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Chapter 5: Problem 9

Find the periodic payment \(R\) required to accumulate a sum of \(S\) dollars over \(t\) yr with interest earned at the rate of \(r \% / y\) year compounded \(m\) times a year. S=20,000, r=4, t=6, m=2

Short Answer

Expert verified
The periodic payment \(R\) required to accumulate a sum of \(20,000\) dollars over \(6\) years with interest earned at a rate of \(4\%\) per year compounded semi-annually is approximately \(1,491.50\).

Step by step solution

01

Write down the annuity formula

\( S = R(\frac{(1 + \frac{r}{m \times 100})^{m \times t} - 1}{\frac{r}{m \times 100}}) \)
02

Plug in the given values

\( 20,000 = R(\frac{(1 + \frac{4}{2 \times 100})^{2 \times 6} - 1}{\frac{4}{2 \times 100}}) \)
03

Simplify the equation

\(20,000 = R(\frac{(1 + \frac{1}{50})^{12} - 1}{\frac{1}{50}})\)
04

Calculate the value inside the brackets

Calculate the value inside the brackets: \( (1+\frac{1}{50})^{12} \approx 1.26824 \) Now subtract 1: \( 1.26824 - 1 = 0.26824 \)
05

Solve for R

Now we just need to solve for R: \( 20,000 = R(\frac{0.26824}{\frac{1}{50}}) \) Multiply both sides by \(\frac{1}{50}\): \( 400,000 = R \times 0.26824 \) Now, divide both sides by 0.26824: \( R = \frac{400,000}{0.26824} \) \( R \approx 1,491.50 \) So, the periodic payment R required to accumulate a sum of \(20,000 over 6 years with interest earned at a rate of 4%/year compounded semi-annually is about \)1,491.50.

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Key Concepts

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

Periodic Payment Calculation
Calculating periodic payments is essential in financial planning. It involves determining the regular amount needed to reach a target sum within a specified time. This is particularly crucial in scenarios like saving for retirement or planning a significant purchase.

To compute the periodic payment, we use the annuity formula:
  • \( S = R \left(\frac{(1 + \frac{r}{m \times 100})^{m \times t} - 1}{\frac{r}{m \times 100}}\right) \)
Here, the aim is to solve for \( R \) (the periodic payment). The formula factors in the compounding effect of interest along with the frequency of payments.

In our example, we set \( S = 20,000 \), \( r = 4 \), \( t = 6 \), and \( m = 2 \). Plugging these values into the annuity formula helps in finding the periodic payment \( R \), which ensures that you accumulate the desired amount by the end of the period.
Compound Interest
Compound interest is interest calculated on the initial principal, which also includes all the accumulated interest from previous periods. It can significantly impact the growth of an investment over time. The power of compound interest lies in its ability to multiply invested money due to the effect of "interest on interest."

In the periodic payment problem:
  • The interest rate is compounded semi-annually, meaning you earn interest on your interest twice a year.
  • The formula incorporates compounding frequency \( m \), ensuring more frequent calculations and impact on the total sum to be accumulated.
  • In our earlier calculation example, the compounded term \( (1 + \frac{1}{50})^{12} \) reflects the periodic application of interest, influencing the overall result.
Understanding compound interest is key to making informed financial decisions, especially when calculating long-term investment returns or payment plans.
Financial Mathematics
Financial mathematics deals with the financial markets and their various complexities. It provides the tools to solve practical problems involving investments and loans. Fundamental concepts include understanding interest rates, investment growth, and cash flow analysis.

In the context of this problem:
  • We employ the annuity formula and compound interest principles to determine the necessary periodic payments.
  • Financial mathematics equips us with the skills to manipulate formulas to derive meaningful financial insights that aid in decision-making.
  • Techniques used here can be applied across different financial products like savings plans, bonds, and mortgages.
Grasping these concepts ensures better financial planning and the ability to foresee the outcomes of various financial strategies, aligning them with personal goals and market expectations.

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Most popular questions from this chapter

Five and a half years ago, Chris invested \(\$ 10,000\) in a retirement fund that grew at the rate of \(10.82 \% /\) year compounded quarterly. What is his account worth today?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If interest is compounded annually, then the effective rate is the same as the nominal rate.

FINANCING A Home Sarah secured a bank loan of \(\$ 200.000\) for the purchase of a house. The mortgage is to be amortized through monthly payments for a term of \(15 \mathrm{yr}\), with an interest rate of \(6 \% /\) year compounded monthly on the unpaid balance. She plans to sell her house in 5 yr. How much will Sarah still owe on her house?

FiNANCING A HomE The Taylors have purchased a \(\$ 270,000\) house. They made an initial down payment of \(\$ 30,000\) and secured a mortgage with interest charged at the rate of \(8 \% / y\) ear on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over \(30 \mathrm{yr}\), what monthly payment will the Taylors be required to make? What is their equity (disregarding appreciation) after 5 yr? After 10 yr? After 20 yr?

A HomE The Johnsons have accumulated a nest egg of \(\$ 40,000\) that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of \(\$ 2400 /\) month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed \(\$ 3000\). If local mortgage rates are \(7.5 \%\) lyear compounded monthly for a conventional 30 -yr mortgage, what is the price range of houses that they should consider?
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