An interest rate is 5% per annum with continuous compounding. What is the equivalent rate with semi-annual compounding?

Question

BlackTom AI · Accepted Answer

We start by identifying the given nominal rate with continuous compounding: 5% per year, r_c = 0.05. The first step is to relate continuous compounding to an effective annual rate: e^{r_c} - 1 = e^{0.05} - 1 ≈ 0.051271.

Next, we seek a nominal rate with semi-annual compounding, r_s, such that the effective annual yield matches the continuous case. With semi-annual compounding, the accumulation factor over one year is (1 + r_s/2)^{2}. We set this equal to the continuous-accumulation factor:
(1 + r_s/2)^{2} = e^{0.05}.

Taking square roots gives 1 + r_s/2 = e^{0.025} ≈ 1.025315. Solving for r_s yields r_s = 2*(e^{0.025} - 1) ≈ 2*(0.025315) ≈ 0.05063, i.e., about 5.063%.

Rounding to two decimals, the equivalent semi-annual rate is approximately 5.06%.

Why option a (5.06%) is correct: it matches the precise calculation above, providing a nominal annual rate compounded semi-annually that yields the same annual growth as the given 5% with continuous compounding.

Why this reasoning excludes other possible interpretations: if one simply annualized the continuous rate without adjusting for compounding, or used a rough approximation, the result would differ from the exact (e^{0.025}-1)×2 value. The key is to equate the annual accumulation factors, not just the nominal rates, which is why the process uses e^{0.05} on one side and (1 + r_s/2)^{2} on the other. Since option a reflects this correctly, it is the appropriate choice.

Note on the provided options: the data shows a. 5.06% as a candidate and another entry labeled 'cross out' which is not a usable numeric choice. The mathematical solution confirms 5.06% as the correct equivalent rate.

An interest rate is 5% per annum with continuous compounding. What is the equivalent rate with semi-annual compounding?

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