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Time Value of Money Concepts

Time Value of Money Concepts. Chapter 6. Simple Interest. 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 ) Balance after 3 years is $1000 + $180 = $1,180.

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Time Value of Money Concepts

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  1. Time Value of Money Concepts Chapter 6

  2. Simple Interest 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) Balance after 3 years is $1000 + $180 = $1,180

  3. Assume we deposit $1,000 in a bank that earns 6% interest compounded annually. Compound Interest What is the balance in our account at the end of three years?

  4. Compound Interest

  5. 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 deposit $1,000 for three years that earns 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

  6. Writing in a more efficient way, we can say . . . . $1,191.02 = $1,000 × [1.06]3 Future Value of a Single Amount Number of Compounding Periods FV = PV × (1 + i)n Future Value Amount Invested at the Beginning of the Period Interest Rate

  7. Future Value of a Single Amount Excel Solution =FV(6%, 3, 0, 1000)

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

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

  10. 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? Present Value of a Single Amount

  11. Present Value of a Single Amount Excel Solution: =PV(8%, 5, 0, 20000) If you deposit $13,611.60 now, at8% annual interest, you will have$20,000 at the end of 5 years.

  12. 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.

  13. 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% Determining the UnknownInterest Rate Excel Solution =RATE(2, 0, -1000, 1092)

  14. Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes. Accounting Applications of Present Value Techniques—Single Cash Amount 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 noninterest-bearing notes.

  15. 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. The present value of the asset or liability is obtained by discounting the cash flow(s) using the company’s credit-adjusted risk-free rate of interest.

  16. Examples: Valuing a future asset/liability ABC Company sells services to XYZ company in the amount of $100,000, with payment to be made five years after the date of sale. The appropriate discount rate for both companies is 10%. How do ABC and XYZ record this transaction: On the date of sale? At the end of years 1-4? At the end of year 5 when payment is due?

  17. Basic Annuities An annuity is a seriesof equal periodic payments. Period 1 Period 2 Period 3 Period 4 $10,000 $10,000 $10,000 $10,000

  18. Today 1 2 3 4 Ordinary Annuity An annuity with payments at the end of the period is known as an ordinary annuity. $10,000 $10,000 $10,000 $10,000 End of year 1 End of year 2 End of year 3 End of year 4

  19. An annuity with payments at the beginning of the period is known as an annuity due. Today 1 2 3 4 Annuity Due $10,000 $10,000 $10,000 $10,000 Beginning of year 1 Beginning of year 2 Beginning of year 3 Beginning of year 4

  20. We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded interest annually, on all invested funds. What will be the fund balance at the end of 10 years? Excel Solution: =FV(8%, 10, 2500) = $36,216 How would the solution/answer change if interest were compounded quarterly or monthly? Future Value of an Ordinary Annuity

  21. Compute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually. Excel Solution: =FV(6%, 4, 10000, 0, 1) = $46,371 Future Value of an Annuity Due

  22. You wish to withdraw $10,000 at the endof 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? Present Value of an Ordinary Annuity

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

  24. If you invest $31,699 today you will be able to withdraw $10,000 at the end of each of the next four years. 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? Excel Solution: =PV(10%, 4, 10000) = $31,699

  25. Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. Excel Solution: =PV(6%, 4, 10000, 0, 1) = $36,730 Present Value of an Annuity Due

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

  27. Present Value? $12,500 $12,500 1/1/13 12/31/13 12/31/14 12/31/15 12/31/16 12/31/17 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2015. If you require a 12% return on your investments, how much are you willing to pay for this investment?

  28. Present Value? $12,500 $12,500 1/1/13 12/31/13 12/31/14 12/31/15 12/31/16 12/31/17 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2015. If you require a 12% return on your investments, how much are you willing to pay for this investment? • Correct Solution Process • Calculate the present value of the annuity as of the beginning of the annuity period. Excel: =PV(12%, 2, 12500) = $21,126 • Discount the single value amount calculated in (1) to its present value as of today. Excel: =PV(12%, 2, 0, 21126) = $16,842

  29. In present value problems involvingannuities, there are four variables: Solving for Unknown Values inPresent Value of Annuity Situations 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.

  30. Present Value $700 Today End ofYear 1 End ofYear 2 End ofYear 3 End ofYear 4 Solving for Unknown Valuesin 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?

  31. Solving for Unknown Valuesin 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? Excel Solution: =PMT(8%, 4, 700) = $211.34

  32. Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. Accounting Applications of Present Value Techniques—Annuities Long-term Bonds Long-term Leases Pension Obligations

  33. Valuation of Long-term Leases Certain long-term leases require the recording of an asset and corresponding liability at the present value of future lease payments.

  34. Valuation of Long-term Leases On January 1, 2013, Todd Furniture Company signed a 20-year lease for a new retail showroom. The lease agreement calls for annual payments of $25,000 for 20 years beginning on January 1, 2013. The appropriate rate of interest for this long-term lease is 8%. Calculate the value of the asset acquired and the liability assumed by Todd (the present value of an annuity due at 8% for 20 years). Excel Solution: =PV(8%, 20, 25000, 0, 1) = $265,090 What journal entries are made?

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