Chapter 1 Appendix Time Value of Money: The Basics Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Time Value of Money = Interest Interest = the cost of money Answers the questions: –“If I deposit $10,000 today, how much will I have for a down payment on a house in 5 years?” –“Will $2,000 saved each year give me enough money when I retire?” –“How much must I save today to have enough for my children’s education?” Time Value of Money App 1-2

Time Value of Money Basic Principles A dollar received today is worth more than a dollar received a year from today A dollar that will be received in the future is worth less than a dollar today Why? –A dollar today could be saved or invested –A dollar in the future is uncertain App 1-3

Time Value of Money Definitions Solving TVM Problems –Types of Problems Interest rate basics - Simple interest Future value - Single amount & Annuity Present value - Single amount & Annuity Calculating Loan payments –Solutions Methods Formulas TVM Tables Financial Calculator App 1-4

TVM Major Components Future Value (FV) –The increased value of money from interest earned –The amount to which a current sum will grow given a certain interest rate and time period –“Compounding” Present Value (PV) –The current value of a future amount given a certain interest rate and time period –“Discounting” App 1-5

Basic TVM Definitions Payment (PMT or annuity) –Amount of annuity deposit or withdrawal Sign Convention: –Applies to PV, PMT and FV –Positive = inflow to YOU Money received as a loan is an inflow –Negative = outflow from YOU Deposit to an account is an outflow App 1-6

Basic TVM Definitions Interest rate (i or I/Y) –Stated as a percent per year –Also called “discount rate” –12% = “0.12” in formulas “12” in financial calculators App 1-7

Basic TVM Definitions Time Periods (n or t) –Expressed in years 3 months = “0.25” years 2 ½ years = “2.5” years –Interest rate and time period must match Annual periods  annual rate Monthly periods  monthly rate App 1-8

Single Amount & Annuities Single Amount: –A single payment made or received at one time –Calculator: PMT=0 Annuity: –Finite series of equal payments that occur at regular intervals –PMT key used –Sign convention is important App 1-9

Basic TVM Formulas Simple Interest:Principal x Rate x Time Future Value: Single Amount FV = PV(1 + i) n Annuity Present Value Single Amount Annuity App 1-10

TVM Calculator Solutions Texas Instruments BA-II Plus FV = future value PV = present value PMT = periodic payment I/Y = interest rate (i) N = number of periods One of these MUST be negative N I/Y PV PMT FV App 1-11

Texas Instruments BA-II Plus I/Y = period interest rate (i) –P/Y must equal 1 –Interest is entered as a percent, not a decimal 5% interest = “5”, not “0.05” Clear the registers before each problem –[2 nd ] [CLR TVM] Or re-enter each field App 1-12

Calculating interest earned: –Principal = dollar amount of savings –Annual rate of interest –Length of time money on deposit (in years) Simple interest: Time Value of Money Interest Rate Basics Amt in Svgs X Annual Interest Rate Time Period Interest X = App 1-13

You borrow $1,000 at 5% annual interest for 1 year: Principal = $1,000 Interest rate = 5% =.05 Time period = 1 Interest Rate Basics Example A $50 $1,000 X.051 X = App 1-14

You deposit $750 at 8% per year for 9 months: Principal = $750 Interest rate = 8% Time period = 9/12 =.75 Interest Rate Basics Example B $45 $750 X X = App 1-15

Principal = $750 Interest rate = 8% Time period = 9/12 =.75 Interest Rate Basics Example B – Calculator* Calculator Solution:.75N 8I/Y -750 PV (enter “750” then “ S ” then “PV”) 0 PMT CPT FV = – 750 = ≈ 45 * Calculator solutions match the TI Business Analyst II+. Keystroke adjustments may need to be made for other financial calculators App 1-16

Interest Rate Basics Example B – Calculator Calculator Solution.75N 8I/Y -750 PV ** 0 PMT CPT FV = – 750 = ≈ 45 ** Remember: when using a financial calculator, either PV or FV must be negative. Outflows (from you) are negative Inflows (to you) are positive Depositing money in an account is an outflow App 1-17

Future Value of a Single Amount Amount to which current savings will increase = Original amount + compounded interest = Compounding Formula Solution: Table Solution: Calculator Solution : N I/Y PV PMT CPT FV App 1-18

Future Value of a Single Amount Formula & TVM Table Solutions Example C Suppose you invest $1 for 3 years at 10% How much would you have? Formula Solution: FV=PV(1+ i ) n =1(1.10) 3 =1(1.331) =1.331 TVM Tables Solution: Exhibit 1-A Periods = 3 Rate = 10% Factor = FV = PV(Factor) FV = 1(1.331) FV = App 1-19

Future Value of a Single Amount Calculator Solution Example C Suppose you invest $1 for 3 years at 10%. How much would you have? Calculator Solution 3 N 10 I/Y -1 PV 0 PMT CPT FV = App 1-20

