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The Time Value of Money and Discounted Cash Flow Analysis

Chapter 4&6. The Time Value of Money and Discounted Cash Flow Analysis. Chapter Outline. Future and Present value for single amount Simple versus compounding and discounting The effective annual Rate Finding number of periods

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The Time Value of Money and Discounted Cash Flow Analysis

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  1. Chapter 4&6 The Time Value of Money and Discounted Cash Flow Analysis

  2. Chapter Outline • Future and Present value for single amount • Simple versus compounding and discounting • The effective annual Rate • Finding number of periods • Finding the Discount Rate (rate of return) for an Investment • Future and Present Values of Multiple Cash Flows • Annuities • Future value • Present value • Finding payment( Annual, quarterly, & monthly) • Finding the discount rate (rate of return). • Perpetuities • Present Value without growth • Present Value with growth • Capital Budgeting: evaluate potential investment projects • Inflation and Discounted Cash Flows • Loan Amortization

  3. Case 1 The Time Value of Money for a single amount

  4. Future Value and Compoundingfor a singeamount • Time is money: • Interest rate • Inflation rate • Uncertainty • FV = PV(1+i)n (1) • Appropriate (i ) depending on the risk involved • Ordinary calculator • Excel • Tables • Double-to-72rule • Example :You invest $100 for 2 years at 10% per year. • FV = 100 (1+i)2 = PV (1.1 x 1.1) = 121 • Four Components :

  5. Future Values and Compounding • The rule of 72: • Example : IF PV = 100 and i=10: • 72/ i = 72/10 = 7.2 years to get 200 and 14.4 years to get 400 • Example : Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years? • FV = 3,000,000(1.15)5 = 6,034,072

  6. The Effective Annual Rate (EFE): • If the compounding is more than once a year: 12times , 4times, 365 times • (2) • Example: if the nominal annual rate is 6%, compounded monthly. What is the effective? • APR/m= .06/12 =.005 EFF=(1.005)12-1= 1,0616-1= 6.16% • Example: you save at the bank for one year, and you got the following alternatives. Which is the best for you? • Bank A: 15% compounded daily Continuously • Bank B: 15.5% ” Quarterly • Bank C: 16% “ Annually • Bank A:pay daily actually paying = 0.15/365 = 0.000411 • FV for 1 kr= 1 x (1.000411)365 = 1.161 -1 , so (EFF ) = 16.18%. • Bank B:paying 0.155/4 = 0.0387 or 3.87% per quarter • FV for 1 kr= 1 x (1.0387)4 = 1.164, (-1). (EFF ) = 16.42 % best • Bank C:No compounding during the year. (EFF) = 16%.

  7. Present Values (Discounting)-Market Price • How much do I have to invest today to have a specific amount in the future? • Solve for PV in Equation (1): • (3) • (1/1-i)ncalled: discount factor or PV factor • (i): Required rate of return, discount rate or cost of capital • Choose the appropriate discount rate • Example :You want to begin saving for you daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? • PV = 150,000 / (1.08)17 = 40,540.34

  8. Present Value – Important Relationships • For a given interest rate – the longer the time period, the lower the present value • What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% • 5 years: PV = 500 / (1.1)5 = 310.46 • 10 years: PV = 500 / (1.1)10 = 192.77 • For a given time period – the higher the interest rate, the smaller the present value • What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? • Rate = 10%: PV = 500 / (1.1)5 = 310.46 • Rate = 15%; PV = 500 / (1.15)5 = 248.58

  9. Finding Discount Rate or the IRR • Often we will want to know what the implied interest rate in an investment • Solve for (i) in Equation (3) • (4) • Example : • You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest? • i = (1200 / 1000)1/5 – 1 = .03714 = 3.714% • Excel: solve (Rate)

  10. Finding Number of periods • Start with PV equation and solve for n (remember logs) • (5) • Ex: You need 50 000. You have 25 000. i = 12% …find (n)

  11. Finding Number of periods • You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?

  12. Case 2 The Time Value of Money for Multiple Cash Flows

  13. Future Value with Multiple Cash flows • Find the value of each year cash flow and add them together. • Ex: you deposit $1000 at the beginning of each of the next 2 yearsi = 10%. How much do you have at the end of Year 2? • FV (first 1000) = 1000 (1.10)2 = 1210 + • FV (second 1000 ) = 1000 (1.10) = 1100 • The answer is: 2310 • Ex: Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? • Year 0 CF: 500 (1.09)2 = 594.05 • Year 1 CF: 600(1.09) = 654.00 • Total FV = 594.05 + 654.00 = 1248.05

  14. Present Value with Multiple Cash flows • Find the PV of each cash flows and add them • Ex: You need 1000 in one year time, and • 2000 in 2 years time. i = 9% • How much do invest today? • PV of 2000 in 2 years: 2000/(1.09)2= 1 683 + • PV of 1000 in 1 year : 1000/(1.09) = 917.43 • Ex: you need 200 in one year, 400 in two years, 600 in 3 years, and 800 in 4 years. How much do you need today? i = 12% • (6)

