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Chapter 5: Problem 63

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 63 to solve Exercises \(64-68\)

Short Answer

Expert verified
To derive the formula for effective rate, \(r_{\text {eff }}\), given an initial investment of \(P\) that grows to an accumulated amount \(A\) in \(t\) years, follow these steps: 1. Express the accumulated amount 'A' as: \[A = P(1 + r_{\text {eff }})^t\] 2. Divide both sides by 'P': \[\frac{A}{P} = (1 + r_{\text {eff }})^t\] 3. Take the 't'-th root of both sides: \[\sqrt[t]{\frac{A}{P}} = 1 + r_{\text {eff }}\] 4. Subtract '1' from both sides to isolate the effective rate: \[r_{\text {eff }} = \sqrt[t]{\frac{A}{P}} - 1\] Thus, the formula for the effective rate is: \(r_{\text {eff }} = (A / P)^{1 / t} - 1\).

Step by step solution

01

Identify the formula for principal, interest, and time.

In this case, we are given an initial investment of \(P\), which is the principal. After 't' years, the accumulated amount becomes 'A' which includes the principal and interest. The effective rate, also known as annual effective yield, is denoted by \(r_{\text {eff }}\).
02

Derive the formula for accumulated amount 'A'

In terms of the effective rate, we can express the accumulated amount 'A' as the following formula: \[A = P(1+r_{\text {eff }})^t\] This formula represents that the accumulated amount 'A' is equal to the initial investment 'P' multiplied by the growth factor \(1+r_{\text {eff }}\) raised to the power 't' years.
03

Solve for the effective rate

Now, our goal is to isolate the effective rate \(r_{\text {eff }}\) in our formula. To do this, we can follow these steps: 1. Divide both sides of the equation by 'P' to eliminate the principal on the left-hand side: \[\frac{A}{P} = (1+r_{\text {eff }})^t\] 2. Take the 't'-th root of both sides to eliminate the exponent 't' on the right-hand side: \[\sqrt[t]{\frac{A}{P}} = 1+r_{\text {eff }}\] 3. Lastly, subtract '1' from both sides to isolate the effective rate: \[r_{\text {eff }} = \sqrt[t]{\frac{A}{P}} - 1\] This final equation is what we are asked to show: \[r_{\text {eff }}=(A / P)^{1 / t}-1\] Now that we have derived the formul for the effective rate, it can be utilized to solve relevant problems involving annual effective yield.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Investment Growth
Investment growth refers to how much an initial investment (principal) increases over time. When you invest money, the goal is often to see your capital grow, meaning the amount you originally invested increases over a period. This growth can occur because of various factors, including interest earnings and capital appreciation.

For instance, if you put \( \\(1000 \) into a savings account, and it grows to \( \\)1100 \) in a year, your investment has grown by \( \$100 \), or a 10"). Investment growth can be calculated using different methods, depending on the nature of the investment.
  • Linear growth: When a fixed amount of interest is earned every period.
  • Exponential growth: When the investment grows by a percentage over each period, leading it to compound over time.

Understanding how your investment is expected to grow helps make informed decisions, ensuring that your money works effectively over the investment period.
Compounded Interest
Compounded interest is one of the keys to understanding investment growth. It refers to the process where the interest earned on an investment is reinvested to earn more interest. This process results in faster growth compared to simple interest, where interest is only earned on the principal.

To calculate compound interest, you often use the formula:
\[ A = P(1 + r/n)^{nt} \]
Where:
  • A is the amount of money accumulated after n years, including interest.
  • P is the principal amount (the initial investment).
  • r is the annual interest rate (decimal).
  • n is the number of times that interest is compounded per year.
  • t is the time the money is invested for in years.

This compounding effect has the potential to massively increase the value of the investment over the years, especially if compounded frequently. The frequency of compounding鈥攚hether yearly, quarterly, monthly, etc.鈥攃an greatly influence the final amount of interest you earn. The more frequently the interest compounds, the more substantial the total amount.
Annual Effective Yield
The annual effective yield, also known as the effective interest rate, is the interest rate on an investment that represents compounding. Unlike nominal rates, which may vary in terms of the compounding frequency, the annual effective yield considers the impact of compounding once per year.

The formula for calculating the annual effective yield is:
\[ r_{\text{eff}} = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1 \]
Here, the formula shows how the accumulated amount (\( A \)) relates to the principal (\( P \)) over time (\( t \) years). It allows investors to convert the nominal rate into an equivalent rate as if it compounds just once per year.
  • A higher effective yield indicates a more significant return on investment when considering compounding effects.
  • This measure is critical for comparing different investment opportunities to see which offers the best return after factoring in how frequently the interest is compounded.

By understanding the concept of annual effective yield, you can more accurately assess and compare the potential profitability of various investments.

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Most popular questions from this chapter

Having received a large inheritance, Jing-mei's parents wish to establish a trust for her college education. If 7 yr from now they need an estimated \(\$ 70,000\), how much should they set aside in trust now, if they invest the money at \(10.5 \%\) compounded quarterly? Continuously?

Mike's Sporting Goods sells elliptical trainers under two payment plans: cash or installment. Under the installment plan, the customer pays \(\$ 22 /\) month over 3 yr with interest charged on the balance at a rate of \(18 \% /\) year compounded monthly. Find the cash price for an elliptical trainer if it is equivalent to the price paid by a customer using the installment plan.

FiNANCIAL ANALYSIS A group of private investors purchased a condominium complex for \(\$ 2\) million. They made an initial down payment of \(10 \%\) and obtained financing for the balance. If the loan is to be amortized over \(15 \mathrm{yr}\) at an interest rate of \(12 \% /\) year compounded quarterly, find the required quarterly pavment

The simple interest formula \(A=P(1+r t)\) [Formula (1b)] can be written in the form \(A=P r t+P\), which is the slope-intercept form of a straight line with slope \(P r\) and A-intercept \(P .\) a. Describe the family of straight lines obtained by keeping the value of \(r\) fixed and allowing the value of \(P\) to vary. Interpret your results. b. Describe the family of straight lines obtained by keeping the value of \(P\) fixed and allowing the value of \(r\) to vary. Interpret your results.

BatLoon PAYMENT MoRTGAGES Emilio is securing a 7 -yr Fannie Mae "balloon" mortgage for \(\$ 280,000\) to finance the purchase of his first home. The monthly payments are based on a 30 -yr amortization. If the prevailing interest rate is \(7.5 \% /\) year compounded monthly, what will be Emilio's monthly payment? What will be his "balloon" payment at the end of 7 yr?
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