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Investing Decision: The Time Value of Money

Investing Decision: The Time Value of Money. Basic Concepts. Future value of a sum Present value of a sum Future value of an annuity Present value of an annuity. Symbols. PV = Present Value (some books use P as principal or present value)

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Investing Decision: The Time Value of Money

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  1. Investing Decision:The Time Value of Money The Time Value of Money

  2. Basic Concepts • Future value of a sum • Present value of a sum • Future value of an annuity • Present value of an annuity The Time Value of Money

  3. Symbols • PV = Present Value (some books use P as principal or present value) • FV = Future Value at time t (some books use S as compound sums) • PMT = Payment, Annuity, or cash amount (some books use R or C) • n = Number of time periods • t = Period • r = Interest rate (Some books use i or k) t The Time Value of Money

  4. Symbols (cont.) • = Future value of interest factor • = Present value of interest factor • = Future value interest factor for an annuity • = Present value interest factor for an annuity The Time Value of Money

  5. Future Value of $100 at 10% The Time Value of Money

  6. Interest on Interest Illustration Future value of $100 at 10% interest The Time Value of Money

  7. Interest on Interest Illustration The Time Value of Money

  8. TVM in Excel To Find Enter This Formula Future Value =FV(rate,nper,pmt,pv) Present Value =PV(rate,nper,pmt,fv) Discount rate =RATE(nper,pmt,pv,fv) Number of periods =NPER(rate,pmt,pv,fv) *In Excel, the rate is entered as a decimal, not a percent. A discussion of “type” (beginning or ending of period) is given in a latter slide. The Time Value of Money

  9. Value in 10 years of $100 received today at 8% interest Formula: Calculator: Enter 10 as n or number of periods Enter 8 as i or interest rate Enter $100 as PV or present value Push the FV (or future value) key Excel: nper = 10 payment periods pmt = $0 payment each period pv = $100 present value type = 0 for period end payments rate = 8% * [optional] = FV(rate,nper,pmt,pv,[type]) = FV(.08,10,0,100,0) = -215.89 The Time Value of Money

  10. Future Value of $1 at Different Periods and Rates The Time Value of Money

  11. Present value of $100 received in 10 years at 10% Formula: Calculator: Enter 10 as n or number of periods Enter 10 as i or interest rate (some calculators require .10) Enter $100 as FV or future value Push the PV (or present value) key Excel: = PV(rate,nper,pmt,fv,[type]) = PV(.1,10,0,100) = - $38.55 type can be omitted The Time Value of Money

  12. Present value of $100 received in 10 years at 10% You may have an older boss (or a text book author) Who uses tables to solve TVM problems Tables: The Time Value of Money

  13. Present Value of $1 received at the end of period The Time Value of Money

  14. Present Value of $1 at Different Periods and Rates The Time Value of Money

  15. Present Value of $300 Annuity for 4 Years at 8% The Time Value of Money

  16. Present Value of $300 Annuity for 4 Years at 8% Formula:Excel: = PV(rate,nper,pmt,[fv],[type]) = PV(.08,4,300) future value and type can be omitted = -$993.64 Calculator: Enter 4 as n or years Enter 8 as i or interest rate Enter $300 as PMT (annuity or payment) Push PV (or present value) key Remember to set the calculator to “End-of-period” The Time Value of Money

  17. Future Value of $1,000 Annuity for 5 Years at 12% The Time Value of Money

  18. Future Value of $1,000 Annuity for 5 Years at 12% Formula:Excel: = FV(rate,nper,pmt,[pv],[type]) = FV(.12,5,1000) present value and type can be omitted = -$6,352.85 Calculator: Enter 5 as n or number Enter 12 as i or interest rate Enter $1,000 as PMT (annuity or payment) Push FV (or future value) key Remember to set the calculator to “End-of-period” The Time Value of Money

  19. Beginning-of-Period vs. End-of-Period Annuities As part of a settlement from your grandmother’s trust, you are given $1,000 today. Additionally, you will also be paid $1,000 one and two years from now. What will be the value of the inheritance exactly three years from today if you invest at 8%? $3,506.11 Also known as an “Annuity Due” *Set calculator to beginning of period mode or set “type” equal to 1 in Excel The Time Value of Money

