## Review Questions

4-17 How can you determine the size of the equal, annual, end-of-period deposits necessary to accumulate a certain future sum at the end of a specified future period at a given annual interest rate?

4-18 Describe the procedure used to amortize a loan into a series of equal periodic payments.

4—19 Which present value interest factors would be used to find (a) the growth rate associated with a series of cash flows and (b) the interest rate associated with an equal-payment loan?

4r~20 How can you determine the unknown number of periods when you know the present and future values—single amount or annuity—and the applicable rate of interest?

Summary

### Focus on Value

Time value of money is an important tool that financial managers and other market participants use to assess the effects of proposed actions. Because firms have long lives and some decisions affect their long-term cash flows, the effective application of time-value-of-rnoney techniques is extremely important. These techniques enable financial managers to evaluate cash flows occurring at different times so as to combine, compare, and evaluate them and link them to the firm's overall goal of share price maximization. It will become clear in Chapters 6 and 7 that the application of time value techniques is a key part of the value determination process needed to make intelligent value-creating decisions.

Review of Learning .Goals

Key definitions, formulas, and equations for this chapter are summarized in Table 4.9.

§ Discuss the role of time value in finance, the use of computational tools, and the basic patterns of cash flow. Financial managers and investors use time-value-of-money techniques when assessing the value of expected cash flow streams. Alternatives can be assessed by either compounding to find future value or discounting to find present value. Financial managers rely primarily on present value techniques. Financial calculators, electronic spreadsheets, and financial tables can streamline the application of time value techniques. The cash flow of a firm can be described by its pattern—single amount, annuity, or mixed stream.

|K| Understand the concepts of future value and present value, their calculate! tion for single amounts, and the relationship between them. Future value (FV) relies on compound interest to measure future amounts: The initial principal or deposit in one period, along with the interest earned on it, becomes the beginning principal of the following period.

The present value (PV) of a future amount is the amount of money today that is equivalent to the given future amount, considering the return that can be earned. Present value is the inverse of future value.

Find the future value and the present value of both an ordinary annuity and an annuity due, and find the present value of a perpetuity. An annuity is a pattern of equal periodic cash flows. For an ordinary annuity, the cash flows occur at the end of the period. For an annuity due, cash flows occur at the beginning of the period.

The future value of an ordinary annuity can be found by using the future value interest factor for an annuity. The present value of an ordinary annuity can be found by using the present value interest factor for an annuity. A simple conversion can be applied to use the FV and PV interest factors for an ordinary annuity to find, respectively, the future value and the present value of an annuity due. The present value of a perpetuity—an infinite-lived annuity—is found using 1 divided by the discount rate to represent the present value interest factor.

§ Calculate both the future value and the present value of a mixed stream of cash flows. A mixed stream of cash flows is a stream of unequal periodic cash flows that reflect no particular pattern. The future value of a mixed stream of cash flows is the sum of the future values of each individual cash flow. Similarly, the present value of a mixed stream of cash flows is the sum of the present values of the individual cash flows.

§ Understand the effect that compounding interest more frequently than annually has on future value and on the effective annual rate of interest. Interest can be compounded at intervals ranging from annually to daily, and even continuously. The more often interest is compounded, the larger the future amount that will be accumulated, and the higher the effective, or true, annual rate (EAR).

Summary of Key Definitions, Formula's, and Equations for Time Value of Money

Definitions of variables e — exponential function = 2.7133 EAR ra effective annual rate FV„ = future value or amount at the end of period n FVA„ = future value of an «-year annuity i = annual rate of interest m = number of times per year interest is compounded n = number of periods—typically years—over which money earns a return PMT = amount deposited or received annually at the end of each year

PV — initial principal or present value PVA„ — present value of an n-year annuity t = period number index

Interest factor formulas

Future value of a single amount with annual compounding:

FVIF,,„ = (1 + /)" [Equarion 4.5; factors in Table A-l]

Present value of a single amount 1

PVlFt „ = —-— [Equation 4.11; factors in Table A-2]

