Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

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1 Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important concept -- Almost everythng from ths pont on n fnance based upon understandng ths concept. You must get ths materal down cold. You can do problems usng formulas, calculators, and spreadsheets. We wll prmarly use fnancal calculators. Please have your fnancal calculators wth you n class. The Concept of TVM s One of Fnance s BIG contrbutons: t provdes: A vald way for valung dfferent patterns of cash flows on a common bass. A way of cuttng through exaggerated clams. The easest way to envson ths s usng Tme Lnes: Types of Interest usmple Interest Interest pad (earned) on only the orgnal amount, or prncpal borrowed (lent). The Tme Value of Money Compoundng and Dscountng Sngle Sums Compound Interest Interest pad (earned) on prevous nterest earned, as well as on the prncpal borrowed (lent). Page 1

2 Why TIME Why s TIME such an mportant element n your decson? TIME allows you the opportunty to postpone consumpton and earn INTEREST. We know that recevng $1 today s worth more than $1 n the future. Ths s due to opportunty costs. The opportunty cost of recevng $1 n the future s the nterest we could have earned f we had receved the $1 sooner. Today Future If we can measure ths opportunty cost, we can: If we can measure ths opportunty cost, we can: Translate $1 today nto ts equvalent n the future (compoundng). If we can measure ths opportunty cost, we can: Translate $1 today nto ts equvalent n the future (compoundng). Today Future? If we can measure ths opportunty cost, we can: Translate $1 today nto ts equvalent n the future (compoundng). Today Future? Translate $1 n the future nto ts equvalent today (dscountng). Page 2

3 If we can measure ths opportunty cost, we can: Translate $1 today nto ts equvalent n the future (compoundng). Today Translate $1 n the future nto ts equvalent today (dscountng). Today? Future? Future Look carefully at the tmelne: The tmelne allows you to vsualze the actual stuaton. nclude all elements (n,,fv,pv,+$,-$) understand the dynamcs. make accurate calculatons, every tme. speed up the process of problem solvng. Why use Tmelnes? useful for accurate TVM calculatons. essental for captal budgetng. Future Value essental for understandng TVM. essental for ncremental cash flows. essental for succeedng n Fnance. Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 1 year? Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 1 year? PV = FV = 0 1 Page 3

4 Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 1 year? PV = -100 FV = 0 1 P/Y = 1 I = 6 N = 1 PV = -100 FV = $106 Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 1 year? PV = -100 FV = P/Y = 1 I = 6 N = 1 PV = -100 FV = $106 Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 1 year? Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 5 years? PV = -100 FV = FV = PV (FVIF, n ) FV = 100 (FVIF.06, 1 ) (use FVIF table, or) FV = PV (1 + ) n FV = 100 (1.06) 1 = $106 Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 5 years? PV = FV = 0 5 Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 5 years? PV = -100 FV = 0 5 P/Y = 1 I = 6 N = 5 PV = -100 FV = $ Page 4

5 Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 5 years? PV = -100 FV = P/Y = 1 I = 6 N = 5 PV = -100 FV = $ Future Value - sngle sums If you depost $100 n an account earnng 6%, how much would you have n the account after 5 years? PV = -100 FV = FV = PV (FVIF, n ) FV = 100 (FVIF.06, 5 ) (use FVIF table, or) FV = PV (1 + ) n FV = 100 (1.06) 5 = $ Future Value - sngle sums If you depost $100 n an account earnng 6% wth quarterly compoundng, how much would you have n the account after 5 years? Future Value - sngle sums If you depost $100 n an account earnng 6% wth quarterly compoundng, how much would you have n the account after 5 years? PV = FV = 0? Future Value - sngle sums If you depost $100 n an account earnng 6% wth quarterly compoundng, how much would you have n the account after 5 years? PV = -100 FV = 0 20 P/Y = 4 I = 6 N = 20 PV = -100 FV = $ Future Value - sngle sums If you depost $100 n an account earnng 6% wth quarterly compoundng, how much would you have n the account after 5 years? PV = -100 FV = P/Y = 4 I = 6 N = 20 PV = -100 FV = $ Page 5

