Chapter 3 Time Value of Money 3 1

Chapter 3 Time Value of Money 3 -1

After studying Chapter 3, you should be able to: 1. 2. 3. 4. 5. 6. 7. 3 -28. Understand what is meant by "the time value of money. " Understand the relationship between present and future value. Describe how the interest rate can be used to adjust the value of cash flows – both forward and backward – to a single point in time. Calculate both the future and present value of: (a) an amount invested today; (b) a stream of equal cash flows (an annuity); and (c) a stream of mixed cash flows. Distinguish between an “ordinary annuity” and an “annuity due. ” Use interest factor tables and understand how they provide a shortcut to calculating present and future values. Use interest factor tables to find an unknown interest rate or growth rate when the number of time periods and future and present values are known. Build an “amortization schedule” for an installment-style loan.

The Time Value of Money u The Interest Rate u Simple Interest u Compound Interest u Stream of Cash Flows u 3 -3 u u Single Amounts u Annuities u Mixed Flows Compounding More than once a year u Semi-annual and other compounding periods u Continuous Compounding u Effective Annual Interest Rate Amortizing a Loan

The Interest Rate Which would you prefer -- $1, 000 today or today $1, 000 in 10 years? $1, 000 in 10 years Obviously, $1, 000 today You already recognize that there is TIME VALUE TO MONEY!! MONEY 3 -4 Note = If all cash flows are certain, the rate of interest can be used to adjust the value of money

Why TIME? Why is TIME such an important TIME element in your decision? TIME allows you the opportunity to TIME postpone consumption and earn INTEREST 3 -5

Types of Interest u Simple Interest paid (earned) on only the original amount, or principal, borrowed (lent). u Compound Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). 3 -6

Simple Interest Formula 3 -7 SI = P 0(i)(n) SI: Simple Interest P 0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods

Simple Interest Example u Assume that you deposit $100 in an account earning 8% simple interest for 10 years. What is the accumulated interest at the end of the 2 nd year? u. SI 3 -8 = P 0(i)(n) = $100(. 08)(10) = $80

Simple Interest (FV) u What is the Future Value ( Future Value FV) of the FV deposit? FV = P 0 + SI = $100 + $80 = $180 u Future Value is the value at some future 3 -9 time of a present amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (PV) u What is the Present Value ( Present Value PV) of the PV previous problem? The Present Value is simply the $100 you originally deposited. That is the value today! u Present Value is the current value of a 3 -10 future amount of money, or a series of payments, evaluated at a given interest rate.

Why Compound Interest? u Future value of $1 investment for various time periods at an 8% annual interest rate Years At SI 2 $1. 16 $1. 17 20 $2. 60 $4. 66 200 $17. 00 3 -11 At Compound Interest $4, 838, 949. 59

Future Value Single Deposit (Graphic) Assume that you deposit $100 at a $100 compound interest rate of 8% for 2 years 0 8% 1 2 $1, 000 FV 2 3 -12

Future Value Single Deposit (Formula) FV 1 = P 0 (1+i)1 = $1, 000 (1. 08) = $1, 080 Compound Interest You earned $80 interest on your $1, 000 deposit over the first year. This is the same amount of interest you would earn under simple interest. 3 -13

Future Value Single Deposit (Formula) FV 1 = P 0 (1+i)1 FV 2 FV 3 3 -14 = $100 (1. 08) $100 = $108 = P 0 (1+i) = $100(1. 08) $100 2 = P 0 (1+i)2 = $100(1. 08) $100 = $1, 16. 64 = P 0 (1+i)(1+i) = $100(1. 08)( 1. 08) $100 3 = P 0 (1+i)3 = $100(1. 08) 100 = $125. 97

General Future Value Formula FV 1 = P 0(1+i)1 FV 2 = P 0(1+i)2 FV 3 = P 0(1+i)3 etc. General Future Value Formula: Future Value FVn = P 0 (1+i)n or FVn = P 0 (FVIFi, n) -- See Table I 3 -15

Valuation Using Table I FVIFi, n is found on Table I at the end of the book. 3 -16

Using Future Value Tables FV 2 3 -17 = $1, 000 (FVIF 7%, 2) = $1, 000 (1. 145) = $1, 145 [Due to Rounding]

Story Problem Example A wants to know how large her deposit of $10, 000 today will become at a compound annual $10, 000 interest rate of 10% for 5 years 0 10% 1 2 3 4 5 $10, 000 FV 5 3 -18

