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**Chapter 1 Appendix Time Value of Money: The Basics McGraw-Hill/Irwin**

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.

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**Time Value 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?” App 1-2

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**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

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**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 Excel Functions App 1-4

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**Basic TVM Definitions Future Value (FV) Present Value (PV)**

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

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**Basic TVM Definitions Payment (PMT or annuity) Sign Convention:**

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 It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts. App 1-6 6

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**Basic TVM Definitions Interest rate (i or I/Y)**

Stated as a percent per year Also called “discount rate” 12% = “0.12” in formulas & in Excel “12” in financial calculators It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts. App 1-7 7

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**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

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**Single Amount & Annuities**

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

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**Basic TVM Formulas Simple Interest: Principal x Rate x Time**

Future Value: Single Amount FV = PV(1 + i)n Annuity Present Value Single Amount App 1-10

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**TVM Calculator Solutions Texas Instruments BA-II Plus**

FV = future value PV = present value PMT = periodic payment I/Y = interest rate N = number of periods One of these MUST be negative I am providing information on the Texas Instruments BA-II Plus – other calculators are similar. If you recommend or require a specific calculator other than this one, you may want to make the appropriate changes. Note: the more information students have to remember to enter the more likely they are to make a mistake. For this reason, I normally tell my students to set P/Y = 1 and leave it that way. Then I teach them to work on a period basis, which is consistent with using the formulas. If you want them to use the P/Y function, remind them that they will need to set it every time they work a new problem and that CLR TVM does not affect P/Y. If students are having difficulty getting the correct answer, make sure they have done the following: Set decimal places to floating point (2nd Format, Dec = 9 enter) or show 4 to 5 decimal places if using and HP Double check and make sure P/Y = 1 Make sure to clear the TVM registers after finishing a problem (or before starting a problem) It is important to point out that CLR TVM clears the FV, PV, N, I/Y and PMT registers. C/CE and CLR Work DO NOT affect the TVM keys The remaining slides will work the problems using the notation provided above for calculator keys. The formulas are presented in the notes section. N I/Y PV PMT FV App 1-11 11

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**Texas Instruments BA-II Plus**

I/Y = period interest rate (i) P/Y must = 1 Interest is entered as a percent, not a decimal 5% interest = “5”, not “0.05” Clear the registers before each problem [2nd] [CLR TVM] Or reenter each field I am providing information on the Texas Instruments BA-II Plus – other calculators are similar. If you recommend or require a specific calculator other than this one, you may want to make the appropriate changes. Note: the more information students have to remember to enter the more likely they are to make a mistake. For this reason, I normally tell my students to set P/Y = 1 and leave it that way. Then I teach them to work on a period basis, which is consistent with using the formulas. If you want them to use the P/Y function, remind them that they will need to set it every time they work a new problem and that CLR TVM does not affect P/Y. If students are having difficulty getting the correct answer, make sure they have done the following: Set decimal places to floating point (2nd Format, Dec = 9 enter) or show 4 to 5 decimal places if using and HP Double check and make sure P/Y = 1 Make sure to clear the TVM registers after finishing a problem (or before starting a problem) It is important to point out that CLR TVM clears the FV, PV, N, I/Y and PMT registers. C/CE and CLR Work DO NOT affect the TVM keys The remaining slides will work the problems using the notation provided above for calculator keys. The formulas are presented in the notes section. App 1-12 12

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**TVM with Excel Spreadsheet Functions**

=FV(Rate,Nper,Pmt,PV) =PV(Rate,Nper,Pmt,FV) =RATE(Nper,Pmt,PV,FV) =NPER(Rate,Pmt,PV,FV) =PMT(Rate,Nper,PV,FV) Use the formula icon (ƒx) when you can’t remember the exact formula Click on the tabs at the bottom of the worksheet to move between examples. 13

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**Time Value of Money Interest Rate Basics**

Calculating interest earned: Principal = dollar amount of savings Annual rate of interest Length of time money on deposit (in years) Simple interest: Annual Interest Rate Amt in Svgs Time Period X = X Interest App 1-14

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**Interest Rate Basics Example A**

You borrow $1,000 at 5% annual interest for 1 year: Principal = $1,000 Interest rate = 5% = .05 Time period = 1 $1,000 X .05 1 = $50 App 1-15

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**Interest Rate Basics Example B**

You deposit $750 at 8% per year for 9 months: Principal = $750 Interest rate = 8% Time period = 9/12 = .75 $750 X .08 0.75 = $45 App 1-16

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**Interest Rate Basics Example B – Calculator***

Principal = $750 Interest rate = 8% Time period = 9/12 = .75 Calculator Solution Keystrokes .75 N 8 I/Y -750 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 1-17

