Chapter 4 Time Value of Money Copyright 2003 Pearson Education, Inc. Slide 4-1 Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. 2. Understand the concept of future value and present value, their calculation for a single amounts, and the relationship of present value to future value. 3. Find the future value and the present value of both an ordinary annuity and an annuity due, and the present value of a perpetuity. Copyright 2003 Pearson Education, Inc.

Slide 4-2 Learning Goals 4. Calculate both the future value and the present value of a mixed stream of cash flows. 5. Understand the effect that compounding interest more frequently than annually has on future value and the effective annual rate of interest. 6. Describe the procedures involved in (1) determining deposits to accumulate to a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods. Copyright 2003 Pearson Education, Inc. Slide 4-3 The Role of Time Value in Finance 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? Copyright 2003 Pearson Education, Inc. Slide 4-4 The Role of Time Value in Finance 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!

Copyright 2003 Pearson Education, Inc. Slide 4-5 Basic Concepts Future Value: compounding or growth over time Present Value: discounting to todays value Single cash flows & series of cash flows can be considered Time lines are used to illustrate these relationships Copyright 2003 Pearson Education, Inc. Slide 4-6 Computational Aids Use the Equations Use the Financial Tables

Use Financial Calculators Use Spreadsheets Copyright 2003 Pearson Education, Inc. Slide 4-7 Computational Aids Copyright 2003 Pearson Education, Inc. Slide 4-8 Computational Aids Copyright 2003 Pearson Education, Inc. Slide 4-9

Computational Aids (not used) Copyright 2003 Pearson Education, Inc. Slide 4-10 Calculator Keys Texas Instruments (HP dont need to hit CPT) Copyright 2003 Pearson Education, Inc. Slide 4-11 Advantages of Computers & Spreadsheets Spreadsheets go far beyond the computational abilities of calculators. Spreadsheets have the ability to program logical decisions.

Spreadsheets display not only the calculated values of solutions but also the input conditions on which solutions are based. Spreadsheets encourage teamwork (but not for your assignments at Lee!) Spreadsheets enhance learning. Spreadsheets communicate as well as calculate. Copyright 2003 Pearson Education, Inc. Slide 4-12 Basic Patterns of Cash Flow 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: Copyright 2003 Pearson Education, Inc.

Slide 4-13 Simple Interest With simple interest, you dont 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 Copyright 2003 Pearson Education, Inc. Slide 4-14 Compound Interest With compound interest, a depositor earns interest on interest! Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00 Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25 Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76 Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55 Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

Copyright 2003 Pearson Education, Inc. Slide 4-15 Time Value Terms PV0 = present value or beginning amount k interest rate = FVn = future value at end of n periods

n = number of compounding periods A = an annuity (series of equal payments or receipts) Copyright 2003 Pearson Education, Inc. Slide 4-16 Equations: Four Basic Models FVn

= PV0(1+k)n = PV(FVIFk,n) PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)

A (1+k)n - 1 k = A(FVIFAk,n) PVA0 = A 1 - [1/(1+k)n] = k A(PVIFAk,n) FVAn = Copyright 2003 Pearson Education, Inc. Slide 4-17 Future Value Example

Equation (Algebraically) and Using FVIF Tables You deposit $2,000 today at 6% interest. How much will you have in 5 years? Using equation: Using tabled value: $2,000 x (1.06)5 = $2,000 x FVIF6%,5 $2,000 x 1.3382 = $2,676.40 Copyright 2003 Pearson Education, Inc. Slide 4-18 Future Value Example Using Excel You deposit $2,000 today at 6%

interest. How much will you have in 5 years? PV k n FV? $ Copyright 2003 Pearson Education, Inc. 2,000 6.00% 5 $2,676 Excel Function =FV (interest, periods, pmt, PV)

=FV (.06, 5, , 2000) Slide 4-19 Future Value Example A Graphic View of Future Value Copyright 2003 Pearson Education, Inc. Slide 4-20 Compounding More Frequently than Annually 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. Copyright 2003 Pearson Education, Inc. Slide 4-21 Compounding More Frequently than Annually For example, what would be the difference in future value if I deposit $100 for 5 years and earn 12% annual interest compounded (a) annually, (b) semiannually, (c) quarterly, an (d) monthly? Annually: 100 x (1 + .12)5 = $176.23 Semiannually:

100 x (1 + .06)10 = $179.09 Quarterly: 100 x (1 + .03)20 = $180.61 Monthly: 100 x (1 + .01)60 = $181.67 Copyright 2003 Pearson Education, Inc. Slide 4-22

Compounding More Frequently than Annually On Excel Annually PV $ Sem iAnnually Quarterly 100.00 k 12.0% n

5 FV $176.23 Copyright 2003 Pearson Education, Inc. $ 100.00 0.06 10 $179.08 $ 100.00

Monthly $ 100.00 0.03 0.01 20 60 $180.61 $181.67 Slide 4-23

Continuous Compounding With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes: FVn (continuous compounding) = PV x (ekxn) where e has a value of 2.7183. Continuing with the previous example, find the Future value of the $100 deposit after 5 years if interest is compounded continuously. Copyright 2003 Pearson Education, Inc. Slide 4-24 Continuous Compounding With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes:

FVn (continuous compounding) = PV x (ekxn) where e has a value of 2.7183. FVn = 100 x (2.7183).12x5 Copyright 2003 Pearson Education, Inc. = $182.22 Slide 4-25 Nominal & Effective Rates 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 + k/m) m -1 Copyright 2003 Pearson Education, Inc.