Future value of an ordinary annuity:

FVIFA^ = ¿(1 + i)''1 [Equation4.13; factors in Table A-3]

Present value of an ordinary annuity:

PVIFAi „ = T, -- 1 [Equation 4.15; factors in Table A-4]

Future value of an annuity due;

FVIFA,-» (annuity due) = FVIFAtj„X(l 4- i) [Equation 4.17] Present value of an annuity due:

PVIFAit„ (annuity due) = PVIFA,^ X (1 + ¿) [Equarion 4.18] Present value of a perpetuity:

Future value with compounding more frequently than annually:

For continuous compounding, m =

FViF(> {continuous compounding) = eiXn [Equarion 4,22] To find the effective annual rate:

Basic equations

Future value (single amount): FV„ = PVX (FV7Fy,„) [Equation 4.6]

Present value (single amount): PV = FV„ X (PV7F,.„) [Equation 4.12]

Future value (annuity): FVA„ = PMTx (FVIFA;,„) [Equation 4.14]

Present value (annuity): PVA„ = PMTX (PVIFA,_n) [Equation 4.16]

The annual percentage rate (APR)—a nominal annual rate—is quoted on credit cards and loans. The annual percentage yield (APY)—an effective annual rate—is quoted on savings products.

ffcj Describe the procedures involved in (1) determining deposits needed to Ipj accumulate a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods. (1) The periodic deposit to accumulate a given future sum can be found by solving the equation for the future value of an annuity for the annual payment. (2) A loan can be amortized into equal periodic payments by solving the equation for the present value of an annuity for the periodic payment. (3) Interest or growth rates can be estimated by finding the unknown interest rate in the equation for the present value of a single amount or an annuity. (4) An unknown number of periods can be estimated by finding the unknown number of periods in the equation for the present value of a single amount or an annuity.

Self-Test Problems (Solutions in Appendix B)

ST4-1 Future values for various compounding frequencies Delia Martin has \$10,000 that she can deposit in any of three savings accounts for a 3-year period. Bank A compounds interest on an annual basis, bank B compounds interest twice each year, and bank C compounds interest each quarter. All three banks have a stated annual interest rate of 4%.

a. What amount would Ms. Martin have at the end of the third year, leaving all interest paid on deposit, in each bank?

b. What effective annual rate {EAR) would she earn in each of the banks?

c. On the basis of your findings in parts a and b, which bank should Ms. Martin deal with? Why?

d. If a fourth bank (bank D), also with a 4% stated interest rate, compounds interest continuously, how much would Ms. Martin have at the end of the third year? Does this alternative change your recommendation in part c? Explain why or why not.

ST4-2 Future values of annuities Ramesh Abdul wishes to choose the better of two equally costly cash flow streams: annuity X and annuity Y. X is an annuity due with a cash inflow of \$9,000 for each of 6 years. Y is an ordinary annuity with a cash inflow of \$10,000 for each of 6 years. Assume that Ramesh can earn 15% on his investments.

a. On a purely subjective basis, which annuity do you think is more attractive? Why?

b. Find the future value at the end of year 6, FVAg, for both annuity X and annuity Y.

c. Use your finding in part b to indicate which annuity is more attractive. Why? Compare your finding to your subjective response in part a.

ST4-3 Present values of single amounts and streams You have a choice of accepting either of two 5-year cash flow streams or single amounts. One cash flow stream is an ordinary annuity, and the other is a mixed stream. You may accept alternative A or B—either as a cash flow stream or as a single amount. Given the cash flow stream and single amounts associated with each (see the following table), and assuming a

9% opportunity cost, which alternative (A or B) and in which form (cash flow stream or single amount) would you prefer?

2 700 900

3 700 700

4 700 500

5 700 300

fJS ST4-4 Deposits needed to accumulate a future sum Judi Janson wishes to accumulate '61 \$8,000 by the end of 5 years by making equal, annual, end-of-year deposits over the next 5 years. If Judi can earn 7% 'on her investments, how much must she deposit at the end of each year to meet this goal?