Grace Lends $\$50,000$ On The Condition That She Is Repaid The Money In 15 Equal Yearly Installments. If She Receives Interest At The Rate Of $4\%$ Per Annum, What Is The Amount Of Each Installment?

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Introduction

Grace lends $50,000\$50,000 to an individual with the condition that the money will be repaid in 15 equal yearly installments. The interest rate is 4%4\% per annum, and we need to calculate the amount of each installment. This problem involves the concept of annuities, which is a series of equal payments made at equal intervals over a fixed period of time.

Understanding Annuities

An annuity is a financial arrangement where a series of equal payments are made at equal intervals over a fixed period of time. In this case, the payments are made yearly, and the fixed period is 15 years. The amount of each installment is the same, and the interest rate is 4%4\% per annum.

Formula for Annuity

The formula for calculating the amount of each installment in an annuity is given by:

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

where:

  • AA is the amount of each installment
  • PP is the principal amount (initial amount lent)
  • ii is the interest rate per period
  • nn is the number of periods

Applying the Formula

In this case, the principal amount PP is $50,000\$50,000, the interest rate ii is 4%4\% or 0.040.04, and the number of periods nn is 15. We can now substitute these values into the formula to calculate the amount of each installment.

A=50000(0.04(1+0.04)15(1+0.04)15−1)A = 50000 \left( \frac{0.04(1+0.04)^{15}}{(1+0.04)^{15}-1} \right)

Calculating the Amount of Each Installment

Using a calculator or a computer program, we can calculate the value of the expression inside the parentheses:

(1+0.04)15=1.8099(1+0.04)^{15} = 1.8099

0.04(1.8099)1.8099−1=0.0804\frac{0.04(1.8099)}{1.8099-1} = 0.0804

Now, we can multiply this value by the principal amount to get the amount of each installment:

A=50000×0.0804=4020A = 50000 \times 0.0804 = 4020

Conclusion

The amount of each installment is $4020\$4020. This means that the individual who borrowed the money will have to repay $4020\$4020 each year for 15 years, with an interest rate of 4%4\% per annum.

Importance of Annuities

Annuities are an important concept in finance, as they help individuals and businesses manage their financial obligations over a fixed period of time. By calculating the amount of each installment, individuals can plan their finances and make informed decisions about their investments.

Real-World Applications

Annuities have many real-world applications, including:

  • Retirement planning: Annuities can help individuals plan for their retirement by providing a steady stream of income over a fixed period of time.
  • Business financing: Annuities can help businesses manage their financial obligations by providing a fixed amount of money each period.
  • Investment planning: Annuities can help individuals plan their investments by providing a steady stream of income over a fixed period of time.

Limitations of Annuities

While annuities are a useful financial tool, they have some limitations. For example:

  • Annuities are typically used for long-term financial planning, and may not be suitable for short-term financial needs.
  • Annuities may have fees and charges associated with them, which can reduce the amount of money available for investment.
  • Annuities may not provide the same level of flexibility as other investment options, such as stocks or bonds.

Conclusion

In conclusion, the amount of each installment in an annuity is calculated using the formula:

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

where:

  • AA is the amount of each installment
  • PP is the principal amount (initial amount lent)
  • ii is the interest rate per period
  • nn is the number of periods

In this case, the amount of each installment is $4020\$4020, and the individual who borrowed the money will have to repay this amount each year for 15 years, with an interest rate of 4%4\% per annum.

Introduction

In our previous article, we discussed how to calculate the amount of each installment in an annuity, where Grace lends $50,000\$50,000 to an individual with the condition that the money will be repaid in 15 equal yearly installments. The interest rate is 4%4\% per annum. In this article, we will answer some frequently asked questions (FAQs) related to annuities and installments.

Q: What is an annuity?

A: An annuity is a financial arrangement where a series of equal payments are made at equal intervals over a fixed period of time. In this case, the payments are made yearly, and the fixed period is 15 years.

Q: What is the formula for calculating the amount of each installment in an annuity?

A: The formula for calculating the amount of each installment in an annuity is given by:

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

where:

  • AA is the amount of each installment
  • PP is the principal amount (initial amount lent)
  • ii is the interest rate per period
  • nn is the number of periods

Q: What is the interest rate per period?

A: The interest rate per period is 4%4\% or 0.040.04.

Q: What is the number of periods?

A: The number of periods is 15 years.

Q: What is the principal amount?

A: The principal amount is $50,000\$50,000.

Q: How do I calculate the amount of each installment?

A: To calculate the amount of each installment, you can use the formula:

A=50000(0.04(1+0.04)15(1+0.04)15−1)A = 50000 \left( \frac{0.04(1+0.04)^{15}}{(1+0.04)^{15}-1} \right)

Using a calculator or a computer program, you can calculate the value of the expression inside the parentheses:

(1+0.04)15=1.8099(1+0.04)^{15} = 1.8099

0.04(1.8099)1.8099−1=0.0804\frac{0.04(1.8099)}{1.8099-1} = 0.0804

Now, you can multiply this value by the principal amount to get the amount of each installment:

A=50000×0.0804=4020A = 50000 \times 0.0804 = 4020

Q: What is the amount of each installment?

A: The amount of each installment is $4020\$4020.

Q: Why is it important to calculate the amount of each installment?

A: It is important to calculate the amount of each installment because it helps individuals and businesses manage their financial obligations over a fixed period of time. By calculating the amount of each installment, individuals can plan their finances and make informed decisions about their investments.

Q: What are some real-world applications of annuities?

A: Annuities have many real-world applications, including:

  • Retirement planning: Annuities can help individuals plan for their retirement by providing a steady stream of income over a fixed period of time.
  • Business financing: Annuities can help businesses manage their financial obligations by providing a fixed amount of money each period.
  • Investment planning: Annuities can help individuals plan their investments by providing a steady stream of income over a fixed period of time.

Q: What are some limitations of annuities?

A: While annuities are a useful financial tool, they have some limitations. For example:

  • Annuities are typically used for long-term financial planning, and may not be suitable for short-term financial needs.
  • Annuities may have fees and charges associated with them, which can reduce the amount of money available for investment.
  • Annuities may not provide the same level of flexibility as other investment options, such as stocks or bonds.

Conclusion

In conclusion, annuities are a useful financial tool that can help individuals and businesses manage their financial obligations over a fixed period of time. By calculating the amount of each installment, individuals can plan their finances and make informed decisions about their investments.