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Chapter 10: Problem 40

Suppose you invest \(\$ P\) on a CD paying \(1.85 \%\) interest compounded continuously for a term of five years. At the end of the term you get \(\$ 1645.37\) from the bank. Find the value of the original principal \(P\).

Short Answer

Expert verified
The original principal \(P\) is calculated to be approximately \$1500.

Step by step solution

01

Identification of Known variables

From the problem, you know the final amount \(A = \$1645.37\), the time \(t = 5 years\), and the interest rate \(r = 1.85\% = 0.0185\). The only variable left to find is the principal, \(P\).
02

Rearrange the formula to solve for P

By rearranging the formula \(A = Pe^{rt}\), you can obtain \(P = A / (e^{rt})\).
03

Substitute the known values and calculate

Substitute \(A = \$1645.37\), \(r = 0.0185\) and \(t = 5\) into the rearranged formula to solve for \(P\). This results in \(P = \$1645.37 / (e^{0.0185 \times 5})\). Evaluating this expression will give you the value of \(P\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Compounding
Continuous compounding is a powerful financial concept where interest is added to the principal amount continuously, instead of at discrete intervals such as annually or monthly. This means that the earnings on interest are constantly being calculated and added back into the principal, resulting in a faster accumulation of wealth.
This concept is different from regular compounding methods:
  • Frequency: Interest is calculated infinitely many times per year.
  • Formula: The amount of money grows according to the formula: \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal investment, and \( r \) is the annual interest rate as a decimal.
Continuous compounding can significantly increase the total amount accrued, especially when interest rates are higher or the investment period is longer. This formula can be used effectively to project future values of investments.
Principal Calculation
Principal calculation involves finding the initial amount of money invested before any interest has been applied. Given a future value after interest is applied, back-calculate the original investment amount using the formula derived for continuous compounding.
To find the principal, formula manipulation is needed. The formula \( P = A / (e^{rt}) \) rearranges the continuous compounding equation:
  • Given: The future value \( A \), interest rate \( r \), and time \( t \).
  • Rearranged: \( P = A / (e^{rt}) \) solves for \( P \).
This transformation helps us easily calculate how much initial money was invested if we know how much it will grow to over time. By substituting known values into this formula, we reveal the principal.
Exponential Growth
Exponential growth refers to an increase that becomes more rapid in proportion to the growing total number or size. In finance, especially with continuous compounding, growth occurs exponentially over time because the interest accumulates on top of interest that has previously been added.
The exponential growth in this context is modeled by the formula \( e^{rt} \):
  • Exponential Function: The base, represented by \( e \), is a constant approximately equal to 2.71828.
  • Compound Effect: The power \( rt \) represents the rate and time, causing exponential growth as these increase.
This exponential model illustrates why investments grow faster when compounded continuously. Understanding exponential growth is crucial to grasp the potential for wealth generation in investment scenarios involving continuous compounding.

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Most popular questions from this chapter

[In all the installment loans, assume that the term of the loan and the APR remain the same.] (a) Explain why, in the amortization formula, the monthly payment \(M\) is proportional to the principal \(P\). In other words, explain why if the monthly payment on a loan with principal \(P\) is \(M,\) then the monthly payment on a loan with principal \(c P\) (where \(c\) is any positive constant) is \(c M\). (Hint: What happens in the amortization formula when you replace \(P\) by \(c P ?)\) (b) Explain why if the monthly payment on a loan with principal \(P\) is \(M,\) and the monthly payment on a second loan with principal \(Q\) is \(N,\) then the monthly payment on a loan with principal \((P+Q)\) is \((M+N)\).

Jefferson Elementary School has 750 students. The Friday before spring break \(22 \%\) of the student body was absent. How many students were in school that day?

Find the value of a retirement savings account paying an APR of \(6.6 \%\) (compounded monthly) after 45 years of monthly contributions (contributions made at the end of each month, including the last month) when the monthly contribution is (a) \(\$ 125\) (b) \(\$ 62.50\) (c) \(\$ 187.50\)

Between 2000 and 2011 the average annual inflation rate was \(3 \%\). Find the salary in 2000 dollars that would be equivalent to a \(\$ 50,000\) salary in 2011 .

What should your monthly contribution be if your goal is to have \(\$ 500,000\) in your retirement savings account after 40 years? Assume the APR is \(6.6 \%\) compounded monthly and that contributions are made at the end of each month, including the last month.
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