To finance the development of product, A borrow 28000$ at 5% compounded monthly. If loan is to be repaid in equal annually payment over 5 year and first payment is due one year after date of loan what is size of annual payment

To solve the problem, we need to calculate the annual payment $ R $ that will fully amortize the loan over 5 years with interest compounded monthly. Here's how to approach the problem:

### Step 1: Identify the given information
- **Loan amount (Principal)** $ P = 28000 $ dollars
- **Annual interest rate** $ r = 5\% $ or $ 0.05 $
- **Compounding frequency**: Monthly (12 times a year)
- **Loan term**: 5 years
- **First payment due**: One year after the loan is taken

### Step 2: Calculate the monthly interest rate
The monthly interest rate $ i $ is calculated by dividing the annual interest rate by 12:
\[
i = \frac{r}{12} = \frac{0.05}{12} = 0.0041667
\]

### Step 3: Calculate the present value of the loan
Since the loan payments are annual but interest is compounded monthly, we need to find the present value of the loan using the formula for the present value of an ordinary annuity:

\[
PV = \frac{R}{i} \left[1 - (1 + i)^{-n}\right]
\]

Where:
- $ PV $ is the loan amount ($28,000).
- $ R $ is the annual payment.
- $ n $ is the total number of compounding periods. Since payments are annual but interest is compounded monthly, we have $ n = 12 \times 5 = 60 $.

However, since payments are annual, we need to adjust for this. The relationship between the monthly and annual payments is modified because the loan effectively "compounds" monthly, but the payments are made annually. This necessitates using the **effective annual rate (EAR)**:

\[
EAR = (1 + i)^{12} - 1 = (1 + 0.0041667)^{12} - 1 \approx 0.0511619
\]

Thus, the effective annual rate is approximately 5.11619%.

### Step 4: Calculate the size of the annual payment
We can now calculate the annual payment $ R $ using the formula for an annuity:

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

Where $ n $ is the number of years (5 in this case):

\[
R = \frac{28000 \times 0.0511619}{1 - (1 + 0.0511619)^{-5}}
\]

### Step 5: Plug in the values and calculate
Let's calculate the exact value for $ R $.

Let me calculate that for you.The size of the annual payment required to repay the $28,000 loan over 5 years, with a 5% interest rate compounded monthly, is approximately **$6,488.08**.

Would you like further details or have any other questions?

### Related Questions:
1. How is the effective annual rate (EAR) different from the nominal rate?
2. What would be the size of the monthly payment instead of annual?
3. How does the payment size change if the interest rate is compounded annually?
4. What if the loan term was extended to 10 years instead of 5?
5. Can you explain how the formula for an annuity is derived?
6. How does increasing the interest rate affect the annual payment?
7. What happens to the payment if the loan is repaid semi-annually instead of annually?
8. How does the time value of money impact loan payments?

**Tip:** Always consider the effective interest rate when dealing with different compounding periods and payment frequencies.

Discover how to calculate the annual payment needed to repay a $28,000 loan over 5 years with a 5% annual interest rate compounded monthly. Get a detailed step-by-step solution with formulas and computations.

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