6 Future Value - sngle sums If you depost $100 n an account earnng 6% wth quarterly compoundng, how much would you have n the account after 5 years? PV = -100 FV = Future Value - sngle sums If you depost $100 n an account earnng 6% wth monthly compoundng, how much would you have n the account after 5 years? 0 20 FV = PV (FVIF, n ) FV = 100 (FVIF.015, 20 ) (can t use FVIF table) FV = PV (1 + /m) m x n FV = 100 (1.015) 20 = $ Future Value - sngle sums If you depost $100 n an account earnng 6% wth monthly compoundng, how much would you have n the account after 5 years? PV = FV = 0? Future Value - sngle sums If you depost $100 n an account earnng 6% wth monthly compoundng, how much would you have n the account after 5 years? PV = -100 FV = 0 60 P/Y = 12 I = 6 N = 60 PV = -100 FV = $ Future Value - sngle sums If you depost $100 n an account earnng 6% wth monthly compoundng, how much would you have n the account after 5 years? PV = -100 FV = P/Y = 12 I = 6 N = 60 PV = -100 FV = $ Future Value - sngle sums If you depost $100 n an account earnng 6% wth monthly compoundng, how much would you have n the account after 5 years? PV = -100 FV = FV = PV (FVIF, n ) FV = 100 (FVIF.005, 60 ) (can t use FVIF table) FV = PV (1 + /m) m x n FV = 100 (1.005) 60 = $ Page 6

7 $300 Compound Interest The Future Value of a Sngle Amount Graphcal Presentaton Dfferent Interest Rates $250 $200 $ k = 8% $150 $100 $50 $ Amount k = 4% k = 0% Year 38 Future Value - contnuous compoundng What s the FV of $1,000 earnng 8% wth contnuous compoundng, after 100 years? Future Value - contnuous compoundng What s the FV of $1,000 earnng 8% wth contnuous compoundng, after 100 years? PV = FV = 0? Future Value - contnuous compoundng What s the FV of $1,000 earnng 8% wth contnuous compoundng, after 100 years? PV = FV = FV = PV (e n ) FV = 1000 (e.08x100 ) = 1000 (e 8 ) FV = $2,980, Future Value - contnuous compoundng What s the FV of $1,000 earnng 8% wth contnuous compoundng, after 100 years? PV = FV = $2.98m FV = PV (e n ) FV = 1000 (e.08x100 ) = 1000 (e 8 ) FV = $2,980, Page 7

8 Compoundng Perods Frequency wth whch nterest s credted for calculatng future nterest, usually annually, semannually, quarterly, or monthly. The shorter the perod, the more nterest s earned on nterest Annually 12% $100 $112 Semannually 6% 6% $100 $106 $ Quarterly 3% 3% 3% 3% $100 $103 $ $ $ The Effectve Annual Rate (EAR) The rate of annually compounded nterest equvalent to the nomnal rate compounded more frequently Table 5-2 Compoundng Fnal balance Annual $ Semannual $ Quarterly $ Monthly $ Year End Balances at Varous Compoundng Perods $100 Intal Depost and k nom = 12% In general: EAR = 1 + k m m nom 1 Quote the annual (nomnal) rate (k nom ) statng the compoundng perod mmedately afterward "12% compounded quarterly" Present Value Present Value - sngle sums If you receve $100 one year from now, what s the PV of that $100 f your opportunty cost s 6%? Present Value - sngle sums If you receve $100 one year from now, what s the PV of that $100 f your opportunty cost s 6%? PV = FV = 0? Present Value - sngle sums If you receve $100 one year from now, what s the PV of that $100 f your opportunty cost s 6%? PV = FV = P/Y = 1 I = 6 N = 1 FV = 100 PV = Page 8

9 Present Value - sngle sums If you receve $100 one year from now, what s the PV of that $100 f your opportunty cost s 6%? PV = FV = P/Y = 1 I = 6 N = 1 FV = 100 PV = Present Value - sngle sums If you receve $100 one year from now, what s the PV of that $100 f your opportunty cost s 6%? PV = FV = PV = FV (PVIF, n ) PV = 100 (PVIF.06, 1 ) (use PVIF table, or) PV = FV / (1 + ) n PV = 100 / (1.06) 1 = $94.34 Present Value - sngle sums If you receve $100 fve years from now, what s the PV of that $100 f your opportunty cost s 6%? Present Value - sngle sums If you receve $100 fve years from now, what s the PV of that $100 f your opportunty cost s 6%? PV = FV = 0? Present Value - sngle sums If you receve $100 fve years from now, what s the PV of that $100 f your opportunty cost s 6%? PV = FV = P/Y = 1 I = 6 N = 5 FV = 100 PV = Present Value - sngle sums If you receve $100 fve years from now, what s the PV of that $100 f your opportunty cost s 6%? PV = FV = P/Y = 1 I = 6 N = 5 FV = 100 PV = Page 9

