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The Time Value of Money

The Time Value of Money. FIN261 Personal Finance Spring 2006 Dana Smith. Outline. Time Value of Money Future Value and Present Value Compounding and Discounting Applications of Time Value of Money Lump sum Evaluating multiple investments Compounding more frequently than annually

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The Time Value of Money

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  1. The Time Value of Money FIN261 Personal Finance Spring 2006 Dana Smith

  2. Outline • Time Value of Money • Future Value and Present Value • Compounding and Discounting • Applications of Time Value of Money • Lump sum • Evaluating multiple investments • Compounding more frequently than annually • Annuities • Evaluate a single investment using required rate of return FIN261 Spring 2006

  3. Time Value of Money • The basic idea behind the concept of time value of money is: • $1 received today is worth more than $1 in the future OR • $1 received in the future is worth less than $1 today • Why? • because interest can be earned on the money • The connecting piece or link between present (today) and future is the interest rate, r or I/Y FIN261 Spring 2006

  4. Time Value of Money • Economic decisions cannot be made without recognizing that cash flows occurring at different times have different values depending on • Time • Interest rates on similar alternatives • Adjustments allow comparisons FIN261 Spring 2006

  5. Future Value and Present Value • Future Value (FV) is what money today will be worth at some point in the future • Present Value (PV) is what money at some point in the future is worth today FIN261 Spring 2006

  6. The value of a lump sum or stream of cash payments at a future point in time: FV = PV x (1+r)n Future Value • Interest rate, r Future Value depends on: Number of periods, n FIN261 Spring 2006

  7. FV4 = $146.41 FV3 = $133.10 FV2 = $121 FV1 = $110 Future Value of $100 (4 Years, 10% Interest ) PV = $100 0 1 2 3 4 End of Year FIN261 Spring 2006

  8. Compounding Compounding: the process of earning interest in each successive year on 1.) the original balance or original principal 2.) past interest payments FIN261 Spring 2006

  9. Year 1: FV1 = $110 • Earns 10% interest on initial $100 • FV1 = $100+$10 = $110 • Earn $10 interest again on $100 principal • Earns $1 on previous year’s interest of $10: $10 x 10% = 1 • FV2 = $110+$10+$1 = $121 Year 2: FV2 = $121 • Earn $10 interest again on $100 principal • Earns $2.10 on previous years’ interest of $21: $21 x 10% = $2.10 • FV3 = $121+$10+$2.10 = $133.10 Year 3: FV3 = $133.10 • Earn $10 interest again on $100 principal • Earns $3.15 on previous years’ interest of $33.10: $33.10 x 10% = $3.31 • FV4 = $133.10+$10+$3.31 = $146.41 Year 4: FV4 = $146.41 Compounding FIN261 Spring 2006

  10. Present Value Today's value of a lump sum or stream of cash payments received at a future point in time: FIN261 Spring 2006

  11. FV4 = $146.41 FV1 = $110 FV2 = $121 FV3 = $133.10 PV = $100 Present Value of $146.41 (4 Years, 10% Interest ) Discounting 0 1 2 3 4 End of Year FIN261 Spring 2006

  12. Discounting Discounting: the process of converting a future cash flow into a present value Discounting: the reverse operation of compounding FIN261 Spring 2006

  13. Lump Sum: Finding PV • Lump sum (single cash flow) decisions • Example: How much would you have to invest today at 10% interest to receive $150,000 in 10 years? • Formula FIN261 Spring 2006

  14. Lump Sum: Business Calculator • TVM third row from top • N is number of periods • I/Y is annual rate of interest (r in formulas) • PV is present value • FV is future value FIN261 Spring 2006

  15. Lump Sum: Finding PV • N = 10 • I/Y = 10 • FV = 150,000 • PV = ? • Computing the present value gives: $57,831.49 FIN261 Spring 2006

  16. Lump Sum: Finding I/Y • Phil Goode wins $3,500 from a local radio station. He estimates that he will need $7,300 in 3 years to backpack across Europe. What must his rate of return (ROR) be on his investments if he wants to meet this financial goal? • N = 3 • PV = -3,500 • FV = 7,300 • I/Y = ? FIN261 Spring 2006

  17. Lump Sum: Finding FV • Xiao wants to put a down payment on an apartment in Beijing within 4 years. How large will such a payment be if he has 22,000 yuan to invest at 17%? • N = 4 • PV = -22,000 • I/Y = 17% • FV = ? FIN261 Spring 2006

  18. Evaluating Multiple Investments • Which one do you buy? • A single share of Axir Inc. is trading at $22.33. Analysts project share prices to be $60 in 5 years. • TackiWall Co. just announced its first IPO. Shares will begin selling at $17. The stock’s estimated share price is $25 in 2 years. • Your uncle wants to retire from his successful business and spend his days on the beach. He is selling ownership in his venture to the tune of $50 a share. In 10 years, shares will be worth $300. FIN261 Spring 2006

  19. Compounding more frequently than annually • Interest is often compounded monthly, quarterly, or semiannually in the real world • Adjustments must be made to calculations and formulas • the number of years is multiplied by the number of compounding periods, m • the annual interest rate is divided by the number of compounding periods, m (or P/Y is adjusted on the business calculator) FIN261 Spring 2006

