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The question asks for the present value of a series of quarterly payments of **$3,000** over **8 years**, with an interest rate of **10% compounded monthly**. It has two parts: (a) doing the calculation using formulas, and (b) performing it in MS Excel.

### **Part (a) Calculation Using Formulas**
The question refers to calculating the present value of an **annuity**. Since the payments are quarterly, but the interest is compounded monthly, we need to adjust the interest rate accordingly.

#### **Key parameters:**
- **Quarterly Payment (PMT)**: 3,000  
- **Number of Years (n)**: 8  
- **Payments per year**: 4 (quarterly)  
- **Total number of payments (N)**: $8 \times 4 = 32$  
- **Nominal Annual Interest Rate (r)**: 10% or 0.10  
- **Monthly interest rate (i_m)**: $ \frac{10\%}{12} = \frac{0.10}{12} \approx 0.008333 $  
- **Quarterly interest rate (i_q)**: $(1 + 0.008333)^3 - 1 \approx 0.0252$ (since three months make one quarter)

#### **Present Value Formula for an Annuity:**

\[
PV = PMT \times \left( 1 - (1 + i_q)^{-N} \right) \div i_q
\]

#### **Substitute the values:**

\[
PV = 3,000 \times \left( 1 - (1 + 0.0252)^{-32} \right) \div 0.0252
\]

\[
PV = 3,000 \times \left( 1 - (1.0252)^{-32} \right) \div 0.0252
\]

\[
PV = 3,000 \times \left( 1 - 0.446 \right) \div 0.0252
\]

\[
PV = 3,000 \times 0.554 \div 0.0252
\]

\[
PV = 3,000 \times 21.98 \approx 65,940
\]

Thus, the **present value** of the series of payments is approximately **$65,940**.

---

### **Part (b) Calculation in MS Excel**
To perform the calculation in **MS Excel**, you can use the `PV` function.

#### **Excel Formula:**
\[
\text{=PV}(0.0252, 32, -3000)
\]

- **Rate**: 0.0252 (quarterly interest rate)  
- **Nper**: 32 (total number of payments)  
- **PMT**: -3000 (negative because it is an outgoing payment)  

Place the formula in **cell E43** as instructed.

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Let me know if you need further clarification or assistance!

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### **Follow-up Questions:**
1. What happens to the present value if the interest rate increases?
2. How would the present value change if the payments were monthly instead of quarterly?
3. Can you explain how to derive the formula for the quarterly interest rate?
4. How does the compounding frequency impact the final result?
5. How would the Excel formula change if the payments were at the beginning of each period (annuity due)?

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**Tip:** In financial calculations, always ensure that the compounding frequency and payment intervals are aligned for accuracy.

Quelle est la valeur présente d'une série de paiements trimestriels égaux de 3 000 $ qui s'étendent sur une période de huit ans si le taux d'intérêt est de 10 % composé mensuellement ?

Learn how to find the present value of a series of $3,000 quarterly payments over 8 years with an interest rate of 10% compounded monthly. This problem involves using annuity formulas and adjusting for different compounding frequencies. Includes detailed steps and Excel formula.

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