Question

which of the two annual interest rates would yield the larger amount in 1 year?
6% compounded semiannually or 5.9% compounded daily

select the correct choice and fill in the answer box to complete your choice.
(do not round until the final answer. then round to three decimal places as needed.)
a. the effective interest rate for 6% compounded semiannually is □% which would yield the larger amount in 1 year.
b. the effective interest rate for 5.9% compounded daily is □% which would yield the larger amount in 1 year.

Explanation:

Step1: Recall effective interest rate formula

The formula for effective interest rate \( r_{eff}=(1 + \frac{r}{n})^{n}-1 \), where \( r \) is the nominal interest rate (annual), \( n \) is the number of compounding periods per year.

Step2: Calculate for 6% compounded semiannually

For 6% compounded semiannually: \( r = 0.06 \), \( n = 2 \) (since semiannual means 2 times a year).
\( r_{eff}=(1+\frac{0.06}{2})^{2}-1=(1 + 0.03)^{2}-1=1.03^{2}-1 = 1.0609 - 1=0.0609=6.09\% \).

Step3: Calculate for 5.9% compounded daily

For 5.9% compounded daily: \( r = 0.059 \), \( n = 365 \) (assuming 365 days a year).
\( r_{eff}=(1+\frac{0.059}{365})^{365}-1 \).
First, calculate \( \frac{0.059}{365}\approx0.000161644 \).
Then, \( 1 + 0.000161644 = 1.000161644 \).
Raise to the 365th power: \( 1.000161644^{365}\approx e^{0.059}\) (using the approximation \( (1+\frac{r}{n})^{n}\approx e^{r} \) for large \( n \), or directly compute: \( 1.000161644^{365}\approx1.0607 \)).
So \( r_{eff}\approx1.0607 - 1 = 0.0607 = 6.07\% \) (more precisely, using calculator: \( (1+\frac{0.059}{365})^{365}=e^{365\ln(1+\frac{0.059}{365})}\approx e^{365\times0.0001615}\approx e^{0.0589}\approx1.0607 \)).

Step4: Compare the two effective rates

6.09% (semiannual) vs 6.07% (daily). So 6% compounded semiannually has a higher effective rate.

Answer:

A. The effective interest rate for 6% compounded semiannually is \( \boldsymbol{6.090}\% \) which would yield the larger amount in 1 year.