Periodic interest rates. You have a savings account in which you leave the funds for one year without adding to or withdrawing from the account. Which would you rather have: a daily compounded rate of \( 0.055 \% \), a weekly compounded rate of \( 0.265 \% \), a monthly compounded rate of \( 1.35 \% \), a quarterly compounded rater of \( 3.75 \% \), a semiannually compounded rate of \( 9 \% \), or an annually compounded rate of \( 15 \% \) ? What is the effective annsal rate (EAR) of a daily compounded rate of \( 0.055 \% \) ? 96 (Round to two decimal places.)

Alright, I need to determine which periodic interest rate is more favorable for a savings account that's left untouched for one year. The options are: 1. Daily compounded at 0.055% 2. Weekly compounded at 0.265% 3. Monthly compounded at 1.35% 4. Quarterly compounded at 3.75% 5. Semiannually compounded at 9% 6. Annually compounded at 15% First, I recall that the effective annual rate (EAR) is a way to compare different compounding frequencies by expressing all rates as if they were compounded annually. The formula for EAR is: \[ EAR = \left(1 + \frac{r}{n}\right)^n - 1 \] where: - \( r \) is the nominal annual interest rate - \( n \) is the number of compounding periods per year However, in this case, the rates provided are not annual but periodic. So, I need to adjust them to an annual basis. Let's calculate the EAR for each option: 1. **Daily Compounded at 0.055%** - Nominal rate \( r = 0.055\% \) - Compounded daily, so \( n = 365 \) - EAR = \( \left(1 + \frac{0.00055}{365}\right)^{365} - 1 \) - Calculating this, EAR ≈ 0.055% (since the daily rate is very small, the EAR is approximately the same as the nominal rate) 2. **Weekly Compounded at 0.265%** - Nominal rate \( r = 0.265\% \) - Compounded weekly, so \( n = 52 \) - EAR = \( \left(1 + \frac{0.00265}{52}\right)^{52} - 1 \) - Calculating this, EAR ≈ 0.265% (similarly, the EAR is approximately the nominal rate) 3. **Monthly Compounded at 1.35%** - Nominal rate \( r = 1.35\% \) - Compounded monthly, so \( n = 12 \) - EAR = \( \left(1 + \frac{0.0135}{12}\right)^{12} - 1 \) - Calculating this, EAR ≈ 1.35% (again, the EAR is approximately the nominal rate) 4. **Quarterly Compounded at 3.75%** - Nominal rate \( r = 3.75\% \) - Compounded quarterly, so \( n = 4 \) - EAR = \( \left(1 + \frac{0.0375}{4}\right)^{4} - 1 \) - Calculating this, EAR ≈ 3.75% (the EAR is still approximately the nominal rate) 5. **Semiannually Compounded at 9%** - Nominal rate \( r = 9\% \) - Compounded semiannually, so \( n = 2 \) - EAR = \( \left(1 + \frac{0.09}{2}\right)^{2} - 1 \) - Calculating this, EAR ≈ 9.2025% (here, the EAR is slightly higher than the nominal rate) 6. **Annually Compounded at 15%** - Nominal rate \( r = 15\% \) - Compounded annually, so \( n = 1 \) - EAR = \( \left(1 + \frac{0.15}{1}\right)^{1} - 1 \) - Calculating this, EAR = 15% (the EAR is the same as the nominal rate) From these calculations, it's clear that the EAR increases with the frequency of compounding. The daily and weekly compounding rates have EARs that are very close to their nominal rates, while the semiannual and annual compounding rates offer higher EARs. Now, for the second part of the question: What is the effective annual rate (EAR) of a daily compounded rate of 0.055%? Using the EAR formula: \[ EAR = \left(1 + \frac{0.00055}{365}\right)^{365} - 1 \] Calculating this: \[ EAR ≈ \left(1 + 0.0000015068\right)^{365} - 1 ≈ 1.055% - 1 = 0.055% \] So, the EAR is approximately 0.055%.