10 Present Value - sngle sums If you receve $100 fve years from now, what s the PV of that $100 f your opportunty cost s 6%? Present Value - sngle sums What s the PV of $1,000 to be receved 15 years from now f your opportunty cost s 7%? PV = FV = PV = FV (PVIF, n ) PV = 100 (PVIF.06, 5 ) (use PVIF table, or) PV = FV / (1 + ) n PV = 100 / (1.06) 5 = $74.73 Present Value - sngle sums What s the PV of $1,000 to be receved 15 years from now f your opportunty cost s 7%? PV = FV = 0 15 Present Value - sngle sums What s the PV of $1,000 to be receved 15 years from now f your opportunty cost s 7%? PV = FV = P/Y = 1 I = 7 N = 15 FV = 1,000 PV = Present Value - sngle sums What s the PV of $1,000 to be receved 15 years from now f your opportunty cost s 7%? PV = FV = P/Y = 1 I = 7 N = 15 FV = 1,000 PV = Present Value - sngle sums What s the PV of $1,000 to be receved 15 years from now f your opportunty cost s 7%? PV = FV = PV = FV (PVIF, n ) PV = 100 (PVIF.07, 15 ) (use PVIF table, or) PV = FV / (1 + ) n PV = 100 / (1.07) 15 = $ Page 10

11 Present Value - sngle sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what s your annual rate of return? Present Value - sngle sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what s your annual rate of return? PV = FV = 0 5 Present Value - sngle sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what s your annual rate of return? PV = FV = 11, P/Y = 1 N = 5 PV = -5,000 FV = 11,933 I = 19% Present Value - sngle sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what s your annual rate of return? PV = FV (PVIF, n ) 5,000 = 11,933 (PVIF?, 5 ) PV = FV / (1 + ) n 5,000 = 11,933 / (1+ ) = ((1/ (1+) 5 ) = (1+) 5 (2.3866) 1/5 = (1+) =.19 Present Value - sngle sums Suppose you placed $100 n an account that pays 9.6% nterest, compounded monthly. How long wll t take for your account to grow to $500? PV = FV = 0 Present Value - sngle sums Suppose you placed $100 n an account that pays 9.6% nterest, compounded monthly. How long wll t take for your account to grow to $500? PV = -100 FV = 500 0? P/Y = 12 FV = 500 I = 9.6 PV = -100 N = 202 months Page 11

12 Present Value - sngle sums Suppose you placed $100 n an account that pays 9.6% nterest, compounded monthly. How long wll t take for your account to grow to $500? PV = FV / (1 + ) n 100 = 500 / (1+.008) N 5 = (1.008) N ln 5 = ln (1.008) N ln 5 = N ln (1.008) = N N = 202 months Hnt for sngle sum problems: In every sngle sum future value and present value problem, there are 4 varables: FV, PV,, and n When dong problems, you wll be gven 3 of these varables and asked to solve for the 4th varable. Keepng ths n mnd makes tme value problems much easer! Present Value - An Example Suppose you have the opportunty to buy a pece of land for $10,000 today, and sell t n eght years for $20,000. Is ths a good deal f you can put your money n a rsk-equvalent that s expected to earn 10 percent a year compounded annually? Present Value - An Example The present value of the $20,000 you expect to receve at the end of eght years s: PV = $20,000 [ 1/(1.10) 8 ] = $ I=10% Facts Tmelne n= 8 Ths s a bad deal snce the present value of return n eght years s less than the cost of the land. The Tme Value of Money Compoundng and Dscountng Cash Flow Streams Annutes Annuty: a sequence of equal cash flows, occurrng at the end of each perod. 4 Page 12

13 Annutes Annuty: a sequence of equal cash flows, occurrng at the end of each perod. 4 Examples of Annutes: If you buy a bond, you wll receve equal sem-annual coupon nterest payments over the lfe of the bond. If you borrow money to buy a house or a car, you wll pay a stream of equal payments. Examples of Annutes: If you buy a bond, you wll receve equal sem-annual coupon nterest payments over the lfe of the bond. If you borrow money to buy a house or a car, you wll pay a stream of equal payments. Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? P/Y = 1 I = 8 N = 3 PMT = -1,000 FV = $3, Page 13

14 Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? P/Y = 1 I = 8 N = 3 PMT = -1,000 FV = $3, Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? FV = PMT (FVIFA, n ) Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? FV = PMT (FVIFA, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? FV = PMT (FVIFA, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) FV = PMT (1 + ) n -1 Page 14

