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Principles of Managerial Finance 9th Edition Chapter 5 Time Value of Money

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Learning Objectives Discuss the role of time value in finance and the use of computational aids used to simplify its application. Understand the concept of future value, its calculation for a single amount, and the effects of compounding interest more frequently than annually. Find the future value of an ordinary annuity and an annuity due and compare these two types of annuities. Understand the concept of present value, its calculation for a single amount, and its relationship to future value.

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Learning Objectives Calculate the present value of a mixed stream of cash flows, an annuity, a mixed stream with an embedded annuity, and a perpetuity. Describe the procedures involved in: –determining deposits to accumulate a future sum, –loan amortization, and –finding interest or growth rates

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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 in one year, or one that would return $500,000 after two years?

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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.

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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

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Computational Aids Use the Equations Use the Financial Tables Use Financial Calculators Use Spreadsheets

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Computational Aids

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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.

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Compound 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 With compound interest, a depositor earns interest on interest!

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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)

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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 FV: future value PV: present value IF: interest factor A: annuity

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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 1.3382 =$2,676.40 Algebraically and Using FVIF Tables

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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)

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Future Value Example A Graphic View of Future Value

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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. 有效利率

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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, 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 FV n =PV 0 ×(1+k/m) m×n

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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: FV n (continuous compounding) = PV x (e kxn ) 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.

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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: FV n (continuous compounding) = PV x (e kxn ) where “e” has a value of 2.7183. FVn = 100 x (2.7183).12x5 = $182.22

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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

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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%

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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 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 0.74758 = $1,494.52 Algebraically and Using PVIF Tables

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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

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Present Value Example A Graphic View of Present Value

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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. The future value of an annuity due will always be greater than the future value of an otherwise equivalent ordinary annuity because interest will compound for an additional period.

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Annuities

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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) = $315.25 Using the FVIFA Tables 0 1 23 100 100X1.05=105 100X(1.05) 2 =110.25

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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 (.05, 3,100, )

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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) = $330.96 Using the FVIFA Tables FVA = 100(3.152)(1.05) = $330.96

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100 100 100 100*1.05=105 100*(1.05) 2 =110.25 100*(1.05) 3 =115.76 100 100 100

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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 (.05, 3,100, )x(1.05) =315.25*(1.05)

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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) = $4,973.70 Using PVIFA Tables 2000 2000÷1.1 2000÷(1.1) 2 2000÷(1.1) 3

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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, )

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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

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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)

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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 … 1000 ………………

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Loan Amortization 6000=AxPVIFA 10%,4 6000=Ax3.170 ∴ A=6000÷3.170=1892.74

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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 first important to note that although there are 7 years show, 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 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:

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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?

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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? Excel Function =Rate(periods, pmt, PV, FV) =Rate(6,,1000, 5525)

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