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### Time value

1. 1. The Time Value of MoneyThe Time Value of Money Lohithkumar B
2. 2. What is Time Value?What is Time Value? We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return In other words, “a dollar received today is worth more than a dollar to be received tomorrow” That is because today’s dollar can be invested so that we have more than one dollar tomorrow January 30, 2015 © Lohithkumar B
3. 3. Why TIME?Why TIME? TIMETIME allows one the opportunity to postpone consumption and earn INTERESTINTEREST. NOT having the opportunity to earn interest on money is called OPPORTUNITY COST. January 30, 2015 © Lohithkumar B
4. 4. The Time Value of MoneyThe Time Value of Money Which would you rather have -- \$1,000 today\$1,000 today or \$1,000 in 5 years?\$1,000 in 5 years? Obviously, \$1,000 today\$1,000 today. Money received sooner rather than later allows one to use the funds for investment or consumption purposes. This concept is referred to as the TIME VALUE OF MONEYTIME VALUE OF MONEY!! January 30, 2015 © Lohithkumar B
5. 5. The Terminology of Time ValueThe Terminology of Time Value Present Value - An amount of money today, or the current value of a future cash flow Future Value - An amount of money at some future time period Period - A length of time (often a year, but can be a month, week, day, hour, etc.) Interest Rate - The compensation paid to a lender (or saver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate) January 30, 2015 © Lohithkumar B
6. 6. AbbreviationsAbbreviations PV - Present value FV - Future value Pmt - Per period payment amount N - Either the total number of cash flows or the number of a specific period i - The interest rate per period January 30, 2015 © Lohithkumar B
7. 7. TimelinesTimelines 0 1 2 3 4 5 PV FV Today A timeline is a graphical device used to clarify the timing of the cash flows for an investment Each tick represents one time period January 30, 2015 © Lohithkumar B
8. 8. Calculating the Future ValueCalculating the Future Value Suppose that you have an extra \$100 today that you wish to invest for one year. If you can earn 10% per year on your investment, how much will you have in one year? 0 1 2 3 4 5 -100 ? January 30, 2015 © Lohithkumar B
9. 9. January 30, 2015 © Lohithkumar B 9 Single Sum - Future & Present ValueSingle Sum - Future & Present Value Assume can invest PV at interest rate i to receive future sum, FV Similar reasoning leads to Present Value of a Future sum today. 1 2 30 FV1 = (1+i)PV FV3 = (1+i)3 PV PV FV2 = (1+i)2 PV 1 2 30 PV = FV1/(1+i) FV1 PV = FV2/(1+i)2 FV2 PV = FV3/(1+i)3 FV3
10. 10. 1 Calculating the Future Value (cont.)Calculating the Future Value (cont.) Suppose that at the end of year 1 you decide to extend the investment for a second year. How much will you have accumulated at the end of year 2? 0 1 2 3 4 5 -110 ? January 30, 2015 © Lohithkumar B
11. 11. 1 Generalizing the Future ValueGeneralizing the Future Value Recognizing the pattern that is developing, we can generalize the future value calculations as follows: If you extended the investment for a third year, you would have: January 30, 2015 © Lohithkumar B
12. 12. 1 Compound InterestCompound Interest Note from the example that the future value is increasing at an increasing rate In other words, the amount of interest earned each year is increasing • Year 1: \$10 • Year 2: \$11 • Year 3: \$12.10 The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amount January 30, 2015 © Lohithkumar B
13. 13. 1 Compound Interest GraphicallyCompound Interest Graphically 0 500 1000 1500 2000 2500 3000 3500 4000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Years FutureValue 5% 10% 15% 20% 3833.76 1636.65 672.75 265.33 January 30, 2015 © Lohithkumar B
14. 14. 1 The Magic of CompoundingThe Magic of Compounding On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for \$24 worth of beads and other trinkets. Was this a good deal for the Indians? This happened about 371 years ago, so if they could earn 5% per year they would now (in 1997) have: If they could have earned 10% per year, they would now have: \$54,562,898,811,973,500.00 = 24(1.10)371 \$1,743,577,261.65 = 24(1.05)371 That’s about 54,563 Trillion dollars! January 30, 2015 © Lohithkumar B