15 Future Value - annuty If you nvest $1,000 each year at 8%, how much would you have after 3 years? Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? FV = PMT (FVIFA, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) FV = PMT (1 + ) n -1 FV = 1,000 (1.08) 3-1 = $ Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? P/Y = 1 I = 8 N = 3 PMT = -1,000 PV = $2, Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? P/Y = 1 I = 8 N = 3 PMT = -1,000 PV = $2, Page 15

16 Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? PV = PMT (PVIFA, n ) Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? PV = PMT (PVIFA, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? PV = PMT (PVIFA, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) 1 PV = PMT 1 - (1 + ) n Present Value - annuty What s the PV of $1,000 at the end of each of the next 3 years, f the opportunty cost s 8%? The Tme Value of Money PV = PMT (PVIFA, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) 1 PV = PMT 1 - (1 + ) n 1 PV = (1.08 ) 3 = $2, Other Cash Flow Patterns Page 16

17 Perpetutes Suppose you wll receve a fxed payment every perod (month, year, etc.) forever. Ths s an example of a perpetuty. You can thnk of a perpetuty as an annuty that goes on forever. Present Value of a Perpetuty When we fnd the PV of an annuty, we thnk of the followng relatonshp: Present Value of a Perpetuty Mathematcally, When we fnd the PV of an annuty, we thnk of the followng relatonshp: PV = PMT (PVIFA, n ) Mathematcally, (PVIFA, n ) = Mathematcally, (PVIFA, n ) = 1-1 (1 + ) n Page 17

18 Mathematcally, (PVIFA, n ) = 1-1 (1 + ) n When n gets very large, We sad that a perpetuty s an annuty where n = nfnty. What happens to ths formula when n gets very, very large? When n gets very large, When n gets very large, 1-1 (1 + ) n 1-1 (1 + ) n ths becomes zero. When n gets very large, 1-1 (1 + ) n ths becomes zero. Present Value of a Perpetuty So, the PV of a perpetuty s very smple to fnd: So we re left wth PVIFA = 1 Page 18

19 Present Value of a Perpetuty So, the PV of a perpetuty s very smple to fnd: What should you be wllng to pay n order to receve $10,000 annually forever, f you requre 8% per year on the nvestment? PV = PMT What should you be wllng to pay n order to receve $10,000 annually forever, f you requre 8% per year on the nvestment? PV = PMT = $10, What should you be wllng to pay n order to receve $10,000 annually forever, f you requre 8% per year on the nvestment? PV = PMT = $10, = $125,000 Begn Mode vs. End Mode Ordnary Annuty vs. Annuty Due $1000 $1000 $ Page 19

20 Begn Mode vs. End Mode year year year Begn Mode vs. End Mode year year year PV n END Mode Begn Mode vs. End Mode year year year Begn Mode vs. End Mode year year year PV n END Mode FV n END Mode Begn Mode vs. End Mode year year year Begn Mode vs. End Mode year year year PV n BEGIN Mode PV n BEGIN Mode FV n BEGIN Mode Page 20

21 Earler, we examned ths ordnary annuty: Earler, we examned ths ordnary annuty: Earler, we examned ths ordnary annuty: Usng an nterest rate of 8%, we fnd that: Earler, we examned ths ordnary annuty: Usng an nterest rate of 8%, we fnd that: The Future Value (at 3) s $3, Earler, we examned ths ordnary annuty: Usng an nterest rate of 8%, we fnd that: The Future Value (at 3) s $3, The Present Value (at 0) s $2, What about ths annuty? Same 3-year tme lne, Same 3 $1000 cash flows, but The cash flows occur at the begnnng of each year, rather than at the end of each year. Ths s an annuty due. Page 21

22 Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Mode = BEGIN P/Y = 1 I = 8 N = 3 PMT = -1,000 FV = $3, Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Smply compound the FV of the ordnary annuty one more perod: Mode = BEGIN P/Y = 1 I = 8 N = 3 PMT = -1,000 FV = $3, Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Smply compound the FV of the ordnary annuty one more perod: FV = PMT (FVIFA, n ) (1 + ) Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Smply compound the FV of the ordnary annuty one more perod: FV = PMT (FVIFA, n ) (1 + ) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) Page 22

