right)^{365}-1$$
- step3: Rewrite the expression:
  $$\frac{3650449^{365}}{3650000^{365}}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{3650449^{365}}{3650000^{365}}-\frac{3650000^{365}}{3650000^{365}}$$
- step5: Transform the expression:
  $$\frac{3650449^{365}-3650000^{365}}{3650000^{365}}$$
Calculate or simplify the expression $$ (1 + 0.0449/12)^12 - 1 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+\frac{0.0449}{12}\right)^{12}-1$$
- step1: Divide the terms:
  $$\left(1+\frac{449}{120000}\right)^{12}-1$$
- step2: Add the numbers:
  $$\left(\frac{120449}{120000}\right)^{12}-1$$
- step3: Rewrite the expression:
  $$\frac{120449^{12}}{120000^{12}}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{120449^{12}}{120000^{12}}-\frac{120000^{12}}{120000^{12}}$$
- step5: Transform the expression:
  $$\frac{120449^{12}-120000^{12}}{120000^{12}}$$
Round the number $$ 0.04592 $$ to nearest hundredth.
Round the number by following steps:
- step0: Round $$0.04592$$ to $$2$$ decimal places using standard rounding method:
  $$0.04592$$
- step1: Round $$0.04592$$ to $$2$$ decimal places using standard rounding method$$:$$
  $$0.05$$
Round the number $$ 0.045662 $$ to nearest hundredth.
Round the number by following steps:
- step0: Round $$0.045662$$ to $$2$$ decimal places using standard rounding method:
  $$0.045662$$
- step1: Round $$0.045662$$ to $$2$$ decimal places using standard rounding method$$:$$
  $$0.05$$
Round the number $$ 0.045836 $$ to nearest hundredth.
Round the number by following steps:
- step0: Round $$0.045836$$ to $$2$$ decimal places using standard rounding method:
  $$0.045836$$
- step1: Round $$0.045836$$ to $$2$$ decimal places using standard rounding method$$:$$
  $$0.05$$
The effective annual yield (EAY) for the account with a nominal rate of 4.49% is as follows:

1. **Quarterly Compounding:**
   $$
EAY_{quarterly} \approx 0.05 \text{ or } 5.00\%
   $$

2. **Monthly Compounding:**
   $$
EAY_{monthly} \approx 0.05 \text{ or } 5.00\%
   $$

3. **Daily Compounding:**
   $$
EAY_{daily} \approx 0.05 \text{ or } 5.00\%
   $$

### Conclusion
In this case, all three compounding methods yield an effective annual yield of approximately 5.00%.

### Effect of Changing the Compounding Period
While the calculations show the same EAY for this specific nominal rate, generally, increasing the frequency of compounding (from annually to quarterly, monthly, or daily) tends to increase the effective annual yield. However, in this instance, the differences are negligible due to the specific nominal rate and the rounding of results.

In practice, for different nominal rates, you would typically see a gradual increase in EAY with more frequent compounding.
The effective annual yield for a 4.49% nominal rate is approximately 5.00% with quarterly, monthly, and daily compounding. Increasing the compounding frequency generally increases the effective annual yield, but in this case, all methods yield the same result due to the specific interest rate and rounding.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Economics Questions