To finance the development of a new product, a company borrowed $
3
4
,
0
0
0
at
6
%
compounded monthly. If the loan is
to be repaid in equal semi
-
annually payments over eight years and the first payment is due six months after the date of
the loan, what is the size of the semi
-
annual payment?
The size of the semi
-
annual payment is $
3
4
3
0
.
0
5
.
(
Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.
)

To determine the semi-annual payment for the loan, we need to follow these steps:

1. **Identify the variables for the annuity formula**:
   - $ PV $ (Present Value) = $34,000
   - $ r $ (monthly interest rate) = 6% annual interest compounded monthly = $ \frac{0.06}{12} $ = 0.005
   - Convert to the semi-annual interest rate: $ (1 + 0.005)^{6} - 1 $
   - $ n $ (total number of payments) = 8 years $\times$ 2 = 16 semi-annual payments

2. **Calculate the semi-annual interest rate**:
   \[
   r_{semi-annual} = (1 + 0.005)^{6} - 1
   \]
   \[
   r_{semi-annual} = (1.005)^{6} - 1
   \]
   \[
   r_{semi-annual} \approx 1.030301 - 1 = 0.030301
   \]

3. **Use the annuity formula to calculate the semi-annual payment (PMT)**:
   The annuity formula is:
   \[
   PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}
   \]
   Solving for $ PMT $:
   \[
   PMT = PV \times \frac{r}{1 - (1 + r)^{-n}}
   \]
   Substitute the values:
   \[
   PMT = 34000 \times \frac{0.030301}{1 - (1 + 0.030301)^{-16}}
   \]

4. **Calculate the denominator**:
   \[
   (1 + 0.030301)^{-16} \approx (1.030301)^{-16}
   \]
   \[
   (1.030301)^{-16} \approx 0.631775
   \]
   \[
   1 - 0.631775 = 0.368225
   \]

5. **Calculate the semi-annual payment**:
   \[
   PMT = 34000 \times \frac{0.030301}{0.368225}
   \]
   \[
   PMT = 34000 \times 0.082286
   \]
   \[
   PMT \approx 2797.724
   \]

There seems to be an inconsistency with the provided solution. Let's re-evaluate the calculations to ensure accuracy.

### Re-evaluated Steps

1. **Semi-annual interest rate calculation**:
   \[
   r_{semi-annual} = (1 + 0.005)^6 - 1
   \]
   \[
   r_{semi-annual} = (1.005)^6 - 1
   \]
   \[
   r_{semi-annual} \approx 0.030416
   \]

2. **Substitute the values back into the annuity formula**:
   \[
   PMT = 34000 \times \frac{0.030416}{1 - (1 + 0.030416)^{-16}}
   \]

3. **Calculate the denominator**:
   \[
   (1 + 0.030416)^{-16} \approx (1.030416)^{-16} \approx 0.632
   \]
   \[
   1 - 0.632 = 0.368
   \]

4. **Calculate the semi-annual payment**:
   \[
   PMT = 34000 \times \frac{0.030416}{0.368}
   \]
   \[
   PMT = 34000 \times 0.082632
   \]
   \[
   PMT \approx 2809.488
   \]

This is more consistent with typical loan calculations, but there might be minor rounding differences. Therefore, to match the provided answer exactly, a more precise calculation or verification might be needed.

Would you like any more details or explanations on any of the steps?

---

Here are 8 related questions that expand on this information:

1. How does the compounding frequency affect the size of the payments?
2. What would be the monthly payment if the loan were repaid monthly instead of semi-annually?
3. How does changing the interest rate impact the payment amount?
4. What if the loan term was different, say 10 years instead of 8 years?
5. How would a balloon payment at the end of the term affect the semi-annual payments?
6. Can you derive the formula for the present value of an annuity?
7. How would you account for a grace period in loan repayment calculations?
8. What happens if the borrower makes extra payments periodically?

**Tip:** Always ensure to use precise rounding in each step to minimize the error in final financial calculations.

Learn how to calculate semi-annual loan payments with compound interest using the annuity formula. This problem involves a $34,000 loan at 6% compounded monthly, repaid semi-annually over 8 years. Detailed step-by-step solution provided.

Math Problem Statement

To finance the development of a new product, a company borrowed $ 3 4 , 0 0 0 at 6 % compounded monthly. If the loan is to be repaid in equal semi

annually payments over eight years and the first payment is due six months after the date of the loan, what is the size of the semi

annual payment? The size of the semi

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation