Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. 09 The Time Value of Money Block, Hirt, and Danielsen Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Time value associated with money Determining future value at given interest rate Present value based on current value of funds to be received Determining yield on investment. Compounding or discounting occurring on less than annual basis Chapter Outline 9-2

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Time value of money used to determine whether future benefits sufficiently large to justify current outlays Mathematical tools of time value of money used in making capital allocation decisions Relationship to The Capital Outlay Decision 9-3

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Measuring value of amount allowed to grow at given interest over period of time Assuming worth of $1,000 needs to be calculated after 4 years at 10% interest per year 1 st year……$1,000 × 1.10 = $1,100 2 nd year.....$1,100 × 1.10 = $1,210 3 rd year……$1,210 × 1.10 = $1,331 4 th year……$1,331 × 1.10 = $1,464 Future Value – Single Amount 9-4

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value – Single Amount 9-5

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value of $1(FV IF ) 9-6

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value – Single Amount In determining future value, following can be used: FV = PV × FV IF Where FV IF = interest factor If $10,000 were invested for 10 years at 8%, future value would be: FV = PV × FV IF (n = 10, i = 8%) FV = $10,000 × = $21,

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Present Value – Single Amount 9-8

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Present Value of $1(PV IF ) 9-9

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Relationship of Present and Future Value 9-10

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Interest Rate – Single Amount 9-11

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Number of Periods – Single Amount 9-12

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value – Annuity Annuity A series of consecutive payments or receipts of equal amount Future value of an annuity Calculated by compounding each individual payment into the future and then adding up all of these payments 9-13

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value – Annuity 9-14

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Compounding Process for Annuity 9-15

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value of an Annuity of $1(FV IFA ) 9-16

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Present Value – Annuity 9-17

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Present Value of an Annuity of $1(PV IFA ) 9-18

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Comparisons include Relationship between present value and future value Inverse relationship exists between present value and future value of single amount Relationship between present value of single amount and present value of annuity Present value of annuity is sum of present values of single amounts payable at end of each period Relationship between future value and future value of annuity Future value of annuity is sum of future values of single amounts receivable at end of each period Time Value Relationships 9-19

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Review of variables involved in time value of money FV/PV — Future/present value of money N — Number of years I — Interest or yield A — Annuity value/payment per period in annuity Given first three variables and determining fourth variable, A (unknown ) Determining the Annuity Value 9-20

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Annuity Equaling a Future Value 9-21

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Annuity Equaling a Present Value 9-22

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Relationship of Present Value to Annuity Annual interest based on beginning balance for each year 9-23

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Loan Amortization Mortgage loan to be repaid over 20 years at 8% interest 9-24

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Loan Amortization Table Part of payments to mortgage company for interest payment, remainder applied to debt reduction 9-25

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Six Formulas 9-26

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Compounding frequency Certain contractual agreements may require semiannual, quarterly, or monthly compounding periods In such cases N = No. of years × No. of compounding periods during year I = Quoted annual interest / No. of compounding periods during year Compounding over Additional Periods 9-27

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Patterns of Payment Problems may evolve around number of different payment or receipt patterns Not every situation involves single amount or annuity Contract may call for payment of different amount each year over stated period or period of annuity Compounding over Additional Periods 9-28

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Compounding Frequency— Cases Case 1: Determine the future value of a $1,000 investment after 5 years at 8% annual interest compounded semiannually Where n = 5 × 2 = 10; i = 8%/2 = 4% FV = PV × (1 + i) n FV = $1,000 × (1.04) 10 = $1,

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Compounding Frequency— Cases 9-30

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Assume a contract involving payments of different amounts each year for a three-year period To determine the present value, each payment is discounted to the present and then totaled (Assuming 8% discount rate) Patterns of Payment with a Deferred Annuity 9-31

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Situations involving combination of single amounts and annuity When annuity is paid sometime in future Deferred Annuity 9-32

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Deferred Annuity Case Assuming a contract involving payments of different amounts each year for three year period Annuity of $1,000 paid at end of each year from fourth through eighth year To determine present value of cash flows at 8% discount rate: 9-33

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. To determine annuity: Deferred Annuity Case 9-34

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. To discount $3,993 back to present, which falls at beginning of fourth period, discount back three periods at 8% interest rate Deferred Annuity Case 9-35

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Deferred Annuity Case 9-36

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Alternate Method to Compute Deferred Annuity 1.Determine present value factor of annuity for total time period, where n = 8, i = 8%, PV IFA = Determine present value factor of annuity for total time period (8) minus deferred annuity period (5) 8 – 5 = 3; n = 3; i = 8% Value = Subtract the value of step 2 from the value of step 1, and multiply by A 9-37

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Alternate Method to Compute Deferred Annuity 4.$3,170 is same answer for present value of annuity as that reached by first method 5.Present value of five-year annuity is added to present value of inflows over first three years 9-38