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Chapter 1: Problem 6

Certificate of Deposit A CD is bought for \(\$ 2500\) and held 3 years until maturity. What is the furure value of the \(\mathrm{CD}\) at the end of the 3 years if it earns interest compounded quarterly at a nominal rate of \(6.6 \% ?\)

Short Answer

Expert verified
The future value of the CD is approximately \( \$3053.48 \).

Step by step solution

01

Understand the formula

To find the future value of the Certificate of Deposit (CD), use the compound interest formula: \ \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \] \ where \( FV \) is the future value, \( P \) is the principal amount, \( r \) is the nominal annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years.
02

Identify the known values

From the problem, \( P = 2500 \) (the initial amount), \( r = 0.066 \) (the nominal interest rate as a decimal), \( n = 4 \) because interest is compounded quarterly, and \( t = 3 \) years.
03

Plug the values into the formula

Substitute the known values into the compound interest formula: \ \[ FV = 2500 \left(1 + \frac{0.066}{4}\right)^{4 \times 3} \] \ Now, calculate inside the parentheses first.
04

Calculate inside the parentheses

Calculate the expression \( \frac{0.066}{4} \): \ \[ \frac{0.066}{4} = 0.0165 \] \ Then, add 1: \ \[ 1 + 0.0165 = 1.0165 \]
05

Compute the exponent

Calculate the exponent term \( 4 \times 3 = 12 \). The expression now becomes: \ \[ FV = 2500 \times (1.0165)^{12} \]
06

Evaluate the power

Compute \( (1.0165)^{12} \): \ \[ (1.0165)^{12} \approx 1.22139 \]
07

Calculate the future value

Multiply the result by the principal:\ \( 2500 \times 1.22139 = 3053.475 \). \ Thus, the future value is approximately \( \$ 3053.48 \).

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Key Concepts

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

Future Value
The future value (FV) is central in finance, particularly when dealing with investments such as a certificate of deposit (CD). It represents what an initial investment will grow to over a period, considering the effect of compound interest. When you buy a CD, the bank promises you a fixed interest rate over the term of the deposit, allowing you to predict its future value.

To calculate the future value of an investment, you'd typically use the compound interest formula: \( FV = P \left(1 + \frac{r}{n}\right)^{nt} \).This formula considers:
  • Your initial investment or principal (\( P \)).
  • The annual interest rate (\( r \)) as a decimal.
  • How many times interest is compounded each year (\( n \)).
  • The number of years (\( t \)) the money is invested.
Future value calculations are crucial because they help in understanding how much an initial investment will be worth and the actual purchasing power you might have once the investment matures.
Interest Rate
The interest rate is a powerful tool in growing your investments through compound interest. It signifies how much your money can earn over time, expressed as a percentage.

In the context of a certificate of deposit, the nominal interest rate is used to calculate how frequently any earned interest is compounded over the term of the deposit. For instance, a nominal rate of 6.6% indicates how much interest accumulates on your principal annually before considering additional compounding effects.
  • When interest is compounded more frequently (such as quarterly), your investment earns interest on previously earned interest sooner, boosting the overall growth.
  • The effective annual rate considers these compounding effects and may thus be slightly higher than the stated nominal rate, reflecting real earnings.
An understanding of the interest rate is essential, as it directly impacts how fast and efficiently your investment will grow. It's also a key factor to compare different investment instruments and choose the best option for your financial goals.
Certificate of Deposit
A certificate of deposit (CD) is a savings product offered by banks or financial institutions, providing a fixed interest rate for a fixed term.

When you invest in a CD, you commit a certain amount of money to the bank for a predefined period, ranging from a few months to several years. In return, the bank guarantees a specified rate of interest, which is often higher than a regular savings account. This makes CDs an attractive option for those looking to securely grow their savings:
  • CDs are ideal for risk-averse investors who prefer stability and predictable returns without market risks.
  • The fixed rate and term provide certainty, as you know exactly how much your investment will grow over time.
  • However, early withdrawal can incur penalties, so it's key to ensure you won't need those funds before maturity.
With the right strategy, CDs can be a valuable tool in a diversified investment portfolio, enhancing financial growth while minimizing risks.

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Most popular questions from this chapter

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