  20. Compounding Intervals m compounding periods The more frequent the compounding period, the larger the FV! FIN261 Spring 2006

  21. For semiannual compounding, m equals 2: Compounding More Frequently Than Annually FV at end of 2 years of $125,000 deposited at 5% interest FIN261 Spring 2006

  22. For quarterly compounding, m equals 4: Compounding More Frequently Than Annually FV at end of 2 years of $125,000 deposited at 5% interest FIN261 Spring 2006

  23. Compounding more frequently • EXAMPLE: Juan Garza invested $20,000 10 years ago at 12 percent, compounded quarterly. How much has he accumulated? • Using formula: • Using business calculator FIN261 Spring 2006

  24. FV at end of 2 years of $125,000 at 5 % annual interest, compounded continuously: Continuous Compounding • In extreme case, interest compounded continuously: FV = PV x (e r x n) FIN261 Spring 2006

  25. Effective annual rate: the annual rate actually paid or earned Stated Rate vs. the Effective Annual Rate Stated rate: the contractual annual rate charged by lender or promised by borrower FIN261 Spring 2006

  26. Stated Rate vs. the Effective Annual Rate • FV of $100 at end of 1 year, invested at 5% stated annual interest, compounded: • Annually: FV = $100 (1.05)1 = $105 • Semiannually: FV = $100 (1.025)2 = $105.06 • Quarterly: FV = $100 (1.0125)4 = $105.09 Stated rate of 5% does not change.What about the effective rate? FIN261 Spring 2006

  27. For annual compounding, effective = stated • For semiannual compounding • For quarterly compounding Effective Rates: Always Greater Than or Equal to Stated Rates FIN261 Spring 2006

  28. Special considerations – Effective Annual Rate (EAR) • Effective Annual Rate (EAR) • I CONV • EXAMPLE: What rate does a depositor actually earn if a 6% annual interest rate is compounded monthly? • Using the business calculator FIN261 Spring 2006

  29. Annuity • Annuity: • a stream or series of equal payments, equally spaced • The payments are assumed to be received at the end of each period • A good example of an annuity is an installment loan, where loan payments are paid over a number of years (ex. home mortgage loan) • PMT = is equal, periodic amounts (payments) FIN261 Spring 2006

  30. Future Value and Present Value of an Ordinary Annuity Compounding FutureValue $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 End of Year Present Value FIN261 Spring 2006 Discounting

  31. Annuity • EXAMPLE: What is the value of $1,000 to be received at the end of each of the next 5 years if an investor can earn only a 5.5% annual return? • Using calculator FIN261 Spring 2006

  32. Present Value of Ordinary Annuity(5 Years, 5.5% Interest Rate) 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $947.87 $898.45 $851.61 $807.22 $765.13 Sum of present value payments: $4,270.28 FIN261 Spring 2006

  33. Future Value of Ordinary Annuity(End of 5 Years, 5.5% Interest Rate) $1,238.82 $1,174.24 $1,113.02 $1,055.00 $1,000.00 $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 End of Year Sum of future value payments: 5,581.08 FIN261 Spring 2006

  34. Annuity • Installment Loans • EXAMPLE: Jim and Kathy want to buy a house costing $100,000. They plan to put $20,000 down and finance the rest. What will their monthly loan payments be if they can get a loan at 7% for 30 years? FIN261 Spring 2006

  35. Annuity • Using the business calculator • EXAMPLE: What is the outstanding balance on an installment loan with 3 years to maturity, annual loan payments of $1,200 and an interest rate of 10%? FIN261 Spring 2006

  36. Special considerations – Annuities Due • Annuities received at the beginning rather than the end of the period. • Annuities due • EXAMPLE: How much will a landlord have accumulated if he deposits his monthly $1000 rent payments and earns 5.5% for 5 years? Rent payments are received and deposited at the beginning of each month. FIN261 Spring 2006

  37. Special considerations – Annuities Due • Annuities received at the beginning rather than the end of the period. • Annuities due • EXAMPLE: How much will a landlord be willing to pay for a potential investment property that has $1000 monthly rent payments and earns 5.5% for 5 years? Rent payments are received and deposited at the beginning of each month. FIN261 Spring 2006

  38. Required Rate of Return • What is your required rate of return? • Businesses and investors use their required ROR to analyze investment opportunities • Their required rate of return is used in the I/Y input of the calculator in TVM calculations • Can be thought of as opportunity cost • This is how to analyze an investment by itself FIN261 Spring 2006

  39. 2 Questions to Ask in Time Value of Money Problems • Future Value or Present Value? • Finding Future Value: Present (Now)  Future • Finding Present Value: Present  Future • Single amount or Annuity? • Single amount: one-time (or lump) sum • Annuity: same amount per period for a number of periods FIN261 Spring 2006

  40. For Next Week • Read chapter 11 in the textbook • Quiz on the reading • Homework assignment 3: construct a personal income statement for the year ending 1/31/2006; refer to worksheet 2.3 FIN261 Spring 2006

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