15. 15. 1 The Magic of Compounding (cont.)The Magic of Compounding (cont.) The Wall Street Journal (17 Jan. 92) says that all of New York city real estate is worth about \$324 billion. Of this amount, Manhattan is about 30%, which is \$97.2 billion At 10%, this is \$54,562 trillion! Our U.S. GNP is only around \$6 trillion per year. So this amount represents about 9,094 years worth of the total economic output of the USA! At 5% it seems the Indians got a bad deal, but if they earned 10% per year, it was the Dutch that got the raw deal Not only that, but it turns out that the Indians really had no claim on Manhattan (then called Manahatta). They lived on Long Island! As a final insult, the British arrived in the 1660’s and unceremoniously tossed out the Dutch settlers. January 30, 2015 © Lohithkumar B
16. 16. 1 Present ValuePresent Value Discounting is the process of translating a future value or a set of future cash flows into a present value. January 30, 2015 © Lohithkumar B
17. 17. 1 Calculating the Present ValueCalculating 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: January 30, 2015 © Lohithkumar B
18. 18. 1 Present Value: An ExamplePresent Value: An Example Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about \$100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal? January 30, 2015 © Lohithkumar B
19. 19. 1 Present Value ExamplePresent Value Example Joann needs to know how large of a deposit to make today so that the money will grow to \$2,500\$2,500 in 55 years. Assume today’s deposit will grow at a compound rate ofcompound rate of 4% annually. 0 1 2 3 4 55 4% PV0 \$2,500\$2,500 January 30, 2015 © Lohithkumar B
20. 20. 2 Present Value Solution Calculation based on general formula: PV0 = FVn / (1+i)n PV0 = \$2,500/(1.04)5 = \$2,054.81 January 30, 2015 © Lohithkumar B
21. 21. 2 Finding “n” or “i” when one knows PV and FV If one invests \$2,000 today and has accumulated \$2,676.45 after exactly five years, what rate of annual compound interest was earned? January 30, 2015 © Lohithkumar B
22. 22. 2 Frequency of Compounding General Formula: FVn = PVPV00(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PVPV00: PV of the Cash Flow today January 30, 2015 © Lohithkumar B
23. 23. 2 Frequency of Compounding Example Suppose you deposit \$1,000 in an account that pays 12% interest, compounded quarterly. How much will be in the account after eight years if there are no withdrawals? PV = \$1,000 i = 12%/4 = 3% per quarter n = 8 x 4 = 32 quarters January 30, 2015 © Lohithkumar B
24. 24. 2 Solution based on formula:Solution based on formula: FV= PV (1 + i)n = 1,000(1.03)32 = 2,575.10 January 30, 2015 © Lohithkumar B
25. 25. 2 January 30, 2015 © Lohithkumar B 25 Annuities 25 Regular or ordinary annuity is a finitefinite set of sequential cash flows, all with the same value A, which has a first cash flow that occurs one period from now. An annuity due is a finitefinite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediatelyimmediately.
26. 26. 2 AnnuitiesAnnuities An annuity is a series of nominally equal payments equally spaced in time Annuities are very common: • Rent • Mortgage payments • Car payment • Pension income The timeline shows an example of a 5-year, \$100 annuity 0 1 2 3 4 5 100 100 100 100 100 January 30, 2015 © Lohithkumar B
27. 27. 2 The Principle of Value AdditivityThe Principle of Value Additivity How do we find the value (PV or FV) of an annuity? First, you must understand the principle of value additivity: • The value of any stream of cash flows is equal to the sum of the values of the components In other words, if we can move the cash flows to the same time period we can simply add them all together to get the total value January 30, 2015 © Lohithkumar B
28. 28. 2 Present Value of an AnnuityPresent Value of an Annuity We can use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components: January 30, 2015 © Lohithkumar B
29. 29. 2 Present Value of an Annuity (cont.)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: 0 1 2 3 4 5 100 100 100 100 100 62.0 968.3 075.1 382.6 490.9 1 379.08 January 30, 2015 © Lohithkumar B
30. 30. 3 Present Value of an Annuity (cont.)Present Value of an Annuity (cont.) Actually, there is no need to take the present value of each cash flow separately We can use a closed-form of the PVA equation instead: January 30, 2015 © Lohithkumar B
31. 31. 3 Present Value of an Annuity (cont.)Present Value of an Annuity (cont.) We can use this equation to find the present value of our example annuity as follows: This equation works for all regular annuities, regardless of the number of payments January 30, 2015 © Lohithkumar B
32. 32. 3 The Future Value of an AnnuityThe Future Value of an Annuity We can also use the principle of value additivity to find the future value of an annuity, by simply summing the future values of each of the components: January 30, 2015 © Lohithkumar B
