If Quarterly Payments Are Made For 15 Years, Find The Value For { N $}$ In The Following Present Value Ordinary Annuity Formula:${ PV = P\left(\frac{1-(1+r)^{-n}}{r}\right) }$a. 45 B. 60 C. 15 D. { \frac{15}{4}$}$

Introduction to Ordinary Annuity Formula

In finance, an ordinary annuity is a series of equal payments made at the end of each period. The present value of an ordinary annuity formula is used to calculate the current value of these future payments. The formula is given by:

${ PV = P\left(\frac{1-(1+r)^{-n}}{r}\right) }$

where:

• PV is the present value of the annuity
• P is the periodic payment
• r is the interest rate per period
• n is the number of periods

In this article, we will focus on finding the value of n in the given formula when quarterly payments are made for 15 years.

Quarterly Payments for 15 Years

To find the value of n, we need to substitute the given values into the formula. Since the payments are made quarterly, we will use a quarterly interest rate and the number of periods will be 15 years multiplied by 4 (since there are 4 quarters in a year).

Let's assume the periodic payment (P) is $100, the interest rate (r) is 5% per quarter, and the number of periods (n) is 60 (15 years multiplied by 4).

Calculating the Present Value

Substituting the given values into the formula, we get:

${ PV = 100\left(\frac{1-(1+0.05)^{-60}}{0.05}\right) }$

To calculate the present value, we need to evaluate the expression inside the parentheses first.

Evaluating the Expression

Using a calculator or a financial calculator, we can evaluate the expression as follows:

${ 1-(1+0.05)^{-60} \approx 1-0.3017 \approx 0.6983 }$

Now, we can substitute this value back into the formula:

${ PV = 100\left(\frac{0.6983}{0.05}\right) }$

Simplifying the Expression

Simplifying the expression, we get:

${ PV = 100\left(13.966\right) }$

${ PV \approx 1396.60 }$

Finding the Value of n

Now that we have calculated the present value, we need to find the value of n. To do this, we can use the formula:

${ PV = P\left(\frac{1-(1+r)^{-n}}{r}\right) }$

Rearranging the formula to solve for n, we get:

${ n = \frac{\log\left(1-\frac{rPV}{P}\right)}{\log(1+r)} }$

Substituting the given values, we get:

${ n = \frac{\log\left(1-\frac{0.05 \times 1396.60}{100}\right)}{\log(1+0.05)} }$

Evaluating the Expression

Using a calculator or a financial calculator, we can evaluate the expression as follows:

${ n \approx \frac{\log(0.6983)}{\log(1.05)} }$

${ n \approx \frac{-0.1543}{0.0212} }$

${ n \approx 7.28 }$

Conclusion

In this article, we used the present value of an ordinary annuity formula to calculate the value of n when quarterly payments are made for 15 years. We assumed a periodic payment of $100, an interest rate of 5% per quarter, and the number of periods was 60. We calculated the present value and then used the formula to find the value of n. The result was approximately 7.28, which is closest to option d. $\frac{15}{4}$.

References

• "Present Value of an Ordinary Annuity Formula" by Investopedia
• "Ordinary Annuity Formula" by Calculator Soup
• "Present Value of an Annuity" by Financial Calculator

Discussion

What do you think about the present value of an ordinary annuity formula? Have you used it in any financial calculations? Share your thoughts and experiences in the comments below.

Related Articles

• "Understanding the Future Value of an Ordinary Annuity Formula"
• "Calculating the Present Value of a Growing Annuity"
• "Using the Present Value of an Ordinary Annuity Formula in Real-World Scenarios"
  Frequently Asked Questions (FAQs) about the Present Value of an Ordinary Annuity Formula =====================================================================================

Q: What is the present value of an ordinary annuity formula?

A: The present value of an ordinary annuity formula is a mathematical formula used to calculate the current value of a series of equal payments made at the end of each period. The formula is given by:

${ PV = P\left(\frac{1-(1+r)^{-n}}{r}\right) }$

Q: What are the variables in the present value of an ordinary annuity formula?

A: The variables in the present value of an ordinary annuity formula are:

• PV: the present value of the annuity
• P: the periodic payment
• r: the interest rate per period
• n: the number of periods

Q: How do I calculate the present value of an ordinary annuity?

A: To calculate the present value of an ordinary annuity, you need to substitute the given values into the formula and evaluate the expression. You can use a calculator or a financial calculator to simplify the calculation.

Q: What is the difference between the present value of an ordinary annuity and the future value of an ordinary annuity?

A: The present value of an ordinary annuity formula calculates the current value of a series of equal payments made at the end of each period, while the future value of an ordinary annuity formula calculates the future value of a series of equal payments made at the beginning of each period.

Q: Can I use the present value of an ordinary annuity formula to calculate the value of a bond?

A: Yes, you can use the present value of an ordinary annuity formula to calculate the value of a bond. The formula can be used to calculate the present value of the bond's coupon payments and the face value of the bond.

Q: What are some common applications of the present value of an ordinary annuity formula?

A: Some common applications of the present value of an ordinary annuity formula include:

• Calculating the present value of a series of equal payments made at the end of each period
• Calculating the value of a bond
• Calculating the present value of a growing annuity
• Calculating the present value of a perpetuity

Q: Can I use the present value of an ordinary annuity formula to calculate the value of a perpetuity?

A: Yes, you can use the present value of an ordinary annuity formula to calculate the value of a perpetuity. A perpetuity is a type of annuity that makes payments indefinitely.

Q: What are some common mistakes to avoid when using the present value of an ordinary annuity formula?

A: Some common mistakes to avoid when using the present value of an ordinary annuity formula include:

• Not substituting the correct values into the formula
• Not evaluating the expression correctly
• Not using the correct interest rate or number of periods
• Not considering the time value of money

Q: Can I use the present value of an ordinary annuity formula to calculate the value of a growing annuity?

A: Yes, you can use the present value of an ordinary annuity formula to calculate the value of a growing annuity. A growing annuity is a type of annuity that makes payments that increase over time.

Q: What are some real-world applications of the present value of an ordinary annuity formula?

A: Some real-world applications of the present value of an ordinary annuity formula include:

• Calculating the present value of a series of equal payments made at the end of each period
• Calculating the value of a bond
• Calculating the present value of a growing annuity
• Calculating the present value of a perpetuity
• Calculating the value of a retirement account

Conclusion

The present value of an ordinary annuity formula is a powerful tool used to calculate the current value of a series of equal payments made at the end of each period. The formula is widely used in finance and accounting to calculate the value of bonds, growing annuities, and perpetuities. By understanding the variables and applications of the present value of an ordinary annuity formula, you can make informed decisions about your financial investments and planning.

References

• "Present Value of an Ordinary Annuity Formula" by Investopedia
• "Ordinary Annuity Formula" by Calculator Soup
• "Present Value of an Annuity" by Financial Calculator

Discussion

What do you think about the present value of an ordinary annuity formula? Have you used it in any financial calculations? Share your thoughts and experiences in the comments below.

Related Articles

• "Understanding the Future Value of an Ordinary Annuity Formula"
• "Calculating the Present Value of a Growing Annuity"
• "Using the Present Value of an Ordinary Annuity Formula in Real-World Scenarios"