Suppose your mom decides to gift you₹10,000 every year starting from today for the next five years. You deposit this amount in a bank as and when you receive and get 10% per annum interest rate compounded annually. What is the present value of this annuity?

To calculate the present value of this annuity, where your mom gifts you ₹10,000 every year for five years, starting today, we need to consider each cash flow and discount it back to the present value using the given interest rate.

### Step 1: Understand the Cash Flows
You will receive ₹10,000 at the beginning of each year for five years. This means it's an **annuity due** because the payments are made at the start of each period.

### Step 2: Present Value Formula for Annuity Due
The present value \( PV \) of an annuity due can be calculated using the formula:
\[
PV = P \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \times (1 + r)
\]
Where:
- \( P \) is the annual payment (₹10,000),
- \( r \) is the interest rate (10% or 0.10),
- \( n \) is the number of periods (5 years).

### Step 3: Calculate the Present Value
Let's break it down:

1. First, calculate the present value factor for an ordinary annuity (without the immediate extra compounding):
   \[
   PV_{\text{ordinary}} = \frac{1 - (1 + 0.10)^{-5}}{0.10}
   \]
   
2. Compute \( (1 + 0.10)^{-5} \):
   \[
   (1.10)^{-5} \approx 0.6209
   \]
   
3. Subtract from 1:
   \[
   1 - 0.6209 = 0.3791
   \]
   
4. Divide by the interest rate:
   \[
   \frac{0.3791}{0.10} = 3.791
   \]

5. Adjust for the annuity due by multiplying by \( 1 + r \):
   \[
   PV = 10,000 \times 3.791 \times 1.10 \approx 10,000 \times 4.1701 \approx 41,701
   \]

### Final Answer
The present value of the annuity, given that the payments start immediately and are compounded annually at 10%, is approximately ₹41,701.

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Do you have any questions or need further details? Here are some related questions that might interest you:

1. How would the present value change if the interest rate were 8% instead of 10%?
2. What would be the future value of this annuity after 5 years?
3. How does the concept of "annuity due" differ from an "ordinary annuity"?
4. What would happen to the present value if the payments were delayed by one year?
5. How would inflation affect the real value of the ₹10,000 you receive each year?

**Tip:** When dealing with annuities, always distinguish between an annuity due (payments at the start of each period) and an ordinary annuity (payments at the end), as this affects the present value calculation.

Learn how to calculate the present value of an annuity due where ₹10,000 is gifted annually for five years, compounded annually at 10%. Understand the financial concepts of annuities, compounding interest, and present value calculations. Suitable for advanced high school and college students.

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