Understanding the Present Value of an Ordinary Annuity Formula

Oct 13, 2025 by recoveryshake.com 32 views

Understanding the Present Value of an Ordinary Annuity Formula

In finance and accounting, the present value of an ordinary annuity formula is a crucial concept used to calculate the current value of a series of future cash flows. The formula is given by:

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

where:

PV is the present value of the annuity
P is the periodic payment
i 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 and the Time Value of Money

When payments are made quarterly, the interest rate per period is typically divided by 4, as there are 4 quarters in a year. This is known as the quarterly interest rate. The number of periods, n, is also affected, as there are 4 quarters in a year.

Let's assume that the interest rate per year is 6%, and the payments are made quarterly. The quarterly interest rate would be:

${ i = \frac{6\%}{4} = 1.5\% }$

The number of periods, n, would be:

${ n = 15 \text{ years} \times 4 \text{ quarters/year} = 60 \text{ quarters} }$

Solving for n in the Present Value Formula

Now, let's substitute the values of P, i, and n into the present value formula:

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

${ PV = P\left(\frac{1-(1+0.015)^{-60}}{0.015}\right) }$

To solve for n, we need to isolate n in the equation. However, this is not a straightforward process, as the equation involves a negative exponent. We can use numerical methods or a financial calculator to find the value of n.

Numerical Solution

Using a financial calculator or a numerical method, we can find the value of n that satisfies the equation:

${ PV = P\left(\frac{1-(1+0.015)^{-n}}{0.015}\right) }$

After performing the calculations, we get:

${ n \approx 60 }$

Conclusion

In this article, we used the present value of an ordinary annuity formula to find the value of n when quarterly payments are made for 15 years. We assumed an interest rate of 6% per year and a quarterly interest rate of 1.5%. Using numerical methods or a financial calculator, we found that the value of n is approximately 60.

Comparison of Options

Now, let's compare our result with the given options:

a. 45 b. 60 c. 15 d. $\frac{15}{4}$

Our result, b. 60, matches one of the options. Therefore, the correct answer is:

b. 60

Final Thoughts

The present value of an ordinary annuity formula is a powerful tool used in finance and accounting to calculate the current value of a series of future cash flows. By understanding the formula and its components, we can make informed decisions about investments and financial planning. In this article, we used the formula to find the value of n when quarterly payments are made for 15 years, and we compared our result with the given options.
Frequently Asked Questions (FAQs) about the Present Value of an Ordinary Annuity Formula

In the previous article, we discussed the present value of an ordinary annuity formula and how to find the value of n when quarterly payments are made for 15 years. In this article, we will answer some frequently asked questions (FAQs) about the formula and its applications.

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 future cash flows. It is given by:

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

where:

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

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

A: An ordinary annuity is a series of equal payments made at the end of each period, while an annuity due is a series of equal payments made at the beginning of each period. The present value of an annuity due formula is given by:

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

Q: How do I calculate the present value of an annuity using a financial calculator?

A: To calculate the present value of an annuity using a financial calculator, follow these steps:

Enter the periodic payment (P) and the interest rate per period (i).
Enter the number of periods (n).
Press the "PV" button to calculate the present value.

Q: What is the time value of money?

A: The time value of money is the concept that money received today is worth more than the same amount of money received in the future. This is because money received today can be invested to earn interest, making it worth more in the future.

Q: How do I determine the interest rate per period?

A: The interest rate per period is typically determined by dividing the annual interest rate by the number of periods per year. For example, if the annual interest rate is 6% and the payments are made quarterly, the interest rate per period would be:

${ i = \frac{6\%}{4} = 1.5\% }$

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

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

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

where:

FV is the future value of the annuity
P is the periodic payment
i is the interest rate per period
n is the number of periods

Q: How do I use the present value of an ordinary annuity formula to calculate the present value of a series of future cash flows?

A: To use the present value of an ordinary annuity formula to calculate the present value of a series of future cash flows, follow these steps:

Determine the periodic payment (P) and the interest rate per period (i).
Determine the number of periods (n).
Plug the values into the formula:

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

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the present value of an ordinary annuity formula and its applications. We discussed the formula, its components, and how to use it to calculate the present value of a series of future cash flows. We also provided examples and explanations to help clarify the concepts.