When payment frequency does not equal compounding frequency

Question

BlackTom AI · Accepted Answer

When payment frequency differs from compounding frequency, compute the effective interest rate that applies to each payment period: i_p = (1 + r_nom/m)^(m/p) − 1.

1) Relevant knowledge points and concepts
- Nominal annual rate (r_nom or APR) convertible m times per year produces a per-compounding-period rate r_nom/m.  
- Effective annual rate (EAR) = (1 + r_nom/m)^m − 1.  
- If payments occur p times per year (payment period = 1/p year) but compounding is m times per year, you must use the effective rate that corresponds to the payment period, not the raw APR/M or the EAR alone.

2) Complete reasoning process (derivation)
- A dollar invested for 1/m year grows by a factor of 1 + r_nom/m.  
- Over a payment period of length 1/p year there are m/p compounding subperiods, so growth over one payment period is (1 + r_nom/m)^(m/p).  
- Therefore the periodic (per payment) effective interest rate is i_p = (1 + r_nom/m)^(m/p) − 1.  
- If you are instead given an effective annual rate EAR, the per-payment rate is i_p = (1 + EAR)^(1/p) − 1.

3) Example with intermediate steps
- Suppose APR r_nom = 12% (0.12) compounded monthly (m = 12), payments quarterly (p = 4).  
- Monthly rate = 0.12/12 = 0.01. Number of months in a quarter = m/p = 12/4 = 3.  
- Per-quarter effective rate = (1 + 0.01)^3 − 1 = 1.01^3 − 1 = 1.030301 − 1 = 0.030301 ≈ 3.0301%.

4) Why the correct answer is correct
- Using an effective rate that matches the payment period correctly captures the accumulation over the exact time between payments. The APR by itself ignores intra-year compounding, so it does not produce correct periodic interest for payments that don’t align with compounding.

5) Why the other options are incorrect
- "You must use the annual percentage rate directly": incorrect because APR/M ignores compounding intervals and won’t give the correct per-payment rate when frequencies differ.  
- "The effective interest rate cannot be calculated": incorrect — it can be calculated as shown.  
- "Compounding has no effect on the payments": incorrect because compounding frequency changes the accumulated growth between payments and thus affects payment amounts.

6) Conclusion
- Therefore you need to find an effective rate that matches the payment period (the provided correct answer).

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