Chapter 9. Time Value of Money

Transcription

1 Chapter 9 Time Value of Money

2 Intro Congratulations!!! You have won a cash prize! You have two payment options: A - Receive $10,000 now OR B - Receive $10,000 in three years. Which option would you choose?

3 In general, normally most people would choose to receive the $10,000 now.

4 Rule: All things being equal, it is better to have money now rather than later

5 Intro But, why is that? Aren t they the same? A BD 20 Today A BD 20 after 3 years

6 Although the bill is the same, you can do much more with the money if you have it now. Over time you can earn more interest on your money

7 Intro If you receive the $10,000 today, you can increase the future value of your money by investing and earning money over time. However, if you chose Option B, whatever you receive is actually the future value.

8 Can we calculate how much exactly your $10,000 would worth in the future?

9 Future Value Let s assume you choose option A, you simply deposited it in a saving account at an annual rate of 4.5%. How much would the $10,000 worth at the end of the first year?

10 Future Value Let s assume you choose option A, you simply deposited it in a saving account at an annual rate of 4.5%. How much would the $10,000 worth at the end of the first day? 10,000 + (10,000*4.5%) = $10,450 OR 10,000 ( %) = $10,450

11 Future Value If you left your amount untouched one more year, how much would it worth? 10,450 + (10,450*4.5%) = $10, Mathematically, It is possible to to rewrite the formula: 10,450 ( %) = $10, ,000 (1+ 4.5%) (1+4.5%) = $10, ,000 (1+4.5)^2 = $10, Interest you receive in year 1 Interest you receive in year 2

12 Future Value Now, can you calculate how much the $10,000 worth after 3 years?

13 Future Value Now, can you calculate how much the $10,000 worth after 3 years? 10,000 (1+4.5)^3 = $11,411.66

14 Future Value General Formula: where: FV = Future Value PV = Present Value i = interest rate n = period Remember, it is compounding of Interest

15 Future Value Using the financial calculator to calculate future values: Use CMPD function: (solving the previous problem) n = 3 PV = 10,000 FV =? I = 4.5% PMT = 0

16 Future Value Question 1: ( without the use of calculator) You invest $4,000 for three years at 7 percent. 1. What is the value of your investment after one year? 2. What is the value of your investment after two years? 3. What is the value of your investment after three years? 4. Confirm that your final answer is correct by using the calculator function.

17 Future Value Question 5: If you invest $19,500 today, how much will you have: a. In 12 years at 11 percent? b. In 18 years at 7 percent? c. In 25 years at 8 percent? d. In 10 years at 6 percent (compounded semiannually)?

18 Present Value If you received $10,000 today, the present value would off course be $10,000. Present value: Is what your investment gives you now if you were to spend it today.

19 Present Value Let s assume, you are planning to study MBA after three years. The MBA course would cost $10,000. How much you should invest today? In other words, how much does the needed $10,000 worth today?

20 Present Value To calculate the present value you must subtract the (hypothetical) accumulated interest from the $10,000. To achieve this, we can discount the future payment amount ($10,000) by the interest rate for the period. In essence, all you are doing is rearranging the future value equation above so that you may solve for P.

21 Present Value Hence, The formula is as following: Remember, in present values, we discount future values

22 Present Value So, back to your MBA. How much should you invest today? PV =? FV = 10,000 i = 4.5% n = 3

23 Present Value So, back to your MBA. How much should you invest today? PV =? FV = 10,000 i = 4.5% n = 3 PV = (10,000)/ ( )^3 = $8,762.9

24 Present Value Using the financial calculator to calculate future values: Use CMPD function: (solving the previous problem) n = 3 PV =? FV = 10,000 I = 4.5% PMT = 0

25 Present Value Say you could receive either $15,000 today or $18,000 in four years. Which would you choose if interest rates are currently 4%?

26 Present Value Say you could receive either $15,000 today or $18,000 in four years. Which would you choose if interest rates are currently 4%? we should compare between Present values: $15,000 today The present value of $18,000 that you will receive in four years? PV = 18,000/ (1+0.04)^4 = $15,386.48

27 Present Value Question 2 What is the present value of: a. $7,900 in 10 years at 11 percent? b. $16,600 in 5 years at 9 percent? c. $26,000 in 14 years at 6 percent?

28 Present Value What is the present value of $270,000 to be received after 40 years with a 19 percent discount rate? b. Would the present value of the funds in part a be enough to buy a $1,300 concert ticket?

