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BCOR 2200 Chapter 4

BCOR 2200 Chapter 4. Introduction to Time Value of Money. Chapter Outline: 4.1 Future Value and Compounding 4.2 Present Value and Discounting 4.3 More on PV and FV Key Concepts and Skills: Be able to compute the future value of an investment made today

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BCOR 2200 Chapter 4

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  1. BCOR 2200Chapter 4 Introduction to Time Value of Money

  2. Chapter Outline: 4.1 Future Value and Compounding 4.2 Present Value and Discounting 4.3 More on PV and FV Key Concepts and Skills: • Be able to compute the future value of an investment made today • Be able to compute the present value of cash to be received at some future date • Be able to compute the return on an investment • Be able to compute the number of periods required for an investment to grow to certain amount given a growth rate • Learn the “Rule of 72’s”

  3. Here is the Idea: • Get $100 today or Get $100 in one year. • Which is better? • Obviously getting the $100 today is better. Why? • If you want to buy something today, you can. • If you want to buy something in one year instead: • You can lend the $100 today for one year • And have more than $100in one year • So if I don’t get the money for one year, I need to get more than $100 • How much more? Talk about that in a soon!

  4. What we will do: Chapter 4: • One Payment • What one payment worth today (PV) if we get it sometime in the future (FV)? • Given the FV, calculate the PV • What is one payment worth at some point in the future (FV) if we get the payment today (PV)? • Given the PV, calculate the FV Later (in Chapter 5): • What if there are a whole bunch of payments?

  5. The Notation: FV = PV(1+r)t OR PV = FV/(1+r)t FV = Future Value PV = Present Value r = The interest rate t = the number of periods (years) • We will solve for each of these 4 variables • If we have the other 3. • And talk about what the variables mean

  6. 4.1 Future Value and Compounding • Future Value: What will a payment made today be worth later? Save $100 for 1 year at 10% interest. What will we have in 1 year? t = 1 r = 10% PV = $100 FV = ? In 1 year the FV= PV( 1 + r)1 = $100(1.1)1 = $110 • Save $100 for 2 years at 10% interest Leave $110 in bank for a second year: $110(1.1) = $121 $100(1.1)(1.1) = $100(1.1)2 • General Notation $100(1 + r)t (1 + r)tis sometimes called the Future Value Interest Factor

  7. Table 4.2 • So $100 in 5 years is worth $100(1.6105) = $161.05 • But nobody uses tables anymore! • Use your calculator! • This is equal to $100(1.10)5 = $100(1.6105) = $161.05

  8. Simple Interest vs. Compound Interest $100 in Five Years at 10% per year: • Interest Factor: (1 + r)t = 1.15 = 1.61051 • Future Value: $100(1.61051) = $161.05 • 10% Interest on the $100 in each year is $10 • Over five years it is $50 • The extra $61.05 - $50 = $11.05 is interest on interest • Also called COMPOUND INTEREST

  9. Using the Calculator • Financial Calculators have a Time Value of Money (TVM) function: N = number of periods I/YR = interest per period (year) PMT = periodic payments (will get to this later) PV = Present Value FV = Future Value

  10. Using the Calculator • Note the negative in front of the PV. • The Idea is you are paying the PV so it is an outflow • Be conscious of this when using the TVM function • Make sure your calculator is set to “1 period per year” • Payments Occur at the end of the period N = 5 I/YR = 10 PV = -100FV = 161.05 OR N = 5 I/YR = 10 PV = 100 FV = -161.05

  11. 4.2 Present Value and Discounting • What is the PV of $1,000 in one year discounted at 7%? • How much put away now so I have $1,000 in one year? • What is $1,000 paid in one year worth today? PV = $1,000/(1.07) = $934.58 Or TVM Function on calculator: FV = 1,000 I/YR = 7% N = 1 PV = -934.58 • $1,000 in two years discounted to today at 7%? PV = $1,000/(1.07)2 = $873.44 Or TVM Function on calculator: FV = 1,000 I/YR = 7% N = 2 PV = -873.44 Calculator instructions on Page 103 or Appendix D on page 620

  12. Clicker Question: • What is the PV of $10,000 if you receive the money in 5 years and it is discounted at 12% per year? • What is the FV in 8 years of $30,000 paid today if it earns 9% per year? Note: Even though the calculator will show negative numbers as FV and PV outputs, we still report positive values. • PV = $16,000 and FV = $49,200 • PV = $10,000 and FV = $49,200 • PV = $4,000 and FV = $49,200 • PV = $4,000 and FV = $59,777 • PV = $5,674 and FV = $59,777

  13. Clicker Answer: PV of $10,000 paid in 5 years at 12%: N = 5 I/YR = 12% FV = 10000 PV = -5,674 FV in 8 years of $30,000 paid today at 9%: N = 8 I/YR = 9% PV = 30000 FV = -59,777 PV = $5,674 and FV = $59,777 The answer is E.

