What is the effective rate of interest per day equivalent to 4.25% p.a. compounding semi-annually?

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Question restatement: What is the effective rate of interest per day equivalent to 4.25% p.a. compounding semi-annually?

Step 1: Interpret the semiannual compounding.
- A nominal annual rate of 4.25% compounded semi-annually means the rate per half-year is 4.25% / 2 = 2.125%.
- The accumulation factor over one year using semi-annual compounding is (1 + 0.02125)^2.
- Compute this: (1.02125)^2 = 1.0429515625, so the annual effective rate is 4.29515625%.

Step 2: Translate to a daily rate under daily compounding.
- Let i_d be the effective daily rate such that compounding daily for 365 days yields the same annual accumulation: (1 + i_d)^365 = (1 + 0.02125)^2 = 1.0429515625.
- Take natural logs to solve for i_d: 365 * ln(1 + i_d) = ln(1.0429515625).
- Therefore, ln(1 + i_d) = ln(1.0429515625) / 365.
- Exponentiate: 1 + i_d = exp[ ln(1.0429515625) / 365 ].
- So i_d = exp[ ln(1.0429515625) / 365 ] - 1.

Step 3: Numerical approximation.
- Using a quick approximation, ln(1.0429515625) ≈ 0.0420286.
- Divide by 365: ≈ 0.00011518.
- Exponentiate: exp(0.00011518) ≈ 1.00011519.
- Subtract 1: i_d ≈ 0.00011519 in decimal form.
- Convert to percentage: 0.00011519 × 100% ≈ 0.011519% per day (roughly).

Step 4: Compare with provided option (if any).
- An exact-like value such as 0.0115225% per day is consistent with the calculation above, given rounding differences in intermediate steps.

Summary note: The daily effective rate that equates to 4.25% p.a. with semiannual compounding is approximately 0.01152% per day. If the answer choices included 0.0115225%, it would be a close match to this computation, acknowledging minor rounding variations.

FINC5001 (ND) 2.8: Module 2 Practice Problems

What is the effective rate of interest per day equivalent to 4.25% p.a. compounding semi-annually?

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