Lecture Note 2 Time Value of Money

Transcription

1 Seg250 Management Prncples for Engneerng Managers Lecture ote 2 Tme Value of Money Department of Systems Engneerng and Engneerng Management The Chnese Unversty of Hong Kong

2 Interest: The Cost of Money In fnancal world money tself s a commodty. Money costs money Cost of Money measured by an nterest rate Interest The cost of havng money avalable

3 The Tme Value of Money Example: Buy today or buy later?

4 The Tme Value of Money (Cont d The prncple of the Tme value of money The economc value of a sum depends on when the sum s receved. Money has both earnng power and purchasng power over tme. A dollar receved today has a greater value than a dollar receved at some future tme. The operaton of nterest and the tme value of money have to be consdered for decdng among alternatve proposals.

5 Elements of Transactons Involvng Interest Types of transactons nvolve nterest: E.g. borrowng money nvestng money or purchasng machnery on credt Common elements n transactons: Prncple (P The ntal amount of money nvested or borrowed Interest rate ( Measures the cost or prce of money Expressed as a percentage per perod of tme. Interest perod (n Determnes how frequently nterest s calculated. umber of nterest perods ( A plan for recepts or dsbursements (A n Yelds a partcular cash flow pattern over a specfed length of tme. E.g. a seres of equal monthly payments that repay a loan. Future amount of money (F Results from the cumulatve effects of the nterest rate over a number of nterest perods.

6 Example of an Interest Transacton An electroncs manufacturng company borrows $20000 from a bank at 9% annual nterest rate to buy a machne. The company also pays a $200 loan orgnaton fee when the loan commences. Two repayment plans are offered by the bank: Prncple amount P s $ Interest rate s 9% Plan : Interest perod n s one year. Duraton of the transacton s fve years. umber of nterest perods = 5 Repayment Plans Offered by the Lender End of Year Recepts Payments Plan Plan 2 Year 0 $ $ $ Year Year Year Year Year Both payment plans are based on a rate of 9% nterest. Plan 2: A sngle future repayment F.

7 Cash Flow Dagrams What s t? The flow of cash! $9800 $20000 Cash flow at n=0 Is a net cash flow After summng $20000 and takng away $200 = 9% $200 0 Years $54.85 $54.85 $54.85 $54.85 $54.85 End-of-perod conventon The assumpton of placng all cash flow transactons at the end of an nterest perod When to use t? Cash flow dagrams s strongly recommended for stuatons n whch the analyst needs to clarfy or vsualze what s nvolved when flows of money occur at varous tme.

8 Smple Interest The total nterest earned s lnearly proportonal to the prncpal (loan. The nterest do not accumulate Mathematcally I = ( P( ( where P = Prncple = number of perods (e.g. year = nterest rate per nterest perod Queston: How much nterest we need to pay f we have borrowed $000 for 3 years wth an annual nterest rate of 0%?

9 Compound Interest The nterest earned for any perod s based on the remanng prncpal amount plus any accumulated nterest charges up to the begnnng of that perod. Mathematcally I = ( P n= n ( = P[( ] Queston: What s the nterest we need to pay f we have borrowed $000 for 3 years wth an annual nterest rate of 0%? Year Year 2 Year 3 Borrowed $000 $00 $20 Interest $00 $0 $2

10 Compound Interest (cont d Suppose that the nterest s accumulated n sx months (twce a year Queston: What s the nterest we need to pay f we have borrowed $000 for 3 years wth a 5% nterest rate for every sx months? Year 0.5 Year Year.5 Year 2 Year 2.5 Year 3 Borrowed $000 $050 $52.5 $207.6 Interest $50 $52.5 $55. $60.4 ote that the nterest that we have to pay s more than 0% a year!

11 Economc Equvalence Consder the followng two opton: (a Receve $20000 today and $50000 ten years from now (b Receve $8000 each year for the next 0 years Whch one to choose?

12 Economc Equvalence (Cont d Economc equvalence exsts between cash flows that have the same economc effect and could therefore be traded for one another. Economc equvalence refers to the fact that a cash flow whether a sngle payment or a seres of payments can be converted to an equvalent cash flow at any pont n tme. Even though the amounts and tmng of the cash flows may dffer the approprate nterest rate makes them equal.

