Ch. 4 - The Time Value of Money

# Ch. 4 - The Time Value of Money

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## Ch. 4 - The Time Value of Money

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1. Ch. 4 - The Time Value of Money

2. Topics Covered • Future Values • Present Values • Multiple Cash Flows • Perpetuities and Annuities • Effective Annual Interest Rate • Inflation & Time Value

3. The Time Value of Money Compounding and Discounting Single Sums

4. Future Values Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment.

5. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of \$100. Interest Earned Per Year = 100 x .06 = \$ 6

6. Future Values Example - Simple Interest Interest earned at a rate of 6% for three years on a principal balance of \$100. Today Future Years 123 Interest Earned 6 6 6 Value 100 106112 118 Value at the end of Year 3 = \$118

7. Future Values Example - Compound Interest Interest earned at a rate of 6% for three years on the previous year’s balance. Interest Earned Per Year =Prior Year Balance x .06

8. Future Values Example - Compound Interest Interest earned at a rate of 6% for three years on the previous year’s balance. Today Future Years 1 2 3 Interest Earned 6.00 6.36 6.74 Value 100 106.00 112.36 119.10 • Future Value of \$100 compounded at 6% for three years = \$119.10

9. Future Value of Single Cash Flow

10. Future Values Example - FV What is the future value of \$100 if interest is compounded annually at a rate of 6% for three years?

11. Future Values with Compounding Interest Rates

12. Example: Mutual Fund Fees and Retirement Savings • Prof. Finance moves to a new university and has \$100,000 in retirement savings to invest (rollover) into a new retirement account. • Prof. Finance wants to invest this money for 25 years into an indexed stock fund, which is expected to return 9% annually. • Prof. has two choices: Vanguard Total Equity Fund with a 0.4% annual expense fee and Onguard Total Fencing Fund with an 1.2% annual expense fee. • What is the difference in Prof. Finance’s expected future retirement savings between the two funds?

13. Present Values Present Value Value today of a future cash flow. Discount Factor Present value of a \$1 future payment. Discount Rate Interest rate used to compute present values of future cash flows.

14. Present Values

15. Example: Paying for Baby’s MBA • Just had a baby. You think the baby will take after you and earn academic scholarships to attend college to earn a Bachelor’s degree. However, you want send your baby to a top-notch 2-year MBA program when baby is 25. You have estimated the future cost of the MBA at \$85,000 for year 1 and \$89,000 for year 2.

16. Example: Paying for Baby’s MBA • Today, you want to finance both years of baby’s MBA program with one payment (deposit) into an account paying 8% interest compounded annually. • How large must this deposit be?

17. Time Value of Money(applications) • Value of Free Credit • Implied Interest Rates • Internal Rate of Return • Time necessary to accumulate funds

18. Example : Finding Rate of Return or Interest Rate • A broker offers you an investment (a zero coupon bond) that pays you \$5,000 five years from now for the cost of \$3,700 today. • What is your annual rate of return?

19. Important Time Value Relationships • Increasing interest rate and time increases future value. POSITIVE RELATIONSHIP. • Increasing interest rate and time decreasespresent value. INVERSE RELATIONSHIP.

20. 0 1 2 3 4 The Time Value of Money Compounding and Discounting Cash Flow Streams

21. Perpetuities • Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.

22. Perpetuities PV of Perpetuity Formula C = cash payment r = interest rate

23. Perpetuities & Annuities Example - Perpetuity You want to create an endowment to fund a football scholarship, which pays \$15,000 per year, forever, how much money must be set aside today in the rate of interest is 5%?

24. Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

25. 0 1 2 3 4 Annuities • Annuity: a sequence of equal cash flows, occurring at the end of each period. This is known as an ordinary annuity. PV FV

26. Examples of Ordinary Annuities: • If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. • If you borrow money to buy a house or a car, you will pay a stream of equal payments.

27. Annuity-due • A sequence of periodic cash flows occurring at the beginning of each period. 0 1 2 3 4 PV FV

28. Examples of Annuities-due • Monthly Rent payments: due at the beginning of each month. • Car lease payments. • Cable TV and most internet service bills.

