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 find the present value (PV) of this annuity, we need to calculate the present value of each payment and sum them up. The formula for the present value of a future payment is:

\[
PV = \frac{FV}{(1 + r)^n}
\]

Where:
- $ FV $ is the future value (the payment amount, which is \$10,000).
- $ r $ is the interest rate (10% or 0.10).
- $ n $ is the number of years until the payment.

Since the payments are made annually for 5 years, the payments happen at $ n = 0, 1, 2, 3, 4 $. We will calculate the present value of each of these payments and sum them up.

### Calculation of Present Value of Each Payment:
1. **Payment at $ n = 0 $** (today):
   \[
   PV_0 = \frac{10,000}{(1 + 0.10)^0} = 10,000
   \]
2. **Payment at $ n = 1 $** (after 1 year):
   \[
   PV_1 = \frac{10,000}{(1 + 0.10)^1} = \frac{10,000}{1.10} = 9,090.91
   \]
3. **Payment at $ n = 2 $** (after 2 years):
   \[
   PV_2 = \frac{10,000}{(1 + 0.10)^2} = \frac{10,000}{1.21} = 8,264.46
   \]
4. **Payment at $ n = 3 $** (after 3 years):
   \[
   PV_3 = \frac{10,000}{(1 + 0.10)^3} = \frac{10,000}{1.331} = 7,513.15
   \]
5. **Payment at $ n = 4 $** (after 4 years):
   \[
   PV_4 = \frac{10,000}{(1 + 0.10)^4} = \frac{10,000}{1.4641} = 6,830.14
   \]

### Total Present Value:
\[
PV_{\text{total}} = PV_0 + PV_1 + PV_2 + PV_3 + PV_4
\]
\[
PV_{\text{total}} = 10,000 + 9,090.91 + 8,264.46 + 7,513.15 + 6,830.14 = 41,698.66
\]

### Conclusion:
The present value of this annuity is approximately **\$41,698.66**.

Would you like further clarification or details on this calculation?

Here are 5 related questions:
1. How would the present value change if the interest rate increased to 12%?
2. What is the future value of this annuity at the end of 5 years?
3. How would the result differ if payments were made at the end of each year instead of the beginning?
4. How does the present value of an annuity differ from the present value of a lump sum?
5. Can the present value formula be adapted for monthly compounding?

**Tip:** In financial problems, increasing the interest rate decreases the present value, while lowering the interest rate increases the present value.

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?

Discover how to calculate the present value of a 5-year annuity where you receive $10,000 annually at a 10% compound interest rate. This solution uses key formulas to find the total present value, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation