What effective interest rate per quarter is equivalent to 8% p.a. compounding semi-annually?

Question

BlackTom AI · Accepted Answer

We start from the given: an 8% annual rate with semi-annual compounding means the nominal rate is 8% per year but interest is credited twice a year, so each half-year period has interest of 8%/2 = 4%.

Step 1: Compute the effective annual growth factor under semi-annual compounding. Over one year the amount grows by (1 + 0.04)^2 = 1.04^2 = 1.0816.

Step 2: Find the quarterly rate i_q that would yield the same annual growth when compounded quarterly. If the quarterly rate is i_q, then over four quarters in a year the growth factor is (1 + i_q)^4. Set this equal to the semi-annual effective-year factor: (1 + i_q)^4 = 1.0816.

Step 3: Solve for i_q. Take the fourth root: 1 + i_q = (1.0816)^(1/4). The fourth root of 1.0816 is the square root of the square root of 1.0816. Since 1.04^2 = 1.0816, the square root of 1.0816 is 1.04, and the square root of 1.04 is about 1.0199. Therefore i_q ≈ 1.0199 − 1 = 0.0199, i.e., about 1.99% per quarter.

Step 4: Compare to the option provided. An option stating 1.98% per quarter is in line with the calculated value, as 1.98% is very close to the precise value of roughly 1.99% and accounts for rounding differences in intermediate steps.

Additional note: If you test 1.98% as a quarterly rate, the quarterly growth factor is 1.0198. Four such periods give (1.0198)^4 ≈ 1.0816, which matches the annual growth factor from the semi-annual compounding, confirming the approximation.

What effective interest rate per quarter is equivalent to 8% p.a. compounding semi-annually?

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