  15. III- Annuity Future Value (annuiteter): • A finite series of equal payments occurs at regular intervals. • 1- Ordinary (end of the period) annuity future value • (7) • Ex: you save 100 each year for the coming 3 years and the interest rate is 10%. How much you accumulate? • Either : 100(1.1)2+(100(1.1)1+100=331 • Or: . • 2- Annuity due (immediate) future value. Beginning • Either 100(1.1)3+(100(1.1)2+100(1.1)=364 • Or (8)

  16. B- Annuity Present Value (Market Price) • 1- Ordinary annuity present value : • A- • B- (9) • C is Payments (=PMT), n : period, i = interest rate • Ex: you get 500 at the end of each of the next 3 years and you use discount rate of 10%. Excel • How much would you pay for it now (PV)? • A- or • B- Calculate PV factor = 1/(1.1)3= 0.751 • Annuity PV factor=(1-PV factor)/i =(1-0.751)/0.10= 2.4868 • Annuity PV = 500 x 2.486 = 1 243.. PV Factor

  17. Annuity Present Value • Ex: you will get 50 000 each year for the next 20 years. If the discount rate is 8% what the value of this annuity? • Calculate PV factor= (1/(1+i)n)= (1/(1.08)20)= 0,2145 • 2-AnnuityPV factor=(1-PVfactor)/i =(1-0.2145)/0.08= 9,818 • 3- Annuity PV = 50 000 x 9.818 = 490900 • Excel: =NUVÄRDE(.08;20;-50000(PMT))= • 2- Annuity due present value: • (10)

  18. Annuity Present Value • EX: You are buying some land from your parents today. You agree to pay them $5,000 a year for six years. The first payment is due today. What is the actual selling price of the land if your parents are only charging you 3% interest? • Calculate PV factor= (1/(1+i)n)= (1/(1.03)6)= 0,83748 • Annuity PV factor=(1-PVfactor)/i =(1- 0,8374)/0.03= 5,41718 • Annuity PV = 5000 (5,41718)(1.03) = 27898,477

  19. C- Find Annuity Payment(C or PMT) • Example • You retire at the age of 65 and then living another 25 years. Your want to have $500,000 in your retirement savings on the day you retire and spend it all by the time you die. During your retirement, you expect to earn 5% on your savings. How much money can you withdraw from your savings each year during your retirement if you withdraw the funds on the last day of each year? • 500 000 = C x 14.0939 • C = 35476,3408 • C AD = 35476,3408/1.05= 33786,88

  20. Annuity payment • Monthly payments (excel) • You currently owe $3,780 on your credit card. The interest rate is 1.5% per month. How much do you have to pay each month if you want to have this bill paid off within two years?

  21. Annuity Payment • Quarterly payments • Your borrowed $12,000 (PV) and you would like to pay the amount on eight equal quarterly payments. The interest rate is 10% What is the amount of each quarterly payment?

  22. D- Annuity time periods • You plan to buy a Motorcycle that cost $7,500(FV). To do this, you are saving $2,000 a year. Your savings account pays 3% interest. How long will you have to wait to buy the Motorcycle if you want to pay cash for the purchase?

  23. E: FindAnnuity Interest Rate(IRR) • A friend of yours wants to borrow SEK 3000 and he offered to repay you SEK 1000 every year for four years. What interest rate here? • Trial And Error: • Try with 10%: • 1- PV factor= (1/(1+i)n)= (1/(1.1)4)= 0,6830 • 2-APV factor= (1-PVfactor)/i =(1- 0,6830)/0.10= 3,17 • 3- APV = 1000 x 3,17 = 3170 • Thus, 10% is to low • We know that PV and discount rate are negatively related • We have to increase (i). The correct answer is 12.59%

  24. IV : Perpetuities • Infinite series of equal payments (forever ) • Perpetuity Present Value • (11) • Growing perpetuity (12) • Ex: consider at stream of 100 per year and forever, i=10% what is PV? • PV(market Price) = 100/.10= 1000 • Ex: you will get 500 each year (with i = 8%) • PV (market price) = 500/0.08 = 6 250

  25. Perpetuities • Perpetuities with Growth • Ex: An investment gives you 1000 at the first year, i = 9% and it will grow at 4% each year. • PV = C1/i-g = 1000/0.05= 20 000 • If you can buy this by < 20 000…..good • Ex: a stock pays cash div. that grows by 3% every year. The next div is $1 per year. How much should you pay for the stock if i =10%? • PV = 1/(.10-.03) = $14.29

  26. V- Capital Budgeting (Investeringsbedömning) • Most important issue in corporate finance • Techniques used to evaluate potential investment projects • 1- Net Present Value (NPV) method or DCF • (13) • Accept any project that has a positive NPV. • Advantage: simple, count all cash flows, time value of M. • The major problem: uncertainty. • Ex: A 100 zero-coupon bond is selling for 75, n = 5 years. If deposit at bank I = 8%. Should you buy the bond? • Calculate PV = 100/(1.08)5 = 68.06Market Price • NPV =-75+ 68.05 = -6.94 …no

