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CON ESAS FORMULAS RESOLVER Y RESPONDER EL SIGUIENTE PROBLEMA Establecer el precio de contado de una casa con las siguientes condiciones:

a) Enganche del 20% de su valor de contado.

b) Amortizaciones trimestrales vencidas por valor de Q 7,500.00 por un plazo de 18 años, las cuales dieron inicio, al principio del cuarto bimestre del tercer año de adquirida la casa, la institución que otorgó el financiamiento cargó una tasa del 10 3/4% anual de interés para los primeros 8 años y por la diferencia de años cargó el 12.5% anual de interés capitalizable en forma trimestral, además se sabe que deberá realizarse un pago único extraordinario por valor de Q 140,000.00 junto con la última amortización.

To solve this problem, we need to determine the cash price (or "precio de contado") of the house using the provided conditions. Here's a breakdown of the steps involved:

### Given Information:
1. **Down Payment (Enganche):** 20% of the cash price.
2. **Installments (Amortizaciones):** 
   - Quarterly payments of Q7,500 for 18 years.
   - Payments start at the beginning of the fourth quarter of the third year.
3. **Interest Rates:**
   - 10.75% annually for the first 8 years.
   - 12.5% annually for the remaining years, compounded quarterly.
4. **Balloon Payment:** A single final payment of Q140,000 is due with the last installment.

### Step-by-Step Solution:

#### 1. **Find the number of total quarters (n):**
   The payment term is 18 years starting from the third year. There are 4 quarters per year:
   \[
   n = 18 \times 4 = 72 \text{ quarters}
   \]

#### 2. **Interest Rates Adjustments:**
   - **First 8 years (32 quarters):** The interest rate is 10.75% annually.
     The quarterly interest rate for this period is:
     \[
     i_1 = \frac{10.75}{4} = 2.6875\% \text{ per quarter}
     \]
   - **Remaining 10 years (40 quarters):** The interest rate is 12.5% annually.
     The quarterly interest rate for this period is:
     \[
     i_2 = \frac{12.5}{4} = 3.125\% \text{ per quarter}
     \]

#### 3. **Calculate the present value of the installments:**
   Since the installments change interest rates partway through, we calculate the present value for each period separately.

   - **Present Value for the First 32 Quarters (at 10.75% annually):**
     The formula for the present value of an annuity is:
     \[
     PV_1 = A \times \left(1 - \frac{1}{(1 + i_1)^{32}}\right) \div i_1
     \]
     where \(A = 7,500\), and \(i_1 = 0.026875\).

   - **Present Value for the Next 40 Quarters (at 12.5% annually):**
     \[
     PV_2 = A \times \left(1 - \frac{1}{(1 + i_2)^{40}}\right) \div i_2
     \]
     where \(i_2 = 0.03125\).

#### 4. **Calculate the present value of the balloon payment:**
   This is simply the present value of Q140,000 paid at the end of the 72 quarters:
   \[
   PV_{\text{balloon}} = \frac{140,000}{(1 + i_2)^{40}}
   \]

#### 5. **Add the Down Payment:**
   Since the down payment is 20% of the house price, we solve for the total price \(P\). The down payment is \(0.20P\), and the remaining 80% is covered by the present value of the installments and balloon payment:
   \[
   0.80P = PV_1 + PV_2 + PV_{\text{balloon}}
   \]

#### 6. **Solve for the Price:**
   \[
   P = \frac{PV_1 + PV_2 + PV_{\text{balloon}}}{0.80}
   \]

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Here are five related questions to expand the information:
1. How does the formula for annuity payments work in amortization schedules?
2. What are the implications of varying interest rates over time on a loan?
3. How does compounding interest quarterly affect long-term loan repayments?
4. Can balloon payments significantly alter the total cost of a mortgage?
5. What factors should be considered when choosing between fixed and variable interest rates?

**Tip:** Always verify the timing of payments and interest changes in a loan agreement, as they can greatly affect the overall cost.

CON ESAS FORMULAS RESOLVER Y RESPONDER EL SIGUIENTE PROBLEMA Establecer el precio de contado de una casa con las siguientes condiciones: a) Enganche del 20% de su valor de contado. b) Amortizaciones trimestrales vencidas por valor de Q 7,500.00 por un plazo de 18 años, las cuales dieron inicio, al principio del cuarto bimestre del tercer año de adquirida la casa, la institución que otorgó el financiamiento cargó una tasa del 10 3/4% anual de interés para los primeros 8 años y por la diferencia de años cargó el 12.5% anual de interés capitalizable en forma trimestral, además se sabe que deberá realizarse un pago único extraordinario por valor de Q 140,000.00 junto con la última amortización.

Solve the problem of determining the cash price of a house with a down payment, quarterly amortizations, compounded interest, and a balloon payment. Learn how to apply the present value of an annuity formula, deal with changing interest rates, and calculate the final price of the house over 18 years.

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