Copyright © 2003 Pearson Education, Inc. Slide 4-0 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 Answer! It depends on the interest rate! 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.

Copyright © 2003 Pearson Education, Inc. Slide 4-5 Basic Concepts 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

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 – don’t 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 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 With simple interest, you don’t earn interest on interest.

Copyright © 2003 Pearson Education, Inc. Slide 4-14 Compound 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= $ With compound interest, a depositor earns interest on interest!

Copyright © 2003 Pearson Education, Inc. Slide 4-15 Time Value Terms PV 0 =present value or beginning amount k= 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)

Copyright © 2003 Pearson Education, Inc. Slide 4-16 Equations: Four Basic Models FV n = PV 0 (1+k) n = PV(FVIF k,n ) PV 0 = FV n [1/(1+k) n ] = FV(PVIF k,n ) FVA n = A (1+k) n - 1= A(FVIFA k,n ) k PVA 0 = A 1 - [1/(1+k) n ] = A(PVIFA k,n ) k

Copyright © 2003 Pearson Education, Inc. Slide 4-17 Future Value Example You deposit $2,000 today at 6% interest. How much will you have in 5 years? $2,000 x (1.06) 5 = $2,000 x FVIF 6%,5 $2,000 x =$2, Equation (Algebraically) and Using FVIF Tables Using equation:Using tabled value:

Copyright © 2003 Pearson Education, Inc. Slide 4-18 Future Value Example You deposit $2,000 today at 6% interest. How much will you have in 5 years? Using Excel Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,, 2000)

Copyright © 2003 Pearson Education, Inc. 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 $100 for 5 years and earn 12% annual interest compounded (a) annually, (b) semiannually, (c) quarterly, an (d) monthly? Annually: 100 x (1 +.12) 5 =$ Semiannually:100 x (1 +.06) 10 = $ Quarterly:100 x (1 +.03) 20 =$ Monthly:100 x (1 +.01) 60 =$181.67

Copyright © 2003 Pearson Education, Inc. Slide 4-22 Compounding More Frequently than Annually

Copyright © 2003 Pearson Education, Inc. 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: FV n (continuous compounding) = PV x (e kxn ) where “e” has a value of 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: FV n (continuous compounding) = PV x (e kxn ) where “e” has a value of FVn = 100 x (2.7183).12x5 = $182.22

Copyright © 2003 Pearson Education, Inc. 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) 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 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 PVIF 6%,5 $2,000 x = $1, Algebraically and Using PVIF Tables

Copyright © 2003 Pearson Education, Inc. Slide 4-29 Present Value Example How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit? Excel Function =PV (interest, periods, pmt, FV) =PV (.06, 5,, 2000) Using Excel

Copyright © 2003 Pearson Education, Inc. 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 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) = $ Year 1 $100 deposited at end of year=$ Year 2 $100 x.05 = $ $100 + $100 =$ Year 3 $205 x.05 = $ $205 + $100=$ Using the FVIFA Tables

Copyright © 2003 Pearson Education, Inc. Slide 4-34 Future Value of an Ordinary Annuity 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. Using Excel Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, )

Copyright © 2003 Pearson Education, Inc. Slide 4-35 Future Value of an Annuity Due 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) = $ Using the FVIFA Tables FVA = 100(3.152)(1.05) = $330.96

Copyright © 2003 Pearson Education, Inc. Slide 4-36 Future Value of an Annuity Due 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. Using Excel Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, ) =315.25*(1.05)

Copyright © 2003 Pearson Education, Inc. Slide 4-37 Present Value of an Ordinary Annuity 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, Using PVIFA Tables

Copyright © 2003 Pearson Education, Inc. Slide 4-38 Present Value of an Ordinary Annuity 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? Using Excel Excel Function =PV (interest, periods, pmt, FV) =PV (.10, 3, 2000, )

Copyright © 2003 Pearson Education, Inc. Slide 4-39 Present Value of a Mixed Stream 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%. Using Tables

Copyright © 2003 Pearson Education, Inc. Slide 4-40 Present Value of a Mixed Stream 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%. Using EXCEL Excel Function =NPV (interest, cells containing CFs) =NPV (.09,B3:B7)

Copyright © 2003 Pearson Education, Inc. 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? It is important to note that although 7 years show, there are only 6 time periods between the initial deposit and the final value.

Copyright © 2003 Pearson Education, Inc. 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? Thus, $1,000 is the present value, $5,525 is the future value, and 6 is the number of periods. Using Excel, we get:

Copyright © 2003 Pearson Education, Inc. 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?

Copyright © 2003 Pearson Education, Inc. 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? Excel Function =Rate(periods, pmt, PV, FV) =Rate(6,,1000, 5525)