29. Perpetuities & Annuities PV of Ordinary Annuity Formula C = cash payment r = interest rate t = Number of years cash payment is received

30. Perpetuities & Annuities PV Annuity Factor (PVAF) - The present value of \$1 a year for each of t years.

31. Perpetuities & Annuities Applications • Value of payments • Implied interest rate for an annuity • Calculation of periodic payments • Mortgage payment • Annual income from an investment payout • Future Value of annual payments

32. Perpetuities & Annuities PV (and FV) of Annuity-dues = PV (or FV) of ordinary annuity x (1 + r) or BGN mode on financial calculator. C = cash payment r = interest rate t = Number of years cash payment is received

33. Example: Invest Early in an IRA • How much would you have at age 65 if you deposit \$2,500 at the end of each year in an account paying 9% annually starting at: • (A) age 41? • (B) age 22?

34. Why an IRA? • Imagine in the last example, you didn’t take advantage of the tax-sheltered environment of an IRA. • Your annual investment return would be taxed! • With a 28% tax rate, our annual after-tax return would fall from 9% to 6.48% (=9%(1-.28)). • At age 65: I would have \$135,519 vs. \$191,975. • You would have \$535,392 vs. \$1,102,114: 52% less!! The IRS killed Kenny,…!

35. Example: Enjoying your Retirement • You go ahead and make the contributions starting at age 22 in the last example, giving you \$1,102,114 at age 65. • You expect to live to age 85. So, you want to make 20 annual withdrawals from your IRA paying 9% at the beginning of each year starting at age 65. • How large can this annual withdrawal be?

36. More annuity fun, enjoying your release from baseball • Bob B. is released from the last year of his guaranteed contract from a New York baseball team. He is due \$5.9 million from the last year of this contract. Bob and the team agree to defer the \$5.9 million at 8% interest for 15 years. At this time (15 years from today), the team will begin the first of 15 equal annual payments at 8% interest. • The press is reporting the payments will total \$30 million. Are they correct?

37. Non-Annual Interest Compounding • When interest is compounded more frequently than once a year. • Important non-annual compounding terms and things to know: • Quoted Annual Rate, or Annual Percentage Rate (APR): Stated nominal annual rate before compounding. • Effective Annual Rate (EAR): the actual (effective) annual interest rate earned or paid. • Periodic Interest Rate: the interest rate paid or charged each interest compounding period = quoted rate/m, where m = number of compounding periods per year.

38. Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

39. FV and PV with non-annual interest compounding • n = number of years • m = number of times interest is paid per year • APR = nominal annual rate (APR) • APR/m = periodic rate Single CF FVnm = PV(1+ARR/m)nm PV = FVnm/(1+APR/m)nm

40. Non-annual annuities Ordinary: • PV= C(PVAFAPR/m,nm) • FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm Annuity-Due: • PV= C(PVAFAPR/m,nm)(1+APR/m) • FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm+1

41. Example: Low Rate or Rebate? • The Frontier family want to buy a sport ut (SUV). They decide on a 4-wheel drive Jeep Grand Cherokee. The purchase price with tax of the vehicle is \$32,500. The Frontiers have \$4,000 as a down payment. • Jeep offers the choice of two incentives on the 4-door Grand Cherokee. • 0% APR Financing for 60 months, or • \$3,000 rebate which would be applied toward the purchase price. If the Frontiers elect to take the rebate, they can get 4.49% APR financing for 60 months. • Question: Which incentive would give the Frontiers the lowest monthly payment?

42. Example: The \$200 national ISP signup credit: good deal for whom? • A national ISP all provide \$200 for new customers to use at a particular electronics store chain if they sign-up for a 2-year internet service contract at \$21.95/month. • What interest rate (APR) are you paying on this “free money” if you wanted internet service and could get it for free? (200 PV, -21.95 PMT, 24 N 0 FV, CPT I/Y = 9.8%/month x 12= 117.8% APR!!) • What interest rate (APR) are you paying on this “free money” if you wanted internet service and could get it for \$9.95/month?(-12 PMT CPT I/Y = 3.15%/mo x 12 = 37.8%! Thanks, but no thanks!

43. Inflation Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases.

44. Inflation approximation formula

45. Inflation Example If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate? Savings Bond

46. Example: Real retirement income • Going back to your retirement in 43 years, you expect 3% inflation along with your 9% nominal investment rate annually and want to withdraw \$32,000 in real terms at the beginning of each year for 20 years once you retire. • How will this change your retirement saving plans?