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**5 Time Value of Money Introduction to Finance Chapter**

Lawrence J. Gitman Jeff Madura Introduction to Finance Time Value of Money

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Learning Goals Discuss the role of time value in finance and the use of computational aids to simplify its application. Understand the concept of future value and its calculation for a single amount; understand the effects on future value and the true rate of interest of compounding more frequently than annually. Understand the concept of present value, its calculation for a single amount, and the relationship of present to future cash flow.

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Learning Goals Find the future value and present value of an ordinary annuity, the future value of an annuity due, and the present value of a perpetuity. Calculate the present value of a mixed stream of cash flows, describe the procedures involved in: Determining deposits to accumulate to a future sum Loan amortization Finding interest or growth rates

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**The Role of Time Value in Finance**

Most financial decisions involve costs and 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 in one year, or one that would return $500,000 after two years? Answer It depends on the interest rate!

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**Basic Concepts Future Value Present Value**

Compounding or growth over time Present Value Discounting to today’s value Single cash flows and series of cash flows can be considered Time lines are used to illustrate these relationships

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**Computational Aids Use the equations Use the financial tables**

Use financial calculators Use spreadsheets

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Computational Aids Figure 5.1

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Computational Aids Figure 5.2

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Computational Aids Figure 5.3

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Computational Aids Figure 5.4

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Simple Interest 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

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Compound Interest With compound interest, a depositor earns interest on interest! Year 1: 5% of $ = $ $ = $105.00 Year 2: 5% of $ = $ $ = $110.25 Year 3: 5% of $ = $ $ = $115.76 Year 4: 5% of $ = $ $ = $121.55 Year 5: 5% of $ = $ $ = $127.63

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**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)

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**Four Basic Models FVn = PV0(1+k)n = PV(FVIFk,n)**

PV0 = FVn[1/(1+k)n] = FV(PVIFk,n) FVAn = A (1+k)n - 1 = A(FVIFAk,n) k PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n)

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**Future Value Example Algebraically and Using FVIF Tables**

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 FVIF6%,5 $2,000 x = $2,676.40

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**Microsoft® Excel Function**

Future Value Example Using Microsoft® Excel You deposit $2,000 today at 6% interest. How much will you have in 5 years? Microsoft® Excel Function = FV(interest, periods, pmt, PV) = FV(.06, 5, , 2000)

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**A Graphic View of Future Value**

Figure 5.5

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**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.

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**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, and (d) monthly? Annually: 100 x ( )5 = $176.23 Semiannually: 100 x ( )10 = $179.09 Quarterly: 100 x ( )20 = $180.61 Monthly: 100 x ( )60 = $181.67

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**Compounding More Frequently Than Annually**

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**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 Continuing with the previous example, find the future value of the $100 deposit after 5 years if interest is compounded continuously. FVn = 100 x (2.7183).12x5 = $182.22

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**Nominal and 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 is greater than the nominal rate whenever compounding occurs more than once per year. EAR = (1 + k/m)m - 1

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**Nominal and 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 = ( /12)12 - 1 EAR = 19.56%

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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. It is also known as discounting, the reverse of compounding. The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, and the cost of capital.

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**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 = $1,494.52

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**Microsoft® Excel Function**

Present Value Example Using Microsoft® 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? Microsoft® Excel Function =PV(interest, periods, pmt, FV) =PV(.06, 5, , 2000)

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**A Graphic View of Present Value**

Figure 5.6

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**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.

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Annuities Table 5.1

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**Future Value of an Ordinary Annuity**

Using the FVIFA Tables An annuity is an 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 = $ $100 + $100 = $205.00 Year 3 $205 x .05 = $ $205 + $100 = $315.25

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**Future Value of an Ordinary Annuity**

Using Microsoft® Excel An annuity is an 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? Microsoft® Excel Function =FV(interest, periods, pmt, PV) =FV(.06,5,100, )

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**Future Value of an Annuity Due**

Using the FVIFA Tables An annuity is an 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) = $330.96

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**Future Value of an Annuity Due**

Using Microsoft® Excel An annuity is an 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. Microsoft® Excel Function =FV(interest, periods, pmt, PV) =FV(.06, 5,100, ) =315.25*(1.05)

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**Present Value of an Ordinary Annuity**

Using PVIFA Tables An annuity is an 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

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**Present Value of an Ordinary Annuity**

Using Microsoft® Excel An annuity is an 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? Microsoft® Excel Function =PV(interest, periods, pmt, FV) =PV(.10, 3, 2000, )

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**Present Value of a Mixed Stream**

Using Microsoft® 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%. Microsoft® Excel Function =NPV(interest, cells containing CFs) =NPV(.09,B3:B7)

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**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

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Loan Amortization Table 5.7

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**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 there are 7 years shown, there are only 6 time periods between the initial deposit and the final value.

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**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 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.

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**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 which grew as shown in the table below?

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**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 which grew as shown in the table below? Microsoft® Excel Function =Rate(periods, pmt, PV, FV) =Rate(6, ,1000, 5525)

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**Using Microsoft® Excel**

The Microsoft® Excel Spreadsheets used in the this presentation can be downloaded from the Introduction to Finance companion web site:

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**5 End of Chapter Introduction to Finance Chapter Lawrence J. Gitman**

Jeff Madura Introduction to Finance End of Chapter

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