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The Time Value of Money

The Time Value of Money. Economics 71a Spring 2007 Mayo, Chapter 7 Lecture notes 3.1. Goals. Compounding and Future Values Present Value Valuing an income stream Annuities Perpetuities Mixed streams Term structure again Compounding More applications. Compounding.

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The Time Value of Money

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  1. The Time Value of Money Economics 71a Spring 2007 Mayo, Chapter 7 Lecture notes 3.1

  2. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More applications

  3. Compounding • PV = present or starting value • FV = future value • R = interest rate • n = number of periods

  4. First example • PV = 1000 • R = 10% • n = 1 • FV = ? FV = 1000*(1.10) = 1,100

  5. Example 2Compound Interest • PV = 1000 • R = 10% • n = 3 • FV = ? FV = 1000*(1.1)*(1.1)*(1.1) = 1,331 • FV = PV*(1+R)^n

  6. Example 3:The magic of compounding • PV = 1 • R = 6% • n = 50 • FV = ? • FV = PV*(1+R)^n = 18 • n = 100, FV = 339 • n = 200, FV = 115,000

  7. Example 4:Doubling times • Doubling time = time for funds to double

  8. Example 5Retirement Saving • PV = 1000, age = 20, n =45 • R = 0.05 • FV = PV*(1+0.05)^45 = 8985 • Doubling 14 • R = 0.07 • FV=PV*(1+0.07)^45 = 21,002 • Doubling = 10 • Small change in R, big impact

  9. Retirement Savings at 5% interest

  10. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More applications

  11. Present Value • Go in the other direction • Know FV • Get PV • Answer basic questions like what is $100 tomorrow worth today

  12. ExampleGiven a zero coupon bond paying $1000 in 5 years • How much is it worth today? • R = 0.05 • PV = 1000/(1.05)^5 = $784 • This is the amount that could be stashed away to give 1000 in 5 years time

  13. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More applications

  14. Annuity • Equal payments over several years • Usually annual • Types: Ordinary/Annuity due • Beginning versus end of period

  15. Present Value of an Annuity • Annuity pays $100 a year for the next 10 years (starting in 1 year) • What is the present value of this? • R = 0.05

  16. Future Value of An Annuity • Annuity pays $100 a year for the next 10 years (starting in 1 year) • What is the future value of this at year 10? • R = 0.05

  17. Why the Funny Summation? • Period 10 value for each • Period 10: 100 • Period 9: 100(1.05) • Period 8: 100(1.05)(1.05) • … • Period 1: 100(1.05)^9 • Be careful!

  18. Application: Lotteries • Choices • $16 million today • $33 million over 33 years (1 per year) • R = 7% • PV=$12.75 million, take the $16 million today

  19. Another Way to View An Annuity • Annuity of $100 • Paid 1 year, 2 year, 3 years from now • Interest = 5% • PV = 100/(1.05) + 100/(1.05)^2 + 100/(1.05)^3 • = 272.32

  20. Cost to Generate From Today • Think about putting money in the bank in 3 bundles • One way to generate each of the three $100 payments • How much should each amount be? • 100 = FV = PV*(1.05)^n (n = 1, 2, 3) • PV = 100/(1.05)^n (n = 1, 2, 3) • The sum of these values is how much money you would have to put into bank accounts today to generate the annuity • Since this is the same thing as the annuity it should have the same price (value)

  21. Perpetuity • This is an annuity with an infinite life

  22. Discounting to infinity • Math review:

  23. Present Value of a Constant Stream

  24. Perpetuity Examples and Interest Rate Sensitivity • Interest rate sensitivity • y=100 • R = 0.05, PV = 2000 • R = 0.03, PV = 3333

  25. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More applications

  26. Mixed StreamApartment Building • Pays $500 rent in 1 year • Pays $1000 rent 2 years from now • Then sell for 100,000 3 years from now • R = 0.05

  27. Mixed StreamInvestment Project • Pays -1000 today • Then 100 per year for 15 years • R = 0.05 • Implement project since PV>0 • Technique = Net present value (NPV)

  28. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More applications

  29. Term Structure • We have assumed that R is constant over time • In real life it may be different over different horizons (maturities) • Remember: Term structure • Use correct R to discount different horizons

  30. Term Structure Discounting payments 1, 2, 3 years from now

  31. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More examples

  32. Frequency and compounding • APR=Annual percentage rate • Usual quote: • 6% APR with monthly compounding • What does this mean? • R = (1/12)6% every month • That comes out to be • (1+.06/12)^12-1 • 6.17% • Effective annual rate

  33. General Formulas • Effective annual rate (EFF) formula • Limit as m goes to infinity • For APR = 0.06 • limit EFF = 0.0618

  34. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More examples

  35. More Examples • Home mortgage • Car loans • College • Calculating present values

  36. Home MortgageAmortization • Specifications: • $100,000 mortgage • 9% interest • 3 years (equal payments) pmt • Find pmt • PV(pmt) = $100,000

  37. Mortgage PV • Find PMT so that • Solve for PMT • PMT = 39,504

  38. Car Loan • Amount = $1,000 • 1 Year • Payments in months 1-12 • 12% APR (monthly compounding) • 12%/12=1% per month • PMT?

  39. Car Loan • Again solve, for PMT • PMT = 88.85

  40. Total Payment • 12*88.85 = 1,066.20 • Looks like 6.6% interest • Why? • Paying loan off over time

  41. Payments and Principal • How much principal remains after 1 month? • You owe (1+0.01)1000 = 1010 • Payment = 88.85 • Remaining = 1010 – 88.85 = 921.15 • How much principal remains after 2 months? • (1+0.01)*921.15 = 930.36 • Remaining = 930.36 – 88.85 = 841.51

  42. CollegeShould you go? • 1. Compare • PV(wage with college)-PV(tuition) • PV(wage without college) • 2. What about student loans? • 3. Replace PV(tuition) with PV(student loan payments) • Note: Some of these things are hard to estimate • Second note: Most studies show that the answer to this question is yes

  43. Calculating Present Values • Sometimes difficult • Methods • Tables (see textbook) • Financial calculator (see book again) • Excel spreadsheets (see book web page) • Java tools (we’ll use these sometimes) • Other software (matlab…)

  44. Discounting and Time: Summary • Powerful tool • Useful for day to day problems • Loans/mortgages • Retirement • We will use it for • Stock pricing • Bond pricing

  45. Goals • Compounding and Future Values • Present Value • Valuing an income stream • Annuities • Perpetuities • Mixed streams • Term structure again • Compounding • More examples

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