13 Economc Equvalence (Cont d Comparson of alternatves Compare the value of alternatves by fndng the equvalent value of alternatves at any common pont n tme Usng compound nterest to establsh economc equvalence:

14 Economc Equvalence (Cont d A smple example We nvest $000 at 2% annual nterest for fve years At the end of the nvestment perod our sums grow to: F = P( = $000( = $ We could trade $000 now for the promse of recevng $ n fve years.

15 Economc Equvalence (Cont d Whch plan you would lke to choose? Plan : Pay nterest at the end of each month and prncpal at the end of fourth month $7000 End of the month $70 $70 $70 $70 $7000 Plan 2: Pay prncpal and nterest n one payment

16 Economc Equvalence (Cont d Whch plan you would lke to choose? Plan 3: Pay the debt n for equal end of month nstallments

17 Types of Cash Flow Sngle Cash Flow $ $00 $00 $00 $00 $00 Equal (Unform Seres Lnear Gradent Seres Geometrc Gradent Seres $503G $504G $50G $502G $ $50(g 4 $50(g 3 $50(g 2 $50(g $ $70 $00 $60 $50 $00 Irregular (Mxed Seres

18 Interest Formulas Lst of factors for major formulas and ther names Sngle Payment Seres Equal-Payment (Unform Seres ( A / P = Example: The sngle-payment compound-amount factor ( F / P Fnd the Future Worth (F gven the Present Worth (P nterest rate (% and the number of perods (. F ( F ( P / F = ( F ( P / A = ( A / F = = / P = ( / A = ( ( ( ( ( ( ( P( = P(F/P Sngle - payment compound - amount Sngle - payment present - worth Equal payment - seres compound Equal payment - seres present - worth Snkng - fund Captal - recovery - amount

19 Formulas (cont d The prevous sx equatons s enough for us to solve most of the problem. Remember the followng steps. Identfy the problem carefully 2. Draw a cash flow dagram 3. Decompose the cash flow dagram such that t can be solved by some of the prevous equatons ote: The nterest rates for all equatons are effectve nterest rates

20 Sngle Cash Flows

21 A Loan Plan Suppose you borrowed $8000 now promsng to repay the loan and accumulated nterest after 9 years at 0% nterest rate how much you need to pay at that tme? Cash flow dagram: What s ths amount? 0 2 $P = $00000 $F 9 In the queston we are gven: - P ($8000 (0% and (9 years And we need to fnd: - F (the future worth So we use: - ( F / P = ( Answer - F = $8000 ( F / P0%9 = $8000( = $8000( 0% = $

22 Sngle Cash Flows

23 An Investment Decson How much money you need to put nto the bank now wth annual nterest rate 7% n order to save $ n 45 years? Cash flow dagram: $F = $ $P Answer: P = $4763

24 Purchasng a Tract of Land An nvestor has an opton to purchase a tract of land that wll be worth $0000 n sx years. If the value of the land ncreases at 8% each year how much should the nvestor be wllng to pay now? Cash flow dagram: Answer: $6302

25 Equal-Payment Seres Compound-amount factor Fnd F gven A and Present-worth factor Fnd P gven A and

26 An Investment Plan If you depost $000 n HSBC every year startng from the next year. How much would ths amount be after 5 years f the nterest rate s 5% p.a.? Cash flow dagram: $F ote When usng the component (F / A % the last depost and F are concdent at the same tme. Also A must begn from the frst year (not now. A = $0000 Answer: F = $

27 Borrowng Money Your frend wants to borrow money from you. He agrees to pay you $20000 each year wth annual nterest rate of 5% for 5 years (very generous!. How much money at most should you lend to hm? Cash flow dagram: Answer: P = $67044

28 Lottery Problem A couple won a lottery game. The couple could choose between a sngle lump sum of $04 mllon jackpot or a total of $98 mllon pad out over 25 years (or $7.92 mllon per year. Suppose the couple could nvest the money at 8% annual nterest whch opton should the couple choose? Cash flow dagram:

29 Equal-Payment Seres Snkng-Fund factor Fnd A gven F and Captal-Recovery factor Fnd A gven P and

30 Savngs Plan To set up a savng plan of $00000 n 8 years how much do one need to save each year f the current rate of nterest s 7%? Cash flow dagram: Answer: A = $

31 Become A Mllonare n 30 Years How much do you need to nvest n a bank every year f you want to yeld $ n 45 years? Assume that the nterest rate s 7% Cash flow dagram: Answer: A = $3500

