Suppose you want to borrow ​$110,000 and you find a bank offering a​ 20-year term for a loan of that​ amount, with an APR of 4​%.
a. Find your regular payments if you make them​ yearly, monthly, biweekly​ (every 2​ weeks), or weekly 
b. Compute the total payment for each case in part​ (a).
c. Compare the total payments computed in part​ (b).

Let's break down the problem into steps and solve each part systematically.

### Part (a): Regular Payments

The formula to calculate the regular payment for a loan is given by the **amortization formula**:

\[
P = \frac{r \times PV}{1 - (1 + r)^{-n}}
\]

Where:
- $ P $ is the regular payment
- $ PV $ is the present value of the loan (the principal)
- $ r $ is the interest rate per period
- $ n $ is the total number of payments

#### Yearly Payments
- **APR**: 4% or 0.04
- **Number of payments per year**: 1
- **Loan term**: 20 years
- **Interest rate per period $ r $**: $ 0.04 / 1 = 0.04 $
- **Total number of payments $ n $**: $ 20 \times 1 = 20 $

Let's calculate:

\[
P_{\text{yearly}} = \frac{0.04 \times 110,000}{1 - (1 + 0.04)^{-20}}
\]

#### Monthly Payments
- **Number of payments per year**: 12
- **Interest rate per period $ r $**: $ 0.04 / 12 $
- **Total number of payments $ n $**: $ 20 \times 12 = 240 $

Let's calculate:

\[
P_{\text{monthly}} = \frac{0.04 / 12 \times 110,000}{1 - (1 + 0.04 / 12)^{-240}}
\]

#### Biweekly Payments
- **Number of payments per year**: 26
- **Interest rate per period $ r $**: $ 0.04 / 26 $
- **Total number of payments $ n $**: $ 20 \times 26 = 520 $

Let's calculate:

\[
P_{\text{biweekly}} = \frac{0.04 / 26 \times 110,000}{1 - (1 + 0.04 / 26)^{-520}}
\]

#### Weekly Payments
- **Number of payments per year**: 52
- **Interest rate per period $ r $**: $ 0.04 / 52 $
- **Total number of payments $ n $**: $ 20 \times 52 = 1040 $

Let's calculate:

\[
P_{\text{weekly}} = \frac{0.04 / 52 \times 110,000}{1 - (1 + 0.04 / 52)^{-1040}}
\]

### Part (b): Total Payments

The total payment over the life of the loan is simply the regular payment multiplied by the total number of payments:

\[
\text{Total Payment} = P \times n
\]

### Part (c): Comparison

Finally, we will compare the total payments for each case calculated in part (b) to see how they differ.

Let's compute these values.Here are the results:

### Part (a): Regular Payments
- **Yearly Payment:** $8,093.99
- **Monthly Payment:** $666.58
- **Biweekly Payment:** $307.47
- **Weekly Payment:** $153.70

### Part (b): Total Payments
- **Total Yearly Payments:** $161,879.85
- **Total Monthly Payments:** $159,978.81
- **Total Biweekly Payments:** $159,885.25
- **Total Weekly Payments:** $159,845.14

### Part (c): Comparison of Total Payments
The total payments differ slightly based on the payment frequency. Here's the ranking from highest to lowest total payment:
1. **Yearly:** $161,879.85
2. **Monthly:** $159,978.81
3. **Biweekly:** $159,885.25
4. **Weekly:** $159,845.14

The more frequent the payments, the lower the total payment over the life of the loan. This is because interest is compounded less frequently, which results in lower overall interest paid.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How does the compounding frequency affect the interest paid on a loan?
2. What is the difference between APR and APY (Annual Percentage Yield)?
3. How can biweekly payments save you money on a mortgage?
4. What are the pros and cons of making weekly versus monthly payments?
5. How do different amortization schedules impact the principal and interest portions of payments?

**Tip:** When considering a loan, always compare the total cost over the loan term rather than just the payment amount. Frequent payments can reduce the overall interest paid.

Learn how to calculate loan payments for a $110,000 loan with a 4% APR over 20 years. Understand the differences in payments made yearly, monthly, biweekly, and weekly. Get detailed formulas and comparison of total payments.

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