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Chapter 5: Problem 43

Jodie invested \(\$ 15,000\) in a mutual fund 4 yr ago. If the fund grew at the rate of \(9.8 \%\) lyear compounded monthly, what would Jodie's account be worth today?

Short Answer

Expert verified
Jodie's account would be worth approximately \(\$21,789.69\) today.

Step by step solution

01

Identify the given values and convert them

We are given: - Principal (P) = $15,000 - Annual interest rate (r) = 9.8% = 0.098 (by dividing by 100) - Compounding frequency (n) = Monthly = 12 times per year - Time (t) = 4 years
02

Calculate the compound interest using the formula

We will use the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\) Where: - A represents the future value of Jodie's account, - P is the principal amount, - r is the annual interest rate, - n is the compounding frequency per year, - t is the time in years.
03

Plug in the given values and solve the equation

Plugging in the given values, we get: \(A = 15000(1 + \frac{0.098}{12})^{12 \times 4}\) Now, we will solve for A: \(A = 15000(1 + \frac{0.098}{12})^{48}\) \(A = 15000(1 + 0.008167)^{48}\) \(A = 15000(1.008167)^{48}\) \(A \approx 15000(1.452646)\) \(A \approx 21789.69\)
04

Interpret the result

The future value of Jodie's account after 4 years, with an annual interest rate of 9.8% compounded monthly, is approximately \(21,789.69. Therefore, Jodie's account would be worth around \)21,789.69 today.

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Key Concepts

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

Mutual Funds
Mutual funds are a popular investment choice that allows individuals like Jodie to pool their money with other investors to purchase a diverse portfolio of stocks, bonds, or other securities. By doing so, mutual funds provide diversification, which helps minimize risk by distributing investments across various asset classes.
They are managed by professional fund managers who make investment decisions on behalf of the investors, aiming to achieve the best possible returns.
  • Individuals can invest in mutual funds with varying levels of risk tolerance, from conservative to aggressive strategies.
  • The funds are accessible with a relatively small amount of capital, making them suitable for a wide range of investors.
  • They offer liquidity, allowing investors to buy or sell shares with relative ease compared to other investment types.
This makes mutual funds an attractive option for long-term financial growth, as they offer a structured way to invest in the market while benefiting from professional management.
Interest Rate Calculation
The calculation of interest rates can greatly impact the amount of money that will grow over time. In the case of Jodie's investment in a mutual fund, the interest rate is compounded monthly, which means it is calculated more frequently than annual compounding.
This compounding frequency affects how fast the investment grows. The formula for calculating compound interest is an essential tool:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
  • P represents the principal amount invested, which in this case is \(\$15,000\).
  • r is the annual nominal interest rate (9.8% expressed as 0.098).
  • n is the number of times interest is compounded per year (12 times for monthly).
  • t is the number of years the money is invested (4 years for Jodie).
Understanding these variables helps investors like Jodie plan for future growth and make informed decisions about the timing and structure of their investments.
Future Value of Investment
The future value of an investment is the amount of money that an original investment will grow to over time, with interest or other forms of return added.
This is crucial for investors who want to know the potential outcome of their investments. In Jodie's scenario, knowing the future value allows her to anticipate the worth of her mutual fund after several years.
  • The future value depends on the principal, the rate of interest, the compounding frequency, and the time invested.
  • By using the compound interest formula, investors can calculate how different rates and compounding periods affect the growth of their investments.
  • This information empowers investors to align investment choices with their financial goals and time horizons.
For Jodie, the detailed calculation showed a future value of approximately \(\$21,789.69\), demonstrating how regular compounding can significantly enhance investment growth over time.

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Most popular questions from this chapter

In the last 5 yr, Bendix Mutual Fund grew at the rate of \(10.4 \% /\) year compounded quarterly. Over the same period, Acme Mutual Fund grew at the rate of \(10.6 \%\) /year compounded semiannually. Which mutual fund has a better rate of return?

BatLoon PAYMENT MoRTGAGES Emilio is securing a 7 -yr Fannie Mae "balloon" mortgage for \(\$ 280,000\) to finance the purchase of his first home. The monthly payments are based on a 30 -yr amortization. If the prevailing interest rate is \(7.5 \% /\) year compounded monthly, what will be Emilio's monthly payment? What will be his "balloon" payment at the end of 7 yr?

BALLOON PAYMENT MoRTGAGES Olivia plans to secure a 5 -yr balloon mortgage of \(\$ 200,000\) toward the purchase of a condominium. Her monthly payment for the 5 yr is calculated on the basis of a 30 -yr conventional mortgage at the rate of \(6 \% /\) year compounded monthly. At the end of the 5 yr, Olivia is required to pay the balance owed (the "balloon" payment). What will be her monthly payment, and what will be her balloon payment?

Lauren plans to deposit \(\$ 5000\) into a bank account at the beginning of next month and \(\$ 200 /\) month into the same account at the end of that month and at the end of each subsequent month for the next \(5 \mathrm{yr}\). If her bank pays interest at the rate of \(6 \%\) /year compounded monthly, how much will Lauren have in her account at the end of 5 yr? (Assume she makes no withdrawals during the 5 -yr period.)

Find the book value of office equipment purchased at a cost \(C\) at the end of the \(n\) th year if it is to be depreciated by the double declining-balance method over 10 yr. $$ C=\$ 20,000, n=4 $$
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