32 Become a Mllonare By Savng $ a Day! If you nvest $ each day how many years would the money become $000000? Assume that the nterest rate s 0%. Cash flow dagram: Answer: = 58.93

33 Mortgage Plan You want to buy an apartment at the prce of $ You wll do ths wth a mortgage from Heng Seng Bank at (P-2% for 30 years. What s your monthly payment? Cash flow dagram: Answer: A = $X (Monthly payment = $X/2

34 Mortgage Plan You want to buy an apartment at the prce of $ You wll do ths wth a mortgage from Heng Seng Bank at (P-2% for 30 years. What s your monthly payment? Cash flow dagram: Answer: A = $X (Monthly payment = $X/2

35 When s large Present Value of Perpetutes For (A / P : When s large the annual payment wll be less and less and eventually: P = A

36 More About the Formulas For most of the problems they are very complex and we have to combne the formulas and reformulate the cash flow dagram to solve these problems. That s why cash flow dagram s very useful.

37 Investment Plan A father on the day hs son s born wshes to determne what amount would have to be put nto an account wth 2% nterest rate so that he can wthdraw $2000 durng hs son s 8 th 9 th 20 th and 2 st brthdays. Cash flow dagram: $A = $2000 $F $A = $2000 Step Step 2 $F $P $P Answer: P = $884.46

38 Mortgage Plans Bank : (P 0.5% for the whole duraton 4% cash rebate Bank 2: (P 2.25% for the fst 3 years (P.75% for the rest of the tme $5000 admnstratve fee Bank 3: (P 2% for the whole duraton Developer: (P 2% for the whole duraton for the frst 70% of the prce (P 2% for the rest of the purchasng prce frst ten years are nterest fee Whch plan should you choose (Borrow $ ?

39 Two Savng Plans Plan : You have to depost $000 each year for 5 years. The annual nterest rate s 0%. You can take your money out when you are 65. You can jon ths plan when you are 22. Plan 2: You have to depost $2000 each year when you are 32 untl 65. The annual nterest rate s 0%. Answer: $50400 versus $444500!

40 Equvalent Cash Flow Dagram Determne Q n terms of H so as to make both cash flow dagram equvalent

41 Lnear Gradent Seres Sometmes engneers encounter stuatons nvolvng perodc payments that ncrease or decrease by a constant amount G. ( - G ( - 2G ( - 3G 3G... 2G G ote Both Tme 0 and Tme DO OT contan any cash flows

42 Lnear Gradent Seres (Cont d The formulas for the unform gradent seres: = = = = = = = = = ( / ( / ( / ( ( / ( / ( / ( ] [( ( ( ( ( / ( 2 / ( / ( / ( n n F A G F G A F P G F G P A F A F A F G F L L

43 A Smple Example Suppose that the expenses are: $000 for the second year $2000 for the thrd year and $3000 for the forth year. What s the present worth of the expenses? Assume that the nterest rate s 5%. Cash flow dagram: Answer: P = $3790 / A = $326.3

44 A Complcated Example Suppose we have a project whch wll expense $8000 $7000 $6000 and $5000 n Year Year 2 Year 3 and Year 4 respectvely. Assume that the nterest rate s 5%. What s the total present expense of the project? Cash flow: $8000 $7000 $6000 $5000 Answer: P = -$9050 $P

45 Geometrc Gradent Seres Smlar to the dea of lnear gradent seres the geometrc gradent seres s used to model the stuaton that a fxed percentage s ncreased or decreased n every perod

46 Geometrc Gradent Seres (cont d Fnd P when you are gven A: = = = = = = = = = = = g F P A g g P F F P A g A g g g A g A g A g A g A A A A A F P A F P A F P A P n n n n n / ( ] / ( / ( [ ( ] ( ( [ ( ( ( ( ( ( ( ( ( ( / ( 2 / ( / ( L L L

47 Geometrc Gradent Seres An Example Suppose the retrement penson s $50000 each year and the cost of lvng ncreases at an annual rate of 5% due to nflaton what s the addtonal savngs requred to meet the future ncrease n cost of lvng over 35 years? (nterest rate at 7%.

48 Geometrc Gradent Seres An Example (cont d

49 References Lecture 4: Money-Tme Relatonshps & Equvalence by Gabrel Fung Chan S. Park Fundamentals of Engneerng Economcs. Prentce Hall