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Chapter 7: Problem 33

An investment pays \(\$ 20\) semiannully for the next 2 years. The investment has a 7 percent nominal interest rate, and interest is compounded quarterly. What is the future value of the investment?

Short Answer

Expert verified
The future value of the investment is approximately $83.04.

Step by step solution

01

Understand the Problem

The problem involves calculating the future value of an investment in the form of annuity payments. The investment pays $20 every half year for 2 years, with a nominal interest rate of 7% compounded quarterly. We need to determine the future value of these payments at the end of the investment period.
02

Identify Annuity Characteristics

Since the payments are made semiannually, the annuity has four periods (2 payments per year for 2 years). The nominal interest rate is 7%, and since the interest is compounded quarterly, we need to adjust our calculations for this compounding frequency.
03

Calculate Effective Interest Rate per Semiannual Period

First, find the effective interest rate for a six-month period: - The quarterly rate is the nominal rate divided by 4: \[ i_q = \frac{0.07}{4} = 0.0175 \] - To find the semiannual rate, we compound this quarterly rate twice (since 6 months includes two quarters): \[ i_{6m} = (1 + i_q)^2 - 1 = (1 + 0.0175)^2 - 1 \approx 0.0353 \] This rate will be used as the effective interest rate per semiannual period for annuity calculations.
04

Calculate Future Value of Annuity

The future value of an annuity formula is: \[ FVA = P \times \frac{(1 + i)^n - 1}{i} \] where, - \( P = 20 \) (annuity payment) - \( i = 0.0353 \) (effective interest rate per period)- \( n = 4 \) (total periods)Substitute the values:\[ FVA = 20 \times \frac{(1 + 0.0353)^4 - 1}{0.0353} \approx 20 \times \frac{1.1465 - 1}{0.0353} \approx 20 \times 4.152 \approx 83.04 \]

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Key Concepts

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

Annuity Payments
An annuity refers to a series of equal payments made at regular intervals. In the context of this exercise, a semiannual annuity payment of $20 is made over 2 years.
This means there are a total of four payments during the investment period. The consistency of the payment amount makes annuities predictable and easy to manage.

Understanding annuities is crucial because they form part of many financial products, like retirement plans and insurance contracts.
  • Types of Annuities: Annuities can be classified based on the timing of the payment, such as ordinary annuities (payments at the end of each period) or annuities due (payments at the beginning of each period).
  • Future Value: The goal of calculating the future value of an annuity is to understand how much the periodic payments will accumulate to, after interest is applied over a set period.
This exercise uses the future value formula for annuities to determine the total amount at the end of the payment periods, ensuring you understand how much the investment will be worth after compounding.
Nominal Interest Rate
The nominal interest rate is the annual interest rate stated without adjusting for compounding within the year. In the exercise, the nominal interest rate is 7% per annum.
This rate serves as a starting point for calculating how much interest will be earned.

However, unlike other rates such as the effective interest rate, the nominal rate does not account for the effects of compounding.
  • Importance: Understanding the nominal rate is important for financial calculations as it is commonly listed on financial products like loans and annuities.
  • Relation to Effective Rate: The effective interest rate gives a more accurate measure by including compounding effects. In this exercise, since interest is compounded quarterly, adjustments are necessary to reflect true earning potential.
Misunderstanding the difference between nominal and effective rates can lead to inaccurate financial planning, which is why effective rate calculations are essential.
Compounding Quarterly
Compounding refers to the process where an asset's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time.
In this scenario, compounding quarterly means that the investment earns interest four times a year.

This frequent compounding impacts the effective interest rate, which offers a more precise understanding of how much the investment is really growing.
  • Calculation: To adjust for quarterly compounding, the nominal rate is divided by the number of compounding periods in a year (in this case, 4).
  • A Real-Life Example: Consider compounding like baking a cake; each quarterly compounding period adds more layers. The more layers or compounding periods, the thicker the cake or higher the investment returns.
This concept is vital for accurately determining the future value of investments which require precise calculations for effective financial planning.

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Most popular questions from this chapter

a. Set up an amortization schedule for a \(\$ 25,000\) loan to be repaid in equal installments at the end of each of the next 5 years. The interest rate is 10 percent, compounded annually. b. How large must each annual payment be if the loan is for \(\$ 50,000\) ? Assume that the interest rate remains at 10 percent, compounded annually, and that the loan is paid off over 5 years. c. How large must each payment be if the loan is for \(\$ 50,000\), the interest rate is 10 percent, compounded annually, and the loan is paid off in equal installments at the end of each of the next 10 years? This loan is for the same amount as the loan in part b, but the payments are spread out over twice as many periods. Why are these payments not half as large as the payments on the loan in part b?

Find the interest rates, or rates of return, on each of the following: a. You borrow \(\$ 700\) and promise to pay back \(\$ 749\) at the end of 1 year. b. You lend \(\$ 700\) and receive a promise to be paid \(\$ 749\) at the end of 1 year. c. You borrow \(\$ 85,000\) and promise to pay back \(\$ 201,229\) at the end of 10 years. d. You borrow \(\$ 9,000\) and promise to make payments of \(\$ 2,684.80\) per year for 5 years.

A 15 -year security has a price of \(\$ 340.4689 .\) The security pays \(\$ 50\) at the end of each of the next 5 years, and then it pays a different fixed cash flow amount at the end of each of the following 10 years. Interest rates are 9 percent. What is the annual cash flow amount between Years 6 and 15 ?

A rookie quarterback is in the process of negotiating his first contract. The team's general manager has offered him three possible contracts. Each of the contracts lasts for 4 years. All of the money is guaranteed and is paid at the end of each year. The terms of each of the contracts are listed below: The quarterback discounts all cash flows at 10 percent. Which of the three contracts offers him the most value?

Your parents are planning to retire in 18 years. They currently have \(\$ 250,000\), and they would like to have \(\$ 1,000,000\) when they retire. What annual rate of interest would they have to earn on their \(\$ 250,000\) in order to reach their goal, assuming they save no more money?
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