Story Problem Solution u Calculation based on general formula: FVn = P 0 (1+i)n FV 5 = $10, 000 (1+ 0. 10)5 = $16, 105. 10 u Calculation based on Table I: FV 5 = $10, 000 (FVIF 10%, 5) = $10, 000 (1. 611) = $16, 110 [Due to Rounding] 3 -19

Present Value Single Deposit (Graphic) Assume that you need $1, 000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 8% compounded annually. 0 8% 1 2 $1, 000 PV 0 3 -20 PV 1

Present Value Single Deposit (Formula) PV 0 = FV 2 / (1+i)2 0 8% = $1, 000 / (1. 08)2 = $857. 34 1 2 $1, 000 PV 0 3 -21

General Present Value Formula PV 0 = FV 1 / (1+i)1 PV 0 = FV 2 / (1+i)2 etc. General Present Value Formula: Present Value PV 0 = FVn / (1+i)n or PV 0 = FVn (PVIFi, n) -- See Table II 3 -22

Valuation Using Table II PVIFi, n is found on Table II at the end of the book. 3 -23

Using Present Value Tables PV 2 3 -24 = $1, 000 (PVIF $1, 000 7%, 2) = $1, 000 (. 873) $1, 000 = $873 [Due to Rounding]

Story Problem Example A wants to know how large of a deposit to make so that the money will grow to $10, 000 in $10, 000 5 years at a discount rate of 5 years 10%. 0 10% 1 2 3 4 5 $10, 000 PV 0 3 -25

Story Problem Solution u Calculation based on general formula: PV 0 = FVn / (1+i)n 5 PV 0 = $10, 000 / (1+ 0. 10) $10, 000 = $6, 209. 21 u Calculation based on Table I: PV 0 = $10, 000 ( $10, 000 PVIF 10%, 5) = $10, 000 (. 621) $10, 000 = $6, 210. 00 [Due to Rounding] 3 -26

Unknown Interest Rate If the future value, present value and number of times are known but interest rate is unknown: Investment Today =$1, 000 Future Value after 8 years =$3, 000 u Calculation based on general formula: FVn = PV 0 (FVIFi, n) FV 8 = PV 0 (FVIFi, 8) $3, 000 = $1, 000 (FVIF $1, 000 i, 8) FVIF i, 8 = $3, 000/ $1, 000 = 3 Note: How we can find this value from table…. . ? ? PTO 3 -27

Table value for unknown Interest rate u. We will go to Table I in row 8 and will find where 3 exists. u. The near value is 3. 059 in row 8 and column 15% but this is approximate value. We can get this rate through solving equation. 3 -28

Unknown Interest Rate through solving equation 3 -29 FVn = PV 0 (1+i)n $3, 000 (1+i)8 (1+i) i 8 = $1, 000 (1+i) $1, 000 = $3, 000/ $1, 000 = 3 = 31/8 = 1. 1472 – 1 =. 1472 or 14. 72%

Unknown Number of Compounding Periods If the future value, present value and interest rate are known but number of times is unknown: Investment Today =$1, 000 Future Value with 10% =$1, 900 u Calculation based on general formula: FVn = PV 0 (FVIFi, n) FVn = PV 0 (FVIF. 10, n) $1, 900 = $1, 000 (FVIF $1, 000 . 10, n) FVIF. 10, n = $1, 900/ $1, 000 = 1. 9 Note: How we can find this value from table…. . ? ? 3 -30

Table value for unknown Interest rate u We will go to Table I in Column 10% and will find where 1. 9 falls in column. u The near value is 1. 949 and it falls in 7 years row but this is approximate value. We can get this rate through solving equation. 3 -31

Unknown Number of Years through solving equation 3 -32 FVn = PV 0 (1+i)n $1, 900 (1+. 10)n n(ln 1. 10) n n n = $1, 000 (1+. 10) $1, 000 = $1, 900/ $1, 000 = 1. 90 = (ln 1. 90)/(ln 1. 10) = 6. 73 years

Double Your Money!!! Quick! How long does it take to double $5, 000 at a compound rate of 12% per year (approx. )? We will use the “Rule-of-72”. 3 -33

The “Rule-of-72” 1. 2. How long does it take to double $5, 000 at a compound rate of 12% per year (approx. )? What rate of interest is required to double $5, 000 in 6 years? Formula to find N 1. N = 72 / i% Formula to find i% 2. i% = 72 / N 3 -34