29 Present Value Your aunt offers you a choice of $20,100 in 20 years or $870 today. If money is discounted at 17 percent, which should you choose?

30 Present Value How much would you have to invest today to receive: a. $15,000 in 8 years at 10 percent? b. $20,000 in 12 years at 13 percent? c. $6,000 each year for 10 years at 9 percent? d. $50,000 each year for 50 years at 7 percent?

31 Present Value Your father offers you a choice of $105,000 in 12 years or $47,000 today. a. If money is discounted at 8 percent, which should you choose? b. If money is still discounted at 8 percent, but your choice is between $105,000 in 9 years or $47,000 today, which should you choose?

32 Interest Rates Given, the two formulas we have learnt, can you find the interest rates if you were asked?

33 Interest Rates We find interest rates, if we know the present and the future values of the investment, but we wish to know the return on the investment we have made.

34 Interest Rates

35 Interest Rates Question: Franklin Templeton has just invested $9,260 for his son (age one). This money will be used for his son s education 18 years from now. He calculates that he will need $71,231 by the time the boy goes to school. What rate of return will Mr. Templeton need in order to achieve this goal?

36 Investment Period We shall use the following formula to find the period to invest a specified amount to and receive returns.

37 Investment Period Question: At a growth (interest) rate of 15 percent annually, how long will it take for a sum to double? To triple? Select the year that is closest to the correct answer.

38 Future Value of Annuity

39 Future Value of Annuity At some point in your life, you may have had to make a series of fixed payments over a period of time - such as rent or car payments

40 Future Value of Annuity OR have received a series of payments over a period of time, such as bond coupons.

41 These received or paid fixed amounts over a period of time are called annuities

42 Future Value of Annuity Formal definition: Annuities are series of consecutive payments or receipts of equal amounts. There are two types of annuities: Ordinary Annuity: Payments are required at the end of each period. Annuity Due: Payments are required at the beginning of each period.

43 Future Value of Annuity Being able to calculate the future value of annuity would help: 1. The inventor to recognize how much his periodic investment would worth in the future. 2. The borrower to know the total cost of a loan based on the required periodic payment he makes.

44 Future Value of Annuity Ordinary Annuity Example: Assume you know you are going to receive $1000 every year for the coming five years. Also, you would invest every received amount at interest 5%. How much you would have at the end of the five-year period?

45 Future Value of Annuity Ordinary Annuity Example: Assume you know you are going to receive $1000 every year for the coming five years. Also, you would invest every received amount at interest 5%. How much you would have at the end of the five-year period?

46 Future Value of Annuity However, the previous calculation can be found via the following formula: Where: FVa = The future value of annuity, A = fixed periodic payment, i = interest rate, n = period

47 Future Value of Annuity However, the previous calculation can be found via the following formula: = = $

48 Future Value of Annuity Using the financial calculator to calculate future values: Use CMPD function: (solving the previous problem) n = 5 PV = 0 FV =? I =5% PMT = 1000

49 Present Value of Annuity If you would like to determine today's value of a future payment series, you need to use the formula that calculates the present value of an ordinary annuity. Meaning, the process is reversed. Hence, the payments are discounted back to the present. Let s go back to our example. You want to know how much would the five $1000 future payments are worth today.

50 Present Value of Annuity Let s go back to our example. You want to know how much would the five $1000 future payments are worth today.

51 Present Value of Annuity Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many future payments. As such, we can use a mathematical shortcut for PV of an ordinary annuity.

52 Present Value of Annuity Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many future payments. As such, we can use a mathematical shortcut for PV of an ordinary annuity. = $

53 Present Value of Annuity Using the financial calculator to calculate future values: Use CMPD function: (solving the previous problem) n = 5 PV =? FV = 0 I =5% PMT = 1000

54 Present Value of Annuity Question: Your grandfather has offered you a choice of one of the three following alternatives: $10,000 now; $4,800 a year for eight years; or $56,000 at the end of eight years. Assuming you could earn 9 percent annually, which alternative should you choose? If you could earn 10 percent annually, would you still choose the same alternative?

55 Determining the Annuity Value Annuity Equaling a Future value Imagine you want to take a trip around the world in four years time. The trip would cost you $20,000. How much you should save every year? Your bank offers you a 4% interest rate

56 Determining the Annuity Value Annuity Equaling a Future value Imagine you want to take a trip around the world in four years time. The trip would cost you $20,000. How much you should save every year? Your bank offers you a 4% interest rate Meaning, you have to save $ every year at 4% for the coming 4 years to be able to take the trip.

