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Chapter 0: Problem 72

The amount of money, \(A(t),\) in \(a\) savings account that pays 3\% interest, compounded quarterly for \(t\) years, with an initial investment of \(P\) dollars, is given by $$ A(t)=P\left(1+\frac{0.03}{4}\right)^{4 t} $$ If \(\$ 800\) is invested at \(3 \%,\) compounded quarterly, how much will the investment be worth after 3 yr?

Short Answer

Expert verified
The investment will be worth approximately $874.75 after 3 years.

Step by step solution

01

Understand the Problem

We need to find the future value of an investment given a principal amount, interest rate, compounding frequency, and time period. The formula provided is for compound interest with quarterly compounding.
02

Identify Given Values

From the problem:- Initial investment (Principal) \( P = 800 \) dollars- Annual interest rate = 3% or 0.03- Compounding period = quarterly = 4 times a year- Time period \( t = 3 \) years.
03

Plug Values into Formula

Substitute the given values into the compound interest formula: \[A(t) = 800\left(1+\frac{0.03}{4}\right)^{4 \cdot 3}\] Simplifying gives: \[A(t) = 800\left(1+0.0075\right)^{12}\]
04

Simplify Base of Exponent

Calculate the base inside the parentheses: \[1 + 0.0075 = 1.0075\] So, the expression becomes: \[A(t) = 800\left(1.0075\right)^{12}\]
05

Calculate the Exponentiation

Calculate \((1.0075)^{12}\). Repeated multiplications can be done manually or using a calculator to find: \[(1.0075)^{12} \approx 1.093443\]
06

Calculate Final Investment Value

Substitute the calculated exponentiation back to find the final amount:\[A(t) = 800 \times 1.093443 \approx 874.75\]
07

Interpret the Result

The amount in the account after 3 years is approximately \(\$874.75\).

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Key Concepts

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

Quarterly Compounding
When we talk about quarterly compounding, we're referring to the process of calculating interest four times a year.
Each time the interest is calculated, it is added to the principal, allowing the investment to grow faster than with just annual compounding.
  • Quarterly compounding divides the annual interest rate by four, since there are four quarters in a year.
  • For example, an interest rate of 3% per year becomes 0.75% per quarter.
  • This added frequency of compounding allows the investment to grow due to earning interest on interest earned in previous periods.
Quarterly compounding is beneficial in contrast to less frequent methods because the more often interest is compounded, the more money you will eventually have. This is why investments with more frequent compounding periods can yield higher returns over time.
Future Value Calculation
The future value of an investment is what an initial sum of money, known as the principal, will grow to after a certain number of years at a given interest rate.
The calculation involves the principal amount, the annual interest rate, the number of compounding periods per year, and the total number of years the money is invested.

The formula for calculating future value with compounding interest is:\[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \] where:
  • \(A(t)\) is the future value of the investment/loan, including interest.
  • \(P\) is the principal investment amount (the initial deposit or loan amount).
  • \(r\) is the annual interest rate (decimal).
  • \(n\) is the number of times that interest is compounded per year.
  • \(t\) is the number of years the money is invested or borrowed for.
Using this formula, you can determine how much your investment will be worth at any given point in the future, provided you reinvest all your interest.
Interest Rate
The interest rate is a percentage that represents the cost of borrowing money or the gain in earning it as interest in an investment.
In the context of savings and investments, a higher interest rate provides a higher return on investment, given the same principal and time frame.

Two important considerations about interest rates:
  • Nominal Interest Rate: This is the stated interest rate, not accounting for compounding effects.
  • Effective Interest Rate: Reflects the impact of compounding over a period, offering a true sense of the cost or profitability of the interest rate.
The effective interest rate can be higher than the nominal rate due to compounding, and it's crucial to understand this distinction when considering investment options.
For the exercise, the nominal rate is 3%, but with quarterly compounding, the effective rate will be slightly higher, thus increasing the investment over time.

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Most popular questions from this chapter

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