Fundamentals of Corporate Finance Third Canadian Edition prepared

Chapter 4 The Time Value of Money n Future Values and Compound Interest n

Introduction w As a financial manager you will often have to compare cash payments

Future Value w Future value is the amount to which an investment will grow

Future Value n. Simple Interest w Interest is earned only on the original investment.

Future Value n. Simple Interest w You earn interest only on the original investment

Future Value n Simple Interest Today Starting balance Year 1 Year 2 Year 3

Future Value n Simple Interest Today Starting balance Interest earned Ending balance Year 1

Future Value n Compound Interest w Interest is earned on the interest. w Example:

Future Value n. Compound Interest w You earn interest on interest w Interest earned

Future Value n Compound Interest Today Starting balance Interest earned Ending balance Year 1

Future Value n Compound Interest w Value at the end of Year 1 =

Future Value n Future Value w In general, value at the end of year

Future Value n Future Value w Example: You invest $100 in an account paying

Future Value Interest rates Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -16

Future Value Interest rates Note: As r increases, FV increases As t increases, FV

Future Value w Future values are higher because of the interest earned. w This

Present Value n Let’s turn things around… w How much do you need to

Present Value Today Year 1 6% Year 2 6% Year 3 6% Year 4

Present Value n Present Value w In general, present value of a future cash

Present Value n Present value w Example: You have been offered $1 million five

Present Value Today Year 1 10% Year 2 10% Year 3 10% Year 4

Present Value vs Future Value n PV and FV are related! w The formula

Present Value vs Future Value n PV and FV are related! FV = PV

Multiple Cash Flows n Future Value w Example: You deposit $1, 200 in your

Multiple Cash Flows 0 1 8% $1, 200 2 8% 8% $1, 400 3

Multiple Cash Flows 0 1 2 8% $1, 200 8% 8% $1, 400 3

Multiple Cash Flows n Present Value w Example: Your auto dealer gives you the

Multiple Cash Flows n Problem definition w Option 1: $15, 500 today w Option

Multiple Cash Flows n Present value of Option 2 0 1 8% 8% $8,

Multiple Cash Flows n Compare the two options at the same point in time

Perpetuities and Annuities n Definitions w Annuity: Equally spaced and a level stream of

Perpetuities and Annuities n Perpetuities w The PV of a perpetuity is calculated by

Perpetuities and Annuities n Perpetuities w Example: In order to create an endowment, which

Perpetuities and Annuities n Perpetuities 0 1 10% PV 2 10% $100, 000 PV

Perpetuities and Annuities n Perpetuities w Example contd. : If the first perpetuity payment

Perpetuities and Annuities n Perpetuities 0 1 2 4 … 10% 5 PV at

Perpetuities and Annuities n Annuities w The PV of a t-period annuity is given

Perpetuities and Annuities n Annuities w Example: You are purchasing a car. You are

Perpetuities and Annuities 0 1 10% 2 10% $4, 000 3 $4, 000 PV

Perpetuities and Annuities n Growing Perpetuities 0 1 r% PV 2 r% C PV

Perpetuities and Annuities n Growing Annuities 0 1 r% PV 2 r% C 3

Inflation n Definitions w Inflation: Rate at which prices as a whole are increasing.

Inflation n Formulae w The exact formula: 1+ real interest rate = 1+ nominal

Inflation w Example: If the interest rate on one year government bonds is 5.

Effective Annual Interest Rates n Definitions w Effective Annual Interest Rate [EAR]: Interest rate

Effective Annual Interest Rates n Formulae where m = number of compounding periods in

Effective Annual Interest Rates w Example: Given a monthly rate of 1%, what is

Summary of Chapter 4 w Future value (FV) is the amount to which an

Summary of Chapter 4 w A level stream of payments which continues forever is

Summary of Chapter 4 w Annual percentage rates (APR) do not recognize the effect

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Chapter 4 The Time Value of Money n Future Values and Compound Interest n Present Values n Multiple Cash Flows n Perpetuities and Annuities n Inflation and Time Value of Money n Effective Annual Interest Rates Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -2 2

Introduction w As a financial manager you will often have to compare cash payments which occur at different dates. w To make optimal decisions, you must understand the relationship between a dollar today [Present value] and a dollar in the future [Future value]. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -3 3

Future Value w Future value is the amount to which an investment will grow after earning interest. w Interest can be of two types: § Simple interest § Compound interest Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -4 4

Future Value n. Simple Interest w Interest is earned only on the original investment. w Example: You invest $100 in an account paying simple interest at the rate of 6% per year. How much will the account be worth in 5 years? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -5 5

