Problem 72 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Martha invested $$\$ 40,000$$ in a boutique 5 yr ago. Her investment is worth $$\$ 70,000$$ today. What is the effective rate (annual effective yield) of her investment?

Step-by-Step Solution

Answer

The annual effective yield (interest rate) of Martha's investment is approximately $$11.97\%$$.

Step 1: Identify the given information

In this problem, we are given the following information: - Initial investment (P): $$\$ 40,000$$ - Final value of the investment (A): $$\$ 70,000$$ - Time (t): 5 years

Step 2: Use the compound interest formula

The compound interest formula is given by: \(A = P(1 + r)^t\) where: - A - the final value of the investment - P - the initial investment - r - the annual effective yield (interest rate) - t - the time in years We will use this formula to find the interest rate (r).

Step 3: Rearrange the formula to find r

We need to isolate r in the compound interest formula. We start by dividing both sides by P: \(\frac{A}{P} = (1 + r)^t\) Now, we need to extract r from the exponent. We can do this by taking the t-th root of both sides: \(\sqrt[t]{\frac{A}{P}} = 1 + r\) Subtract 1 from both sides to get r: \(r = \sqrt[t]{\frac{A}{P}} - 1\)

Step 4: Insert the given values into the formula

Now, we can plug the values given in the problem into the formula: \(r = \sqrt[5]{\frac{70,000}{40,000}} - 1\)

Step 5: Calculate the interest rate

Now, we can calculate the interest rate: \(r = \sqrt[5]{\frac{7}{4}} - 1 \approx 0.1197\) To convert r to a percentage, multiply by 100: \(r \approx 0.1197 \times 100 = 11.97\%\)

Step 6: State the conclusion

The annual effective yield (interest rate) of Martha's investment is approximately $$11.97\%$$.

Key Concepts

Effective Annual Yield

The effective annual yield is a critical measure for understanding the true rate of return on an investment.
It's often referred to as the interest rate when discussing compounded investments. In simple terms, it tells you how much your investment grows annually after taking into account the effect of compounding.

To find the effective annual yield, you use the compound interest formula:

• The final amount (\(A\)) is the amount your investment has grown to, which in this case is \(\\(70,000\).
• The initial investment (\(P\)) is the amount initially invested, here \(\\)40,000\).
• Time (\(t\)) is the duration the money has been invested, which is 5 years.
• The rate (\(r\)) is what we solve for, representing the annual effective yield.

Investors often calculate the effective annual yield to compare different investment opportunities and to understand how much an investment is truly yielding, beyond simple interest calculations.

Investment Growth

Investment growth refers to how much your investment increases in value over time.
Using compounding as a powerful tool, growth can be exponential rather than just linear.

The basic idea is that your investment not only earns interest on the initial amount but also on the interest that accumulates each year.
In our example, \(\\(40,000\) grew to \(\\)70,000\) over five years, illustrating significant growth due to compounding.
Here's how compounding works to your advantage:

• The initial investment begins to grow when interest is first applied.
• Each subsequent year, the interest is calculated on the new total, which includes previously earned interest.
• Over time, this leads to accelerated growth of your investment.

Understanding investment growth helps investors predict potential future returns and make better financial decisions. Monitoring this growth is essential in evaluating whether an investment is meeting financial goals.

Mathematical Finance

Mathematical finance is the field where mathematical tools help solve financial problems and optimize investment strategies.
It heavily relies on concepts like compound interest to determine future values of investments.

In the example given, mathematical finance helps us solve for the annual effective yield using the compound interest formula, which is derived from basic algebraic principles applied to financial scenarios.
Consider these aspects of mathematical finance:

• Using formulas to calculate payments, future values, and yields ensures accuracy and sound financial planning.
• Understanding the mathematics behind finance helps in assessing risks and returns efficiently.
• It allows investors to create models that simulate various investment scenarios and their outcomes.

For students, grasping these concepts is crucial, as it lays the groundwork for analyzing financial opportunities and making informed personal or professional investment decisions.