Find the effective annual rate of interest for an investment that earns 5.5% per year when interest is compounded continuously.

Introduction / Context:
This problem introduces the idea of continuous compounding and asks for the effective annual rate. Continuous compounding means that interest is added at every instant, rather than daily, monthly, or yearly. The effective annual rate is the equivalent simple annual rate that would produce the same growth over one year as the continuous compounding process. This concept is important in more advanced financial mathematics and calculus based applications.

Given Data / Assumptions:

• Nominal rate r = 5.5% per year, compounded continuously.
• We want the effective annual rate, that is, the rate which, if applied once per year, would produce the same growth on a principal over one year.
• The standard continuous compounding formula A = P * e ^ (r * t) applies, where e is the base of natural logarithms.
• Time period t = 1 year for computing the effective annual rate.

Concept / Approach:
For continuous compounding, the amount after time t at nominal rate r (in decimal form) is A = P * e ^ (r * t). The effective annual rate R is defined such that A = P * (1 + R) after one year. Therefore, for t = 1, we have P * (1 + R) = P * e ^ r. Cancelling P from both sides gives 1 + R = e ^ r and hence R = e ^ r - 1. We simply plug in r = 0.055 and compute the result to get the effective annual rate in decimal form and then convert it to a percentage.

Step-by-Step Solution:
Step 1: Convert the nominal rate to decimal form: r = 5.5% = 0.055. Step 2: For continuous compounding, the one year amount is A = P * e ^ r. Step 3: Define the effective annual rate R such that A = P * (1 + R). Step 4: Set P * (1 + R) equal to P * e ^ r. This gives 1 + R = e ^ r. Step 5: Therefore, R = e ^ r - 1 = e ^ 0.055 - 1. Step 6: Compute e ^ 0.055 approximately. It is about 1.05654. Step 7: So R = 1.05654 - 1 = 0.05654, which as a percentage is about 5.654%. Step 8: Rounding to two decimal places, the effective annual rate is approximately 5.65%.

Verification / Alternative check:
We can compare continuous compounding at 5.5% with ordinary annual compounding at 5.5%. Under annual compounding with nominal and effective rate equal at 5.5%, the one year growth factor is 1.055. Under continuous compounding, we found a growth factor of about 1.05654, which is slightly higher. The difference is modest, about 0.15 percentage points, which makes sense because the nominal rate is not very large. The value 5.65% lies just above 5.5%, so it is a reasonable effective rate under continuous compounding.

Why Other Options Are Wrong:
The options 5.75%, 5.85%, and 5.95% are increasingly larger than the correct effective rate and would require a growth factor much bigger than e ^ 0.055, which is not consistent with the given nominal rate. The option 5.50% corresponds exactly to the nominal rate and would be correct only if interest were compounded annually rather than continuously. Since continuous compounding gives a slightly higher effective rate, 5.65% is the only option that matches the proper calculation.

Common Pitfalls:
One common mistake is to confuse the nominal and effective rates and to assume that the effective rate equals the nominal rate, regardless of the compounding method. Others may try to use a discrete compounding formula like (1 + r) ^ n - 1 with n set to a large number and then approximate, which is more complex than necessary. Some candidates also miscalculate e ^ r by using an incorrect approximate value for e or by mixing up logarithms. Remembering the clean relation R = e ^ r - 1 for continuous compounding at time t = 1 helps simplify these problems.

Final Answer:
The effective annual rate of interest for an investment earning 5.5% per year compounded continuously is approximately 5.65%, which corresponds to option A.