Problem 65 - College Algebra

Question

Jamal wants to save \(\$ 54,000\) for a down payment on a home. How much will he need to invest in an account with 8.2\(\% \mathrm{APR}\) , compounding daily, in order to reach his goal in 5 years?

Step-by-Step Solution

Answer

Jamal needs to invest approximately $36,022.

Step 1: Understanding the Components

We need to calculate the present value of the investment needed with daily compounding interest to reach Jamal's goal. The formula for compound interest is \( A = P (1 + \frac{r}{n})^{nt} \), where \( A \) is the amount, \( P \) is the principal, \( r \) is the annual interest rate, \( n \) is the number of times that interest is compounded per year, and \( t \) is the time in years.

Step 2: Assign Known Values

Given in the problem:- \( A = 54,000 \) (the future value he needs)- \( r = 0.082 \) (the annual interest rate of 8.2% expressed as a decimal)- \( n = 365 \) (daily compounding)- \( t = 5 \) years.

Step 3: Set Up the Equation for Present Value

Rearrange the compound interest formula to solve for \( P \):\[ P = \frac{A}{(1 + \frac{r}{n})^{nt}} \]

Step 4: Substitute Values into the Equation

Plug in the known values:\[ P = \frac{54,000}{(1 + \frac{0.082}{365})^{365 \times 5}} \]

Step 5: Calculate the Present Value

First calculate \( \frac{0.082}{365} \) to find the rate per period, then compute \( (1 + \frac{0.082}{365})^{1825} \) to find the total compounding factor over 5 years. Divide \( 54,000 \) by the resulting factor to find \( P \).

Step 6: Interpret the Result

The computed \( P \) will be the amount Jamal needs to initially invest in the account to reach his target savings of \$54,000 in 5 years.

Key Concepts

Understanding Present Value

Present value is a financial concept that helps us determine how much a future sum of money is worth today. This is crucial for making informed investment decisions. When we talk about present value, we're considering factors like interest rates and compounding periods. For Jamal's exercise, we identified that he needs a present value calculation to figure out the initial amount to invest to achieve his goal of \(54,000 in the future. The present value formula rearranges the basic compound interest formula to solve for the initial principal amount \( P \).

• Given: the future value \( A \), or the amount Jamal desires in 5 years, is \)54,000.
• Find: present value \( P \), or how much Jamal should invest now.

Therefore, by correctly using the formulas and understanding the time value of money, Jamal can determine exactly how much to invest today to have enough for his house down payment.

Clarifying Annual Percentage Rate (APR)

The annual percentage rate (APR) is the annual rate charged for earning interest on an investment. In this exercise, Jamal is working with an APR of 8.2%. APR gives a sense of the average percentage over an entire year, making it easier to compare financial products or investments. In the context of compounding interest, APR is critical because it affects how much interest Jamal’s savings will earn over time. APR is expressed as a decimal in calculations. For Jamal's exercise, we converted 8.2% to 0.082. This percentage is part of the formula that determines how his investment grows. When compounded, the interest calculated is reinvested into the principal, allowing the investment to grow at an accelerating rate.

Daily Compounding and its Impact

Daily compounding refers to the process of applying interest to the principal amount and, subsequently, to previously earned interest each day. Compounding daily means that interest is calculated and added to the account balance 365 times a year.

• In this exercise, daily compounding influences how quickly Jamal's initial investment grows to reach the $54,000 mark in 5 years.
• Every day, a small percentage of interest is added, significantly increasing the total over time compared to less frequent compounding.

This concept is pivotal because it highlights how much more beneficial daily compounding is compared to, say, annual or semi-annual compounding. By calculating and adding interest daily, Jamal takes advantage of compound growth, which will help him meet his savings objective more efficiently.