33. 33. 3 The Future Value of an Annuity (cont.)The Future Value of an Annuity (cont.) Using the example, and assuming a discount rate of 10% per year, we find that the future value is: 100 100 100 100 100 0 1 2 3 4 5 146.41 133.10 121.00 110.00 }= 610.51 at year 5 January 30, 2015 © Lohithkumar B
34. 34. 3 The Future Value of an Annuity (cont.)The Future Value of an Annuity (cont.) Just as we did for the PVA equation, we could instead use a closed-form of the FVA equation: This equation works for all regular annuities, regardless of the number of payments January 30, 2015 © Lohithkumar B
35. 35. 3 The Future Value of an Annuity (cont.)The Future Value of an Annuity (cont.) We can use this equation to find the future value of the example annuity: January 30, 2015 © Lohithkumar B
36. 36. 3 Annuities DueAnnuities Due Thus far, the annuities that we have looked at begin their payments at the end of period 1; these are referred to as regular annuities A 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 0 1 2 3 4 5 100 100 100 100 100 100 100 100 100 1005-period Annuity Due 5-period Regular Annuity January 30, 2015 © Lohithkumar B
37. 37. 3 Present Value of an Annuity DuePresent Value of an Annuity Due We can find the present value of an annuity due in the same way as we did for a regular annuity, with one exception Note from the timeline that, if we ignore the first cash flow, the annuity due looks just like a four-period regular annuity Therefore, we can value an annuity due with: January 30, 2015 © Lohithkumar B
38. 38. 3 Present Value of an Annuity Due (cont.)Present Value of an Annuity Due (cont.) Therefore, the present value of our example annuity due is: Note that this is higher than the PV of the, otherwise equivalent, regular annuity January 30, 2015 © Lohithkumar B
39. 39. 3 Future Value of an Annuity DueFuture Value of an Annuity Due To calculate the FV of an annuity due, we can treat it as regular annuity, and then take it one more period forward: 0 1 2 3 4 5 Pmt Pmt Pmt Pmt Pmt January 30, 2015 © Lohithkumar B
40. 40. 4 Future Value of an Annuity Due (cont.)Future Value of an Annuity Due (cont.) The future value of our example annuity is: Note that this is higher than the future value of the, otherwise equivalent, regular annuity January 30, 2015 © Lohithkumar B
41. 41. 4 Deferred AnnuitiesDeferred Annuities A deferred annuity is the same as any other annuity, except that its payments do not begin until some later period The timeline shows a five-period deferred annuity 0 1 2 3 4 5 100 100 100 100 100 6 7 January 30, 2015 © Lohithkumar B
42. 42. 4 PV of a Deferred AnnuityPV of a Deferred Annuity We can find the present value of a deferred annuity in the same way as any other annuity, with an extra step required Before we can do this however, there is an important rule to understand: When using the PVA equation, the resulting PV is always one period before the first payment occurs January 30, 2015 © Lohithkumar B
43. 43. 4 PV of a Deferred Annuity (cont.)PV of a Deferred Annuity (cont.) To find the PV of a deferred annuity, we first find use the PVA equation, and then discount that result back to period 0 Here we are using a 10% discount rate 0 1 2 3 4 5 0 0 100 100 100 100 100 6 7 PV2 = 379.08 PV0 = 313.29 January 30, 2015 © Lohithkumar B
44. 44. 4 PV of a Deferred Annuity (cont.)PV of a Deferred Annuity (cont.) Step 1: Step 2: January 30, 2015 © Lohithkumar B
45. 45. 4 FV of a Deferred AnnuityFV of a Deferred Annuity The future value of a deferred annuity is calculated in exactly the same way as any other annuity There are no extra steps at all January 30, 2015 © Lohithkumar B
46. 46. 4 Uneven Cash FlowsUneven Cash Flows Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity We can find the present or future value of such a stream by using the principle of value additivity January 30, 2015 © Lohithkumar B
47. 47. 4 Uneven Cash Flows: An Example (1)Uneven Cash Flows: An Example (1) Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment? 0 1 2 3 4 5 100 200 300 January 30, 2015 © Lohithkumar B
48. 48. 4 Uneven Cash Flows: An Example (2)Uneven Cash Flows: An Example (2) Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year? 0 1 2 3 4 5 300 500 700 January 30, 2015 © Lohithkumar B
49. 49. 4 Non-annual CompoundingNon-annual Compounding So far we have assumed that the time period is equal to a year However, there is no reason that a time period can’t be any other length of time We could assume that interest is earned semi- annually, quarterly, monthly, daily, or any other length of time The only change that must be made is to make sure that the rate of interest is adjusted to the period length January 30, 2015 © Lohithkumar B