Solution The “Rule-of-72” 1. 2. 3 -35 How long does it take to double $5, 000 at a compound rate of 12% per year (approx. )? What rate of interest is required to double $5, 000 in 6 years? Formula to find N 1. N = 72 / 12 = 6 year Formula to find i% 2. i% = 72 / 6 = 12 Or 12%

Types of Annuities u An Annuity represents a series of equal Annuity payments (or receipts) occurring over a specified number of equidistant periods. u Ordinary Annuity: Payments or receipts Ordinary Annuity occur at the end of each period. u Annuity Due: Payments or receipts Annuity Due occur at the beginning of each period. 3 -36

Examples of Annuities u Student Loan Payments u Car Loan Payments u Insurance Premiums u Mortgage Payments u Retirement Savings 3 -37

Ordinary Annuity (Ordinary Annuity) End of End Period 1 End of End Period 2 End of End Period 3 0 1 2 3 $100 Today 3 -38 Equal Cash Flows Equal Each 1 Period Apart

Annuity Due (Annuity Due) Beginning of Beginning Period 1 Beginning of Beginning Period 2 Beginning of Beginning Period 3 0 1 2 3 $100 Today 3 -39 Equal Cash Flows Equal Each 1 Period Apart

Hint on Annuity Valuation The future value of an ordinary annuity can be viewed as occurring at the end of the last end cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash beginning flow period. 3 -40

Overview of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 n+1 n i%. . . R R R = Periodic Cash Flow FVAn = R(1+i)n-1 + R(1+i)n-2 + . . . + R(1+i)1 + R(1+i)0 3 -41 FVAn

Book Data for an Ordinary Annuity – FVA Cash flows occur at the end of the period 0 1 2 3 4 3 8% $1, 000 $1, 080 $1, 166 FVA 3 = $1, 000(1. 08)2 + $1, 000(1. 08)1 + $1, 000(1. 08)0 $3, 246 = FVA 3 = $1, 166 + $1, 080 + $1, 000 = $3, 246 3 -42

Formula for FVA FVIFAn 3 -43 = (1+i)n - 1 i FVIFA 3 = (1+. 08)3 - 1 = 3. 246 . 08 FVA = R (1+i)n - 1 i n FVA 3 = $1000 x 3. 246 = $3, 246

Valuation Using Table III FVAn = R (FVIFAi%, n) FVA 3 = $1, 000 (FVIFA 8%, 3) = $1, 000 (3. 246) = $3, 264 3 -44

Overview of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 n+1 n i%. . . R R R = Periodic Cash Flow PVAn = R/(1+i)1 + R/(1+i)2 +. . . + R/(1+i)n 3 -45

Example of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 3 4 3 8% $934. 58 $873. 44 $816. 30 $1, 000 $2, 624. 32 = PVA 3 = $1, 000/(1. 08)1 + $1, 000/(1. 08)2 + $1, 000/(1. 08)3 = $925. 93 + $857. 34 + $793. 84 3 -46 = $2, 577. 11

Formula for PVA PVIFAn = 1 -1/(1+i)n i PVIFA 3 = 1 -1/(1+0. 08)3 = 2. 577 PVA n = R 1 -1/(1+i)n I 3 = $1000 x 2. 577 = $2, 577 PVA 3 -47 . 08

Valuation Using Table IV PVAn = R (PVIFAi%, n) PVA 3 = $1, 000 (PVIFA 8%, 3) = $1, 000 (2. 577) = $2, 577 3 -48

Unknown Interest Rate If the future value of annuity, amount of annuity and number of times are known but interest rate is unknown: Amount of Annuity =$1, 000 Future Value after 8 years =$9, 500 u Calculation based on general formula: FVAn = R (FVIFAi, n) FVA 8 = R (FVIFAi, 8) $9, 500 = $1, 000 (FVIFA $1, 000 i, 8) FVIFA i, 8 = $9, 500/ $1, 000 = 9. 5 Note: How we can find this value from table…. . ? ? PTO 3 -49

Table value for unknown Interest rate u. We will go to Table III in row 8 and will find where 9. 5 exists. u. The near value is 9. 549 in row 8 and column 5% but this is approximate value. We can get accurate answer through interpolation. 3 -50

