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Chapter 5: Problem 11

What principal should you deposit at \(5 \frac{1}{2} \%\) per annum compounded semiannually so as to have \(\$ 6000\) after 10 years?

Short Answer

Expert verified
Deposit approximately $3654.66.

Step by step solution

01

Understand the Formula

When dealing with compound interest, we use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where:- \(A\) is the amount of money accumulated after n years, including interest.- \(P\) is the principal amount (initial deposit).- \(r\) is the annual interest rate (decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time the money is invested for in years.
02

Identify Given Values

From the exercise, we know the following values:- \(A = 6000\) (the amount after interest)- \(r = 5.5\% = 0.055\) (annual interest rate)- \(n = 2\) (compounded semiannually)- \(t = 10\) (investment period in years).We need to find \(P\), the initial principal.
03

Rearrange the Formula

To find the principal \(P\), we rearrange the formula:\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]
04

Substitute the Known Values

Now substitute the known values into the formula:\[P = \frac{6000}{\left(1 + \frac{0.055}{2}\right)^{2 \times 10}}\]
05

Calculate the Compound Interest Factor

First, calculate the inside of the parentheses:\[ 1 + \frac{0.055}{2} = 1 + 0.0275 = 1.0275 \]Then raise this to the power of \(2 \times 10 = 20\):\[ (1.0275)^{20} \]
06

Compute the Exponentiation

Calculate \[ (1.0275)^{20} \approx 1.6416 \]
07

Divide to Find the Principal

Finally, divide the amount by the compound interest factor to find \(P\):\[P = \frac{6000}{1.6416} \approx 3654.66\]
08

Conclusion

The principal that should be deposited is approximately $3654.66.

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Key Concepts

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

Principal Amount
The principal amount is the initial sum of money you invest or deposit. It acts as the foundational base from which any accumulated interest is calculated. In the context of our problem, the principal is what we are trying to determine. Understanding the principal amount is crucial as it directly impacts the total amount accrued at the end of an investment period. To calculate the principal when you know the future value, the compound interest formula is often rearranged. The main components required for this calculation involve knowing how frequently the interest compounds and for how long the investment is held. This allows one to solve for the initial amount needed to reach a future goal, such as $6000 in our problem.
Annual Interest Rate
The annual interest rate is a critical element in calculating compound interest. It represents the percentage of the principal that is paid as interest over a year. In the given exercise, the annual interest rate is 5.5%, which is given in percentage form. To use it in calculations, it must be converted into decimal form: - Convert by dividing the percentage by 100. For instance, 5.5% becomes 0.055. This rate is then applied in the compound interest formula to determine how quickly the investment grows. Understanding how the rate affects growth is vital. Higher rates mean faster growth, but they're also crucial in calculating the strength of your investment growth over time.
Compounded Semiannually
Compounding is the process where interest is earned on both the initial principal and the accumulated interest from previous periods. When interest is compounded semiannually, it means the interest is calculated and added to the principal twice a year. In simpler terms, this means every six months, new interest is added to the principal, increasing the amount that will earn interest in the next period. This frequency impacts how much money you鈥檒l accumulate by the end of the investment period. Generally, more frequent compounding leads to higher returns, since interest begins to generate more interest sooner. In our problem, knowing that the interest is compounded semiannually allows us to set the value of n to 2 in the formula, reflecting these two compounding periods per year.
Investment Period
The investment period is the length of time the money is intended to be invested or borrowed for. This term is important in understanding how compound interest functions over extended periods. In the exercise problem, the investment period is 10 years. This value is one of the variables in the compound interest formula, denoted as "t," which accounts for how long the principal amount, along with reinvested interest, will continue to grow. A longer investment period often results in more compound interest being accrued, given that the principal remains invested over more intervals. The term 鈥渢鈥� is a multiplier in the exponent of the compound interest formula, so it helps determine how many times compounding will occur, directly influencing the future value of the investment.

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Most popular questions from this chapter

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