50. 50. 5 Non-annual Compounding (cont.)Non-annual Compounding (cont.) Suppose that you have \$1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds? January 30, 2015 © Lohithkumar B
51. 51. 5 Non-annual Compounding (cont.)Non-annual Compounding (cont.) To solve this problem, you need to determine which bank will pay you the most interest In other words, at which bank will you have the highest future value? To find out, let’s change our basic FV equation slightly: In this version of the equation ‘m’ is the number of compounding periods per year January 30, 2015 © Lohithkumar B
52. 52. 5 Non-annual Compounding (cont.)Non-annual Compounding (cont.) We can find the FV for each bank as follows: First National Bank: Second National Bank: Third National Bank: Obviously, you should choose the Third National Bank January 30, 2015 © Lohithkumar B
53. 53. 5 January 30, 2015 © Lohithkumar B 53 Perpetuities Perpetuity is a series of constant payments, A, each period forever. 1 2 3 4 5 6 7 A 0 A A A A A A Intuition: Present Value of a perpetuity is the amount that must invested today at the interest rate i to yield a payment of A each year without affecting the value of the initial investment. PVperpetuity = ∑[A/(1+i)t ] = A ∑[1/(1+i)t ] = A/i PV1 = A/(1+r) PV2 = A/(1+r)2 PV3 = A/(1+r)3 PV4 = A/(1+r)4 etc. etc.
54. 54. 5 PerpetuitiesPerpetuities A perpetuity is an annuity that has no definite end, or a stream of cash payments that continues forever. A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely. Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. January 30, 2015 © Lohithkumar B
55. 55. 5 Methods of Calculating Interest  Simple interest: the practice of charging an interest rate only to an initial sum (principal amount).  Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn. January 30, 2015 © Lohithkumar B
56. 56. 5 Simple Interest Formula January 30, 2015 © Lohithkumar B ( ) where = Principal amount = simple interest rate = number of interest periods = total amount accumulated at the end of period F P iP N P i N F N = + \$1,000 (0.08)(\$1,000)(3) \$1,240 F = + =
57. 57. 5 Compound Interest Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn. January 30, 2015 © Lohithkumar B
58. 58. 5 Compound Interest Formula January 30, 2015 © Lohithkumar B 1 2 2 1 0: 1: (1 ) 2: (1 ) (1 ) : (1 )N n P n F P i n F F i P i n N F P i = = = + = = + = + = = + M
59. 59. 5 Compound Interest January 30, 2015 © Lohithkumar B 0 \$1,000 \$1,259.71 1 2 3 3 \$1,000(1 0.08) \$1,259.71 F = + =
60. 60. 6 Amortized loanAmortized loan Installment loan in which the monthly payments are applied first toward reducing the interest balance, and any remaining sum towards the principal balance. As the loan is paid off, a progressively larger portion of the payments goes toward principal and a progressively smaller portion towards the interest. Also called amortizing loan. January 30, 2015 © Lohithkumar B
61. 61. 6 Continuous CompoundingContinuous Compounding There is no reason why we need to stop increasing the compounding frequency at daily We could compound every hour, minute, or second We can also compound every instant (i.e., continuously): Here, F is the future value, P is the present value, r is the annual rate of interest, t is the total number of years, and e is a constant equal to about 2.718 January 30, 2015 © Lohithkumar B
62. 62. 6 Continuous Compounding (cont.)Continuous Compounding (cont.) Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. What is the future value of your \$1,000 investment? This is even better than daily compounding The basic rule of compounding is: The more frequently interest is compounded, the higher the future value January 30, 2015 © Lohithkumar B
63. 63. 6 Continuous Compounding (cont.)Continuous Compounding (cont.) Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. If you plan to leave the money in the account for 5 years, what is the future value of your \$1,000 investment? January 30, 2015 © Lohithkumar B
64. 64. 6 Congratulations! You obviously understand this material. Now try the next problem. The Interactive Exercises are found in book. January 30, 2015 © Lohithkumar B
65. 65. 6 Comparing PV to FVComparing PV to FV Remember, both quantities must be present value amounts or both quantities must be future value amounts in order to be compared. January 30, 2015 © Lohithkumar B
66. 66. 6 How to solve a time value of money problem.. The “value four years from today” is a future value amount. The “expected cash flows of \$100 per year for four years” refers to an annuity of \$100. Since it is a future value problem and there is an annuity, you need to solve for a FUTURE VALUE OF AN ANNUITY. January 30, 2015 © Lohithkumar B