Question

Compound Interest In Exercises 49 and 50 , find the time necessary for \(\$ 1000\) to double when it is invested at a rate of \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$ r=5.5 \% $$

Short answer

The time necessary for $1000 to double with the interest rate of 5.5% compounded annually is 12.6 years, monthly - 12.45 years, daily - 12.43 years, and continuously - 12.6 years.

Step-by-step solution

Find the time using Compound Interest formula

Let's find the time (t) for the given \(r=0.055\) (converted 5.5% into decimal form 5.5/100) annually compounded. The formula would look like: \(2000 = 1000 (1+ \frac{0.055}{1})^{1*t}\). Solving for t we get t = \(\frac{log(\frac{2000}{1000})}{log(1+0.055)} = 12.6\) years. The calculation assumes $1000 needs to grow to $2000 to double.

Repeat Step 1 for Monthly Compounding

For monthly compounding, \(n=12\) since interest is compounded 12 times in a year. So, \(2000 = 1000(1+ \frac{0.055}{12})^{12*t}\). Solving for t, we get \(t = \frac{log(\frac{2000}{1000})}{12*log(1 + \frac{0.055}{12})} = 12.45\) years.

Repeat Step 1 for Daily Compounding

For daily compounding, \(n=365\) since interest is compounded 365 times in a year. So, \(2000 = 1000(1+ \frac{0.055}{365})^{365*t}\). Solving for t, we get \(t = \frac{log(\frac{2000}{1000})}{365*log(1 + \frac{0.055}{365})} = 12.43\) years.

Find the time using Continuous Compounding formula

Finally, for continuous compounding, we use the formula: \(2000 = 1000 * e^{0.055*t}\). Solving for t, we get \(t = \frac{log(\frac{2000}{1000})}{0.055}= 12.6\) years.

Key concepts

Time Value of Money

Imagine you have the option to receive \(1000 today or a year from now; which would you choose? The right answer, from a financial standpoint, should be to take the money today. This preference is due to a fundamental principle known as the 'Time Value of Money' (TVM), which asserts that money available now is worth more than the identical sum in the future due to its potential earning capacity. This core financial concept implies that money can earn interest or investment return, thus the earlier you have it, the more income it can generate over time.

Let's solidify this concept with a practical illustration: the doubling time problem with compound interest. When you invest \)1000 at a rate of 5.5% compounded in various frequencies (annually, monthly, daily, or continuously), you're essentially allowing your money to make 'more money' over time. Our exercise focuses on calculating exactly how many years will pass before your investment doubles. Accurately understanding TVM is crucial because it helps us make informed decisions on investments, loans, and savings, ultimately ensuring our money works efficiently for our future selves.

Exponential Growth

Compounding interest is a beautiful example of exponential growth, a process where a quantity increases at a rate proportional to its current value. In the context of our exercise, the quantity is the initial investment of $1000, which grows over time due to the interest that is earned on both the original amount and on the accumulated interest from previous periods. This concept, fundamentally different from linear growth—where values increase by a constant amount—can lead to the investment's value ballooning over time.

The compound interest formula gives us a precise toolkit to calculate the exponential growth of an investment. Whether we’re compounding annually, monthly, daily, or continuously, each approach has nuanced differences in how rapidly the investment will grow. By solving the formula for the respective compounding period, we can understand not only how the money will double but also the incredible impact of compounding frequency on our investment's growth trajectory.

Financial Mathematics

To solve financial problems, we rely on financial mathematics, a field that applies mathematical methods to solve problems in finance. It includes concepts such as the present and future value of money, annuities, loans, and investments, among others. In our doubling time problem, we used a critical tool from financial mathematics: the compound interest formula. It provides a mathematical representation of how compound interest works and is fundamental in computing the growth of investments over time, as seen in the step-by-step solutions for the various compounding intervals.

Understanding the intricate calculations is essential for anyone managing personal finances or working in finance and investments. As per the exercise, changing the frequency of compounding (from annually to daily or continuous, for example) affects how quickly your money grows. Recognizing these subtleties is a testament to the importance of financial mathematics in decision-making, whether for individual savings plans or corporate investment strategies.