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## Chapter 4 Time Value of Money

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**Chapter 4**Time Value of Money**Learning Goals**1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. • Understand the concept of future value and present value, their calculation for a single amounts, and the relationship of present value to future value. • Find the future value and the present value of both an ordinary annuity and an annuity due, and the present value of a perpetuity.**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.**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?**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!**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**Computational Aids**• Use the Equations • Use the Financial Tables • Use Financial Calculators • Use Spreadsheets**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. • Spreadsheets enhance learning. • Spreadsheets communicate as well as calculate.**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:**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**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**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)**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) k**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 1.3382 = $2,676.40**Future Value Example**Using Excel You deposit $2,000 today at 6% interest. How much will you have in 5 years? Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5, , 2000)**Future Value Example**A Graphic View of Future Value**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.**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**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.**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 = $182.22**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**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%**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.**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**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? Excel Function =PV (interest, periods, pmt, FV) =PV (.06, 5, , 2000)**Present Value Example**A Graphic View of Present Value**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.**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**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. Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, )**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) = $330.96**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. Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, ) =315.25*(1.05)**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**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? Excel Function =PV (interest, periods, pmt, FV) =PV (.10, 3, 2000, )**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%.**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%. Excel Function =NPV (interest, cells containing CFs) =NPV (.09,B3:B7)**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**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.**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:**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?**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)