Compounding Frequency and Interest Rate

Oct 13, 2025 by leolavanderia.com.br 65 views

Introduction

In finance, 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 $$PV = P \left(\frac{1-(1+i)^{-2}}{i}\right)$$, where $PV$ is the present value, $P$ is the periodic payment, $i$ is the interest rate per period, and $n$ is the number of periods. In this article, we will focus on finding the exact value of $i$ in the present value ordinary annuity formula when the account is increasing at a rate of $3.3 \%$ compounded semiannually.

Compounding Frequency and Interest Rate

Compounding frequency refers to the number of times interest is compounded per year. In this case, the account is compounded semiannually, which means that interest is compounded twice a year. The interest rate per period is given as $3.3 \%$ , but we need to find the exact value of $i$ in the present value ordinary annuity formula.

Present Value Ordinary Annuity Formula

The present value ordinary annuity formula is given by $$PV = P \left(\frac{1-(1+i)^{-2}}{i}\right)$$. To find the exact value of $i$ , we need to substitute the given values into the formula.

Substituting Values into the Formula

Let's substitute the given values into the present value ordinary annuity formula:

PV = P \left(\frac{1-(1+i)^{-2}}{i}\right)

We are given that the account is increasing at a rate of $3.3 \%$ compounded semiannually. This means that the interest rate per period is $3.3 \%$ divided by 2, which is $1.65 \%$ . However, we need to find the exact value of $i$ in the present value ordinary annuity formula.

Finding the Exact Value of $i$

To find the exact value of $i$ , we need to simplify the present value ordinary annuity formula. We can start by substituting $i = 0.0165$ into the formula:

PV = P \left(\frac{1-(1+0.0165)^{-2}}{0.0165}\right)

Simplifying the formula, we get:

PV = P \left(\frac{1-0.9835}{0.0165}\right)

PV = P \left(\frac{0.0165}{0.0165}\right)

PV = P

This means that the present value of the ordinary annuity is equal to the periodic payment $P$ . Therefore, the exact value of $i$ in the present value ordinary annuity formula is $0.0165$ .

Conclusion

In this article, we have discussed the present value of an ordinary annuity formula and how to find the exact value of $i$ when the account is increasing at a rate of $3.3 \%$ compounded semiannually. We have simplified the present value ordinary annuity formula and found that the exact value of $i$ is $0.0165$ . This value can be used in financial calculations to determine the present value of an ordinary annuity.

References

[1] Investopedia. (n.d.). Present Value of an Ordinary Annuity. Retrieved from https://www.investopedia.com/terms/p/presentvalueofordinaryannuity.asp
[2] Khan Academy. (n.d.). Present Value of an Ordinary Annuity. Retrieved from https://www.khanacademy.org/economics-finance-personal-finance/financial-calculations/annuities/v/present-value-of-an-ordinary-annuity

Mathematical Derivations

Derivation of the Present Value Ordinary Annuity Formula

The present value ordinary annuity formula can be derived using the following steps:

The present value of an ordinary annuity is given by the formula:
$PV = P \left(\frac{1-(1+i)^{-n}}{i}\right)$
where $PV$ is the present value, $P$ is the periodic payment, $i$ is the interest rate per period, and $n$ is the number of periods.
To find the present value of an ordinary annuity with semiannual compounding, we need to substitute $n = 2$ into the formula:
$PV = P \left(\frac{1-(1+i)^{-2}}{i}\right)$
This is the present value ordinary annuity formula with semiannual compounding.

Derivation of the Exact Value of $i$

To find the exact value of $i$ , we need to substitute $i = 0.0165$ into the present value ordinary annuity formula:

PV = P \left(\frac{1-(1+0.0165)^{-2}}{0.0165}\right)

Simplifying the formula, we get:

PV = P \left(\frac{1-0.9835}{0.0165}\right)

PV = P \left(\frac{0.0165}{0.0165}\right)

PV = P

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions 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 future cash flows. It is given by the formula:

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

where $PV$ is the present value, $P$ is the periodic payment, $i$ is the interest rate per period, and $n$ is the number of periods.

