Chuck Ponzi has talked an elderly woman into loaning him ​$30 comma 000 for a new business venture.  She​ has, however, successfully passed a finance class and requires Chuck to sign a binding contract on repayment of the ​$30 comma 000 with an annual interest rate of 10​% over the next 15 years.  Determine the cash flow to the woman under a fully amortized​ loan, in which Ponzi will make equal annual payments at the end of each year so that the final payment will completely retire the original ​$30 comma 000 loan.


To calculate the cash flow to the woman under a fully amortized loan, we need to determine Chuck Ponzi's equal annual payments, which will include both the principal and interest, and ensure that the entire loan is repaid over 15 years. This type of problem is solved using the amortization formula for a loan, which is given by:

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

Where:
- $A$ = annual payment
- $P$ = principal amount (loan), which is $30,000
- $r$ = annual interest rate (as a decimal), which is 10% or 0.10
- $n$ = number of periods (years), which is 15

### Step-by-Step Calculation:
1. Principal $P = 30,000$
2. Annual interest rate $r = 0.10$
3. Number of years $n = 15$

Substitute these values into the formula and calculate the annual payment $A$.

\[
A = \frac{30,000 \cdot 0.10 \cdot (1 + 0.10)^{15}}{(1 + 0.10)^{15} - 1}
\]

Let me calculate that for you.The annual payment that Chuck Ponzi must make to fully repay the $30,000 loan over 15 years at an annual interest rate of 10% is approximately **$3,944.21**.

This means that the elderly woman will receive **$3,944.21** at the end of each year for 15 years.

Would you like further details or clarifications on this? Here are some additional questions to consider:

1. How is the interest portion of each annual payment calculated?
2. How does the principal portion change with each payment?
3. What would happen if the interest rate increased or decreased?
4. What is the total interest paid over the life of the loan?
5. What is the balance after 5 years of payments?

**Tip:** In a fully amortized loan, the interest portion of each payment decreases over time, while the principal portion increases.

Chuck Ponzi has talked an elderly woman into loaning him $30,000 for a new business venture. She has, however, successfully passed a finance class and requires Chuck to sign a binding contract on repayment of the $30,000 with an annual interest rate of 10% over the next 15 years. Determine the cash flow to the woman under a fully amortized loan, in which Ponzi will make equal annual payments at the end of each year so that the final payment will completely retire the original $30,000 loan.

This problem demonstrates how to calculate the equal annual payments for a fully amortized loan of $30,000 over 15 years with a 10% annual interest rate. It involves key mathematical concepts like loan amortization and interest calculation, using the amortization formula to ensure complete repayment. Suitable for university-level finance or math students.

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