TIME VALUE OF MONEY Chapter 5

The Role of Time Value in Finance Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-2 Most financial decisions involve costs & benefits that are spread out over time. Time value of money allows comparison of cash flows from different periods. Question Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 after one year, or one that would return $220,000 after two years?

The Role of Time Value in Finance (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-3 Most financial decisions involve costs & benefits that are spread out over time. Time value of money allows comparison of cash flows from different periods. Answer It depends on the interest rate!

Basic Concepts Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-4 Future Value : compounding or growth over time Present Value : discounting to today’s value Single cash flows & series of cash flows can be considered Time lines are used to illustrate these relationships

Computational Aids (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-5

Computational Aids (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-6

Computational Aids (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-7

Basic Patterns of Cash Flow Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-8 The cash inflows and outflows of a firm can be described by its general pattern. The three basic patterns include a single amount, an annuity, or a mixed stream:

Simple Interest Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-9 With simple interest, you don’t earn interest on interest. Year 1: 5% of $100=$5 + $100 = $105 Year 2: 5% of $100=$5 + $105 = $110 Year 3: 5% of $100=$5 + $110 = $115 Year 4: 5% of $100=$5 + $115 = $120 Year 5: 5% of $100=$5 + $120 = $125

Compound Interest Copyright © 2006 Pearson Addison-Wesley. All rights reserved With compound interest, a depositor earns interest on interest! Year 1: 5% of $100.00= $ $100.00= $ Year 2: 5% of $105.00= $ $105.00= $ Year 3: 5% of $ = $5.51+ $110.25= $ Year 4: 5% of $115.76= $ $115.76= $ Year 5: 5% of $121.55= $ $121.55= $127.63

Time Value Terms Copyright © 2006 Pearson Addison-Wesley. All rights reserved PV 0 =present value or beginning amount i= interest rate FV n =future value at end of “n” periods n=number of compounding periods A=an annuity (series of equal payments or receipts)

Four Basic Models Copyright © 2006 Pearson Addison-Wesley. All rights reserved FV n = PV 0 (1+i) n = PV x (FVIF i,n ) PV 0 = FV n [1/(1+i) n ] = FV x (PVIF i,n ) FVA n = A (1+i) n - 1= A x (FVIFA i,n ) i PVA 0 = A 1 - [1/(1+i) n ] = A x (PVIFA i,n ) i

Future Value of a Single Amount Copyright © 2006 Pearson Addison-Wesley. All rights reserved Future Value techniques typically measure cash flows at the end of a project’s life. Future value is cash you will receive at a given future date. The future value technique uses compounding to find the future value of each cash flow at the end of an investment’s life and then sums these values to find the investment’s future value. We speak of compound interest to indicate that the amount of interest earned on a given deposit has become part of the principal at the end of the period.

Future Value of a Single Amount: Using FVIF Tables Copyright © 2006 Pearson Addison-Wesley. All rights reserved If Fred Moreno places $100 in a savings account paying 8% interest compounded annually, how much will he have in the account at the end of one year? $100 x (1.08) 1 = $100 x FVIF 8%,1 $100 x 1.08 =$108

Future Value of a Single Amount: The Equation for Future Value Copyright © 2006 Pearson Addison-Wesley. All rights reserved Jane Farber places $800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of five years. FV 5 = $800 X ( ) 5 = $800 X (1.338) = $1,070.40

Future Value of a Single Amount: A Graphical View of Future Value Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-16

Present Value of a Single Amount Copyright © 2006 Pearson Addison-Wesley. All rights reserved Present value is the current dollar value of a future amount of money. It is based on the idea that a dollar today is worth more than a dollar tomorrow. It is the amount today that must be invested at a given rate to reach a future amount. Calculating present value is also known as discounting. The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, or the cost of capital.

Present Value of a Single Amount: Using PVIF Tables Copyright © 2006 Pearson Addison-Wesley. All rights reserved Paul Shorter has an opportunity to receive $300 one year from now. If he can earn 6% on his investments, what is the most he should pay now for this opportunity? $300 x [1/(1.06) 1 ] =$300 x PVIF 6%,1 $300 x = $283.02

Present Value of a Single Amount: The Equation for Future Value Copyright © 2006 Pearson Addison-Wesley. All rights reserved Pam Valenti wishes to find the present value of $1,700 that will be received 8 years from now. Pam’s opportunity cost is 8%. PV = $1,700/( ) 8 = $1,700/1.851 = $918.42

Present Value of a Single Amount: A Graphical View of Present Value Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-20

Annuities Copyright © 2006 Pearson Addison-Wesley. All rights reserved Annuities are equally-spaced cash flows of equal size. Annuities can be either inflows or outflows. An ordinary (deferred) annuity has cash flows that occur at the end of each period. An annuity due has cash flows that occur at the beginning of each period. An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

Types of Annuities Copyright © 2006 Pearson Addison-Wesley. All rights reserved Fran Abrams is choosing which of two annuities to receive. Both are 5-year $1,000 annuities; annuity A is an ordinary annuity, and annuity B is an annuity due. Fran has listed the cash flows for both annuities as shown in Table 4.1 on the following slide. Note that the amount of both annuities total $5,000.

