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Chapter 6

Chapter 6. Accounting and the Time Value of Money. 1. Basics. Study of the relationship between time and money Money in the future is not worth the same as it is today because if had money today could invest it and earn interest not because of risk or inflation Based on compound interest

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Chapter 6

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  1. Chapter 6 Accounting and the Time Value of Money

  2. 1. Basics • Study of the relationship between time and money • Money in the future is not worth the same as it is today • because if had money today could invest it and earn interest • not because of risk or inflation • Based on compound interest • not simple interest

  3. 1. Basics • Examples of where TVM used in accounting • Notes Receivable & Payable • Leases • Pensions and Other Postretirement Benefits • Long-Term Assets • Shared-Based Compensation • Business Combinations • Disclosures • Environmental Liabilities

  4. 1I. Future Value of Single Sum • The amount a sum of money will grow to in the future assuming compound interest • Can be compute by • formula: • tables: • calculator: TVM keys FV = PV ( 1 + i )n FV = PV x FVIF(n,i) (Table 6-1) FV = future value n = periods PV = present value i = interest rate FVIF = future value interest factor

  5. 1I. Future Value of Single Sum • Example • If you deposit $1,000 today at 5% interest compounded annually, what is the balance after 3 years?

  6. 1I. Future Value of Single Sum • Calculate by hand

  7. 1I. Future Value of Single Sum • Calculate by formulaFV = 1,000 (1 + . 05)3 = 1,000 x 1.15763 = 1,157.63

  8. 1I. Future Value of Single Sum • Calculate by tableFV = 1,000 x Table factor for FVIF(3, .05) = 1,000 x 1.15763 = 1,157.63

  9. 1I. Future Value of Single Sum • Calculate by calculatorClear calculator: 2nd RESET; ENTER; CE|Cand/or: 2nd CLR TVM3 N5 I/Y1,000 +/- PVCPT FV = 1,157.63

  10. 1I. Future Value of Single Sum • Additional example • If you deposit $2,500 at 12% interest compounded quarterly, what is the balance after 5 years? • less than annual compounding so adjust n and i • n = 20 periods • i = 3% 2,500 x 1.80611 = 4,515.28 20N; 3 I/Y; -2500 PV; CPT FV = 4,515.28

  11. 1II. Present Value of Single Sum • Value now of a given amount to be paid or received in the future, assuming compound interest • Can be compute by • formula: • tables: • calculator: TVM keys PV = FV · 1/( 1 + i )n PV = FV x PVIF(n,i) (Table 6-2) FV = future value n = periods PV = present value i = interest rate PVIF = present value interest factor

  12. 1II. Present Value of Single Sum • Example • If you will receive $5,000 in 12 years and the discount rate is 8% compounded annually, what is it worth today?

  13. 1II. Present Value of Single Sum • Calculate by formulaPV = 5,000 · 1/(1 + . 08)12 = 5,000 x .39711 = 1,985.57

  14. 1II. Present Value of Single Sum • Calculate by tablePV = 5,000 x Table factor for PVIF(12, .08) = 5,000 x .39711 = 1,985.57

  15. 1II. Present Value of Single Sum • Calculate by calculatorClear calculator12 N8 I/Y5,000 FVCPT PV = 1,985.57

  16. 1II. Present Value of Single Sum • Additional example • If you receive $1,157.63 in 3 years and the discount rate is 5%, what is it worth today? • n = 3 periods • i = 5% 1,157.63 x .863838 = 1,000.003 N; 5 I/Y; 1157.63 FV; CPT PV = -1,000.00

  17. 1V. Unknown n or i • Example 1 • If you believe receiving $2,000 today or $2,676 in 5 years are equal, what is the interest rate with annual compounding? PV = FV x PVIF(n, i)2,000 = 2,676 x PVIF(5, i)PVIF(5, i) = 2,000/2,676 = .747384find above factor in Table 2: i ≈ 6%5 N; -2,000 PV; 2,676 FV; CPT 1/Y = 6.00%

  18. 1V. Unknown n or i • Example 2 • Same as last problem but assume 10% interest with annual compounding is the appropriate rate and calculate n. PV = FV x PVIF(n, i)2,000 = 2,676 x PVIF(n, 10%)PVIF(n, 10%) = 2,000/2,676 = .747384find above factor in Table 2: n ≈ 3 years 10 I/Y; -2,000 PV; 2,676 FV; CPT N = 3.06 years

  19. V. Annuities • Basics • annuity • a series of equal payments that occur at equal intervals • ordinary annuity • payments occur at the end of the period • annuity due • payments occur at the beginning of the period

  20. V. Annuities • Ordinary annuity – payments at endPresent Value |_____|_____|_____|_____|_____| Year 5 Year 4 Year 3 Year 1 Year 2 Pmt 2 Pmt 4 Pmt 3 Pmt 1 Evaluate PV

  21. V. Annuities • Annuity due – payments at beginningPresent value |_____|_____|_____|_____|_____| Year 5 Year 2 Year 4 Year 1 Year 3 Pmt 1 Pmt 3 Pmt 4 Pmt 2 Evaluate PV

  22. V. Annuities • For Future Valueof an annuity • more difficult • Determine whether the annuity is ordinary or due based on the last period • if evaluate right after last pmt – ordinary • if evaluate one period after last pmt – due • An important part of annuity problems is determining the type of annuity

  23. V. Annuities • Ordinary annuity – payments at endFuture Value |_____|_____|_____|_____|_____| Year 5 Year 4 Year 3 Year 1 Year 2 Pmt 2 Pmt 4 Pmt 3 Pmt 1 Evaluate FV

