Chapter 5, Problem 42IC

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15th Edition

Eugene F. Brigham + 1 other

ISBN: 9781337395250

Textbook Problem

**TIME VALUE OF MONEY ANALYSIS** You have applied for a job with a local bank. As part of its evaluation process, you must take an examination on time value of money analysis covering the following questions:

- a. Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2; (2) an ordinary annuity of $100 per year for 3 years; and (3) an uneven cash flow stream of −$50, $100, $75, and $50 at the end of Years 0 through 3.
- b. 1. What’s the future value of $100 after 3 years if it cams 4%, annual compounding?
2. What’s the present value of $100 to be received in 3 years if the interest rate is 4%, annual compounding?

- c. What annual interest rate would cause $100 to grow to $119.10 in 3 years?
- d. If a company’s sales are growing at a rate of 10% annually, how long will it take sales to double?
- e. What’s the difference between an ordinary annuity and an annuity due? What type of annuity is shown here? How would you change it to the other type of annuity?

- f. 1. What is the future value of a 3-year, $100 ordinary annuity if the annual interest rate is 4%?
2. What is its present value?

3. What would the future and present values be if it was an annuity due?

- g. A 5-year $100 ordinary annuity has an annual interest rate of 4%.
1. What is its present value?

2. What would the present value be if it was a 10-year annuity?

3. What would the present value be if it was a 25-year annuity?

4. What would the present value be if this was a perpetuity?

- h. A 20-year-old student wants to save $5 a day for her retirement. Every day she places $5 in a drawer. At the end of each year, she invests the accumulated savings ($1,825) in a brokerage account with an expected annual return of 8%.
1. If she keeps saving in this manner, how much will she have accumulated at age 65?

2. If a 40-year-old investor began saving in this manner, how much would he have at age 65?

3. How much would the 40-year-old investor have to save each year to accumulate the same amount at 65 as the 20-year-old investor?

- i. What is the present value of the following uneven cash flow stream? The annual interest rate is 4%.

- j. 1. Will the future value be larger or smaller if we compound an initial amount more often than annually (e.g., semiannually, holding the stated (nominal) rate constant)? Why?
2. Define (a) the stated (or quoted or nominal) rate, (b) the periodic rate, and (c) the effective annual rate (EAR or EFF%).

3. What is the EAR corresponding to a nominal rate of 4% compounded semiannually? Compounded quarterly? Compounded daily?

4. What is the future value of $100 after 3 years under 4% semiannual compounding? Quarterly compounding?

- k. k. When will the EAR equal the nominal (quoted) rate?
- l. 1. What is the value at the end of Year 3 of the following cash flow stream if interest is 4% compounded semiannually? (Hint: You can use the EAR and treat the cash flows as an ordinary annuity or use the periodic rate and compound the cash flows individually.)

2. What is the PV?

3. What would be wrong with your answer to parts l(1) and l(2) if you used the nominal rate, 4%, rather than the EAR or the periodic rate, I_{NOM}/2 = 4%/2 = 2%, to solve the problems?

- m. 1. Construct an amortization schedule for a $1,000, 4% annual interest loan with three equal installments.
2. What is the annual interest expense for the borrower and the annual interest income for the lender during Year 2?

**a.(1)**

Summary Introduction

**To prepare:** Time line for a $100 lump sum cash flow at the end of the year 2.

**Introduction:**

**Time Line:** A line representing the cash flow of a company over a period of time is called time line. It shows the cash flow by representing them diagrammatically.

Explanation

Time line is drawn representing lump sum cash flow,

**(2)**

Summary Introduction

**To prepare:** Time line for ordinary annuity of $100 per year for 3 years.

**(3)**

Summary Introduction

**To prepare:** Time line for an uneven cash flow stream of $50, $100, $75 and, $50 at the end of years 0 through 3.

**b.1.**

Summary Introduction

**To calculate:** Future value of $100 after 3 years if it earns 4%, annual compounding.

**Future Value of Cash Flow:**

The future value of cash flow is the value of cash which it gains after receiving interest for a number of periods. It is also known as the terminal value.

