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Time Value of Money Discounted Cash Flow Analysis

Time Value of Money Discounted Cash Flow Analysis. Time Value of Money. A dollar received today is not worth the same amount as a dollar to be received in the future WHY? You should receive Interest on the dollar received today if it is invested. A Simple Example.

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Time Value of Money Discounted Cash Flow Analysis

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  1. Time Value of MoneyDiscounted Cash Flow Analysis

  2. Time Value of Money A dollar received today is not worth the same amount as a dollar to be received in the future WHY? You should receive Interest on the dollar received today if it is invested.

  3. A Simple Example You deposit $100 today in an account that earns 5% interest annually for one year. How much will you have in one year? Value in one year = Current Value + Interest Earned = $100 + 100(.05) = $100(1+.05) = $105 The $100 today has a Future Value of $105 or The $105 next year has a Present Value of $100

  4. Using a Time Line An easy way to represent this is on a time line Time 0 1 year 5% $100 $105 Beginning of First Year End of First year

  5. What would the $100 be worth in 2 years? You would receive interest on the interest you received in the first year (the interest compounds) Value in 2 years = Value in 1 year + interest = $105 + 105(.05)= $105(1+.05) = $110.25 Or substituting $100(1+.05) for $105 = [$100(1+.05)](1+.05) = $100(1+.05)2 =$110.25

  6. On the time line Time 0 1 2 Cash -$100 $105 $110.25 Flow Beginning of year 1 End of Year 1 Beginning of Year 2 End of Year 2

  7. Generalizing the Formula 110.25 = (100)(1+.05)2 This can be written more generally: Let t = The number of periods = 2 i = The interest rate per period =.05 PV = The Present Value = $100 FV = The Future Value = $110.25 FV = PV(1+i)n ($110.25) = ($100)(1 + 0.05)2 This works for any combination of n, i, and PV

  8. Future Value Interest Factor FV = PV(1+i)n (1+i)n is called the Future Value Interest Factor (FVIFi,n) FVIF’s can be found in tables or calculated Interest Rate 4.0 4.5 5.0 5.5 Periods 1 2 3 1.1025 OR (1+.05)2 = 1.1025 Either way original equation can be rewritten: FV = PV(1+i)n = PV(FVIFi,n)

  9. Calculation MethodsFV = PV(1+i)n • Tables using the Future Value Interest Factor (FVIF) • Regular Calculator • Financial Calculator • Spreadsheet

  10. Using the tables FVIF5%,2 = 1.1025 Plugging it into our equation FV = PV(FVIFi,n) FV = $100(1.1025) = $110.25

  11. Using a Regular Calculator • Calculate the FVIF using the yx key (1+.05)2=1.1025 • Proceed as Before Plugging it into our equation FV = PV(FVIFi,n) FV = $100(1.1025) = $110.25

  12. Financial Calculator Financial Calculators have 5 TVM keys N = Number of Periods = 2 I = interest rate per period =5 PV = Present Value = -$100 FV = Future Value =? PMT = Payment per period = 0 After entering the portions of the problem you know, the calculator will provide the answer

  13. Financial Calculator Example On an HP-10B calculator you would enter: 2 N 5 I -100 PV 0 PMT FV and the screen shows 110.25

  14. Spreadsheet Example • Excel has a FV command • Excel command =FV(rate,nper,pmt,pv,type) • =FV(0.05,2,0,100,0) • =$110.25 • notes: • The inputs needed are basically the same as on the financial calculator • Type refers to whether the payment is at the beginning (type =1) or end (type=0) of the year

  15. Practice Problem If you deposit $3,000 today into a CD that pays 4% annually for a period of five years, what will it be worth at the end of the five years? FV = PV(1+i)n = PV(FVIFi,n) FVIF0.04,5 = (1+0.04)5 = 1.216652 FV = $3,000(1+0.04)5=$3,000(1.216652) FV = $3,649.9587

  16. Compounding Interest • Assume that 100 years ago your ancestors invested $5 at 6%. In the first year there would have been $0.30 in interest. • If you took out the interest each year you would have received a total of $0.30(100) or $30 in interest • How much would the $5 be worth if the interest reinvests? 5(1.06)100 = $1,696.51

  17. Compounding Interest • Leaving the interest in the account allows you to earn interest upon the interest. The impact of the interest compounds or increases over time. • The more periods interest is allowed to accumulate, the greater the impact of the compounding will be.