  27. Capital Budgeting • II- Future Value Rule • If I deposit 75 in the bank I will get : • FV= 75 (1.08)5= 110.20 (bank) • III Yield to maturity (IRR): • We should find a yield to maturity for the bond: • i = (100/75)1/5-1 = 5.92% < 8% • Accept any investment if its return > the opportunity cost of capital

  28. Capital Budgeting • IV-Finding the number of periods (n): Payback- method • If I deposit 75 at the bank with 8% interest rate: • If I buy the Bond : Payback period is 5 Years. • Choose investment with the lowest payback period

  29. Other Application of the Payback- method • Ex: a project costs -50 000, • Payback: 30 000(Y1) + 20 000(Y2) + 10000 (Y3) • The payback period = 2 years exact. • Ex(fraction of the year): • a project costs - 60 000, • Payback: 20 000 (Y1) + 90 000(Y2) • The period = 1+ 40 000/90 000= 1.4/9 y

  30. Advantageous & Disadvantageous of the Payback Rule • Advantageous: • Simple and biasedtowardsliquidity • Problems: • No discounting • Risk is ignored. • How to pick up the cutoff period? • No single period for some projects (E) • Flows after the payback period is ignored. Important?

  31. The payback rule • Ex: Two projects : Required rate of return 15% • Y A (long) B (short) • 0 -250 -250 • 1 100 100 2 100 200 3 100 0 4 100 0 • Payback (A) = 2.5 years • Payback (B)= 1 +(150/200) = 1.75 year • NPV (A) = -250 + 100 x ((1-(1/1.15)4)/0.15 = $35.50 • NPV (B) = -250 + 100 /(1.15) + 200/(1.15)2 = -$11.81 • Accepting (B) the value of shareholder equity • Accepting (A) the value of shareholder equity • Accepting the wrong project. • Ignoring flows behind the cutoff →reject a good project.

  32. The discounted payback • It deals with the time value of money. • Ex: we require 12.5%, costs = -300, cash flows = 100 per year for 5 years. What is the discount payback? • Accumulated Cash Flows: • Rarely used in practice. Why?

  33. VI- Inflation and Discounted Cash Flows • How to Calculate Real FV? • A- Real FV = Nominal FV/Price FV • B- Real FV= PV(1+r)n • Example : You save $100 for 1 year. If the interest rate =10% and the inflation rate is 5%, how much do you have in real term after 1 year? • A- Real FV = 110/(1.05)1=104.761 or • B- • Real FV= 100(1.0476)1=104.76 • Campare nominal with nominal Or real with real • Don’t compare nominal with real

  34. Examples –Calculating Real Future Value • Ex: You are 20 years and you saved 100. How much would you have in real term when you are 65 y? If = 5% and i = 8% per year. n=45 • Calculate • Real FV= 100(1.0285)45=355 • Or compare nominal with nominal: • Nominal FV = 100 x (1.08)45 = 3192 • Price FV = 1 (1.05)45 = 8.985 • Real FV = Nominal FV/Price FV = 3192/8.985 = 355 • Ex. Your kid is 10Y, you like to save for the college. The fees = 15 000 and  = 5% and i = 8%. If you put 8000 today, is that enough? n=8 • Nominal with nominal _ • Investment 8000 (1.08)8 = 14 897 nom • Expenses 15000 (1.05)8 = 22 162 No • Or Real FV= 8000(1.0285)8=10016 no

  35. VII- Loan Amortization • To find Payments C (=PMT): • Apply APV and consider loan amount as the APV: • (14) • Alternative equation: • Solve for C (15) • Ex: You take a 100 000 home mortgage loan, i = 9% a year repaid with interest in 3 installments. Calculate the installments (PMT) • Calculate PV factor = 1/(1.09)3= 0,77218 • Annuity PV factor=(1-PVfactor)/i =(1-.0,77218)/0.09= 2,5313

  36. Amortization Schedule First Year: PMT = Betalning ( 9%; 3;-100 000)= 39505.475 • Interest = 100 000 (.09) = 9000 - Principle = 39505.475-9000 = 30 505 • Remaining = 100 000 - 30 505 = 69495 • Second Year: PMT = Betalning ( 9%; 3;-100 000)= 39505.475 • Interest = 69495 (.09) = 6254,55 - Principle = 39505.475-6254,55 = 33251 • Remaining = 69495- 33251= 36244 • Third Year: PMT = Betalning ( 9%; 3;-100 000)= 39505.475 • Interest = 36244 (.09) = 3262 - Principle = 39505-3262= 36244 • Remaining = 69495- 33251= 36244

  37. Summary

  38. Summary • Find Annuity Payment; PMT(C) • Annuity time periods (n): • Find Annuity Interest Rate(IRR): try and error • Perpetuities • PV= C/i or • PV = C1/(i-g) with Growth • Capital Budgeting: • NPV • FV • Payback method (finding number of periods) • Yiled to maturity • Real Future value • Loan amortization

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