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Chapter 5: Problem 42

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?

Short Answer

Expert verified
The current value of Chris's investment is approximately \(\$ 18,106.60\).

Step by step solution

01

Identify the given values

In this problem, we are given: - P: Principal investment amount = \(\$ 10,000\) - r: Annual interest rate = \(10.82 \%\) - n: Number of times interest is compounded per year (quarterly compounding) = \(4\) - t: Time (years) = \(5.5\)
02

Convert the percentage interest rate to decimal

To convert the given annual interest rate from a percentage to a decimal, divide it by \(100\). In this case, we have: \(r = \frac{10.82}{100} = 0.1082\)
03

Apply the compound interest formula to solve for A

Plug the given values into the compound interest formula and solve for A: \(A = P(1 + \frac{r}{n})^{nt}\) \(A = 10000(1 + \frac{0.1082}{4})^{4 \cdot 5.5}\)
04

Perform the calculations

Carry out the calculations: \(A = 10000(1 + \frac{0.1082}{4})^{22}\) \(A = 10000(1 + 0.02705)^{22}\) \(A = 10000(1.02705)^{22}\) \(A 鈮� 18106.60\)
05

Interpret the result

The current value of Chris's investment is approximately \(\$ 18,106.60\).

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Key Concepts

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

Financial Mathematics
Financial mathematics is the application of mathematical methods to solve problems related to finance. The subject covers a wide range of topics, including interest calculations, investments, and annuities.

When dealing with investments, understanding financial mathematics helps in evaluating the growth of an investment over time. It takes into account various interest rates and compounding periods.
  • The principal amount is the initial sum of money invested.
  • Interest rate refers to the percentage increase per period.
  • The compounding frequency is how often the interest is applied to the principal.
By applying financial formulas, like the compound interest formula, investors can estimate the potential value of their investments after a certain period, aiding in better financial decision-making.
Quarterly Compounding
Quarterly compounding is a concept where the interest is applied to the principal four times a year. This means every quarter, or every three months, the investment gets a boost of interest.

The significance of quarterly compounding lies in its impact on investment growth.
  • More frequent compounding leads to faster growth of investments compared to yearly compounding.
  • With each quarter, the previously compounded interest starts earning interest too, creating a compounding effect.
Using the compound interest formula:\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]where \(A\) is the future value of the investment, \(P\) is the principal, \(r\) is the annual interest rate, \(n\) is the number of times interest is compounded per year, and \(t\) is the number of years. For quarterly compounding, \(n\) would be 4, highlighting the compounding effect's benefits.
Investment Growth
Investment growth refers to the increase in the value of an investment over time due to interest earned. The growth rate of an investment is crucial for forecasting future value.Investments like Chris's retirement fund often rely on compounding to enhance growth prospects.
  • The formula \(A = P\left(1 + \frac{r}{n}\right)^{nt}\) helps calculate the anticipated growth over a specified period.
  • The concept of time, expressed as \(t\), plays a significant role. A longer investment duration can exponentially increase the final amount due to compounding.
  • The rate of return or interest rate also dictates how swiftly an investment amplifies.
Recognizing these elements helps investors like Chris make informed decisions about how long to invest and what interest structures most benefit their financial aims. In practice, applying the compound interest formula allows one to predict the investment's worth at any given future point.

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Most popular questions from this chapter

A state lottery commission pays the winner of the "Million Dollar" lottery 20 installments of \(\$ 50,000 /\) year. The commission makes the first payment of \(\$ 50,000\) immediately and the other \(n=19\) payments at the end of each of the next 19 yr. Determine how much money the commission should have in the bank initially to guarantee the payments, assuming that the balance on deposit with the bank earns interest at the rate of \(8 \% /\) year compounded vearly.

IRAs Martin has deposited \(\$ 375\) in his IRA at the end of each quarter for the past 20 yr. His investment has earned interest at the rate of \(8 \% /\) year compounded quarterly over this period. Now, at age 60 , he is considering retirement. What quarterly payment will he receive over the next 15 yr? (Assume that the money is earning interest at the same rate and that payments are made at the end of each quarter.) If he continues working and makes quarterly payments of the same amount in his IRA until age 65, what quarterly payment will he receive from his fund upon retirement over the following \(10 \mathrm{yr}\) ?

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 the future value of an annuity consisting of \(n\) payments of \(R\) dollars each-paid at the end of each investment period into an account that earns interest at the rate of \(i\) per period-is \(S\) dollars, then $$ R=\frac{i S}{(1+i)^{n}-1} $$

Determine which of the sequences are geometric progressions. For each geometric progression, find the seventh term and the sum of the first seven terms. $$ 0.004,0.04,0.4,4, \ldots $$

Auro FiNANCING Paula is considering the purchase of a new car. She has narrowed her search to two cars that are equally appealing to her. Car A costs \(\$ 28,000\), and car \(B\) costs \(\$ 28,200\). The manufacturer of car A is offering \(0 \%\) financing for 48 months with zero down, while the manufacturer of car \(\mathrm{B}\) is offering a rebate of \(\$ 2000\) at the time of purchase plus financing at the rate of \(3 \%\) year compounded monthly over 48 mo with zero down. If Paula has decided to buy the car with the lower net cost to her, which car should she purchase?
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