  24. V. Annuities • Annuity due – payments at beginningFuture Value (evaluate 1 period after last payment) |_____|_____|_____|_____|_____| Year 5 Year 2 Year 4 Year 1 Year 3 Pmt 1 Pmt 3 Pmt 4 Pmt 2 Evaluate FV

  25. V. Annuities • Tables available in book for • Future Value of Ordinary Annuity (Table 6-3) • Present Value of Ordinary Annuity (Table 6-4) • Present Value of Annuity Due (Table 6-5) • So no table for FV of annuity due

  26. V. Annuities • Annuity table factors conversion • to calculate FV of annuity due • look up factor for FV of ordinary annuity for 1 more period and subtract 1.0000 • to calculate PV of annuity due (can use table) • look up factor for PV of ordinary annuity for 1 less period and add 1.0000 • Use calculator • change calculator to annuity due mode • 2nd BEG; 2nd SET; 2ndQUIT • to change back to ordinary annuity mode • 2nd BEG; 2ndCLR WORK; 2nd QUIT (or 2nd RESET)

  27. V1. Future Value of Annuity • Can be calculated by • formula: • table: • calculator: TVM keys (1 + i)n - 1 FVA(ord) = Pmt ----------------- i FVA(ord or due) = Pmtx FVIFA(ord or due) (n, i) FV = future value n = periods PV = present value i= interest rate FVIF = future value interest factor

  28. V1. Future Value of Annuity • Can be calculated by • formula: (1 + i)n - 1 FVA(due) = Pmt --------------- x (1 + i) i FV = future value n = periods PV = present value i= interest rate FVIF = future value interest factor

  29. V1. Future Value of Annuity • Example • Find the FV of a 4 payment, $10,000, ordinary annuity at 10% compounded annually. (You could treat this as 4 FV of single sum problems and would get correct answer but that method is omitted.)

  30. V1. Future Value of Annuity • Calculate by formulaFVA-ord = 10,000 ----------- = 10,000 x 4.6410 = 46,410 (1 + .1)4 - 1 .1

  31. V1. Future Value of Annuity • Calculate by table (Table 6-3)FVA-ord = 10,000 x FVIFA-ord (4, .10) = 10,000 x 4.64100 = 46,410

  32. V1. Future Value of Annuity • Calculate by calculator4 N; 10 I/Y; -10000 PMT; CPT FV46,410

  33. V1. Future Value of Annuity • Additional examples • Find the FV of a $3,000, 15 payment ordinary annuity at 15%.FVA-ord = 3,000 x FVIFA-ord(15, .15) = 3,000 x 47.58041 = 142,74115 N; 15 I/Y; -3000 PMT; CPT FV = 142,741

  34. V1. Future Value of Annuity • Additional examples • Find the FV of a $3,000, 15 payment annuity due at 15%. (table – look up 1 more period -1.0000)FVA-ord= 3,000 x FVIFA-due (15, .15) = 3,000 x 54.71747= 164,1522nd BGN; 2nd SET; 2nd QUIT15 N; 15 I/Y; -3000 PMT; CPT FV = 164,152

  35. VI1. Present Value of Annuity • Can be calculated by • formula: • table: • calculator: TVM keys 1 – (1/(1 + i)n) PVA(ord) = Pmt --------------------- i PVA(ordor due) = Pmtx PVIFA(ordor due) (n, i) FV = future value n = periods PV = present value i= interest rate PVIF = present value interest factor

  36. VI1. Present Value of Annuity • Can be calculated by • formula: 1 – (1/(1 + i)n) PVA(due) = Pmt --------------------- x (1 + i) i FV = future value n = periods PV = present value i= interest rate PVIF = present value interest factor

  37. VI1. Present Value of Annuity • Example • What is the PV of a $3,000, 15 year, ordinary annuity discounted at 10% compounded annually?

  38. VI1. Present Value of Annuity • Calculate by formulaPVA-ord = 3,000 ---------------- = 3,000 x 7.60608 = 22,818 1 – (1/(1 + .10)15 .10

  39. VI1. Present Value of Annuity • Calculate by table (Table 6-4)PVA-ord= 3,000 x PVIFA-ord (15, 10) = 3,000 x 7.60608 = 22,818

  40. VI1. Present Value of Annuity • Calculate by calculator15 N; 10 I/Y; -3000 PMT; CPT PV22,818

  41. VI1. Present Value of Annuity • Additional examples • Find the PV of a $3,000, 15 payment annuity due discounted at 15%.PVA-due= 3,000 x PVIFA-due (15, .15) = 3,000 x 6.72488= 20,1752nd BGN; 2nd SET; 2nd QUIT15 N; 15 I/Y; -3000 PMT; CPT PV = 20,173

  42. VI1. Present Value of Annuity • Additional examples • If you were to be paid $1,800 every 6 months (at the end of the period) for 5 years, what is it worth today discounted at 12%? PVA-ord= 1,800 x PVIFA-ord(10, .06) = 1,800 x 7.36009= 13,24810 N; 6 I/Y; -1800 PMT; CPT PV = 13,248

  43. VI1. Present Value of Annuity • Additional examples • If you consider receiving $12,300 today or $2,000 at the end of each year for 10 years equal, what is the interest rate? 12,300A-ord = 2,000 x PVIFA-ord(10, i)PVIFA-ord(10, i) = 12,300/2,000 = 6.15000 i ≈ 10%10 N; -2000 PMT; PV = 12300; CPT I/Y = 9.98%

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