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Chapter 3FM

Time value of money

HURRUMAA
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Chapter 3FM

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  1. CHAPTER 3THE TIME VALUE OF MONEY

  2. Learning Objectives • Understand the concept of the time value of money. 3.1 Why money has a time value? 3.2 Future Value of a single sum amount 3.3 Present value of a single sum amount 3.4 Future value of annuity 3.5 Present value of annuity 3.6 Special Case Annuities • Perpetuities Vs Deferred Annuity 3.7 Uneven Cash Flow Streams 3.8 More frequent compounding • Compounding for less than a year • Effective annual rate and the nominal Rate

  3. The Time Value of Money Would you prefer to have 1 million dollar now or 1 million dollar 10 years from now? Of course, we would all prefer the money now! This illustrates that there is an intrinsic monetary value attached to time.

  4. What Does Time Value of Money Mean? • The concept that money available today is worth more than the same amount of money in the future • Thus, a money received today is worth more than a money received tomorrow, why? • This is because that • a money received today can be invested to earn interest • Due to money's potential to grow in value over time. • Because of this potential, money that's available in the present is considered more valuable than the same amount in the future

  5. Take a simple example: • You need to sell your car and you receive offers from three different buyers. • The first offer gives 4,000 Birr to be paid now • The second offer gives 4,100 Birr to be paid one year from now • The third offer gives 4,600 Birr to be paid after five years. • Assume that the second and third offers have no credit risk. • The risk free interest rate is 5%. • Which offer would you accept?

  6. Simple interest is money paid only on the principal. Rate of interest is the percent charged or earned. Time that the money is borrowed or invested (in years). Principal is the amount of money borrowed or invested. Simple Interest and Compound Interest Simple Interest Interest paid (earned) on only the original amount borrowed (lent) I = Prt

  7. COMPOUND INTEREST • Implies that interest paid (earned) on a loan (an investment) is periodically added to the principal • Interest is earned on interest as well as the initial principal • Formulae for a single future amount: • “FV”is the amount of final dollar value, • “P” is the principal, • “r” is the rate of interest, and • “t”is the number of compounding periods per year.

  8. Compound Interest with non-Annual Periods • When compounding occurs more than once in a year ─Possible compounding timings: ─Semi-annually ─Quarterly ─Monthly ─Weekly ─Daily • Formulae for a single future amount can be rearranged as: FV = P × (1 + i/m)mn • Where, m = No. of times compounding occurs during a year

  9. Uses of Time Value of Money • Time Value of Money, or TVM, is a concept that is used in all aspects of finance including: • Bond valuation • Stock valuation • Accept/reject decisions for project management • Financial analysis of firms

  10. Types of TVM Calculations • There are 2 types of TVM calculations • 1.Future Value • Future value of a lump sum • Future value of cash flow stream • Future value of annuities 2. Present value • Present value of a lump sum • Present of cash flow stream • Present value of annuities

  11. Future Value of a Lump Sum • Future value determines the amount that a sum of money invested today will grow to in a given period of time • You can think of future value as the opposite of present value • The process of finding a future value is called “compounding” FVn = PV(1 + i)n • FV = the future value of the investment at the end of n year • i = the annual interest rate • PV = the present value, in today’s dollars, of a sum of money

  12. Example of FV of a Lump Sum How much money will you have in 5 years if you invest $100 today at a 10% rate of return? • Draw a timeline • Write out the formula using symbols: FVt = CF0 * (1+r)t i = 10% ? $100 2 0 1 3 4 5

  13. Example of FV of a Lump Sum • Substitute the numbers into the formula: FV = $100 * (1+.1)5 • Solve for the future value: FV = $161.05

  14. Future Value of a Cash Flow Stream • The future value of a cash flow stream is equal to the sum of the future values of the individual cash flows. • The FV of a cash flow stream can also be found by taking the PV of that same stream and finding the FV of that lump sum using the appropriate rate of return for the appropriate number of periods.

  15. The following equation can be used to find the Future Value of a Cash Flow Stream at the end of year t. FVt Where • FVt = the Future Value of the Cash Flow Stream at the end of year t, • CFt = the cash flow which occurs at the end of year t, • r = the discount rate, • t = the year, which ranges from zero to n, and • n = the last year in which a cash flow occurs

  16. For example, consider an investment which promises to pay $100 one year from now, $300 two years from now, 500 three years from now and $1,000 four years from now. • How much will be the future value of the cash flow streams at the end of year 4, given that the interest rate is 10%,? • The Future Value at the end of year 4 of the investment can be found as follows in the next slide:

  17. $100 $300 $500 $1000 • Draw a timeline: 4 1 2 3 0 ? i = 10% ? ? ?

  18. Write out the formula using symbols n FV = S [CFt * (1+r)n-t] t=0 OR FV = [CF1*(1+r)n-1]+[CF2*(1+r)n-2]+ [CF3*(1+r)n-3] + [CF4*(1+r)n-4] • Substitute the appropriate numbers: FV = [$100*(1+.1)3]+[$300*(1+.1)2]+[$500*(1+.1)1] +[$1,000*(1+.1)0] • Solve for the Future Value: FV = $133.10 + $363.00 + $550.00 + $1000 FV = $2,046.10

  19. The Present Value of Money

  20. What does mean present value of money? • It is a process of discounting the future value of money to obtain its value at zero time period (at present) • Present values tell you the amount you must invest today to accumulate a certain amount at some future time

  21. To determine present values, we need to know: • The amount of money to be received in the future • The interest rate to be earned on the deposit • The number of years the money will be invested Discounting and Compounding The mechanism for factoring in the present value of money element is the discount rate. • The process of finding the equivalent value today of a future cash flow is known as discounting. • Compounding converts present cash flows into future cash flows.

