Question

What is the interest rate if a deposit subject to annual compounding is doubled after 10 years?

Short answer

The interest rate is approximately 7.18%.

Step-by-step solution

Identify the Given Information

The principal amount doubles in 10 years. This means, if the initial amount is P, then after 10 years, the amount will be 2P.

Use the Compound Interest Formula

The compound interest formula is \[ A = P(1 + r)^n \]where A is the amount, P is the principal, r is the annual interest rate, and n is the number of years.

Substitute Known Values

Given that A = 2P and the number of years n = 10, substitute these into the formula:\[ 2P = P(1 + r)^{10} \].

Simplify the Equation

Divide both sides of the equation by P:\[ 2 = (1 + r)^{10} \].

Solve for r

Take the 10th root on both sides to isolate (1 + r):\[ 1 + r = 2^{\frac{1}{10}} \].Subtract 1 from both sides to find r:\[ r = 2^{\frac{1}{10}} - 1 \].

Calculate the Interest Rate

Calculate the value of \[ 2^{\frac{1}{10}} \] using a calculator:\[ 2^{\frac{1}{10}} \approx 1.0718 \].Thus, \[ r \approx 1.0718 - 1 = 0.0718 \].

Convert to Percentage

Convert the decimal to a percentage by multiplying by 100:\[ 0.0718 \times 100 \approx 7.18\text{\textpercent} \].

Key concepts

Interest Rate Calculation

Interest rate calculation is a fundamental concept in financial mathematics. An interest rate is basically the cost of borrowing money or the reward for saving it. Understanding how to calculate the interest rate is essential for managing finances effectively. In the original exercise, we determine the interest rate when a deposit doubles in value over a period of 10 years with annual compounding. Here are the main steps involved:

• Identify the given information: In this case, the initial principal amount (P) and the final amount (2P).
• Apply the compound interest formula: \( A = P(1 + r)^n \).
• Substitute the known values into the formula: Since A is 2P and n is 10, we get \( 2P = P(1 + r)^{10} \).

By simplifying and solving this equation, we isolate the interest rate (r). Ultimately, this calculation helps us find out that the interest rate needed is around 7.18%.
Proper interest rate calculations are crucial for assessing loans, investments, and savings effectively.

Annual Compounding

Annual compounding is a method of calculating interest where the interest is added to the principal at the end of each year. This means that every year, the interest is calculated on the initial principal as well as on the accumulated interest from previous years.
Using the compound interest formula \( A = P(1 + r)^n \), where:

• P is the principal amount.
• r is the annual interest rate.
• n is the number of years.
• A is the amount of money accumulated after n years, including interest.

Annual compounding adds a powerful growth factor to investments and savings. For example, if you invest $1,000 at an annual interest rate of 5%, the amount grows more significantly each year compared to simple interest calculations. In the original exercise, the principal doubles in 10 years due to annual compounding, which is a clear indication of the power of compounded growth.
Understanding annual compounding allows individuals to make more informed financial decisions by appreciating how investments grow over time.

Financial Mathematics

Financial mathematics involves the application of mathematical methods and models to solve problems related to finance. It includes various concepts like interest calculation, time value of money, annuities, and investment appraisals.
In the provided exercise, we use financial mathematics to determine the interest rate required for an investment to double in 10 years with annual compounding. Here’s why financial mathematics is crucial:

• It helps in understanding the growth and decay of investments.
• It aids in calculating the present value and future value of cash flows.
• It provides tools for comparing different financial products and decisions.

The step-by-step solution in the exercise demonstrates the practical application of financial mathematics. By isolating the interest rate, we used logarithms and nth-root calculations, which are common techniques in financial math.
Mastery of financial mathematics is essential for anyone looking to manage personal finances, work in finance-related fields, or pursue studies in economics or business. The ability to calculate and understand interest rates, compounding, and related concepts empowers individuals to make sound financial decisions.