What are the effective annual rates for an account paying an annual interest rate of $$ 6 \% $$ which is compounded: (a) annually? (b) quarterly? (c) daily (assuming there are 365 days in the year)? (d) continuously?

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(a) For annual compounding, the effective annual rate is the same as the nominal rate. That is,
$$
\text{EAR} = 0.06 = 6\%.
$$

(b) For quarterly compounding the effective annual rate is given by
$$
\text{EAR} = \left(1+\frac{0.06}{4}\right)^{4} - 1.
$$
Since $$\frac{0.06}{4} = 0.015$$, we have:
$$
\text{EAR} = (1.015)^{4} - 1.
$$
Calculating,
$$
(1.015)^4 \approx 1.06136,
$$
so,
$$
\text{EAR} \approx 0.06136 \text{ or } 6.14\%.
$$

(c) For daily compounding (with 365 days in a year), the effective rate is
$$
\text{EAR} = \left(1+\frac{0.06}{365}\right)^{365} - 1.
$$
This expression, when computed, gives approximately
$$
\text{EAR} \approx 0.06184 \text{ or } 6.18\%.
$$

(d) For continuous compounding, the effective rate is computed by
$$
\text{EAR} = e^{0.06} - 1.
$$
Computing,
$$
e^{0.06} \approx 1.06184,
$$
so,
$$
\text{EAR} \approx 0.06184 \text{ or } 6.18\%.
$$

Thus, the effective annual rates are:
- (a) 6.00%
- (b) Approximately 6.14%
- (c) Approximately 6.18%
- (d) Approximately 6.18%
- Annually: 6.00% - Quarterly: 6.14% - Daily: 6.18% - Continuously: 6.18%

What are the effective annual rates for an account paying an annual interest rate of \( 6 \% \) which is compounded: (a) annually? (b) quarterly? (c) daily (assuming there are 365 days in the year)? (d) continuously?

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