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+t)^{-n}}{t}\right) \\]a. (45) B. (60) C. (15) D.

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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. This formula is essential in determining the value of investments, loans, and other financial instruments. In this article, we will explore the present value of an ordinary annuity formula and find the value of $n$ when quarterly payments are made for 15 years.

The Present Value of an Ordinary Annuity Formula

The present value of an ordinary annuity formula is given by:

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

where:

  • PV$ is the present value of the annuity

  • P$ is the periodic payment

  • t$ is the interest rate per period

  • n$ is the number of periods

Quarterly Payments for 15 Years

We are given that quarterly payments are made for 15 years. To find the value of $n$, we need to convert the number of years to the number of periods. Since there are 4 quarters in a year, the number of periods is:

n=15×4=60n = 15 \times 4 = 60

However, we are asked to find the value of $n$ in the formula. To do this, we need to substitute the values of $P$, $t$, and $n$ into the formula.

Substituting Values into the Formula

Let's assume that the periodic payment $P$ is $100, and the interest rate per period $t$ is 0.05 (5%). We can now substitute these values into the formula:

PV=100(1−(1+0.05)−600.05){ PV = 100\left(\frac{1-(1+0.05)^{-60}}{0.05}\right) }

Simplifying the Formula

To simplify the formula, we can use the fact that $(1+x)^n = e^{n\ln(1+x)}$. Applying this to the formula, we get:

PV=100(1−e−60ln⁡(1.05)0.05){ PV = 100\left(\frac{1-e^{-60\ln(1.05)}}{0.05}\right) }

Evaluating the Formula

Using a calculator to evaluate the formula, we get:

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 a crucial concept in finance and is used to determine the value of investments, loans, and other financial instruments.

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

  • t$: 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 values of $P$, $t$, and $n$ into the formula:

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

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.

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

A: To calculate the present value of an annuity due, you need to use the formula:

PV=P(1−(1+t)−nt)×(1+t){ PV = P\left(\frac{1-(1+t)^{-n}}{t}\right) \times (1+t) }

Q: What is the significance of the number of periods (n) in the present value of an ordinary annuity formula?

A: The number of periods (n) in the present value of an ordinary annuity formula represents the total number of payments made over a given period of time. It is an essential variable in determining the present value of the annuity.

Q: How do I determine the number of periods (n) in the present value of an ordinary annuity formula?

A: To determine the number of periods (n) in the present value of an ordinary annuity formula, you need to consider the frequency of payments and the total duration of the annuity. For example, if the annuity is paid quarterly for 15 years, the number of periods (n) would be:

n=15×4=60n = 15 \times 4 = 60

Q: What is the importance of the interest rate (t) in the present value of an ordinary annuity formula?

A: The interest rate (t) in the present value of an ordinary annuity formula represents the rate at which interest is earned on the annuity. It is an essential variable in determining the present value of the annuity.

Q: How do I determine the interest rate (t) in the present value of an ordinary annuity formula?

A: To determine the interest rate (t) in the present value of an ordinary annuity formula, you need to consider the market interest rate and the risk-free rate of return. For example, if the market interest rate is 5% and the risk-free rate of return is 3%, the interest rate (t) would be:

t=0.05−0.03=0.02t = 0.05 - 0.03 = 0.02

Q: What is the significance of the periodic payment (P) in the present value of an ordinary annuity formula?

A: The periodic payment (P) in the present value of an ordinary annuity formula represents the amount of each payment made over a given period of time. It is an essential variable in determining the present value of the annuity.

Q: How do I determine the periodic payment (P) in the present value of an ordinary annuity formula?

A: To determine the periodic payment (P) in the present value of an ordinary annuity formula, you need to consider the total amount of the annuity and the number of periods. For example, if the total amount of the annuity is $100,000 and the number of periods is 60, the periodic payment (P) would be:

P=100,00060=1,666.67P = \frac{100,000}{60} = 1,666.67

Conclusion

In conclusion, the present value of an ordinary annuity formula is a crucial concept in finance that is used to calculate the current value of a series of future cash flows. The variables in the formula are the present value of the annuity, the periodic payment, the interest rate per period, and the number of periods. By understanding the present value of an ordinary annuity formula, you can make informed decisions about investments, loans, and other financial instruments.