Question

You have \(\$ 350\), which a friend would like to borrow. If you don't lend it to your friend, you could invest it in an opportunity that would pay out \(\$ 392\) at the end of the year. What annual interest rate should your friend offer you to make you indifferent between these two options?

Short answer

The friend should offer a 12% annual interest rate to make you indifferent.

Step-by-step solution

Identify the Future Value (FV) and Present Value (PV)

The Future Value (FV) is the amount you would receive if you invest the money, which is \( \\( 392 \). The Present Value (PV) is the amount you currently have, which is \( \\) 350 \).

Use the Simple Interest Formula

The simple interest formula is \( FV = PV \times (1 + r) \), where \( r \) is the annual interest rate. Substitute the known values into this formula: \[ 392 = 350 \times (1 + r) \]

Solve for the Interest Rate \( r \)

Rearrange the formula to solve for \( r \):\[ 1 + r = \frac{392}{350} \]\[ r = \frac{392}{350} - 1 \]

Calculate \( r \)

Perform the division and subtraction to find \( r \):\[ r = 1.12 - 1 = 0.12 \] Convert this decimal to a percentage:\( r = 0.12 \times 100\% = 12\% \).

Key concepts

Simple Interest Formula

The simple interest formula is a crucial tool for calculating the growth of an investment over a period of time. It allows you to determine how much extra money you will earn from an initial investment or loan. The formula is expressed as:\[ FV = PV \times (1 + r) \]where:

• **FV** is the future value, which is what you will have after the investment period.
• **PV** is the present value, or the amount of money you are investing now.
• **r** is the interest rate expressed as a decimal.

to calculate using this formula, you need to have your investment's present value, the desired future value, and solve for the unknown interest rate, if that is the missing variable. It is useful because it provides a straightforward way to see how money will grow due to interest over time. The formula assumes the interest accumulates linearly without considering compounding.

Future Value

Future Value (FV) refers to the amount of money an investment will grow to after a specified period, given a particular interest rate. Understanding future value is essential for evaluating different investment options. If you look at the original exercise, the FV is the amount projected from the opportunity you were considering, which was $392.Determining future value through simple interest relies on the formula \( FV = PV \times (1 + r) \). This equation gives investors a clear picture of what their money will become if left to grow over the investment period without any withdrawals. It's especially handy for short-term investments or calculating returns on loans with a fixed interest rate over a single period.Understanding future value helps investors make informed comparisons between different investment vehicles by providing a quantifiable goal or outcome.

Present Value

Present Value (PV) is the current worth of a future sum of money given a specified rate of return. Essentially, it answers the question: How much is a future amount worth today? This calculation is crucial for comparing investment opportunities or deciding whether to lend money, as seen in the exercise example where the PV is $350. The present value is used in the simple interest formula as the initial principal of an investment. By knowing PV, investors can decide whether their current funds should be invested in an alternative or kept as they are. Using PV in calculations ensures that you're not just looking at the future amount blindly but are instead weighting it against what you have now. Present value calculations help in performing discounting, a technique for valuing money that is expected to be received in the future by accounting for interest rates.

Investment Decision Analysis

Investment Decision Analysis is a process used to evaluate different financial opportunities to determine the best course of action. By analyzing potential returns on investment, individuals and companies can make sound financial decisions based on the desirability of varying prospects. It leverages formulas like the simple interest formula to make these evaluations. In our example exercise, the analysis involves comparing the potential returns from lending money to a friend versus investing in an opportunity with a known return. You apply the simple interest formula to determine the interest rate that would make you indifferent between the two choices. Here is how you might conduct such an analysis:

• Identify all your options, such as lending money or investing in stocks.
• Evaluate the potential future values using formulas like the simple interest formula.
• Compute the interest rate required for alternatives to match more appealing investments.
• Consider your financial goals, risk tolerance, and timelines.

Investment decision analysis is crucial as it ensures that funds are allocated towards the most beneficial opportunities, maximizing returns and minimizing risks.