Slide 4-26 Nominal & Effective Rates For example, what is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly? EAR = (1 + .18/12) 12 -1 EAR = 19.56% Copyright 2003 Pearson Education, Inc. Slide 4-27 Present Value 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. Copyright 2003 Pearson Education, Inc. Slide 4-28 Present Value Example Algebraically and Using PVIF Tables How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit? $2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5

$2,000 x 0.74758 = $1,494.52 Copyright 2003 Pearson Education, Inc. Slide 4-29 Present Value Example Using Excel How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit? FV k n PV? $ Copyright 2003 Pearson Education, Inc.

2,000 6.00% 5 $1,495 Excel Function =PV (interest, periods, pmt, FV) =PV (.06, 5, , 2000) Slide 4-30 Present Value Example A Graphic View of Present Value Copyright 2003 Pearson Education, Inc. Slide 4-31

Annuities 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. Copyright 2003 Pearson Education, Inc. Slide 4-32 Annuities Copyright 2003 Pearson Education, Inc.

Slide 4-33 Future Value of an Ordinary Annuity Using the FVIFA Tables Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years. FVA = 100(FVIFA,5%,3) = $315.25 Year 1 $100 deposited at end of year = $100.00 Year 2 $100 x .05 = $5.00 + $100 + $100 =

$205.00 Year 3 $205 x .05 = $10.25 + $205 + $100 = $315.25 Copyright 2003 Pearson Education, Inc. Slide 4-34 Future Value of an Ordinary Annuity Using Excel Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years. PMT k

n FV? $ 100 5.0% 3 $ 315.25 Copyright 2003 Pearson Education, Inc. Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, ) Slide 4-35 Future Value of an Annuity Due Using the FVIFA Tables

Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years. FVA = 100(FVIFA,5%,3)(1+k) = $330.96 FVA = 100(3.152)(1.05) = Copyright 2003 Pearson Education, Inc. $330.96 Slide 4-36 Future Value of an Annuity Due Using Excel Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years.

PMT $ 100.00 k 5.00% n 3 FV $315.25 FVA? $ 331.01 Copyright 2003 Pearson Education, Inc. Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, ) =315.25*(1.05) Slide 4-37 Present Value of an Ordinary Annuity Using PVIFA Tables

Annuity = Equal Annual Series of Cash Flows Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest? PVA = 2,000(PVIFA,10%,3) = $4,973.70 Copyright 2003 Pearson Education, Inc. Slide 4-38 Present Value of an Ordinary Annuity Using Excel Annuity = Equal Annual Series of Cash Flows Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest?

PMT I n PV? $ 2,000 10.0% 3 $4,973.70 Copyright 2003 Pearson Education, Inc. Excel Function =PV (interest, periods, pmt, FV) =PV (.10, 3, 2000, ) Slide 4-39

Present Value of a Mixed Stream Using Tables A mixed stream of cash flows reflects no particular pattern Find the present value of the following mixed stream assuming a required return of 9%. Year Cash Flow PVIF9%,N PV 1 400 0.917 $ 366.80

2 800 0.842 $ 673.60 3 500 0.772 $ 386.00 4

400 0.708 $ 283.20 5 300 0.650 $ 195.00 PV $1,904.60 Copyright 2003 Pearson Education, Inc.

Slide 4-40 Present Value of a Mixed Stream Using EXCEL A mixed stream of cash flows reflects no particular pattern Find the present value of the following mixed stream assuming a required return of 9%. Year Cash Flow 1 400 2 800 3

500 4 400 5 300 NPV $1,904.76 Copyright 2003 Pearson Education, Inc. Excel Function =NPV (interest, cells containing CFs)

=NPV (.09,B3:B7) Slide 4-41 Future Value of a Mixed Stream Copyright 2003 Pearson Education, Inc. Slide 4-42 Future Value of a Mixed Stream Copyright 2003 Pearson Education, Inc. Slide 4-43 Present Value of a Perpetuity A perpetuity is a special kind of annuity. With a perpetuity, the periodic annuity or cash flow stream continues forever.

PV = Annuity/k 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 = $1,000/.08 = $12,500 Copyright 2003 Pearson Education, Inc. Slide 4-44 Loan Amortization Copyright 2003 Pearson Education, Inc. Slide 4-45 Determining Interest or Growth Rates At times, it may be desirable to determine the compound interest rate or growth rate implied by a

series of cash flows. For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below? 1994 $ 1,000 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525 Copyright 2003 Pearson Education, Inc. It is important to note

that although 7 years show, there are only 6 time periods between the initial deposit and the final value. Slide 4-46 Determining Interest or Growth Rates At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows. For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below? 1994 $ 1,000 1995 1,127 1996 1,158 1997 2,345

1998 3,985 1999 4,677 2000 5,525 Copyright 2003 Pearson Education, Inc. Thus, $1,000 is the present value, $5,525 is the future value, and 6 is the number of periods. Using Excel, we get: Slide 4-47 Determining Interest or Growth Rates At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows.

For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below; at what compound interest rate? 1994 $ 1,000 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525 Copyright 2003 Pearson Education, Inc. PV FV

n k? $ $ 1,000 5,525 6 33.0% Slide 4-48 Determining Interest or Growth Rates At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows. For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below; at what compound annual growth rate? 1994 $ 1,000

1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525 Copyright 2003 Pearson Education, Inc. Excel Function =Rate(periods, pmt, PV, FV) =Rate(6, ,1000, 5525) Slide 4-49