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Chapter 12: Problem 6

Saving How much money should be invested every quarter at 10\(\%\) per year, compounded quarterly, in order to have \(\$ 5000\) in 2 years?

Short Answer

Expert verified
The quarterly investment should be approximately \$572.89.

Step by step solution

01

Understand the Problem

We are given the task of figuring out an equal quarterly investment for 2 years, at an annual interest rate of 10\(\%\), compounded quarterly, to accumulate a future value of \$5000. The formula to use is the Future Value of an Annuity formula, considering quarterly compounding.
02

Set Up the Future Value of Annuity Formula

The future value of an annuity formula for periodic payments is \( FV = P \times \frac{(1 + r)^n - 1}{r} \), where \( FV \) is the future value, \( P \) is the periodic payment, \( r \) is the interest rate per period, and \( n \) is the total number of periods. Here, \( FV = 5000 \), \( r = \frac{0.10}{4} = 0.025 \), and \( n = 4 \times 2 = 8 \).
03

Rearrange the Formula to Solve for P

To find \( P \), rewrite the formula as \( P = \frac{FV \times r}{(1 + r)^n - 1} \). This allows us to solve for the periodic payment needed.
04

Substitute Values and Calculate

Substitute \( FV = 5000 \), \( r = 0.025 \), and \( n = 8 \) into the rearranged formula: \( P = \frac{5000 \times 0.025}{(1 + 0.025)^8 - 1} \). Compute the values to find \( P \).
05

Compute the Final Answer

Calculate \( (1.025)^8 = 1.218402 \). Plug back into the equation: \( P = \frac{5000 \times 0.025}{1.218402 - 1} = \frac{125}{0.218402} \approx 572.89 \). Therefore, the quarterly payment should be approximately \$572.89.

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Key Concepts

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

Quarterly Compounding
When dealing with investments or savings accounts, compound interest is a fundamental concept. It essentially represents interest on interest, leading to growth in the total investment over time. Quarterly compounding specifically means that the interest is applied to the account balance every quarter (three months).
This increases the frequency with which interest is calculated compared to an annual compounding which only applies once each year.
  • Quarterly compounding results in four compounding periods within a year.
  • More frequent compounding results in higher future values, as each period has the opportunity to generate additional interest on accumulated interest.
Understanding this concept is crucial in recognizing how small changes in compounding frequency can significantly impact the overall growth of an investment.
Periodic Payment Calculation
A central aspect of planning a consistent investment plan is the calculation of periodic payments. This refers to how much money needs to be deposited or invested at regular intervals over a set period to achieve a financial goal. For those making regular investments, it's the amount they need to allocate each period.
For example, in the problem where we aim to have $5000 in 2 years with quarterly investments, we need to calculate how much money should be invested every quarter.
  • The goal is to determine the periodic payment (denoted as P), which is calculated through a rearrangement of the future value of an annuity formula.
  • This formula takes into account the periodic interest rate and the total number of investment periods.
By understanding and calculating periodic payments, investors can systematically save to meet their future financial targets.
Interest Rate Quarterly Conversion
Before calculating anything related to interest rates in a quarterly context, it's essential to convert an annual interest rate to a quarterly one. This adjustment reflects the frequency with which interest is compounded within a year.
  • The annual interest rate is divided by the number of compounding periods in a year.
  • For quarterly compounding, this means dividing the annual rate by 4.
For instance, a given annual interest rate of 10% would be converted to a quarterly rate of 2.5%. This reduced interest rate is then used in calculations for each quarter, allowing the investor to accurately determine growth based on the specific compounding frequency.
Investment Future Value Calculation
Calculating the future value of an investment is a key aspect of financial planning. It involves determining how much a series of investments will grow over time, taking into account the effects of compounding interest.
The future value of an annuity formula is specifically useful when dealing with periodic investments.
  • The main components are the future value (FV), the periodic payment amount (P), the interest rate per period (r), and the total number of periods (n).
  • The formula is given by: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
This formula helps in predicting how much periodic investments will accumulate after a given duration. By substituting values like $5000 as the future value target, along with the adjusted interest rate and periods, we can solve for the necessary periodic investment to achieve the desired outcome.

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Most popular questions from this chapter

Amortizing a Mortgage When they bought their house, John and Mary took out a \(\$ 90,000\) mortgage at 9\(\%\) interest, repayable monthly over 30 years. Their payment is \(\$ 724.17\) per month (check this using the formula in the text). The bank gave them an amortization schedule, which is a table showing how much of each payment is interest, how much goes toward the principal, and the remaining principal after each payment. The table below shows the first few entries in the amortization schedule. $$ \begin{array}{|c|c|c|c|c|}\hline \text { Payment } & {\text { Total }} & {\text { Interest }} & {\text { Principal }} & {\text { Remaining }} \\\ {\text { number }} & {\text { payment }} & {\text { payment }} & {\text { payment }} & {\text { principal }} \\ \hline 1 & {724.17} & {675.00} & {49.54} & {89,950.83} \\ {2} & {724.17} & {674.63} & {49.54} & {89,901.29} \\\ {3} & {724.17} & {674.26} & {49.91} & {89,851.38} \\ {4} & {724.17} & {673.89} & {50.28} & {89,801.10} \\ \hline\end{array} $$ After 10 years they have made 120 payments and are wondering how much they still owe, but they have lost the amortization schedule. (a) How much do John and Mary still owe on their mortgage? [Hint: The remaining balance is the present value of the 240 remaining payments. (b) How much of their next payment is interest and how much goes toward the principal? [Hint: Since 9\(\% \div\) \(12=0.75 \%\) , they must pay 0.75\(\%\) of the remaining principal in interest each month.

In this exercise we prove the identity $$ \left(\begin{array}{c}{n} \\\ {r-1}\end{array}\right)+\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)=\left(\begin{array}{c}{n+1} \\ {r}\end{array}\right) $$

Bouncing Ball A ball is dropped from a height of 9 \(\mathrm{ft.}\) The elasticity of the ball is such that it always bounces up one-third the distance it has fallen. (a) Find the total distance the ball has traveled at the instant it hits the ground the fifth time. (b) Find a formula for the total distance the ball has traveled at the instant it hits the ground the \(n\)th time.

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (\sqrt{a}+\sqrt{b})^{6} $$

An Annuity That Lasts Forever An annuity in perpetuity is one that continues forever. Such annuities are useful in setting up scholarship funds to ensure that the award continues. (a) Draw a time line (as in Example 1 ) to show that to set up an annuity in perpetuity of amount \(R\) per time period, the amount that must be invested now is $$ A_{p}=\frac{R}{1+i}+\frac{R}{(1+i)^{2}}+\frac{R}{(1+i)^{3}}+\cdots+\frac{R}{(1+i)^{n}}+\cdots $$ where \(i\) is the interest rate per time period. (b) Find the sum of the infinite series in part (a) to show that $$ A_{p}=\frac{R}{i} $$ (c) How much money must be invested now at 10\(\%\) per year, compounded annually, to provide an annuity in perpetuity of \(\$ 5000\) per year? The first payment is due in one year. (d) How much money must be invested now at 8\(\%\) per year, compounded quarterly, to provide an annuity in perpetuity of \(\$ 3000\) per year? The first payment is due in one year.
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