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Time Value of Money Concepts

Time Value of Money Concepts. 6. Time Value of Money. That’s right! A dollar today is more valuable than a dollar to be received in one year. Interest is the rent paid for the use of money over time. Learning Objectives. Explain the difference between simple and compound interest. LO1.

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Time Value of Money Concepts

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  1. Time Value of Money Concepts 6

  2. Time Value of Money That’s right! A dollar today is more valuable than a dollar to be received in one year. Interest is the rent paid for the use of money over time.

  3. Learning Objectives Explain the difference between simple and compound interest. LO1

  4. Interest amount = P × i × n Assume you invest $1,000 at 6% simple interest for 3 years. You would earn $180 interest. ($1,000 × .06 × 3 = $180) (or $60 each year for 3 years) Simple Interest

  5. Compound interest includes interest not only on the initial investment but also on the accumulated interest in previous periods. Compound Interest Principal Interest

  6. Compound Interest Assume we will save $1,000 for three years and earn 6% interest compounded annually. What is the balance in our account at the end of three years?

  7. Compound Interest

  8. Learning Objectives Compute the future value of a single amount. LO2

  9. The future value of a single amount is the amount of money that a dollar will grow to at some point in the future. Assume we will save $1,000 for three years and earn 6% interest compounded annually. Future Value of a Single Amount $1,000.00 × 1.06 = $1,060.00 and $1,060.00 × 1.06 = $1,123.60 and $1,123.60 × 1.06 = $1,191.02

  10. Future Value of a Single Amount Writing in a more efficient way, we can say . . . . $1,000 × 1.06 × 1.06 × 1.06 = $1,191.02 or $1,000 × [1.06]3 = $1,191.02

  11. Future Value of a Single Amount $1,000 × [1.06]3 = $1,191.02 We can generalize this as . . . Number of Compounding Periods FV = PV (1 + i)n Future Value Present Value Interest Rate

  12. Future Value of a Single Amount Find the Future Value of $1 table in your textbook. Find the factor for 6% and 3 periods.

  13. Find the factor for 6% and 3 periods. Solve our problem like this. . . FV = $1,000 × 1.19102 FV = $1,191.02 FV $1 Future Value of a Single Amount

  14. Learning Objectives Compute the present value of a single amount. LO3

  15. Present Value of a Single Amount Instead of asking what is the future value of a current amount, we might want to know what amount we must invest todayto accumulate a known future amount. This is apresent valuequestion. Present value of a single amount is today’s equivalent to a particular amount in the future.

  16. FV (1 + i)n PV = Present Value of a Single Amount Remember our equation? FV = PV (1 + i)n We can solve for PV and get . . . .

  17. Present Value of a Single Amount Find thePresent Value of $1table in your textbook. Hey, it looks familiar!

  18. Present Value of a Single Amount Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time. What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?

  19. Present Value of a Single Amount i = .08, n = 5 Present Value Factor = .68058 $20,000 × .68058 = $13,611.60 If you deposit $13,611.60 now, at 8% annual interest, you will have $20,000 at the end of 5 years.

  20. Learning Objectives Solving for either the interest rate or the number of compounding periods when present value and future value of a single amount are known. LO4

  21. Solving for Other Values FV = PV (1 + i)n Number of Compounding Periods Future Value Present Value Interest Rate There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined.

  22. Determining the Unknown Interest Rate Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? a. 3.5% b. 4.0% c. 4.5% d. 5.0%

  23. Determining the Unknown Interest Rate Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? a. 3.5% b. 4.0% c. 4.5% d. 5.0% Present Value of $1 Table $1,000 = $1,092 × ? $1,000 ÷ $1,092 = .91575 Search the PV of $1 table in row 2 (n=2) for this value.

  24. Accounting Applications of Present Value Techniques—Single Cash Amount Monetary assets and monetary liabilities are valued at the present value of future cash flows. Monetary Assets Monetary Liabilities Money and claims to receive money, the amount which is fixed or determinable Obligations to pay amounts of cash, the amount of which is fixed or determinable

  25. No Explicit Interest Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes. Even though the agreement states it is a noninterest-bearing note, the note does, in fact, include interest. We impute an appropriate interest rate for a loan of this type to use as the interest rate.