  22. Calculating the Present Value • So far, we have seen how to calculate the future value of an investment • But we can turn this around to find the amount that needs to be invested to achieve some desired future value: • Using the Present Value Table • Present value interest factor (PVIF): a factor multiplied by a future value to determine the present value of that amount (PV = FV(PVIFA) • Notice that PVIF is lower as the number of years increases and as the interest rate increases • It can also be calculated using a financial calculator PV = [FV/(1+r)n]

  23. Cash Flow Types and Discounting Method • There are 4 types of cash flows - • single cash flows (Lump sum cash flows), • Cash flows stream • Annuities, • Perpetuities

  24. Single Cash Flows • A single cash flow is a single cash flow in a specified future time period. • Cash Flow: CFt _______________________| Time Period: t • The present value of this cash flow is- PV of Single Cash Flow =

  25. Example: Present Value of a single cash flow: • You would like to accumulate Birr 50,000 in five years by making a single investment today. You believe you can achieve a return from your investment of 8% annually. • What is the amount that you need to invest today to achieve your goal?

  26. PV = 50,000(PVIF)5,8% =50,000 x 0.681 = 34050 Or We can use table

  27. Valuing a Stream of Cash Flows • Valuing a lump sum (single) amount is easy to evaluate because there is one cash flow. • What do we need to do if there are multiple cash flow? • Equal Cash Flows: Annuity or Perpetuity • Unequal/Uneven Cash Flows

  28. 0 1 2 3 Valuing a Stream of Cash Flows ? 1000 2000 3000 • Uneven cash flows exist when there are different cash flow streams each year • Treat each cash flow as a Single Sum problem and add the PV amounts together.

  29. What is the present value of the preceding cash flow stream using a 12% discount rate? = Yr 1 $1,000 / (1.12)1 = $ 893 Yr 2 $2,000 / (1.12)2 = 1,594 Yr 3 $3,000 / (1.12)3 = 2,135 $4,622

  30. Annuities: Ordinary and Annuity Due • Annuity −A series of equal payments for a specified number of periods. Basic Types of Annuities: • Ordinary Annuity: An annuity in which the payments occur at the end of each period • Annuity Due: An annuity in which the payments occur at the beginning of each period. • A deferred annuity is one where the payments do not commence until period of times have elapsed • A perpetuity is an annuity in which the payments continue indefinitely.

  31. Where CF = Cash flow per period r = interest rate n = number of payments

  32. Example: What amount will accumulate  if we deposit $5,000 at the end of each year for the next 5 years?  Assume an interest of 6% compounded annually PV = 5,000 i = .06 n = 5

  33. Annuity Due • A form of annuity where periodic receipts or payments are made at the beginning of the period and one period of the annuity term remains after the last payment.

  34. The Future Value of an Annuity Due is identical to an ordinary annuity except that  each payment occurs at the beginning of a period rather than at the end. • Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (1 + i). Where: FVad = Future Value of an Annuity Due FVoa = Future Value of an Ordinary Annuity i = Interest Rate Per Period

  35. Example: What amount will accumulate  if we deposit $5,000 at the beginning of each year for the next 5 years?  Assume an interest of 6% compounded annually. PV = 5,000, i = .06, n = 5 FVoa = 28,185.46 (1.06) = 29,876.59

  36. Example: What amount will accumulate  if we deposit $1,000 at the beginning of each year for the next 5 years?  Assume an interest of 5% compounded annually. • PV = 1,000i = .05n = 5

  37. Or

  38. Application • The future value annuity formula can be applied in different contexts. • Its important applications are • To know how much we have in the future • FV= A(1+r)n • To know how much to save annually • To find out the interest rate • To know how long should wait to get the accumulated money

  39. 0 1 2 3 4 5 Present value of Annuities • The present value of an annuity is the value now of a series of equal amounts to be received (or paid out) for some specified number of periods in the future. • It is computed by discounting each of the equal periodic amounts. • An annuity is a series of nominally equal payments equally spaced in time • The timeline shows an example of a 5-year, $100 annuity 100 100 100 100 100

  40. Present Value of an Annuity • The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. • We can use the principle of value additively to find the present value of an annuity, by simply summing the present values of each of the components:

  41. Present Value of an Annuity (cont.) • Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years] • Thus, there is no need to take the present value of each cash flow separately • The closed-form of the PVA equation is:

  42. 0 1 2 3 4 5 Present Value of an Annuity (cont.) • Using the example, and assuming a discount rate of 10% per year, we find that the present value is: 62.09 68.30 75.13 82.64 90.91 100 100 100 100 100 379.08

  43. We can use this equation to find the present value of our annuity example as follows: • This equation works for all regular (ordinary) annuities, regardless of the number of payments

  44. 0 1 2 3 4 5 Present Value of an Ordinary Annuity • The annuities that begin their payments at the end of period are referred as regular annuities (ordinary annuities) Present Value of an Annuity Due • An annuity due is the same as a regular annuity, except that its cash flows occur at the beginning of the period rather than at the end. 5-period Annuity Due 100 100 100 100 100 100 100 100 100 100 5-period Regular Annuity

  45. Present Value of an Annuity Due • The present value of an annuity due, sometimes referred to as an immediate annuity, is used to calculate a series of periodic payments, or cash flows, that start immediately • Therefore, we can value an annuity due with: • Note that this is higher than the PV of the regular annuity

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