Determine whether the following statement makes sense or does not make sense, and explain your reasoning. My bank advertises a compound interest rate of 2.4%, although, without making deposits or withdrawals, the balance in my account increased by 2.43% in one year. Question content area bottom Part 1 Select the correct choices that complete the sentence below. The statement ▼ does not make sense. makes sense. The rate, r, is 2.4% or ▼ 0.024 0.24 2.4 and the number of compounding periods, n, is ▼ 12. 4. 1. 2. Substituting these values in the formula for ▼ the effective annual yield the present value for simple interest produces Y= ▼ left parenthesis 1 plus StartFraction 0.024 Over 12 EndFraction right parenthesis Superscript 12 Baseline minus 1 comma left parenthesis 1 plus 0.024 times 12 right parenthesis Superscript StartFraction 12 Over 0.024 EndFraction Baseline minus 1 comma left parenthesis 1 plus StartFraction 0.024 Over 12 EndFraction right parenthesis Superscript 12 Baseline comma which is approximately ▼ 0.329, 1.02, 0.0243, or ▼ 32.9%. 102%. 2.43%.
To determine whether the statement makes sense, we need to analyze the information provided about the bank's compound interest rate and the actual increase in the account balance.
Understanding the Interest Rate: The bank advertises a compound interest rate of 2.4%. This means that if you leave your money in the account for one year, it should grow by approximately 2.4% due to compounding.
Actual Increase: The statement mentions that the balance in the account increased by 2.43% in one year. This is slightly higher than the advertised rate of 2.4%.
Effective Annual Yield: The effective annual yield (EAY) can be calculated using the formula for compound interest. If the nominal interest rate (r) is 2.4% (or 0.024) and it is compounded monthly (which is a common practice), we can use the following formula:
[ EAY = \left(1 + \frac{r}{n}\right)^n - 1 ]
where ( n ) is the number of compounding periods per year. If we assume monthly compounding, then ( n = 12 ).
Substituting the values:
[ EAY = \left(1 + \frac{0.024}{12}\right)^{12} - 1 ]
Calculating this:
[ EAY = \left(1 + 0.002\right)^{12} - 1 \approx (1.002)^{12} - 1 ]
Using a calculator, we find:
[ (1.002)^{12} \approx 1.0243 ]
Therefore:
[ EAY \approx 1.0243 - 1 = 0.0243 \text{ or } 2.43% ]
Conclusion: The effective annual yield calculated (approximately 2.43%) matches the actual increase in the account balance (2.43%). This means that the bank's advertised rate of 2.4% is consistent with the observed increase in the account balance when considering the effects of compounding.
The statement makes sense. The rate, ( r ), is 2.4% or 0.024, and the number of compounding periods, ( n ), is 12. Substituting these values in the formula for the effective annual yield produces ( Y = \left(1 + \frac{0.024}{12}\right)^{12} - 1 ), which is approximately 2.43%.