Future Value n. Simple Interest w You earn interest only on the original investment w Interest earned per year = $100 x 6% = $6. 00 w Total interest earned over 5 -year period = $6. 00 x 5 = $30. 00 w Balance in account at end of Year 5 = $100 + $30 = $130 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -6 6

Future Value n Simple Interest Today Starting balance Interest earned Ending balance Year 1 Year 2 Year 3 Year 4 Year 5 $100 $6 $6 $6 $106 $112 $118 $124 $130 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -9 9

Future Value n Compound Interest w Interest is earned on the interest. w Example: You invest $100 in an account paying compound interest at the rate of 6% per year. How much will the account be worth in 5 years? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -10 10

Future Value n. Compound Interest w You earn interest on interest w Interest earned per year = Previous year’s balance x interest rate w Interest Interest w w earned earned in in in Year Year 1 2 3 4 5 = = = $100. 00 $106. 00 $112. 36 $119. 10 $126. 25 x x x Copyright © 2006 Mc. Graw Hill Ryerson Limited 6% 6% 6% = = = $6. 00 $6. 36 $6. 74 $7. 15 $7. 57 2 -11 11

Future Value n Compound Interest Today Starting balance Interest earned Ending balance Year 1 Year 2 Year 3 Year 4 Year 5 $7. 15 $7. 57 $126. 25 $133. 82 $100 $6 $106 $6. 36 $6. 74 $112. 36 $119. 10 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -12 12

Future Value n Compound Interest w Value at the end of Year 1 = $100. 00 +[$100 x 6%] = $100 x (1+r) w Value at the end of Year 2 = $106. 00 + [$106 x 6%] = $100 x (1+r)2 ……. Value at the end of Year 5 = $100 x (1+r)5 w w Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -13 13

Future Value n Future Value w Example: You invest $100 in an account paying compound interest at the rate of 6% per year. How much will the account be worth in 5 years? FV = $100 x (1+r)t = $100 x (1+0. 06)5 = $133. 82 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -15 15

Future Value w Future values are higher because of the interest earned. w This leads to a basic financial principle: § A dollar received today is worth more than a dollar received tomorrow. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -18 18

Present Value n Let’s turn things around… w How much do you need to invest today into an account paying compound interest at the rate of 6% per year, in order to receive $133. 82 at the end of five years? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -19 19

Present Value Today Year 1 6% Year 2 6% Year 3 6% Year 4 6% Year 5 6% $133. 82 ? Present value = $133. 82 (1+ 0. 06)5 = $100 Simply invert the FV formula to get the PV formula! Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -22 22

Present Value n Present value w Example: You have been offered $1 million five years from now. If the interest rates is expected to be 10% per year, how much is this prize worth to you in today’s dollars? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -24 24

Present Value vs Future Value n PV and FV are related! w The formula for PV is simply the formula for FV inverted! PV at 10% $620, 921 $1, 000 FV at 10% Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -26 26

Present Value vs Future Value n PV and FV are related! FV = PV x (1 + r)t PV = FV x 1/(1 + r)t = $620, 921 x (1 + 0. 10)5 = $1 million x 1/ (1 + 0. 10)5 = $1 million = $620, 921 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -27 27

Multiple Cash Flows n Future Value w Example: You deposit $1, 200 in your bank account today; $1, 400 one year later; and $1, 000 two years from today. If your bank offers you an 8% interest rate on your account, how much money will you have in the account three years from today? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -28 28

Multiple Cash Flows 0 1 2 8% $1, 200 8% 8% $1, 400 3 $1, 000 x(1+0. 08)1=$1, 080. 00 $1, 400 x(1+0. 08)2=$1, 632. 96 $1, 200 x(1+0. 08)3=$1, 511. 65 FV = $4, 224. 61 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -30 30

Multiple Cash Flows n Present Value w Example: Your auto dealer gives you the choice to pay $15, 500 cash now, or make three payments: $8, 000 now and $4, 000 at the end of the following two years. If your cost of money is 8%, which do you prefer? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -31 31

Multiple Cash Flows n Problem definition w Option 1: $15, 500 today w Option 2: $8, 000 today; $4, 000 at the end of one year; and $4, 000 at the end of two years Cash flows can be compared only at the same point in time. Thus, we need to find the PV of Option 2. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -32 32

Multiple Cash Flows n Present value of Option 2 0 1 8% 8% $8, 000 2 $4, 000/(1+0. 08)1=$3, 703. 7 0 $4, 000/(1+0. 08)2=$3, 429. 3 6 PV = $15, 133. 06 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -34 34

Multiple Cash Flows n Compare the two options at the same point in time w PV of Option 1: $15, 500. 00 w PV of Option 2: $15, 133. 06 Option 1 is better for the buyer. Option 2 is better for the auto dealer. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -35 35