  20. Beginning-of-Period vs. End-of-Period Annuities Your grandmother’s trust fund is being distributed. You will receive three $1,000 payments with the first annual payment being made early next September. What will be the value of the inheritance three years from now, just after receiving the third payment? You can invest at 8%. $3,246.40 or $259.71 less than the annuity-due The Time Value of Money

  21. Formulas PV FV Single Payment Annuity The Time Value of Money

  22. Variable Cash Flows Contract (Bond) = 10% interest for 10 years on a $1,000 bond Current market interest rate is 8% Part 1 Part 2 Total PV = $509.51 + $624.69 = $1,134.20 The Time Value of Money

  23. Unknown Payment Borrow $90,000 for a home at a 9% interest rate for 30 years (assume annual payment for simplicity): The Time Value of Money

  24. Unknown Problem The Time Value of Money

  25. Unknown Rate Lend $700 now and receive $1,000 in 5 years The Time Value of Money

  26. Present Value of Deferred Annuity Two-Step Process $3,993 (single amount) $3,170 The Time Value of Money

  27. Variable Cash FlowsPro Football Player Example $2,000,000 $3,800,000 $2,500,000 What is the value of the contract in 1991 (the day the player signs the contract and receives the $1,500,000) if interest rates are at 10%, and if interest rates are at 15%? The Time Value of Money

  28. Pro Football Player Example The Time Value of Money

  29. Loan Amortization Schedule Example: $10,000, 5 year loan with i=15%. PMT=$2,983.15 per year The Time Value of Money

  30. Loan Amortization Schedule (cont.) Total Payment The Time Value of Money

  31. Effective Annual Rates and Compounding The Time Value of Money

  32. Effective Annual Rates (EAR)vs.Quoted Rate The Time Value of Money

  33. Effective Annual Rates (EAR) A: The Time Value of Money

  34. Annual Percentage Rate (APR)vs.Effective Annual Rate (EAR) APR is a U.S. Truth-in-landing creation APR = interest rate per period x number of periods EAR is a finance reality Q: Credit card agreement quotes an interest rate of 18 percent APR. Monthly payments are required. What is the EAR on the card? The Time Value of Money

  35. APR vs. EAR A: An APR of 18 percent with monthly payments is really .18/12 = .015 or 1.5 percent per month. The Time Value of Money

  36. How do you get to Carnegie Hall? • Practice, practice, practice. • It’s easy to watch Olympic gymnasts and convince yourself that you are a leotard purchase away from a triple back flip. • It’s also easy to watch your finance professor do time value of money problems and convince yourself that you can do them too. • There is no substitute for getting out the calculator and flogging the keys until you can do these correctly and quickly. The Time Value of Money

  37. Investment Selection Tools • Payback period • Net Present Value (NPV) • Internal Rate of Return (IRR) The Time Value of Money

  38. Investment Criterion The Time Value of Money

  39. Payback Period Time when cash outflows equal cash inflows Payback between 3 & 4 years The Time Value of Money

  40. Problems with Payback: • No consideration of inflows after the cutoff period • No consideration of the time value of money Why payback is used: • Easy • Does tell how long original investment is at risk The Time Value of Money

  41. Other Bad Investment Criteria • Discounted Payback • Average Accounting Return The Time Value of Money

  42. Net Present Value NPV = PV (cash inflows) -PV (cash outflows) The Time Value of Money

  43. Internal Rate of Return IRR = rate of return where NPV=0 The Time Value of Money

  44. Calculation Practice Find NPV at 20%. Find IRR The Time Value of Money

  45. NPV and IRR Calculation Practice Calculation of NPV at 20% by formula NPV = $369 The Time Value of Money

  46. NPV and IRR Calculation Practice (cont.) Calculation of NPV at 20% by Calculator The Time Value of Money

  47. NPV and IRR Calculation Practice (cont.) Calculation of IRR by formula The Time Value of Money

  48. NPV and IRR Calculation Practice (cont.) Calculation of IRR by Calculator The Time Value of Money

  49. NPV Profile Calculate NPV at 0%, 10%, 20%, … 50% IRR = 39.2% The Time Value of Money

  50. NPV Profile Calculate NPV at 0%, 10%, 20%, … 50% The Time Value of Money

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