  26. Expected Cash Flow Approach Statement of Financial Accounting Concepts No. 7 “Using Cash Flow Information and Present Value in Accounting Measurements” The objective of valuing an asset or liability using present value is to approximate the fair value of that asset or liability.

  27. Learning Objectives Explain the difference between an ordinary annuity and an annuity due. LO5

  28. Basic Annuities An annuity is a series of equal periodic payments.

  29. Ordinary Annuity An annuity with payments at the end of the period is known as an ordinary annuity. End End

  30. Annuity Due An annuity with payments at the beginning of the period is known as an annuity due. Beginning Beginning Beginning

  31. Learning Objectives Compute the future value of both an ordinary annuity and an annuity due. LO6

  32. Future Value of an Ordinary Annuity To find the future value of an ordinary annuity, multiplythe amount of a single payment or receipt by the future value of an ordinary annuity factor.

  33. Future Value of an Ordinary Annuity We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded annually, on all invested funds. What will be the fund balance at the end of 10 years?

  34. Future Value of an Annuity Due To find the future value of an annuity due, multiplythe amount of a single payment or receipt by the future value of an ordinary annuity factor.

  35. Future Value of an Annuity Due Compute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually.

  36. Learning Objectives Compute the present value of an ordinary annuity, an annuity due, and a deferred annuity. LO7

  37. Present Value of an Ordinary Annuity You wish to withdraw $10,000 at the end of each of the next 4 years from a bank account that pays 10% interest compounded annually. How much do you need to invest today to meet this goal?

  38. Today 1 2 3 4 Present Value of an Ordinary Annuity $10,000 $10,000 $10,000 $10,000 PV1 PV2 PV3 PV4

  39. If you invest$31,698.60today you will be able to withdraw $10,000 at the end of each of the next four years. Present Value of an Ordinary Annuity

  40. Present Value of an Ordinary Annuity Can you find this value in the Present Value of Ordinary Annuity of $1 table? More Efficient Computation$10,000 × 3.16986 = $31,698.60

  41. Present Value of an Ordinary Annuity How much must a person 65 years old invest today at 8% interest compounded annually to provide for an annuity of $20,000 at the end of each of the next 15 years? a. $153,981 b. $171,190 c. $167,324 d. $174,680

  42. Present Value of an Ordinary Annuity How much must a person 65 years old invest today at 8% interest compounded annually to provide for an annuity of $20,000 at the end of each of the next 15 years? a. $153,981 b. $171,190 c. $167,324 d. $174,680 PV of Ordinary Annuity $1 Payment $ 20,000.00 PV Factor × 8.55948 Amount $171,189.60

  43. Present Value of an Annuity Due Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually.

  44. Present Value of a Deferred Annuity In a deferred annuity, the first cash flow is expected to occur more than one period after the date of the agreement.

  45. Present Value? $12,500 $12,500 1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2008. If you require a 12% return on your investments, how much are you willing to pay for this investment?

  46. Present Value? $12,500 $12,500 1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2008. If you require a 12% return on your investments, how much are you willing to pay for this investment? • More Efficient Computation • Calculate the PV of the annuity as of the beginning of the annuity period. • Discount the single value amount calculated in (1) to its present value as of today.

  47. Present Value? $12,500 $12,500 1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2008. If you require a 12% return on your investments, how much are you willing to pay for this investment?

  48. Learning Objectives Solve for unknown values in annuity situations involving present value. LO8

  49. Solving for Unknown Values in Present Value Situations In present value problems involving annuities, there are four variables: Present value of an ordinary annuity or Present value of an annuity due The amount of the annuity payment The number of periods The interest rate If you know any three of these, the fourth can be determined.

  50. Present Value $700 Today End ofYear 1 End ofYear 2 End ofYear 3 End ofYear 4 Solving for Unknown Values in Present Value Situations Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made (the annuity amount) to repay the loan in four years?

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