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Chapter 5: Problem 3

At what interest rate (compounded annually) will a sum of \(\$ 4000\) grow to \(\$ 6000\) in 5 years?

Short Answer

Expert verified
The interest rate needed is approximately 8.447%.

Step by step solution

01

Identify the Given Variables

We know the present value (PV) is $4000, the future value (FV) is $6000, and the time period (t) is 5 years. We need to find the interest rate, denoted by r, that is compounded annually.
02

Apply the Compound Interest Formula

The general formula for compound interest is: \[ FV = PV \times (1 + r)^t \]Substituting the known values into the formula gives:\[ 6000 = 4000 \times (1 + r)^5 \]
03

Isolate for the Interest Rate

To find \(r\), we first divide both sides of the equation by $4000:\[ \frac{6000}{4000} = (1 + r)^5 \]Which simplifies to the equation:\[ 1.5 = (1 + r)^5 \]
04

Solve for the Base "1 + r"

To remove the exponent, take the fifth root of both sides:\[ (1.5)^{1/5} = 1 + r \]Calculate the fifth root of 1.5 using a calculator:\[ 1.5^{1/5} \approx 1.08447 \]
05

Determine the Interest Rate

Subtract 1 from the result to find the interest rate:\[ r = 1.08447 - 1 = 0.08447 \]Convert this to a percentage by multiplying by 100:\[ r \approx 8.447\% \]

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Key Concepts

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

Interest Rate Calculation
Calculating the interest rate is fundamental in understanding how investments grow over time. Interest, whether simple or compound, represents the charge for the privilege of borrowing money. When dealing with compound interest, the rate of interest is crucial as it defines how the investment grows each period.
The interest rate can be calculated using the compound interest formula:
  • The Future Value (FV) indicates the amount of money your investment will grow to.
  • The Present Value (PV) is the initial investment or principal.
  • The number of compounding periods in years is denoted as "t".
  • The interest rate per period is "r".
The formula used: \[ FV = PV imes (1 + r)^t \]To find the rate, you need to solve the equation for "r" using algebraic manipulation, often involving roots or powers. The given problem illustrates this by isolating "1 + r" from the compound interest formula and calculating the fifth root to find the interest rate. This calculation enables us to determine how quickly an investment can grow to its future value.
Financial Mathematics
Financial mathematics is an essential tool that helps in understand complex financial decisions by leveraging mathematical formulas. It assists in calculating things like loan payments, investment growth, and savings.
Compound interest is a key element in financial mathematics as it introduces the concept of reinvested gains.
Unlike simple interest where interest is calculated only on the principal amount, compound interest applies the interest on both the initial principal and accumulated interest from previous periods.
  • It allows investments to grow at a faster rate because the calculated interest increases with each period.
  • Usually expressed annually, semi-annually, monthly, or even daily, depending on the terms.
  • Mastering compound interest is crucial for making informed financial choices.
By understanding these concepts, one can make informed choices, ensuring financial stability and growth. The exercise problem shows how understanding basic principles like these in financial mathematics can lead to smart investment decisions.
Future Value and Present Value
Future value and present value are foundational concepts in finance that allow us to evaluate the worth of money at different points in time.
The **Present Value (PV)** is the current value of a future sum of money, given a specific rate of return.
The **Future Value (FV)** is how much a present sum will grow to over a defined period when invested at a certain interest rate.
  • PV and FV help predict how investment decisions made today will affect one's future financial position.
  • Understanding these concepts can help identify necessary investment amounts to achieve future financial goals.
  • They are used in various applications such as retirement planning, loan amortization, and more.
In the context of the exercise, we started with a PV of $4000 and calculated it could grow into an FV of $6000 over 5 years, with the correct interest rate. Knowing how to manipulate these variables can significantly enhance one's ability to plan and track financial goals.

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