Question content area top Part 1 Suppose you want to borrow ​$90 comma 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 that ​is, for n equals​1, ​12, 26, 52. b. Compute the total payment for each case in part​ (a). c. Compare the total payments computed in part​ (b)

To solve this problem, we need to use the formula for the payment on an amortized loan:

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

Where:
- $ P $ is the regular payment
- $ PV $ is the loan amount (present value)
- $ r $ is the periodic interest rate
- $ n $ is the number of payment periods per year
- $ t $ is the total number of years

### Given:
- Loan amount, $ PV = 90,000 $
- Annual Percentage Rate (APR) = 4%, or 0.04 as a decimal
- Term of the loan, $ t = 20 $ years

#### Part (a): Calculating the Regular Payments

We'll calculate the regular payments for different payment frequencies:
1. **Yearly payments ($ n = 1 $)**
2. **Monthly payments ($ n = 12 $)**
3. **Biweekly payments ($ n = 26 $)**
4. **Weekly payments ($ n = 52 $)**

For each case, the periodic interest rate $ r $ will be $ \frac{0.04}{n} $, and the total number of payments will be $ n \times t $.

#### Step 1: Yearly Payments ($ n = 1 $)

\[
r = \frac{0.04}{1} = 0.04
\]
\[
nt = 1 \times 20 = 20
\]
\[
P = \frac{0.04 \times 90,000}{1 - (1 + 0.04)^{-20}}
\]

#### Step 2: Monthly Payments ($ n = 12 $)

\[
r = \frac{0.04}{12} = 0.003333
\]
\[
nt = 12 \times 20 = 240
\]
\[
P = \frac{0.003333 \times 90,000}{1 - (1 + 0.003333)^{-240}}
\]

#### Step 3: Biweekly Payments ($ n = 26 $)

\[
r = \frac{0.04}{26} = 0.001538
\]
\[
nt = 26 \times 20 = 520
\]
\[
P = \frac{0.001538 \times 90,000}{1 - (1 + 0.001538)^{-520}}
\]

#### Step 4: Weekly Payments ($ n = 52 $)

\[
r = \frac{0.04}{52} = 0.000769
\]
\[
nt = 52 \times 20 = 1040
\]
\[
P = \frac{0.000769 \times 90,000}{1 - (1 + 0.000769)^{-1040}}
\]

I will calculate these values now.The regular payments for each case are as follows:
1. **Yearly payments**: $6,622.36
2. **Monthly payments**: $545.38
3. **Biweekly payments**: $251.57
4. **Weekly payments**: $125.75

### Part (b): Calculating the Total Payment for Each Case
To find the total payment, we multiply the regular payment by the total number of payments over 20 years.

1. **Yearly payments**: $ P_1 \times 20 $
2. **Monthly payments**: $ P_2 \times 240 $
3. **Biweekly payments**: $ P_3 \times 520 $
4. **Weekly payments**: $ P_4 \times 1040 $

Let's compute these values.The total payments for each case over the 20-year term are as follows:
1. **Yearly payments**: $132,447.15
2. **Monthly payments**: $130,891.75
3. **Biweekly payments**: $130,815.21
4. **Weekly payments**: $130,782.39

### Part (c): Comparison of Total Payments
- The total payment decreases as the frequency of payments increases. Paying weekly results in the lowest total payment, while paying yearly results in the highest.
- This difference arises due to the way interest accrues. More frequent payments reduce the principal balance faster, leading to lower overall interest charges.

Would you like further details or have any questions?

Here are five related questions:
1. How would the total payments change if the APR increased to 5%?
2. What would be the impact on payments if the loan term were reduced to 15 years?
3. How do biweekly payments affect the overall interest paid compared to monthly payments?
4. What is the effect of making extra payments toward the loan principal?
5. How would the payments change if the loan amount were increased to $120,000?

**Tip:** Making payments more frequently than required can significantly reduce the amount of interest you pay over the life of a loan.

Suppose you want to borrow $90,000, 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—that is, for n equals 1, 12, 26, 52. b. Compute the total payment for each case in part (a). c. Compare the total payments computed in part (b).

Learn how to calculate yearly, monthly, biweekly, and weekly payments for a $90,000 loan at 4% APR over 20 years. This step-by-step guide covers loan amortization, total payment comparisons, and the effect of payment frequency on interest. Suitable for high school seniors and college-level finance students.

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