Unknown Amount of Annuity If the future value annuity, interest rate and number of periods are given but amount of Annuity is unknown. Future value of annuity =$10, 000 Rate of interest = 5% Number of Year = 8 u Calculation based on general formula: FVAn = R (FVIFA i, n) FVA 8 = R (FVIFA. 05, 8) $10, 000 = R (FVIFA R . 05, 8) R = $10, 000/ 9. 549 = $1047. 23 3 -51

Perpetuity An ordinary annuity whose payments or receipts continue for ever. PVA∞ 3 -52 = R 1 -1/(1+i)∞ i = R/I

Example of Perpetuity If $100 received each year forever with rate of interest of 8%. PVA∞ = R 1 -1/(1+i)∞ i PVA∞ = R/I PVA∞ = $100/. 08 = $1, 250 3 -53

Overview View of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 n-1 n i%. . . R R FVADn = R(1+i)n + R(1+i)n-1 + . . . + R(1+i)2 + R(1+i)1 = FVA (1+i) n 3 -54 FVADn

Example of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 4 3 8% $1, 000 $1, 080 $1, 166 $1, 260 FVAD 3 = $1, 000(1. 08)3 + $3, 506 = FVAD 3 2 1 $1, 000(1. 08) + $1, 000(1. 08) = $1, 260 + $1, 166+ $1, 080 = $3, 506 3 -55

Formula for FVAD FVIFADn = FVAD 3 -56 (1+i)n - 1 x(1+i) i = R (1+i)n - 1 x(1+i) n i FVIFAD 3 = (1+. 08)3 - 1 x 1. 08= 3. 506 . 08 FVAD 3 $1, 000 x 3. 506 = $3, 506 =

Valuation Using Table III FVADn = R (FVIFAi%, n)(1+i) FVAD 3 = $1, 000 (FVIFA 8%, 3)(1. 08) = $1, 000 (3. 246)(1. 08) = $3, 506 3 -57

Overview of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 n-1 n i%. . . R R PVADn R: Periodic Cash Flow PVADn = R/(1+i)0 + R/(1+i)1 +. . . + R/(1+i)n-1 = PVA (1+i) n 3 -58

Example of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 3 4 8% $1, 000. 00 $1, 000 $ 925. 93 $ 857. 33 $2, 783. 26 = $2, 783. 26 PVADn = $1, 000/(1. 08)0 + $1, 000/(1. 08)1 + $1, 000/(1. 08)2 = $2, 783. 27 3 -59

Formula for PVAD PVIFADn = PVAD n = PVIFAD 3 = PVAD 3 -60 3 1 -1/(1+i)n x(1+i) i R 1 -1/(1+i)n x(1+i) i 1 -1/(1+0. 08)3 x(1+. 08) = 2. 783 . 08 = $1, 000 x 2. 783 = $2, 783

Valuation Using Table IV PVADn = R (PVIFAi%, n)(1+i) PVAD 3 = $1, 000 (PVIFA 8%, 3)(1. 08) = $1, 000 (2. 577)(1. 08) = $2, 783 3 -61

Mixed Flows A stream of cash flow which are uneven/does not flow any pattern of cash flow 3 -62

Mixed Flows Example What is the present value of $5, 000 to be received annually at the end year 1 and 2, followed by $6, 000 annually at the year 3 and 4 and concluding $1, 000 at the end of year 5, all discounted at 5 percent? 0 2 3 4 5 5% $5, 000 $6, 000 $1, 000 PV 0 3 -63 1

How to Solve? 1. Solve a “piece-at-a-time” by piece-at-a-time discounting each piece back to t=0. piece 2. Solve a “group-at-a-time” by first group-at-a-time breaking problem into groups of annuity streams and any single cash flow groups. Then discount each group back to t=0. group 3 -64

“Piece-At-A-Time” 0 1 2 3 4 5% $5, 000 $6, 000 $1, 000 $4, 760 $4, 535 $5, 184 $4, 938 $ 784 3 -65 5 $20, 201 = PV 0 of the Mixed Flow

“Group-At-A-Time” 0 1 2 3 4 $6, 000 $21, 272 0 Minus $1, 859 Plus $784 3 -66 0 1 2 PV 0 equals $21, 272 $1, 000 1 2 3 4 5 $1, 000

Frequency of Compounding General Formula: FVn = PV 0(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn, m: FV at the end of Year n PV 0: PV of the Cash Flow today 3 -67

