Problem 71 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose an initial investment of $$\$ P$$ grows to an accumulated amount of $$\$ A$$ in \(t\) yr. Show that the effective rate (annual effective yield) is $$ r_{\text {eff }}=(A / P)^{1 / t}-1 $$ Use the formula given in Exercise 71 to solve Exercises \(72-76 .\)

Step-by-Step Solution

Answer

To show that the effective rate (annual effective yield) can be calculated using the formula \( r_{\text {eff }}=(A / P)^{1 / t}-1 \), we start with the formula from Exercise 71: \( A = P(1 + r/n)^{nt} \). After manipulating and simplifying the equations, we achieve the desired formula for the effective rate as \( r_{\text {eff }} =(A / P)^{(1 / t)}-1 \).

Step 1: Review the formula given in Exercise 71

The formula given in Exercise 71 is: \( A = P(1 + r/n)^{nt} \) This formula represents the accumulated amount (\(A\)) of an initial investment (\(P\)) after a certain number of years (\(t\)), with the interest rate (\(r\)) compounded \(n\) times per year.

Step 2: Isolate the term that involves the interest rate

We want to find a formula for the effective rate. To do so, we need to isolate the term that involves the annual effective yield, which is the term with the interest rate. Divide both sides of the equation by \(P\): \( A / P = (1 + r/n)^{nt} \)

Step 3: Simplify the equation for the effective rate

Now, we want to find a formula for the effective rate, \(r_{\text {eff }}\), that can be applied to the equation for A. Since the effective rate represents the annual interest rate that would produce the same accumulation as the given compounding scenario, we can rewrite the equation as: \( A / P = (1 + r_{\text {eff}})^t \) Notice that the right side of this equation looks very similar to the right side of the equation from step 2.

Step 4: Compare the two equations

Now, we need to equate the right sides of the two simplified equations: \( (1 + r_{\text {eff}})^t = (1 + r/n)^{nt} \) Our goal is to isolate the term with \(r_{\text {eff}}\), so we can solve for it.

Step 5: Solve for the effective rate

First, take the t-th root of both sides to remove the exponent t: \( (1 + r_{\text {eff}}) = [ (1 + r/n)^{nt} ]^{\frac{1}{t}} \) Simplify the right side of the equation: \( (1 + r_{\text {eff}}) = (1 + r/n)^{n} \) Now, we want to solve for \(r_{\text {eff}}\): \( r_{\text {eff}} = (1 + r/n)^{n} - 1 \)

Step 6: Replace r with the given formula for the effective rate

Now, we have a formula for the annual effective yield in terms of the interest rate compounded \(n\) times per year: \( r_{\text {eff }} =(A / P)^{(1 / t)}-1 \) This is the desired formula.

Key Concepts

Compound Interest Formula

The compound interest formula is a fundamental concept in finance used to calculate the future value of an investment based on its initial principal, the interest rate, and time. The formula is given as:\[A = P(1 + \frac{r}{n})^{nt}\]Here's what each variable stands for:

• \(A\): The total amount of money accumulated after the investment period, including interest.
• \(P\): The principal or initial amount of money invested.
• \(r\): The annual nominal interest rate (as a decimal).
• \(n\): The number of times that interest is compounded per year.
• \(t\): The time the money is invested for, in years.

This formula accounts for the effects of compounded interest, which means interest is calculated on the initial principal and also on the accumulated interest of previous periods.This compounding effect can significantly increase the future value of an investment, especially over long periods. Each compounding period could be quarterly, monthly, daily, or any other frequency that the interest might be applied.

Annual Effective Yield

The Annual Effective Yield, also known as the effective interest rate, is a reflection of the actual annual return from an investment, considering the effects of compounding.In contrast to nominal rates that don't account for compounding within the year, the effective yield shows how much interest an investor earns or pays over a year.The formula for calculating the effective annual rate when interest is compounded multiple times a year is:\[r_{\text{eff}} = \left(1 + \frac{r}{n}\right)^{n} - 1\]In this formula:

• \(r\): The annual nominal interest rate.
• \(n\): The number of compounding periods per year.

Taking the number of compounding periods into account, the annual effective yield offers a true indicator of financial growth over a period.If interest is compounded more frequently, the effective yield will be higher than the nominal rate, demonstrating the power of compounded interest.This concept is important for investors evaluating different options, as it provides a consistent standard to compare returns.

Investment Growth Calculation

Understanding how an investment grows over time is crucial for making informed financial decisions. The effective interest rate plays a key role in determining this growth.By combining the initial principal amount, the interest rate, and the time horizon, one can project the future value of an investment.Using the relationship:\[r_{\text{eff}} = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1\]We estimate how much the investment will grow.Here's how it works:* Calculate the ratio \(\frac{A}{P}\), which represents how much the investment grows over the specified time period.* Apply the exponent \(\frac{1}{t}\) to this ratio to standardize the growth rate on an annual basis.* Subtract 1 from this result to get the effective rate, which reflects the actual annual growth considering compounding.This formula tells us that, given the initial investment and the total accumulated value after a certain period, we can infer how efficient the investment is annually.Investment growth calculation is essential when planning future investments or evaluating the performance of current assets.