57 Determining the Annuity Value Annuity Equaling a Present value Suppose your wealthy uncle present you with $10,000 now to help you get through the next four years of college.if you are able to earn 6% on deposited funds, what is the value of the equal payments can you withdraw at the end of each year for four years?

58 Determining the Annuity Value Annuity Equaling a Present value Suppose your wealthy uncle present you with $10,000 now to help you get through the next four years of college.if you are able to earn 6% on deposited funds, what is the value of the equal payments can you withdraw at the end of each year for four years?

59 Determining the Annuity Value Annuity Equaling a Present value The table below shows the relationship of present value to annuity.

60 Determining the Annuity Value Loan Amortization: The same process is used to indicate necessary repayment on a loan. Suppose a homeowner signs an $80,000 mortgage to be repaid over 20 years at 8% interest rate. How much must she pay annually to payoff the loan?

61 Determining the Annuity Value Loan Amortization: The same process is used to indicate necessary repayment on a loan. Suppose a homeowner signs an $80,000 mortgage to be repaid over 20 years at 8% interest rate. How much must she pay annually to payoff the loan?

62 Determining the Annuity Value Loan Amortization: The same process is used to indicate necessary repayment on a loan. Suppose a homeowner signs an $80,000 mortgage to be repaid over 20 years at 8% interest rate. How much must she pay annually to payoff the loan? $8, Principle Interest But how much exactly?

63 Determining the Annuity Value Loan Amortization: Payoff table for loan

64 Determining the Annuity Value Example: Betty Bronson has just retired after 25 years with the electric company. Her total pension funds have an accumulated value of $320,000, and her life expectancy is 20 more years. Her pension fund manager assumes he can earn a 11 percent return on her assets. What will be her yearly annuity for the next 20 years?

65 Determining the Annuity Value Example: If your uncle borrows $51,000 from the bank at 9 percent interest over the seven-year life of the loan, what equal annual payments must be made to discharge the loan, plus pay the bank its required rate of interest (round to the nearest dollar)? How much of his first payment will be applied to interest? To principal? How much of his second payment will be applied to each?

66 Patterns of Payments with a Deferred Annuity Not every situation will involve a single amount or an annuity. Let s consider the following: Assume your uncle would support your university tuition for the coming three years. He would pay you $1,000, $2,000, and $3,000 respectively at 8% interest rate. Calculate the present value. n Payment PV Factor PV 1 $1, $ , , , ,382 Total PV = $5,022

67 Patterns of Payments with a Deferred Annuity What if the amounts paid/ received include a combination of single amounts and annuity. For example, if the annuity will be paid at some time in the future, it is referred to as a deferred annuity. It requires a special treatment.

68 Patterns of Payments with a Deferred Annuity Reffer to the previous example but with an annuity of $1,000 that will be paid at the end of each year from the fourth to the eighth year. With a discount rate of 8%. What is the present value of the cash flow? 1. $1, $2, $3, $1, $1, $1, $1, $1,000 Present Value = $5,022 Five-year annuity We have to calculate the PV of the annuities.

69 Total PV = $5, ,170 = $8,192 Patterns of Payments with a Deferred Annuity Reffer to the previous example but with an annuity of $1,000 that will be paid at the end of each year from the fourth to the eighth year. With a discount rate of 8%. What is the present value of the cash flow? 1. $1, $2, $3, $1, $1, $1, $1, $1,000 Present Value = $5,022 Five-year annuity Solving for the five-year annuity, n=5, i = 8%, FV=0, PMT = PV = $3,993. Note that the present value is calculated at year three, because the first annuity payment is at year 4. So, we have to take the PV to time =0, i = 8%, FV = 3,993, PMT =0, n= 3 PV = $3,170

70 Annuities Due Annuities due: Fixed payments or receipts that come at the beginning of each period. As it shown below:

71 Annuities Due: Present Value Assume you receive 5 due annuities.

72 Annuities Due: Present Value The following formula can be used to calculate present value of annuities due. Where C = A = Annuity payments.

73 Annuities Due: Future Value For the same example, let s assume you would like to calculate the future value of annuities.

74 Annuities Due: Present Value The following formula can be used to calculate future value of annuities due. Where C = A = Annuity payments.

75 Annuities Due Using the financial calculator to calculate annuities : Use CMPD function, make sure you choose the beginning of year settings.

76 End of Chapter 9