Future Value of a Single Amount Example C App 1-21

Future Value of a Single Amount Formula & TVM Tables Example D Your savings of $400 earns 12% compounded monthly (=1% per month) How much would you have after 18 months? Table Hint: Use 1% and 18 periods Formula Solution: FV=PV(1+ i ) n =400(1.01) 18 =400(1.196) = TVM Tables Solution: Appendix Exhibit 1-A Periods = 18 Rate = 1% Factor = FV = 400(1.196) FV = App 1-22

Future Value of a Single Amount Calculator Solution Example D Suppose you invest $400 for 18 months at 12% compounded monthly. How much would you have? Calculator Solution 18 N 1 I/Y -400 PV 0 PMT CPT FV = App 1-23

Future Value of a Single Amount Example D App 1-24

Future Value of a Series of Equal Amounts “Annuity” = a series of equal deposits at equal intervals earning a constant rate –Equal annuity deposit amounts = PMT Formula Solution: Table Solution: Calculator Solution: N I/Y PV PMT CPT FV App 1-25

Future Value of a Series of Equal Amounts Formula & TVM Tables Example E What is the future value of three $1 deposits made at the end of the next three years, earning 10% interest? Formula Solution: TVM Tables Solution: Appendix Exhibit 1-B Periods = 3 Rate = 10% Factor = 3.31 FV = 1(3.31) FV = 3.31 App 1-26

Future Value of a Series of Equal Amounts Calculator Solution Example E Calculator Solution 3 N 10 I/Y 0 PV -1 PMT* CPT FV = 3.31 * Note that the PMT value is negative since it is an outflow/deposit. App 1-27

Future Value of a Series of Equal Amounts Example E App 1-28

Future Value of a Series of Equal Amounts Formula & TVM Tables Example F What is the future value of ten $40 deposits earning 8% compounded annually? Formula Solution: TVM Tables Solution: Appendix Exhibit 1-B Periods = 10 Rate = 8% Factor = FV = 40(14.487) FV = App 1-29

Future Value of a Series of Equal Amounts Calculator Solution Example F Calculator Solution 10 N 8 I/Y 0 PV -40 PMT CPT FV = App 1-30

Future Value of a Series of Equal Amounts Example F App 1-31

Present Value Single Amount - Basic Equation FV = PV(1 + i) n Rearrange to solve for PV “Discounting” = finding the present value of one or more future amounts App 1-32

Present Value of a Single Amount Formula Solution: Table Solution: Calculator Solution:N I/Y PMT FV CPT PV App 1-33

Present Value of a Single Amount Formula & TVM Tables Example Example G What is the present value of $1 to be received in 3 years at a 10% interest rate? Formula Solution: PV=FV/(1+ i ) n =1/(1.10) 3 =1*(.7513) = TVM Tables Solution: Appendix Exhibit 1-C Periods = 3 Rate = 10% Factor =.751 PV = FV*(Factor) PV = 1*(0.751) PV = App 1-34

Present Value of a Single Amount Example G Formula Solution: PV=FV/(1+i) n =1/(1.10)3 =1*(0.7513) = TVM Tables Solution: Appendix Exhibit 1-C Periods = 3 (down left column) Rate = 10% (across top) Factor =.751 PV = FV(Factor) PV = 1(0.751) PV = Calculator Solution 3 N 10 I/Y CPTPV = PMT 1 FV App 1-35

Present Value of a Single Amount Example H Formula Solution: PV=FV/(1+i) n =300/(1.05) 14 =300/(1.9799) = TVM Tables Solution: Appendix Exhibit 1-C Periods = 14 (down left column) Rate = 5% (across top) Factor =.505 PV = FV(Factor) PV = 300 x (0.505) PV = $ Calculator Solution 14 N 5 I/Y CPTPV = PMT 300FV You want to have $300 seven years from now. Your savings earns 10% compounded semiannually. How much must you deposit today? App 1-36

Present Value of a Single Amount App 1-37

Present Value of a Series of Equal Amounts Formula Solution: Table Solution: Calculator Solution:N I/Y PMT FV CPT PV Annuity Table Factors = Appendix Exhibit 1-D App 1-38

Present Value of an Annuity Example I You wish to withdraw $1 at the end of each of the next 3 years. (Note: this is an inflow) The account earns 10% compounded annually. How much do you need to deposit today to be able to make these withdrawals? 3 N; 10 I/Y; 1 PMT; CPT PV = FV 0 Exhibit 1-D: Row 3, column 10% Factor = PV = PMT*(Factor) = 1*(2.487) PV = $2.49 App 1-39

Present Value of an Annuity Example J You wish to withdraw $100 at the end of each of the next 10 years. (Inflow) The account earns 14% compounded annually. How much do you need to deposit today to be able to make these withdrawals? 10 N; 14 I/Y; 100 PMT; CPT PV = FV 0 Exhibit 1-D: Factor = PV = PMT*(Factor) = 100*(5.216) PV = $ App 1-40

Present Value of an Annuity App 1-41

Using Present Value to Determine Loan Payments Example K If you borrow $1,000 with a 6% interest rate to be repaid in three equal payments at the end of the next three years, what will the annual payment be? Table Solution: Calculator Solution 3 N; 6 I/Y; CPT PMT = PV = 1000 FV 0 App 1-42

Using Present Value to Determine Loan Payments Example K App 1-43