Questions: Your best friend consults you for investment advice. You learn that his tax rate is 35%, and he has the following current investments and debts: - A car loan with an outstanding balance of 5,000 and a 4.75% APR (monthly compounding) - Credit cards with an outstanding balance of 10,000 and a 14.93% APR (monthly compounding) - A regular savings account with a 30,000 balance, paying a 5.48% effective annual rate (EAR) - A money market savings account with a 100,000 balance, paying a 5.19% APR (daily compounding) - A tax-deductible home equity loan with an outstanding balance of 25,000 and a 4.92% APR (monthly compounding) a. Which savings account pays a higher after-tax interest rate? b. Should your friend use his savings to pay off any of his outstanding debts? The effective after-tax interest rate on the regular savings account is 3.56%. The effective after-tax interest rate on the money market savings account is 3.46%. So, the regular savings account pays a higher after-tax interest rate. The effective after-tax interest rate on the car loan is 4.85%. The effective after-tax interest rate on the credit cards is 15.99%.

Solution Steps

1. To determine which savings account pays a higher after-tax interest rate, calculate the effective after-tax interest rate for both accounts. For the regular savings account, use the given effective annual rate and adjust for taxes. For the money market account, convert the APR to an effective annual rate considering daily compounding, then adjust for taxes.
2. To decide if your friend should use savings to pay off debts, calculate the effective after-tax interest rate for each debt. Compare these rates to the after-tax interest rates of the savings accounts. If a debt's rate is higher than the savings rate, it may be beneficial to pay off that debt.

For the regular savings account, the effective after-tax interest rate is calculated as follows:

\[ \text{Regular Savings After-Tax} = 0.0548 \times (1 - 0.35) = 0.03562 \]

For the money market savings account, we first calculate the effective annual rate (EAR) using daily compounding:

\[ \text{Money Market EAR} = \left(1 + \frac{0.0519}{365}\right)^{365} - 1 \approx 0.0533 \]

Then, the after-tax interest rate is:

\[ \text{Money Market After-Tax} = 0.0533 \times (1 - 0.35) \approx 0.03462 \]

Comparing the after-tax interest rates:

\[ \text{Regular Savings After-Tax} = 0.03562 \quad \text{and} \quad \text{Money Market After-Tax} = 0.03462 \]

Since \(0.03562 > 0.03462\), the regular savings account pays a higher after-tax interest rate.

For the car loan:

\[ \text{Car Loan EAR} = \left(1 + \frac{0.0475}{12}\right)^{12} - 1 \approx 0.04855 \] \[ \text{Car Loan After-Tax} = 0.04855 \times (1 - 0.35) \approx 0.03156 \]

For the credit card:

\[ \text{Credit Card EAR} = \left(1 + \frac{0.1493}{12}\right)^{12} - 1 \approx 0.15995 \] \[ \text{Credit Card After-Tax} = 0.15995 \times (1 - 0.35) \approx 0.10397 \]

For the home equity loan:

\[ \text{Home Equity Loan EAR} = \left(1 + \frac{0.0492}{12}\right)^{12} - 1 \approx 0.05032 \] \[ \text{Home Equity Loan After-Tax} = 0.05032 \times (1 - 0.35) \approx 0.03271 \]

Now we compare the after-tax interest rates of the debts with the after-tax interest rates of the savings accounts:

• Car Loan After-Tax: \(0.03156\) (less than \(0.03562\) and \(0.03462\)) → Do not pay off.
• Credit Card After-Tax: \(0.10397\) (greater than both savings rates) → Should pay off.
• Home Equity Loan After-Tax: \(0.03271\) (less than \(0.03562\) and \(0.03462\)) → Do not pay off.

Final Answer