  14. 4.3 More on PV and FV Some examples: • Example: Your company can pay $800for an asset it believes it can sell for $1,200 in 5 yrs. Similarinvestments pay 10%. • What does similar mean? • Other investments with the same risk (a stock or bond issued by another company) pay 10% • So is paying $800 for something that can be sold in 5 yrs for $1,200 a good idea? PV = FV/(1+r)t = $1,200/(1.1)5 = $745.11 Or N = 5 FV = 1,200 I/YR = 10 PV = 745.11 FV = PV(1+r)t = $800(1.1)5= $1,288.41 Or N = 5 PV = 800 I/YR = 10 FV = 1,288.41 • Bad Idea! Either: • You should pay less than $800 (pay $745.11) to get $1,200 • You should pay $800 to get more than $1,200 (get $1,288.41)

  15. Clicker Question: • An investment costs $20,000 and you expect to hold it for 10 years. • Investments with similar risk earn 12% per year. • If the projected sale price is $75,000, is this a good investment idea? • YES. • NO. • MAYBE.

  16. Clicker Answer: • If the projected sale price exceeds the calculated FV, then it is a good idea: • FV = PV(1+r)t = $20,000(1.12)10 = $20,000(3.10585) • = $62,117 • Or using the Calculator TVM function: • N = 10 I/YR = 12 PV = 20,000 FV = -62,117 $75,000 > $62,117 so invest. Here’s the idea: • Over 10 years, the invest, which costs $20k and pays $75k, has a return greater than 12% • If it paid only $62,117, the return would be 12% • Since it pays more ($75k) it must have a higher return than the required return

  17. Determine the Discount Rate: Solve for r (aka I/YR) PV(1 + r)t = FV (1 + r)t = FV/PV 1 + r = (FV/PV)(1/t) r = (FV/PV)(1/t) – 1 • What rate is need to increase $200 to $400 in 10 years? r = ($400/$200)(1/10) – 1 = 0.071 = 7.18% • What rate is need to increase $200 to $400 in 8 years? r = ($400/$200)(1/8) – 1 = 0.0905 = 9.05% Calculator: N = 8 PV = $200 FV = $400 I/YR = not found N = 8 PV = $200 FV = -$400I/YR = 9.05 N = 8 PV = -$200 FV = $400 I/YR = 9.05

  18. Clicker Question: • What rate is needed to increase $20,000 to $80,000 in 10 years? • Rate not Found • 7.18% • 14.87% • 20.00% • 30.00%

  19. Clicker Answer: r = (FV/PV)(1/t) – 1 = (80/20)(1/10) – 1 = (4)(1/10) - 1 = 0.1487 = 14.87% Or N = 10 PV = -20 FV = 80 I/YR = 14.87 The answer is C.

  20. Determine the Number of Periods: Solve for t (or N) PV(1 + r)t = FV (1 + r)t = FV/PV ln(1 + r)t = ln(FV/PV) t[ln(1 + r)] = ln(FV/PV) t = [ln(FV/PV)]/[ln(1 + r)] But we’ll just use the machine: • How many years are needed to increase $200 to $400 at 7.18% Calculator: PV = -$200 FV = $400 I/YR = 7.18% N = 9.9964 ≈ 10 Calculator: PV = $200 FV = -$400 I/YR = 7.18% N = 9.9964 ≈ 10 • Note: –PV and +FV or +PV and –FV both work • How many years are needed to increase $200 to $400 at 9.05% Calculator: PV = -$200 FV = $400 I/YR = 9.05% N = 8.0007 ≈ 8 See page 110 for Excel tips for calculating FV, PV, r & N in Excel

  21. Clicker Question: • How many years are needed to get $1,000,000 if you invest $22,095 and earn 10% per year? • 20.26 • 22 • 22.26 • 40 • Not Found

  22. Clicker Answer: t = [ln(FV/PV)]/[ln(1 + r)] = ln(1,000,000/22,0950)/ln(1.1) = 40 Or PV = -22,095 FV = 1,000,000 I/YR = 10 N = 40 The answer is D.

  23. Rule of 72’s If FV/PV = 2 (your money doubles) then (r)(t) ≈ 72 • Example: Start with $200 and get $400 then FV/PV = 2 $200 to $400 in 10 years at 7.18% (10)(7.18) = 71.8 ≈ 72 $200 to $400 in 8 years at 9.05% (8)(9.05) = 72.4 ≈ 72 If the value of your stock doubles in 5 years, what is the approximate annualized compounded return? (r)(t) ≈ 72  (r)(5) ≈ 72  (r) ≈ 72/5 = 14.4%

  24. Clicker Question: • You own a house that you believe has doubled in value over the last 20 years. • Using the Rule of 72’s, estimate the approximate annual return on the house. • 3.60% • 5.25% • 7.20% • 10.00% • 20.00%

  25. Clicker Answer: (r)(t) ≈ 72 (r)(20) ≈ 72  (r)(20) ≈ 72/20 = 3.6 That answer is A Check this answers using the calculator’s TVM function: N= 20 PV = -1 FV = 2 I/YR = 3.53 ≈ 3.6 Bonus Question: • Assume you will earn 12%? How long to quadruple in price? • Quadruple is double twice: (r)(t) ≈ 72  72/r ≈ t  72/12 = 6 years • So double in 6, double twice in approximately 12 years • Check this: r = 12 PV = -1 FV = 4 N = 12.23 ≈ 12

  26. Recap: FV = PV(1+r)t • Solve for any of the four variables • FV • PV • r (also called I/YR) • t (also called N) • Use the TVM function on the calculator to solve for the one variable not given • Be sure to understand the economic meaning of the values

  27. Table 4.4 Page 113

  28. What’s Next… Chapter 5: Annuity Payments • Multiple Payments • Payments Occur each period • Every Year • Every Month

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