  18. Compounding at Different Rates of Interest $4,338.58 7% $1,696.51 6% $657.51 5%

  19. Calculating Present Value • We just showed that FV=PV(1+i)n • This can be rearranged to find PV given FV, i and n. • Divide both sides by (1+i)n which leaves PV = FV/(1+i)n

  20. Example If you wanted to have $110.25 at the end of two years and could earn 5% interest on any deposits, how much would you need to deposit today? PV = FV/(1+i)n $110.25 PV = = $100 (1+0.05)2

  21. Present Value Interest Factor PV = FV/(1+i)n 1/(1+i)nis called the Present Value Interest Factor (PVIFi,n) PVIF’s can be found in tables or calculated Interest Rate 4.0 4.5 5.0 5.5 Periods 0 1 2 3 0.907029 OR 1/(1+.05)2 = 0.907029 Either way original equation can be rewritten: PV = FV/(1+i)n = FV(PVIFi,n)

  22. Calculation MethodsPV = FV/(1+i)n • Tables using the Present Value Interest Factor (PVIF) • Regular Calculator • Financial Calculator • Spreadsheet

  23. Using the tables PVIF5%,2 = .9070 Plugging it into our equation PV = FV(PVIFi,n) PV = $110.25(0.9070) = $100.00

  24. Using a Regular Calculator • Calculate the PVIF using the yx key (1/(1+.05))2=.9070 Make sure to divide first then square • Proceed as Before Plugging it into our equation PV = FV(PVIFi,n) PV = $110.25(0.9070) = $100.00

  25. Financial Calculator Financial Calculators have 5 TVM keys N = Number of Periods = 2 I = interest rate per period =5 PV = Present Value = ? FV = Future Value =$110.25 PMT = Payment per period = 0 After entering the portions of the problem you know, the calculator will provide the answer

  26. Financial Calculator Example On an HP-10B calculator you would enter: 2 N 5 I 110.25 FV 0 PMT PV and the screen shows -$100.00

  27. Spreadsheet Example • Excel has a PV command • Excel command =PV(rate,nper,pmt,fv,type) • =FV(0.05,2,0,110.25,0) • =-$100.00 • notes: • The inputs needed are basically the same as on the financial calculator • Type refers to whether the payment is at the beginning (type =1) or end (type=0) of the year

  28. Example • Assume you want to have $1,000,000 saved for retirement when you are 65 and you believe that you can earn 10% each year. • How much would you need in the bank today if you were 25?

  29. Put the problem on a time line Age 25 35 45 55 65 Years 0 10 20 30 40 $1,000,000 PV? PVIF40,10% = 1/(1.1)40 = 0.02209493 • PV = 1,000,000/(1+.10)40=1,000,000(.02209493) • PV = $22,094.93

  30. What if you are currently 35? Or 45? If you are 35 you would need PV = $1,000,000/(1+.10)30 = $57,308.55 If you are 45 you would need PV = $1,000,000/(1+.10)20 = $148,643.63 • This process is called discounting(it is the opposite of compounding)

  31. Example 2 • You decide to attend law school after completing your MBA. You believe that you will need $100,000 when you start Law School in three years. How much would you need in the bank today at 7% to have enough for tuition? $100,000/(1.07)3 = $81,629.7878 PVIF7%,3 =.8163 $100,000(.8163) =$81,630

  32. PV and FV Practice Problem • You hope to buy a new car when you graduate in two years, you believe the car will cost $25,000. If you can earn 9% each year, how much would you need to put in the bank today to be able to buy the car in two years? Interest Rate? FV or PV? Number of Periods? PV = $25,000/1.092 PV = $21,041.99

  33. Solving for the interest rate PV = FVt/(1+i)n or PV(1+i)n=FV Rearrange the above equation FV/PV = (1+i)tn (FV/PV)1/n = 1+i (FV/PV)1/n-1 = i

  34. An Example • What interest rate would you need to double your investment of $1,000 over the next five years? 2,000 = 1,000(1+i)5 2,000/1,000 =2 = (1+i)5 2(1/5)= [(1+i)5](1/5)=1+i 1.1468 – 1=.1468

  35. Rule of 72 – A shortcut • How long does it take for a sum of money to double in value from compounding at a given rate? • If the interest rate is between 5% and 20% then the sum will double in approximately 72/r% • If you are earning 8% interest your money would double in approximately 72/8 = 9 years

  36. An Introduction to determining the “Correct” Interest Rate • So far we have just assumed a level of interest rate for our problems. • How should the correct interest rate be determined? • Interest rates are also linked to the level of risk (we will see this in detail later in the semester). Generally, greater risk results in greater return.