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**Interest Rate Basics Example B – Calculator**

Calculator Solution .75 N 8 I/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 1-18

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**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 Excel Function: FV(Rate,Nper,Pmt,PV ) App 1-19

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**Suppose you invest $1 for 3 years at 10% How much would you have?**

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 = 1.331 FV = PV(Factor) FV = 1(1.331) FV = 1.331 It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-20 20

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**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 = 1.331 Excel Function: =FV(.10,3,0,-1) =1.331 It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-21 21

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**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 TVM Tables Solution: Exhibit 1-A Periods = 18 Rate = 1% Factor = 1.196 FV = 400(1.196) FV = Formula Solution: FV =PV(1+i)n =400(1.01)18 =400(1.196) =478.46 It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-22 22

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**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 = Excel Function: =FV(.01,18,0,-400) =478.46 It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-23 23

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**Future Value of a Series of Equal Amounts**

“Annuity” = 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 Excel Function: FV(Rate,Nper,Pmt,PV) App 1-24

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**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? TVM Tables Solution: Exhibit 1-B Periods = 3 Rate = 10% Factor = 3.31 FV = 1(3.31) FV = 3.31 Formula Solution: It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-25 25

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*** Note that the PMT value is negative since it is an outflow/deposit.**

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 Excel Function: =FV(.10,3,-1,0) =3.31 It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = * Note that the PMT value is negative since it is an outflow/deposit. App 1-26 26

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**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? TVM Tables Solution: Exhibit 1-B Periods = 10 Rate = 8% Factor = FV = 40(14.487) FV = Formula Solution: It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-27 27

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**Future Value of a Series of Equal Amounts Calculator Solution Example F**

8 I/Y 0 PV -40 PMT CPT FV = Excel Function: =FV(.08,10,-40,0) =579.46 It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-28 28

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**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 Point out that the PV interest factor = 1 / (1 + r)t App 1-29 29

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**Present Value of a Single Amount**

Formula Solution: Table Solution: Calculator Solution: N I/Y PMT FV CPT PV Excel Function: =PV(Rate,Nper,Pmt,FV) App 1-30

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**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? TVM Tables Solution: Exhibit 1-C Periods = 3 Rate = 10% Factor = .751 PV = FV*(Factor) PV = 1*(0.751) PV = 0.751 Formula Solution: PV =FV/(1+i)n =1/(1.10)3 =1*(.7513) =0.7513 It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before. Formula: FV = 1000(1.05)5 = 1000( ) = App 1-31 31

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**Present Value of a Single Amount Example G**

Formula Solution: PV =FV/(1+i)n =1/(1.10)3 =1*(0.7513) =0.7513 TVM Tables Solution: Exhibit 1-C Periods = 3 (down left column) Rate = 10% (across top) Factor = .751 PV = FV(Factor) PV = 1(0.751) PV = 0.751 Calculator Solution 3 N 10 I/Y CPT PV = 0 PMT 1 FV Excel Function: =PV(.10,3,0,1) = -0.75 App 1-32

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**Present Value of a Single Amount Example H**

You want to have $300 seven years from now. Your savings earns 10% compounded semiannually. How much must you deposit today? Formula Solution: PV =FV/(1+i)n =300/(1.05)14 =300/(1.9799) =151.52 TVM Tables Solution: Exhibit 1-C Periods = 14 (down left column) Rate = 5% (across top) Factor = .505 PV = FV(Factor) PV = 300 x (0.505) PV = $151.50 Calculator Solution 14 N 5 I/Y CPT PV = 0 PMT 300 FV Excel Function: =PV(.05,14,0,300) = App 1-33

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**Present Value of a Series of Equal Amounts**

Annuity Table Factors = Exhibit 1-D Formula Solution: Table Solution: Calculator Solution: N I/Y PMT FV CPT PV Excel Function: =PV(Rate,Nper,Pmt,FV) App 1-34

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**Present Value of an Annuity Example I**

You wish to withdraw $1 at the end of each of the next 3 years. (= 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 The students can read the example in the book. After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow? Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book. Exhibit 1-D: Row 3, column 10% Factor = 2.487 PV = PMT*(Factor) = 1*(2.487) PV = $2.49 Excel Function: =PV(.10,3,1,0) = -2.49 App 1-35 35

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**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 The students can read the example in the book. After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow? Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book. Exhibit 1-D: Factor = 5.216 PV = PMT*(Factor) = 100*(5.216) PV = $521.60 Excel Function: =PV(.14,10,100,0) = App 1-36 36

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**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 Excel Function: =PMT(.06,3,1000,0) = App 1-37 37

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