Future Value of an Ordinary Annuity Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-23

Future Value of an Ordinary Annuity (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved Fran Abrams wishes to determine how much money she will have at the end of 5 years if he chooses annuity A, the ordinary annuity and it earns 7% annually. Annuity a is depicted graphically below:

Future Value of an Ordinary Annuity: Using the FVIFA Tables Copyright © 2006 Pearson Addison-Wesley. All rights reserved FVA = $1,000 (FVIFA,7%,5) = $1,000 (5.751) = $5,751

Future Value of an Annuity Due: Using the FVIFA Tables Copyright © 2006 Pearson Addison-Wesley. All rights reserved Fran Abrams now wishes to calculate the future value of an annuity due for annuity B in Table 4.1. Recall that annuity B was a 5 period annuity with the first annuity beginning immediately. FVA = $1,000(FVIFA,7%,5)(1+.07) = $1,000 (5.751) (1.07) = $6,154

Present Value of an Ordinary Annuity Copyright © 2006 Pearson Addison-Wesley. All rights reserved Braden Company, a small producer of plastic toys, wants to determine the most it should pay to purchase a particular annuity. The annuity consists of cash flows of $700 at the end of each year for 5 years. The required return is 8%.

Present Value of an Ordinary Annuity: The Long Method Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-28

Present Value of an Ordinary Annuity: Using PVIFA Tables Copyright © 2006 Pearson Addison-Wesley. All rights reserved PVA =$700 (PVIFA,8%,5) =$700 (3.993) =$2,795.10

Present Value of an Annuity Due: Using PVIFA Tables Copyright © 2006 Pearson Addison-Wesley. All rights reserved In the earlier example, we found that the value of Braden Company’s $700, 5 year ordinary annuity discounted at 8% to be about $2,795. If we now assume that the cash flows occur at the beginning of the year, we can find the PV of the annuity due. PVA = $700 (PVIFA,8%,5) (1.08) =$700 (3.993) (1.08) =$3,018.40

Present Value of a Perpetuity Copyright © 2006 Pearson Addison-Wesley. All rights reserved A perpetuity is a special kind of annuity. With a perpetuity, the periodic annuity or cash flow stream continues forever. For example, how much would I have to deposit today in order to withdraw $1,000 each year forever if I can earn 8% on my deposit? PV = Annuity/Interest Rate PV = $1,000/.08 = $12,500

Future Value of a Mixed Stream Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-32

Future Value of a Mixed Stream (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-33

Present Value of a Mixed Stream Copyright © 2006 Pearson Addison-Wesley. All rights reserved Frey Company, a shoe manufacturer, has been offered an opportunity to receive the following mixed stream of cash flows over the next 5 years.

Present Value of a Mixed Stream Copyright © 2006 Pearson Addison-Wesley. All rights reserved If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity? This situation is depicted on the following time line.

Present Value of a Mixed Stream Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-36

Compounding Interest More Frequently Than Annually Copyright © 2006 Pearson Addison-Wesley. All rights reserved Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently. As a result, the effective interest rate is greater than the nominal (annual) interest rate. Furthermore, the effective rate of interest will increase the more frequently interest is compounded.

Compounding Interest More Frequently Than Annually (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved Fred Moreno has found an institution that will pay him 8% annual interest, compounded quarterly. If he leaves the money in the account for 24 months (2 years), he will be paid 2% interest compounded over eight periods.

Compounding Interest More Frequently Than Annually (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-39

Compounding Interest More Frequently Than Annually (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-40

Compounding Interest More Frequently Than Annually (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved A General Equation for Compounding More Frequently than Annually

Compounding Interest More Frequently Than Annually (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved A General Equation for Compounding More Frequently than Annually  Recalculate the example for the Fred Moreno example assuming (1) semiannual compounding and (2) quarterly compounding.

Nominal & Effective Annual Rates of Interest Copyright © 2006 Pearson Addison-Wesley. All rights reserved The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. The effective interest rate is the rate actually paid or earned. In general, the effective rate > nominal rate whenever compounding occurs more than once per year EAR = (1 + i/m) m - 1

Nominal & Effective Annual Rates of Interest (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved Fred Moreno wishes to find the effective annual rate associated with an 8% nominal annual rate (I =.08) when interest is compounded (1) annually (m=1); (2) semiannually (m=2); and (3) quarterly (m=4).

Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-45

Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum (cont.) Copyright © 2006 Pearson Addison-Wesley. All rights reserved Suppose you want to buy a house 5 years from now and you estimate that the down payment needed will be $30,000. How much would you need to deposit at the end of each year for the next 5 years to accumulate $30,000 if you can earn 6% on your deposits? PMT = $30,000/5.637 = $5,321.98

Special Applications of Time Value: Loan Amortization Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-47