978-0133546408 Chapter 5 Solution Manual Part 1

Chapter 5

Time Value of Money

 Instructor’s Resources

Overview

This chapter introduces an important financial concept: the time value

of money. The present value and future of a sum, as well as the present and future values of an annuity, are

explained. Special applications of the concepts include intra-year compounding, mixed cash flow streams, mixed

cash flows with an embedded annuity, perpetuities, deposits to accumulate a future sum, and loan amortization.

Numerous business and personal financial applications are used as examples. The chapter drives home the need to

understand time value of money at the professional level because funding for new assets and programs must be

justified using these techniques. Decisions in a student’s personal life should also be acceptable on the basis of

applying time-value-of-money techniques to anticipated cash flows.

 Answers to Review Questions

1.Future value (FV), the value of a present amount at a future date, is calculated by applying compound interest

over a specific time period. Present value (PV) represents the dollar value today of a future amount, or the

2.A single amount cash flow refers to an individual standalone value occurring at one point in time. An annuity

consists of an unbroken series of cash flows of equal dollar amount occurring over more than one period. A

3.Compounding of interest occurs when an amount is deposited into a savings account and the interest paid after

FVn  PV(1  r)n

4.A decrease in the interest rate lowers the future amount of a deposit for a given holding period because the

deposit earns less at the lower rate. An increase in the holding period for a given interest rate would increase

5.Present value is the current dollar value of a future amount. It indicates how much money today would be

2 Gitman/Zutter Principles of Managerial Finance, Brief, Seventh Edition

PV  FVn  (1  r)n

6.An increasing required rate of return would reduce the present value of a future amount because future dollars

7.Present value calculations are the exact inverse of compound interest calculations. Using compound interest,

10. An ordinary annuity is one for which payments occur at the end of each period. An annuity due is one for

11.The most efficient ways to calculate present value of an ordinary annuity are using an algebraic equation, a

18. The future value of a mixed stream of cash flows is calculated by multiplying each year’s cash flow by

20. As interest is compounded more frequently than once a year, both (a) the future value for a given holding

21. Continuous compounding assumes interest will be compounded an infinite number of times per year, at

Chapter 3: Financial Statements and Ratio Analysis 3

22.The nominal annual rate is the contractual rate that is quoted to the borrower by the lender. The effective annual

APR is the annual percentage rate and is required by “truth-in-lending laws” to be disclosed to consumers. This

rate is calculated by multiplying the periodic rate by the number of periods in one year. The periodic rate is

26. The size of the equal annual end-of-year deposits needed to accumulate a given amount over a certain time

27. Amortizing a loan into equal annual payments involves finding the future payments whose present value at

the loan interest rate just equals the amount of the initial principal borrowed. Amortizing a loan involves

28. The best way to determine an unknown number of periods is through the use of a calculator or spreadsheet. In

both instances, you enter the cash flows and interest rate and then compute the number of periods needed to

 Suggested Answer to Focus on Practice Box: New Century Brings

Trouble for Subprime Mortgages

As a reaction to problems in the subprime area, lenders tightened lending standards. What effect will this

have on the housing market?

The tightening of lending standards following the subprime fiasco further depressed home prices, which in 2007

were already undergoing their steepest, widest decline in history. As the housing-market slump persists, companies

4 Gitman/Zutter Principles of Managerial Finance, Brief, Seventh Edition

 Answers to Warm-Up Exercises

E5-1. Future value of a lump-sum investment

E5-2. Finding the future value

Answer: Because the interest is compounded monthly, the number of periods is 4  12  48 and the

monthly interest rate is 1/12th of the annual rate.

Note: Not all financial calculators work in the same manner. Some require the user to use the

CPT (Compute) button. Others require the user to calculate the monthly interest rate and input

If using a spreadsheet, the solution is:

Column A Column B

E5-3. Comparing a lump sum with an annuity

Answer: This problem can be solved in either of two ways. Both alternatives can be compared as lump

Chapter 3: Financial Statements and Ratio Analysis 5

Method 1: Perform a lump sum comparison. Compare $1.3 million now with the present value of

the 25 payments of $100,000 per year. In this comparison, the present value of the $100,000

Method 2: Compare two annuities. Because the $100,000 per year is already an annuity, all that

remains is to convert the $1.3 million into a 25-year annuity.

E5-4. Comparing the present value of two alternatives

Answer: To solve this problem you must first find the present value of the expected savings over the 5-year

life of the software.

Year Savings Estimate

Present Value

of Savings

Because the $136,402 present value of the savings exceeds the $130,000 cost of the software, the

6 Gitman/Zutter Principles of Managerial Finance, Brief, Seventh Edition

E5-5. Compounding more frequently than annually

Answer: Partners’ Savings Bank:

1

2

1

2

1

$12,000 (1 0.03/2)

$12,000 (1 0.03/2) $12,000 1.030225 $12,362.70

m n

r

FV PV m

FV

´

æ ö

= ´ +

ç ÷

è ø

= ´ +

= ´ + = ´ =

Selwyn’s:

rxn 0.0275 1

1

FV PV (e ) $12,000 (2.7183 )

$12,000 1.027882 $12,334.58

´

= ´ = ´

= ´ =

Joseph should choose the 3% rate with semiannual compounding.

E5-6. Determining deposits needed to accumulate a future sum

Answer: The financial calculator input is as follows:

 Solutions to Problems

P5-1. Using a time line

LG 1; Basic

a, b, and c

Chapter 3: Financial Statements and Ratio Analysis 7

d. Financial managers rely more on present value than future value because they typically make

P5-2. Future value calculation

LG 2; Basic

Case

P5-3. Time to double

LG 1; Basic

Case B: Computer Inputs: I  6%, PV  $100; FV  $200

 N  11.90 years

P5-4. Future values

LG 2; Intermediate

Case Case

P5-5. Personal finance: Time value

LG 2; Intermediate

a. (1) N  3, I  7%, PV  $1,500 b. (1) Interest

(2) N  6, I  7%, PV  $1,500 (2) Interest earned  FV6 – FV3

(3) N  9, I  7%, PV  $1,500 (3) Interest earned  FV9  FV6

8 Gitman/Zutter Principles of Managerial Finance, Brief, Seventh Edition

c. The fact that the longer the investment period is, the larger the total amount of interest collected will

be, is not unexpected and is due to the greater length of time that the principal sum of $1,500 is

P5-6. Personal finance: Time value

LG 2; Challenge

a. (1) N  5, I  2%, PV  $14,000 (2) N  5, I

b. The car will cost $1,576.01 more with a 4% inflation rate than an inflation rate of 2%. This increase is

c. Future value at end of first 2 years:

P5-7. Personal finance: Time value

LG 2; Challenge

Deposit Now: Deposit in 10 Years:

P5-8. Personal finance: Time value

LG 2; Challenge

P5-9. Personal finance: Single-payment loan repayment

LG 2; Intermediate

Chapter 3: Financial Statements and Ratio Analysis 9

P5-10. Present value calculation:

1

PVIF (1 )n

i

=+

LG 2; Basic

Case

P5-11. Present values

LG 2; Basic

Case PV

P5-12. Present value concept

LG 2; Intermediate

a. N  6, I  12%, FV  $6,000 b. N  6, I  12%, FV  $6,000

c. N  6, I  12%, FV  $6,000

d. The answer to all three parts is the same. In each case, the same question is being asked but in a

P5-13. Personal finance: Time Value

LG 2; Basic

a. N  3, I  7%, FV  $500