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

A: The present value of an ordinary annuity and the present value of an annuity due are two different formulas used to calculate the current value of a series of future cash flows. The main difference between the two formulas is the timing of the cash flows. The present value of an ordinary annuity assumes that the cash flows are received at the end of each period, while the present value of an annuity due assumes that the cash flows are received at the beginning of each period.

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 follow these steps:

Determine the periodic payment $P$ .
Determine the interest rate per period $i$ .
Determine the number of periods $n$ .
Substitute the values into the present value ordinary annuity formula:
$PV = P \left(\frac{1-(1+i)^{-n}}{i}\right)$

Q: What is the formula for the present value of an ordinary annuity with semiannual compounding?

A: The formula for the present value of an ordinary annuity with semiannual compounding is:

PV = P \left(\frac{1-(1+i)^{-2}}{i}\right)

where $PV$ is the present value, $P$ is the periodic payment, $i$ is the interest rate per period, and $n$ is the number of periods.

Q: How do I find the exact value of $i$ in the present value ordinary annuity formula?

A: To find the exact value of $i$ in the present value ordinary annuity formula, you need to substitute the given values into the formula and simplify it. For example, if the account is increasing at a rate of $3.3 \%$ compounded semiannually, you can substitute $i = 0.0165$ into the formula:

PV = P \left(\frac{1-(1+0.0165)^{-2}}{0.0165}\right)

Simplifying the formula, you get:

PV = P \left(\frac{1-0.9835}{0.0165}\right)

PV = P \left(\frac{0.0165}{0.0165}\right)

PV = P

This means that the present value of the ordinary annuity is equal to the periodic payment $P$ . Therefore, the exact value of $i$ in the present value ordinary annuity formula is $0.0165$ .

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

A: The present value of an ordinary annuity formula has many common applications in finance, including:

Calculating the present value of a series of future cash flows
Determining the value of a bond or other fixed-income security
Evaluating the financial performance of a company or project
Calculating the present value of an annuity due

Conclusion

In this article, we have answered some of the most frequently asked questions about the present value of an ordinary annuity formula. We have discussed the formula, its applications, and how to calculate the present value of an ordinary annuity. We hope that this article has been helpful in understanding the present value of an ordinary annuity formula.

References

[1] Investopedia. (n.d.). Present Value of an Ordinary Annuity. Retrieved from https://www.investopedia.com/terms/p/presentvalueofordinaryannuity.asp
[2] Khan Academy. (n.d.). Present Value of an Ordinary Annuity. Retrieved from https://www.khanacademy.org/economics-finance-personal-finance/financial-calculations/annuities/v/present-value-of-an-ordinary-annuity

Mathematical Derivations

Derivation of the Present Value Ordinary Annuity Formula

The present value ordinary annuity formula can be derived using the following steps:

The present value of an ordinary annuity is given by the formula:
$PV = P \left(\frac{1-(1+i)^{-n}}{i}\right)$
where $PV$ is the present value, $P$ is the periodic payment, $i$ is the interest rate per period, and $n$ is the number of periods.
To find the present value of an ordinary annuity with semiannual compounding, we need to substitute $n = 2$ into the formula:
$PV = P \left(\frac{1-(1+i)^{-2}}{i}\right)$
This is the present value ordinary annuity formula with semiannual compounding.

Derivation of the Exact Value of $i$

To find the exact value of $i$ , we need to substitute $i = 0.0165$ into the present value ordinary annuity formula:

PV = P \left(\frac{1-(1+0.0165)^{-2}}{0.0165}\right)

Simplifying the formula, we get:

PV = P \left(\frac{1-0.9835}{0.0165}\right)

PV = P \left(\frac{0.0165}{0.0165}\right)

PV = P

This means that the present value of the ordinary annuity is equal to the periodic payment $P$ . Therefore, the exact value of $i$ in the present value ordinary annuity formula is $0.0165$ .