Understanding Time Value of Money in Financial Decisions

1. Ch. 2 - Time Value of Money

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

3. Example : Finding Rate of Return or Interest Rate • A broker offers you an investment (a zero coupon bond) that pays you $1,000 five years from now for the cost of $740 today. • What is your annual rate of return?

5. 0 1 2 3 4 The Time Value of Money Compounding and Discounting Cash Flow Streams

6. 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

7. 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 re-pay the loan with a stream of equal payments.

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

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

10. Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT What is the difference between an ordinary annuity and an annuity due?

11. Solving for FV:3-year ordinary annuity of $100 at 10% • $100 payments occur at the end of each period, but there is no PV. 3 10 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 331

12. Solving for PV:3-year ordinary annuity of $100 at 10% • $100 payments still occur at the end of each period, but now there is no FV. 3 10 100 0 INPUTS N I/YR PV PMT FV OUTPUT -248.69

13. Solving for FV:3-year annuity due of $100 at 10% • Now, $100 payments occur at the beginning of each period. • FVAdue= FVAord(1+I) = $331(1.10) = $364.10. • Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity: BEGIN 3 10 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 364.10

14. Solving for PV:3-year annuity due of $100 at 10% • Again, $100 payments occur at the beginning of each period. • PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55. • Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity: BEGIN 3 10 100 0 INPUTS N I/YR PV PMT FV OUTPUT -273.55

15. 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

16. Example: Invest Early in an IRA • How much would you have at age 65 if you deposit $2,400 at the end of each year in an investment account with a 9% expected annual return starting at: • (A) age 44? • (B) age 22?

17. A) Start at age 44

18. B) Start at age 22

19. Solving for PMT:How much must the 44-year old deposit annually to catch the 22-year old? • To find the required annual contribution, enter the number of years until retirement and the final goal of $1,058,030 and solve for PMT. 21 9 0 1,058,030 INPUTS N I/YR PV PMT FV OUTPUT -18,639

20. Now about this? • Let’s assume that the 44-year old has already accumulated $120,000 in the IRA account. How much would he have to deposit on an annual basis at the 9% expected annual return to catch up with the 22-year old and be a millionaire at age 65?

22. More Annuity Fun! • Springfield mogul Montgomery Burns, age 85, wants to retire at age 100 so he can steal candy from babies full time. Once Mr. Burns retires, he wants to withdraw $100 million at the beginning of each year for 10 years from a special off-shore account that will pay 20% annually. In order to fund his retirement, Mr. Burns will make 15 equal end-of-the-year deposits in this same special account that will pay 20% annually. How large of an annual deposit must be made to fund Mr. Burns’ retirement plans?

24. Perpetuities • Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. • PV of Perpetuity Formula PMT = periodic cash payment i = interest rate

25. 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 if the rate of interest is 5%?

26. 4 0 1 2 3 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV What is the PV of this uneven cash flow stream?

27. Solving for PV:Uneven cash flow stream • Input cash flows in the calculator’s “CF” register: • CF0 = 0 • CF1 = 100 • CF2 = 300 • CF3 = 300 • CF4 = -50 • Under NPV, enter I = 10, down arrow, and press CPT button to get NPV = $530.087. (Here NPV = PV.)

28. The Time Value of Money Non-annual Interest Compounding and Discounting

29. Classifications of interest rates • Nominal rate (INOM) – also called the quoted or state rate. An annual rate that ignores compounding effects. • INOM is stated in contracts. Periods must also be given, e.g. 8% Quarterly or 8% Daily interest. • Periodic rate (IPER) – amount of interest charged each period, e.g. monthly or quarterly. • IPER = INOM / M, where M is the number of compounding periods per year. M = 4 for quarterly and M = 12 for monthly compounding.

30. Classifications of interest rates • Effective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned, accounting for compounding. • EFF% for 10% semiannual investment EFF% = ( 1 + INOM / M )M - 1 = ( 1 + 0.10 / 2 )2 – 1 = 10.25% • Should be indifferent between receiving 10.25% annual interest and receiving 10% interest, compounded semiannually.

31. Why is it important to consider effective rates of return? • Investments with different compounding intervals provide different effective returns. • To compare investments with different compounding intervals, you must look at their effective returns (EFF% or EAR). • See how the effective return varies between investments with the same nominal rate, but different compounding intervals. EARANNUAL 10.00% EARQUARTERLY 10.38% EARMONTHLY 10.47% EARDAILY (365) 10.52%

32. When is each rate used? • INOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. • IPER Used in calculations and shown on time lines. If M = 1, INOM = IPER = EAR. • EAR Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match compounding periods.

33. FV and PV with non-annual interest compounding • n = number of years • m = number of times interest is paid per year • inom = stated annual rate (APR) • inom /m = periodic rate Single CF FVnm = PV(1 + inom/m)nm PV = FVnm/(1 + inom/m)nm Annuities: • Use periodic rate and number of annuity payment and compounding periods if interest compounding period and annuity payment period are the same. • Otherwise, need to find effective interest rate for each annuity payment period.

34. What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

35. Futurama Value Revisited • How much money would Fry have in his bank account in the year 3000 from the $0.93 deposited in the year 2000 if the 2.25% annual rate was compounded quarterly?

36. Let’s buy a car! • Prof. Outback decides to purchase a brand-new 2007 Jeep Liberty Limited 4WD with heated premium leather seats, sunroof, and satellite radio for $28,800. After paying tax and license, Prof. Outback has $4,000 as a down payment. Jeep offers Prof the choice of 3.9% APR financing for 60 months or a $3,000 rebate. Prof. Outback can receive 6.25% APR financing for 60 months through E-Loan if the rebate option is selected. • Which option would result in the lower monthly payment? • At what APR along without the rebate would the Prof. be indifferent between the two options?

37. Monthly Payments

38. Indifference APR