A savings account earns 10% per annum, compounded semi-annually. Calculate the interest rate per 6-month.
储蓄账户每年收益为10%，每半年复利一次。计算每6个月的利率。

Question

BlackTom AI · Accepted Answer

Since the nominal annual interest rate is 10% with semi-annual compounding, the interest applied each 6-month period is the nominal rate divided by the number of compounding periods per year.

1) Relevant knowledge points and concepts
- Nominal annual rate (r_nom) versus periodic rate (r_period): when interest is compounded m times per year, r_period = r_nom / m.
- Effective annual rate (EAR) differs: EAR = (1 + r_period)^m − 1. EAR is not the same as the periodic rate.

2) Step-by-step reasoning
- Given r_nom = 10% = 0.10 per year.
- Compounding semi-annually means m = 2 periods per year.
- Compute periodic (6-month) rate: r_period = r_nom / m = 0.10 / 2.

3) Intermediate calculation
- 0.10 / 2 = 0.05
- Convert to percentage: 0.05 × 100% = 5.00%

4) Why this is correct
- Each compounding period applies one half of the nominal annual rate when compounding periods are equal length. Thus every 6 months the account accrues 5% interest. The effective annual interest would be larger (because of compounding within the year), specifically (1 + 0.05)^2 − 1 = 0.1025 = 10.25%, but that is the annual effective rate, not the per-6-month rate asked.

5) Why the other options are incorrect
- 10.00%: this is the nominal annual rate, not the rate for each 6-month period.
- 10.25%: this is the effective annual rate resulting from semi-annual compounding (EAR), not the 6-month periodic rate.
- 9.76%: not relevant here; it does not represent the per-period rate or the correct effective annual rate for 10% semi-annually, so it is incorrect.

Conclusion: The interest rate per 6-month period is 5.00%, which matches the provided correct answer.

A savings account earns 10% per annum, compounded semi-annually. Calculate the interest rate per 6-month. 储蓄账户每年收益为10%，每半年复利一次。计算每6个月的利率。单项选择题

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