23 Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Smply compound the FV of the ordnary annuty one more perod: FV = PMT (FVIFA, n ) (1 + ) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + ) n -1 (1 + ) Future Value - annuty due If you nvest $1,000 at the begnnng of each of the next 3 years at 8%, how much would you have at the end of year 3? Smply compound the FV of the ordnary annuty one more perod: FV = PMT (FVIFA, n ) (1 + ) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + ) n -1 (1 + ) FV = 1,000 (1.08) 3-1 = $3, (1.08).08 Present Value - annuty due What s the PV of $1,000 at the begnnng of each of the next 3 years, f your opportunty cost s 8%? Present Value - annuty due What s the PV of $1,000 at the begnnng of each of the next 3 years, f your opportunty cost s 8%? Mode = BEGIN P/Y = 1 I = 8 N = 3 PMT = 1,000 PV = $2, Present Value - annuty due What s the PV of $1,000 at the begnnng of each of the next 3 years, f your opportunty cost s 8%? Present Value - annuty due Mode = BEGIN P/Y = 1 I = 8 N = 3 PMT = 1,000 PV = $2, Page 23

24 Present Value - annuty due Smply compound the FV of the ordnary annuty one more perod: Present Value - annuty due Smply compound the FV of the ordnary annuty one more perod: PV = PMT (PVIFA, n ) (1 + ) Present Value - annuty due Smply compound the FV of the ordnary annuty one more perod: PV = PMT (PVIFA, n ) (1 + ) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) Present Value - annuty due Smply compound the FV of the ordnary annuty one more perod: PV = PMT (PVIFA, n ) (1 + ) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) 1 PV = PMT 1 - (1 + ) n (1 + ) Present Value - annuty due Smply compound the FV of the ordnary annuty one more perod: PV = PMT (PVIFA, n ) (1 + ) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) 1 PV = PMT 1 - (1 + ) n (1 + ) 1 PV = (1.08 ) 3 (1.08) = $2, Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 4 Is ths an annuty? How do we fnd the PV of a cash flow stream when all of the cash flows are dfferent? (Use a 10% dscount rate). Page 24

25 Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 4 Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 4 Sorry! There s no qucke for ths one. We have to dscount each cash flow back separately. Sorry! There s no qucke for ths one. We have to dscount each cash flow back separately. Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 4 Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 4 Sorry! There s no qucke for ths one. We have to dscount each cash flow back separately. Sorry! There s no qucke for ths one. We have to dscount each cash flow back separately. Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 4 Sorry! There s no qucke for ths one. We have to dscount each cash flow back separately. -10,000 2,000 4,000 6,000 7,000 4 perod CF PV (CF) 0-10,000-10, ,000 1, ,000 3, ,000 4, ,000 4, PV of Cash Flow Stream: $ 4, Page 25

26 Annual Percentage Yeld (APY) Annual Percentage Yeld (APY) Whch s the better loan: 8% compounded annually, or 7.85% compounded quarterly? We can t compare these nomnal (quoted) nterest rates, because they don t nclude the same number of compoundng perods per year! We need to calculate the APY. Annual Percentage Yeld (APY) quoted rate m APY = ( 1 + ) m - 1 Annual Percentage Yeld (APY) quoted rate m APY = ( 1 + ) m - 1 Fnd the APY for the quarterly loan: Annual Percentage Yeld (APY) quoted rate m APY = ( 1 + ) m - 1 Fnd the APY for the quarterly loan:.0785 APY = ( 1 + ) Annual Percentage Yeld (APY) quoted rate m APY = ( 1 + ) m - 1 Fnd the APY for the quarterly loan:.0785 APY = ( 1 + ) APY =.0808, or 8.08% Page 26

27 Annual Percentage Yeld (APY) quoted rate m APY = ( 1 + ) m - 1 Fnd the APY for the quarterly loan: Practce Problems.0785 APY = ( 1 + ) APY =.0808, or 8.08% The quarterly loan s more expensve than the 8% loan wth annual compoundng! Example Cash flows from an nvestment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you requre a 20% rate of return, what s the PV of these cash flows? Example Cash flows from an nvestment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you requre a 20% rate of return, what s the PV of these cash flows? $ $ Ths type of cash flow sequence s often called a deferred annuty. $ How to solve: 1) Dscount each cash flow back to tme 0 separately. Page 27

28 $ $ How to solve: 1) Dscount each cash flow back to tme 0 separately. How to solve: 1) Dscount each cash flow back to tme 0 separately. $ $ How to solve: 1) Dscount each cash flow back to tme 0 separately. How to solve: 1) Dscount each cash flow back to tme 0 separately. $ $ How to solve: 1) Dscount each cash flow back to tme 0 separately. How to solve: 1) Dscount each cash flow back to tme 0 separately. Or, Page 28

