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Future value Present value Rates of return Amortization CHAPTER 2 Time Value of Money

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Introduction In fact, of all the concepts used in finance, none is more important than the time value of money, which is also called discounted cash flow (DCF) analysis. PV : present value, or beginning amount, in your account i : interest rate INT : dollars of interest you earn FV : future value n : number of periods involved in the analysis

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Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 0123 i% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

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Time line for a $100 lump sum due at the end of Year 2. 100 012 Year i%

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Time line for an ordinary annuity of $100 for 3 years. 100 0123 i%

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Time line for uneven CFs: -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 100 50 75 0123 i% -50

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What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? 0123 10% Finding FVs (moving to the right on a time line) is called compounding. 100

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After 1 year: FV 1 = PV + INT 1 = PV + PV (i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years: FV 2 = PV(1 + i) 2 = $100(1.10) 2 = $121.00.

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After 3 years: FV 3 = PV(1 + i) 3 = $100(1.10) 3 = $133.10. In general, FV n = PV(1 + i) n.

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Three Ways to Find FVs Solve the equation with a regular calculator. Use a spreadsheet.

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10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 0123 PV = ?

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Solve FV n = PV(1 + i ) n for PV: PV= $100 1 1.10 = $1000.7513 = $75.13. 3

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Finding the Time to Double 20% 2 012? FV= PV(1 + i) n $2= $1(1 + 0.20) n (1.2) n = $2/$1 = 2 nLN(1.2)= LN(2) n= LN(2)/LN(1.2) n= 0.693/0.182 = 3.8.

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8 - 14 Copyright © 2002 by Harcourt, Inc.All rights reserved. Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due? PVFV

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What’s the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

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ordinary annuity

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8 - 17 Copyright © 2002 by Harcourt, Inc.All rights reserved. FV Annuity Formula The future value of an annuity with n periods and an interest rate of i can be found with the following formula:

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What’s the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.69 = PV

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8 - 19 Copyright © 2002 by Harcourt, Inc.All rights reserved. PV Annuity Formula The present value of an annuity with n periods and an interest rate of i can be found with the following formula:

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Special Function for Annuities For ordinary annuities, this formula in cell A3 gives 248.96: =PV(10%,3,-100) A similar function gives the future value of 331.00: =FV(10%,3,-100)

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Find the FV and PV if the annuity were an annuity due. 100 0123 10% 100

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8 - 22 Copyright © 2002 by Harcourt, Inc.All rights reserved. PV and FV of Annuity Due vs. Ordinary Annuity PV of annuity due: = (PV of ordinary annuity) (1+i) = (248.69) (1+ 0.10) = 273.56 FV of annuity due: = (FV of ordinary annuity) (1+i) = (331.00) (1+ 0.10) = 364.1

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8 - 23 Copyright © 2002 by Harcourt, Inc.All rights reserved. annuity due

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Excel Function for Annuities Due Change the formula to: =PV(10%,3,-100,0,1) The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due: =FV(10%,3,-100,0,1)

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8 - 25 Copyright © 2002 by Harcourt, Inc.All rights reserved. Uneven Cash Flow Streams We will use Payment (PMT) for annuity situations where the cash flows are equal amounts, and we will use the term Cash flow (CF) to denote uneven cash flows.

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What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

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8 - 27 Copyright © 2002 by Harcourt, Inc.All rights reserved. How to find PV of this uneven cash 1- We could find the PV of each individual cash flow using the numerical. 2- using NPV in excel.

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8 - 28 Copyright © 2002 by Harcourt, Inc.All rights reserved. Spreadsheet Solution Excel Formula in cell A3: =NPV(10%,B2:E2) ABCDE 101234 2100300300-50 3530.09

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HOME WORK Find the future value of the following annuities. The first payment in these annuities is made at the end of Year 1; a. $400 per year for 10 years at 10 percent. b. $200 per year for 5 years at 5 percent. c. $400 per year for 5 years at 0 percent. d. Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

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HOME WORK Find the present value of the following ordinary annuities: a. $400 per year for 10 years at 10 percent. b. $200 per year for 5 years at 5 percent. c. $400 per year for 5 years at 0 percent. d. Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

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