Perpetuities and Annuities n Definitions w Annuity: Equally spaced and a level stream of cash flows [for a finite number of periods] w Perpetuity: Stream of level cash payments that never ends [for an infinite number of periods] Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -36 36

Perpetuities and Annuities n Perpetuities w The PV of a perpetuity is calculated by dividing the level cash flow by the interest rate 0 1 r% PV 2 r% C PV of a perpetuity = 3 r% C ……. r% C ∞ r% C C C r Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -37 37

Perpetuities and Annuities n Perpetuities w The PV of a perpetuity is calculated by dividing the level cash flow by the interest rate 0 1 r% PV 2 r% C PV of a perpetuity = r% C C r 3 ……. r% C ∞ r% C C Note: This formula gives you the present value of a perpetuity starting one period from now Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -38 38

Perpetuities and Annuities n Perpetuities w Example: In order to create an endowment, which pays $100, 000 per year, forever, how much money must be set aside today if the rate of interest is 10%? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -39 39

Perpetuities and Annuities n Perpetuities w Example contd. : If the first perpetuity payment will not be received until four years from today, how much money needs to be set aside today? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -41 41

Perpetuities and Annuities n Perpetuities 0 1 2 4 … 10% 5 PV at end of Year 3 = …. $100, 000 0. 10 $1, 000 (1. 10)3 ∞ 10% $100, 000 PV PV today = 3 $100, 000 = $1, 000 = $751, 315 Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -42 42

Perpetuities and Annuities n Annuities w The PV of a t-period annuity is given by: 0 1 r% PV 2 r% C 3 r% C ……. r% C t r% C C PV of t-period annuity = Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -43 43

Perpetuities and Annuities n Annuities w The PV of a t-period annuity is given by: 0 1 r% PV 2 r% C 3 r% C ……. r% C t r% C C PV of t-period annuity = Note: This formula gives you the present value of an annuity starting one period from now Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -44 44

Perpetuities and Annuities n Annuities w The PV of a t-period annuity is given by: PV of t-period annuity = Note: The term in the square brackets is called the present value annuity factor or PVAF. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -45 45

Perpetuities and Annuities n Annuities w The PV of a t-period annuity is given by: PV of t-period annuity = Note: The term in the square brackets is called the present value annuity factor or PVAF can also be computed by using the annuity tables at the end of the book [Appendix A. 3]. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -46 46

Perpetuities and Annuities n Annuities w Example: You are purchasing a car. You are scheduled to make 3 annual installments of $4, 000 per year, with the first payment one year from now. Given a rate of interest of 10%, what is the price you are paying for the car? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -47 47

Inflation n Definitions w Inflation: Rate at which prices as a whole are increasing. w Nominal interest rate: Rate at which money invested grows. w Real interest rate: Rate at which the purchasing power of an investment increases. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -51 51

Inflation n Formulae w The exact formula: 1+ real interest rate = 1+ nominal interest rate 1 + inflation rate w The approximation: Real interest rate ≈ nominal interest rate – inflation rate Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -52 52

Inflation w Example: If the interest rate on one year government bonds is 5. 0% and the inflation rate is 2. 2%, what is the real interest rate? Real interest rate = [(1+ 0. 05)/(1+0. 022)]-1 = 1. 027 -1 = 2. 7% Real interest rate ≈ 5% - 2. 2% = 2. 8% Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -53 53

Effective Annual Interest Rates n Definitions w Effective Annual Interest Rate [EAR]: Interest rate that is annualized using compound interest w Annual Percentage Rate [APR]: Interest rate that is annualized using simple interest Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -54 54

Effective Annual Interest Rates w Example: Given a monthly rate of 1%, what is the Effective Annual Rate [EAR]? What is the Annual Percentage Rate [APR]? Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -56 56

Effective Annual Interest Rates w Example: Given a monthly rate of 1%, what is the Effective Annual Rate [EAR]? What is the Annual Percentage Rate [APR]? APR = 1% × 12 = 12% Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -57 57

Summary of Chapter 4 w Future value (FV) is the amount to which an investment will grow after earning interest. w The present value (PV) of a future cash payment is the amount you would need to invest today to create that future cash payment. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -58 58

Summary of Chapter 4 w A level stream of payments which continues forever is called a perpetuity. w One which continues for a limited number of years is called an annuity. w You can use the FV and PV formulas to calculate their value or you can use the shortcut formulae. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -59 59

Summary of Chapter 4 w Annual percentage rates (APR) do not recognize the effect of compound interest, that is, they annualize assuming simple interest. w Effective Annual Rates (EAR) annualize using compound interest. Copyright © 2006 Mc. Graw Hill Ryerson Limited 2 -60 60