29 $ $ ) Fnd the PV of the annuty: PV: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PV = $119,624 2) Fnd the PV of the annuty: PV3: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PV3= $119,624 $ $ , ,624 Then dscount ths sngle sum back to tme 0. PV: End mode; P/YR = 1; I = 20; N = 3; FV = 119,624; Solve: PV = $69,226 $ $ , ,624 69, ,624 The PV of the cash flow stream s $69,226. Page 29

30 Retrement Example After graduaton, you plan to nvest $400 per month n the stock market. If you earn 12% per year on your stocks, how much wll you have accumulated when you retre n 30 years? Retrement Example After graduaton, you plan to nvest $400 per month n the stock market. If you earn 12% per year on your stocks, how much wll you have accumulated when you retre n 30 years? Usng your calculator, P/YR = 12 N = 360 PMT = -400 I%YR = 12 FV = $1,397, Retrement Example If you nvest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Retrement Example If you nvest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Page 30

31 Retrement Example If you nvest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Retrement Example If you nvest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? FV = PMT (FVIFA, n ) FV = PMT (FVIFA, n ) FV = 400 (FVIFA.01, 360 ) (can t use FVIFA table) Retrement Example If you nvest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Retrement Example If you nvest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? FV = PMT (FVIFA, n ) FV = 400 (FVIFA.01, 360 ) FV = PMT (1 + ) n -1 (can t use FVIFA table) FV = PMT (FVIFA, n ) FV = 400 (FVIFA.01, 360 ) (can t use FVIFA table) FV = PMT (1 + ) n -1 FV = 400 (1.01) = $1,397, House Payment Example If you borrow $100,000 at 7% fxed nterest for 30 years n order to buy a house, what wll be your monthly house payment? House Payment Example If you borrow $100,000 at 7% fxed nterest for 30 years n order to buy a house, what wll be your monthly house payment? Page 31

32 ???? ???? Usng your calculator, P/YR = 12 N = 360 I%YR = 7 PV = $100,000 PMT = -$ House Payment Example House Payment Example PV = PMT (PVIFA, n ) House Payment Example House Payment Example PV = PMT (PVIFA, n ) 100,000 = PMT (PVIFA.07, 360 ) (can t use PVIFA table) PV = PMT (PVIFA, n ) 100,000 = PMT (PVIFA.07, 360 ) (can t use PVIFA table) 1 PV = PMT 1 - (1 + ) n Page 32

33 House Payment Example PV = PMT (PVIFA, n ) 100,000 = PMT (PVIFA.07, 360 ) (can t use PVIFA table) 1 PV = PMT 1 - (1 + ) n 1 100,000 = PMT 1 - ( ) 360 PMT=$ Team Assgnment Upon retrement, your goal s to spend 5 years travelng around the world. To travel n style wll requre $250,000 per year at the begnnng of each year. If you plan to retre n 30 years, what are the equal monthly payments necessary to acheve ths goal? The funds n your retrement account wll compound at 10% annually How much do we need to have by the end of year 30 to fnance the trp? PV30 = PMT (PVIFA.10, 5) (1.10) = = 250,000 (3.7908) (1.10) = = $1,042, Usng your calculator, Mode = BEGIN PMT = -$250,000 N = 5 I%YR = 10 P/YR = 1 PV = $1,042, ,042,466 Now, assumng 10% annual compoundng, what monthly payments wll be requred for you to have $1,042,466 at the end of year 30? ,042,466 Usng your calculator, Mode = END N = 360 I%YR = 10 P/YR = 12 FV = $1,042,466 PMT = -$ Page 33

34 Formulas PV FV So, you would have to place $ n your retrement account, whch earns 10% annually, at the end of each of the next 360 months to fnance the 5-year world tour. Sngle Payment Annuty ( ) 1 (1 + ) n n n ( 1+ ) ( 1 + ) n 1 Effectve Annual Rates (EAR) vs. Quoted Rate Quoted Rate EAR = m Q : You are offered 12 percent compounded monthly, what s m the EAR? EAR A: Effectve Annual Rates (EAR) = 1 + = 1 + = = = % Quoted (.12) m 12-1 rate - 1 m - 1 APR vs. EAR A: An APR of 18 percent wth monthly payments s really.18/12 =.015 or 1.5 percent per month..18 EAR = = = = % Page 34

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