Impact of Frequency A has deposited $100 in a saving account for one $100 at an annual interest rate of 8%. Assume semiannual compounding: 1. Future Value at the end of six months 2. Future Value at the end of the year At the end of six months (2)(. 5) FV 0. 5 = $100(1+ [. 08/2]) 100 = $100 x (1. 04)1 = $104 (2)(1) FV 1 = $100(1+ [. 08/2]) 100 = $100 x 1. 0816 = $108. 16 3 -68

Impact of Frequency A has deposited $100 in a saving account for one year at an annual interest rate of 8%. Assume quarterly, monthly and daily compounding. (4)(1) Qrtly FV 1 = $100(1+ [. 08/4]) 100 = $100 x 1. 0824 = $108. 24 (12)(1) Monthly FV 1 = $100(1+ [. 08/12]) 100 = $100 x 1. 0830 = $108. 30 Daily 3 -69 FV 1 = $100(1+ 100 [. 08/365])(365)(1) = $100 x 1. 0833 = $108. 33

Impact of Frequency A has deposited $100 in a saving account at an annual $100 interest rate of 8%. Future value at the end of three years assume quarterly, semi annual and annual compounding. (4)(3) Qrtly FV 3 = $100(1+ [. 08/4]) 100 = $100 x 1. 2682 = $126. 82 (2)(3) Semi Ann. FV 3 = $100(1+ [. 08/2]) 100 = $100 x 1. 2653 = $126. 53 Annual FV 3 = $100(1+ 100. 08)(3) = $100 x 1. 2597 = $125. 97 3 -70

Impact of Frequency The present value of $100 at the end of three years with a discounting rate of 8%. 1) Present Value if discounted quarterly 2) Present Value if discounted annually (4)(3) 1) PV 0 = $100/(1+ [. 08/4]) 100 = $100/1. 2682 = $78. 85 2) PV 0 = $100/(1. 08)3 = $100/1. 2597 = $79. 38 3 -71

Impact of Continuous Compounding A has deposited $100 in a saving account at an annual interest rate of 8%. Future value at the end of three year assuming continuous compounding: FVn = PV 0(1 + [i/m])mn 3 -72 FV 3 = PV 0(e)in FV 3 = $100(2. 71828)(0. 08)(3) FV 3 = $100 x 1. 2712 = $127. 12

Impact of Continuous Compounding The present value of $1, 000 to be received at the end of 10 $1, 000 years with a discount rate of 20%. At the end of 10 years with continuous compounding: PV 0 = FVn /(1 + [i/m])mn PV 0 = FVn /(e)in PV 0 = $1, 000/(2. 71828)(0. 20)(10) = $1, 000/7. 389 = $135. 34 3 -73

Effective Annual Interest Rate The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year. (1 + [ i / m ] )m – 1 Suppose Interest rate = 8% Effective Interest Rate if 8% compounding on quarterly: (1 + [ 0. 08/4])4 – 1 = 0. 08243 or 8. 24% 3 -74

Steps to Amortizing a Loan 1. 2. 3. 4. 5. 3 -75 Calculate the payment period i. e. R Determine the interest at the end of year. = Loan Amount x i% Compute principal payment: principal payment (Loan Amount - Interest from Step 2) Determine ending balance in Period t. (Loan Amount - principal payment from Step 3) Start again at Step 2 and repeat.

Amortizing a Loan Example A borrowed $22, 000 at a compound annual $22, 000 interest rate of 12%. Amortize the loan if annual payments are made for 6 years. Step 1: Payment PV 0 = R (PVIFA i%, n) $22, 000 = R (PVIFA 12%, 6) $22, 000 = R (4. 111) $22, 000 R = $22, 000 / 4. 111 = $5, 351 $22, 000 3 -76

Amortizing a Loan Example End of Year Payment Interest 0 1 2 3 4 5 6 3 -77 $5, 351 $5, 351 $32, 106 Principal Ending Bal $2, 640 2, 315 1, 951 1, 542 1, 085 573 $10, 106 $2, 711 $3, 036 $3, 400 $3, 809 $4, 266 $4, 778 $22, 000 $19, 289 $16, 253 $12, 853 $9, 044 $4, 778 $0 ($22, 000)

Steps to Solve Time Value of Money Problems 1. Read problem thoroughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a PV or FV problem 5. Determine stream of Cash Flow: Ø Single Cash Flow Ø Annuity stream of Cash Flow Ø Mixed stream of Cash Flow 6. Solve the problem 3 -78