If the bank is quoting an Annual Percentage Rate (APR) of 0.076 compounded continuously, what is the Effective Annual Rate for the year in percent (0.052 is 5.2%, so answer 5.2000) ?

Question

BlackTom AI · Accepted Answer

We are given an APR of 0.076 with continuous compounding and need the Effective Annual Rate (EAR).

Step 1: Recall the formula for EAR with continuous compounding: EAR = e^(r) - 1, where r is the annual nominal rate expressed as a decimal.

Step 2: Identify r = 0.076. Compute e^(0.076).

Step 3: Use a Taylor expansion or a calculator for e^(0.076): e^(0.076) ≈ 1 + 0.076 + (0.076)^2/2 + (0.076)^3/6 + (0.076)^4/24 + ...
- (0.076)^2/2 = 0.005776/2 = 0.002888
- (0.076)^3/6 = 0.000438976/6 ≈ 0.000073163
- (0.076)^4/24 ≈ 0.000033? wait, (0.076)^4 = 0.000033253; /24 ≈ 0.000001385

Step 4: Sum the terms: 1 + 0.076 + 0.002888 + 0.000073163 + 0.000001385 ≈ 1.078962548

Step 5: Subtract 1 to get EAR: EAR ≈ 0.078962548, which as a percentage is 7.8962548%.

Step 6: Rounding to four decimal places yields 7.8963% (the conventional presentation for EAR in percent form).

In summary, for a continuously compounded APR of 7.6% (0.076), the effective annual rate is approximately 7.8963%.

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If the bank is quoting an Annual Percentage Rate (APR) of 0.076 compounded continuously, what is the Effective Annual Rate for the year in percent (0.052 is 5.2%, so answer 5.2000) ?

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