Find The Future Value Of The Following Annuity Due. Assume That Interest Is Compounded Annually, There Are N N N Payments Of R R R Dollars, And The Interest Rate Is I I I .Given: - R = 800 R = 800 R = 800 - I = 0.06 I = 0.06 I = 0.06 - $n =

Mar 10, 2025 by ADMIN 241 views

**Finding the Future Value of an Annuity Due: A Comprehensive Guide**

Introduction

An annuity due is a type of financial instrument that involves a series of periodic payments made at the beginning of each period. In this article, we will explore the concept of finding the future value of an annuity due, which is a crucial aspect of financial mathematics. We will assume that interest is compounded annually, and we will use the given values of the payment amount, interest rate, and number of payments to calculate the future value of the annuity due.

Understanding Annuity Due

An annuity due is a type of annuity that involves a series of periodic payments made at the beginning of each period. This is in contrast to an ordinary annuity, where payments are made at the end of each period. The future value of an annuity due is the total amount that will be accumulated at the end of the payment period, assuming that interest is compounded annually.

Given Values

In this problem, we are given the following values:

Payment Amount (R): $800
Interest Rate (i): 0.06 (or 6%)
Number of Payments (n): To be determined

Calculating the Future Value of an Annuity Due

The formula for calculating the future value of an annuity due is given by:

FV = R x (((1 + i)^n - 1) / i) x (1 + i)

where FV is the future value of the annuity due, R is the payment amount, i is the interest rate, and n is the number of payments.

However, since the payments are made at the beginning of each period, we need to adjust the formula to account for this. The adjusted formula is given by:

FV = R x (((1 + i)^n - 1) / i) x (1 + i) x (1 + i)

Simplifying the formula, we get:

FV = R x (((1 + i)^n - 1) / i) x (1 + i)^2

Calculating the Future Value

Now that we have the adjusted formula, we can plug in the given values to calculate the future value of the annuity due.

FV = 800 x (((1 + 0.06)^n - 1) / 0.06) x (1 + 0.06)^2

To calculate the future value, we need to determine the value of n, which represents the number of payments. Since the problem does not specify the value of n, we will assume that n is a variable that we need to solve for.

Solving for n

To solve for n, we can use the formula for the future value of an annuity due, which is given by:

FV = R x (((1 + i)^n - 1) / i) x (1 + i)^2

Rearranging the formula to solve for n, we get:

n = log(FV / (R x (1 + i)^2)) / log(1 + i)

Numerical Solution

To find the numerical solution for n, we can use a calculator or a computer program to evaluate the expression.

Assuming that FV = 100,000, R = 800, i = 0.06, and using the formula above, we get:

n ≈ 20.47

Conclusion

In this article, we have explored the concept of finding the future value of an annuity due, which is a crucial aspect of financial mathematics. We have assumed that interest is compounded annually, and we have used the given values of the payment amount, interest rate, and number of payments to calculate the future value of the annuity due. We have also solved for the number of payments, n, using the formula for the future value of an annuity due.

Future Value of Annuity Due Formula

The formula for calculating the future value of an annuity due is given by:

FV = R x (((1 + i)^n - 1) / i) x (1 + i)^2

where FV is the future value of the annuity due, R is the payment amount, i is the interest rate, and n is the number of payments.

Annuity Due vs Ordinary Annuity

Real-World Applications

The concept of finding the future value of an annuity due has numerous real-world applications in finance, accounting, and economics. For example, it can be used to calculate the future value of a series of payments made by a business or an individual, or to determine the present value of a future stream of payments.

Limitations

The formula for calculating the future value of an annuity due assumes that interest is compounded annually, and that the payment amount and interest rate remain constant over the payment period. In reality, interest rates and payment amounts may vary over time, which can affect the accuracy of the calculation.

Conclusion

Q: What is an annuity due?

A: An annuity due is a type of financial instrument that involves a series of periodic payments made at the beginning of each period. This is in contrast to an ordinary annuity, where payments are made at the end of each period.

Q: What is the formula for calculating the future value of an annuity due?

A: The formula for calculating the future value of an annuity due is given by:

FV = R x (((1 + i)^n - 1) / i) x (1 + i)^2

where FV is the future value of the annuity due, R is the payment amount, i is the interest rate, and n is the number of payments.

Q: What is the difference between an annuity due and an ordinary annuity?

A: The main difference between an annuity due and an ordinary annuity is the timing of the payments. In an annuity due, payments are made at the beginning of each period, while in an ordinary annuity, payments are made at the end of each period.

Q: How do I calculate the future value of an annuity due?

A: To calculate the future value of an annuity due, you need to use the formula above and plug in the values of R, i, and n. You can use a calculator or a computer program to evaluate the expression.

Q: What is the significance of the interest rate (i) in the formula?

A: The interest rate (i) is a crucial component of the formula, as it determines the rate at which the future value of the annuity due grows over time. A higher interest rate will result in a higher future value, while a lower interest rate will result in a lower future value.

Q: Can I use the formula for an ordinary annuity to calculate the future value of an annuity due?

A: No, you cannot use the formula for an ordinary annuity to calculate the future value of an annuity due. The formula for an ordinary annuity assumes that payments are made at the end of each period, while the formula for an annuity due assumes that payments are made at the beginning of each period.

Q: What are some real-world applications of the concept of annuity due?

A: The concept of annuity due has numerous real-world applications in finance, accounting, and economics. For example, it can be used to calculate the future value of a series of payments made by a business or an individual, or to determine the present value of a future stream of payments.

Q: What are some limitations of the formula for calculating the future value of an annuity due?

A: The formula for calculating the future value of an annuity due assumes that interest is compounded annually, and that the payment amount and interest rate remain constant over the payment period. In reality, interest rates and payment amounts may vary over time, which can affect the accuracy of the calculation.

Q: Can I use the formula for annuity due to calculate the present value of a future stream of payments?

A: No, the formula for annuity due is used to calculate the future value of a series of payments, not the present value of a future stream of payments. To calculate the present value of a future stream of payments, you need to use a different formula, such as the present value of an annuity formula.

Q: What is the relationship between the future value of an annuity due and the present value of an annuity due?

A: The future value of an annuity due and the present value of an annuity due are related, but they are not the same thing. The future value of an annuity due is the total amount that will be accumulated at the end of the payment period, while the present value of an annuity due is the total amount that is required to be paid at the beginning of the payment period to accumulate a certain amount at the end of the payment period.