**2.**

Summary Introduction

**To calculate:** The present value of $100 if after 3 years at 4% annual compounding.

**c.**

Summary Introduction

**To calculate:** Annual interest rate to grow $100 to $119.10 in 3 years.

**d.**

Summary Introduction

**To calculate:** The time period to grow sales double at 10% annually growth rate.

**e.**

Summary Introduction

**To explain:** Difference between annuity and ordinary annuity, type of annuity shown in table line and changed it to other type of annuity.

**Annuity:**

It is an agreement under which a person pays the lump sum payment or the number of small transactions and in return he gets the amount at later date or upon annuitization. The purpose of the annuity is not to break the flow of income after retirement.

**f. 1.**

Summary Introduction

**To calculate:** Future value of $100 ordinary annuity in 3 years at 4% annual interest rate.

**Future value of cash flow:**

The future value of cash flow is that value of cash which it gains after receiving interest for a number of periods. It is also known as the terminal value.

**2.**

Summary Introduction

**To calculate:** Present value of $100 ordinary annuity in 3 years at 4% annual interest rate.

**3.**

Summary Introduction

**To calculate:** Present and futurevalue of $100 annuity due in 3 years at 4% annual interest rate.

**g.1.**

Summary Introduction

**To calculate:** Present value of ordinary annuity with an annual interest rate of 4% for 5 years.

**2.**

Summary Introduction

**To calculate:** Present value of ordinary annuity with an annual interest rate of 4% for 10 years.

**3.**

Summary Introduction

**To calculate:** Present value of ordinary annuity with an annual interest rate of 4% for 25 years.

**4.**

Summary Introduction

**To calculate:** Present value of ordinary annuity with an annual interest rate of 4% with perpetuity.

**h.1.**

Summary Introduction

**To calculate:** Future value of $5 daily after 45years with 8% annual returns.

**2.**

Summary Introduction

**To calculate:** Future value of $5 daily after 25 years with 8% annual returns.

**3.**

Summary Introduction

**To calculate:** Amount to be saved in by 40 years old to get the return of $58,254.82.

**i.**

Summary Introduction

**To calculate:** The present value of uneven cash flow with annual interest rate 4%.

**j.1.**

Summary Introduction

**To calculate:** Future value with an initial amount more often than annually.

**2.a.**

Summary Introduction

**To explain:** Nominal interest rate.

**b.**

Summary Introduction

**To Explain:** Periodic interest rate

**c.**

Summary Introduction

**To Explain:** Effective annual rate

**3**

Summary Introduction

**To calculate:** Effective annual rate correspondingto a nominal rate of 4% compounded semiannually, quarterly and daily.

**4.**

Summary Introduction

**To calculate:** Future value of $100 after 3 years under 4% semiannual and quarterly compounding.

**k.**

Summary Introduction

**To explain:** When the effective annual rate will equal nominal rate.

**l.1.**

Summary Introduction

**To calculate:** The future value of cash flow stream at the end of 3 year at 4 % compounded annually.

**2.**

Summary Introduction

**To calculate:** The present value of cash flow stream at the end of 3 year at 4 % compounded annually.

**3.**

Summary Introduction

**To explain:** The answer, if instead of 4%/2 or 2%/2 it is 4% or 2%.

**m.**

Summary Introduction

**To prepare:** Amortization schedule for a $1,000 at 4% annual interest loan with three equal installments

**2.**

Summary Introduction

**To explain:** Annual interest expense and annual interest income during year 2.

Ch-5 P-3PCh-5 P-4PCh-5 P-5PCh-5 P-6PCh-5 P-7PCh-5 P-8PCh-5 P-9PCh-5 P-10PCh-5 P-11PCh-5 P-12PCh-5 P-13PCh-5 P-14PCh-5 P-15PCh-5 P-16PCh-5 P-17PCh-5 P-18PCh-5 P-19PCh-5 P-20PCh-5 P-21PCh-5 P-22PCh-5 P-23PCh-5 P-24PCh-5 P-25PCh-5 P-26PCh-5 P-27PCh-5 P-28PCh-5 P-29PCh-5 P-30PCh-5 P-31PCh-5 P-32PCh-5 P-33PCh-5 P-34PCh-5 P-35PCh-5 P-36PCh-5 P-37PCh-5 P-38PCh-5 P-39PCh-5 P-40PCh-5 P-41SPCh-5 P-42IC

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