Question

A bank offers you \(£ 1.10\) next year for every \(£ 0.90\) you give it today. What is the implicit interest rate?

Short answer

The implicit interest rate is 22.22%.

Step-by-step solution

Understanding the Problem

We are asked to find the implicit interest rate when the bank offers £1.10 next year for every £0.90 given today. This means for a principal amount of £0.90, the future value is £1.10.

Setting Up the Formula

The formula to find the interest rate is \( FV = PV(1 + r) \), where \( FV \) is the future value, \( PV \) is the present value, and \( r \) is the interest rate. In this problem, \( FV = 1.10 \) and \( PV = 0.90 \).

Calculating the Interest Rate

To find the interest rate, rearrange the formula: \( 1 + r = \frac{FV}{PV} \). Substituting the values, we get \( 1 + r = \frac{1.10}{0.90} \).

Solving the Equation

Calculate \( \frac{1.10}{0.90} = 1.2222 \). This implies \( 1 + r = 1.2222 \). To find \( r \), subtract 1 from both sides: \( r = 1.2222 - 1 = 0.2222 \).

Final Conversion

Convert the decimal interest rate into a percentage by multiplying by 100, resulting in \( 0.2222 \times 100 = 22.22\% \).

Key concepts

Implicit Interest Rate

The concept of implicit interest rate often comes into play when dealing with agreements where interest isn't explicitly stated but can be inferred from the conditions. In the example provided, the bank offers £1.10 next year for every £0.90 given today, suggesting a form of interest differentiated through the change in value over time.
Implicit interest rate helps to quantify this change or gain. By understanding how much more you receive in the future compared to what you invest today, you're able to make better financial decisions.
Calculating the implicit interest rate involves unraveling the implied yield from the difference between future and present values without an overtly declared rate. It essentially characterizes the "hidden" or inherent earnings on an investment or loan.

Present Value

Present Value (PV) is a fundamental concept in finance, representing how much a future sum of money is worth today. This is essential for comparing the value of money received at different times, allowing you to make informed financial decisions.
In the given exercise, £0.90 is the present value. This is the amount of money you invest or deposit now.
By investing this present value, you expect it to grow to a higher amount in the future. This growth is influenced by interest rates, making PV a starting point for calculating future value and assessing the profitability of an investment.

• Remember, present value helps you compare various financial options by bringing future cash flows to today's terms.
• This allows individuals and businesses to assess the return on investments, considering all future prospects as if they occurred today.

Future Value

Future Value (FV) reflects how much an investment is expected to grow over time, factoring in interest or returns earned. Understanding future value is crucial as it lets you see the potential growth of your current investments, which can help plan future financial goals.
In our exercise, £1.10 represents the future value, indicating what the £0.90 will become after one year.
Calculating future value is essential when analyzing savings accounts, bonds, and other investment products, as it gives a projection of future earnings.

• Future Value is influenced by the interest rate and the time period of the investment.
• It helps you estimate the financial benefits of investing your money instead of letting it stay idle.

Interest Rate Formula

The interest rate formula is pivotal in finance to determine how your investments grow or what your borrowing costs will amount to over time. This is defined as the rate at which your present value grows to become a future value. The formula is denoted as: \[ FV = PV(1 + r) \] where:- \( FV \) stands for Future Value,- \( PV \) is Present Value,- \( r \) is the interest rate.In the exercise, rearranging the formula to find \( r \) involves these steps: 1. Isolate \( 1 + r \) through division: \( 1 + r = \frac{FV}{PV} \)
2. Substitute the known values, \( FV = 1.10 \) and \( PV = 0.90 \), yielding \( 1 + r = 1.2222 \)
3. Subtract 1 to solve for \( r \), leading to \( r = 0.2222 \).
4. To express \( r \) as a percentage, multiply by 100, arriving at the interest rate of 22.22%.
Understanding this formula helps decode various financial products and is instrumental in personal finance planning and investment analysis.