  37. Opportunity Cost • An opportunity cost represents the cost of the best foregone alternative. • When calculating Time Value problems the correct rate combines the idea of risk and return and opportunity cost. • Opportunity Cost Rate – the rate of return on the best available alternative investment of equal risk.

  38. Solving for the number of periods FV = PV(1+i)n Rearrange FV/PV = (1+i)n Take the natural log of both sides ln(FV/PV) = n(ln(1+i)) n = ln(FV/PV)/(ln(1+i))

  39. Questions • What happens to the PV of a future sum as the level of interest rate (discount rate) increases (or decreases)? • What happens to the FV as the interest rate increases (or decreases)? • What happens to the PV of a future sum if the number of periods increases (or decreases)? • What happens to the FV of a current sum if the number of periods increases (or decreases)?

  40. Annuities • Annuity: A series of equal payments made over a fixed amount of time. An ordinary annuity makes a payment at the end of each period. • Example A 4 year annuity that makes $100 payments at the end of each year. • Time 0 1 2 3 4 • CF’s 100 100 100 100

  41. Future Value of an Annuity The FV of the annuity is the sum of the FV of each of its payments. Assume 6% a year Time 0 1 2 3 4 100100100100FV of CF 100(1+.06)0=100.00 100(1+.06)1=106.00 100(1+.06)2=112.36 100(1+.06)3=119.10 FV = 437.4616

  42. FV of An Annuity This could also be written FV=100(1+.06)0+100(1+.06)1+100(1+.06)2+ 100(1+.06)3 FV=100[(1+.06)0+(1+.06)1+(1+.06)2+(1+.06)3] or for any n, i, payment, and t

  43. FVIF of an Annuity (FVIFAr,t) • Just like for the FV of a single sum there is a future value interest factor of an annuity This is the FVIFAi,n FVannuity=PMT(FVIFAi,n)

  44. FVIFA The FVIFA can be approximated by FVIFA = [(1+i)n-1]/i=[FVIFi,n-1]/i

  45. Calculation Methods • Tables - Look up the FVIFA FVIFA6%,4 = 4.374616 FV = 100(4.374616) =437.4616 • Regular calculator -Approximate FVIFA FVIFA = [(1+i)t-1]/i FV = 100(4.374616) =437.4616 • Financial Calculator 4 N 6 I 0 PV -100 PMT FV = 437.4616 • Spreadsheet Excel command =FV(rate,nper,pmt,pv,type) Excel command =FV(.06,4,100,0,0)=437.4616

  46. Practice Problem • Your employer has agreed to make yearly contributions of $2,000 to your Roth IRA. Assuming that you have 30 years until you retire, and that your IRA will earn 8% each year, how much will you have in the account when you retire?

  47. Put the problem on a time line Age 35 36 37 64 65 Years 0 1 2 29 30 2,000 2,000 2,000 2,000 FVIFA30,8% = [(1+0.08)30-1]/0.08 =113.28

  48. Alternative Solution Methods • Financial Calculator 30 N 8 I 0 PV -2000 PMT FV = $226,533.42 • Spreadsheet Excel command =FV(rate,nper,pmt,pv,type) Excel command =FV(.08,30,0,-2000,0)=$226,566.42

  49. Practice Problem 2 • Assume you want to have $1,000,000 for retirement at age 65. If you deposit the same amount each year and are 20 years old today how much will you need to deposit each year if you earn 9%? 1,000,000 = PMT(FVIFA45,9%) 1,000,000 = PMT(525.8587345) $1,901.6514 What if you wait until you are 30 to start saving? 1,000,000 = PMT(FVIFA35,9%) PMT = $4,635.83

  50. Present Value of an Annuity The PV of the annuity is the sum of the PV of each of its payments Time 0 1 2 3 4 100100100100 100/(1+.06)1=94.3396 100/(1+.06)2=88.9996 100/(1+.06)3=83.9619 100/(1+.06)4=79.2094 PV = 346.5105

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