1. Slide 1 The Time Value of Money Time Value of Money Concept Future and Present Values of single payments Future and Present values of periodic payments (Annuities) Present value of perpetuity Future and Present values of annuity due Annual Percentage Yield (APY)

2. Slide 2 The Time Value of Money Concept We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner

3. Slide 3 The Future Value Future Value equation:

4. Slide 4 Future Value – Single Sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Mathematical Solution: FV n = $1 x (1 + i) n FV n = $100 x (1 + 0.06) 1 $106 FV n = $106

5. Slide 5 Future Value – Single Sums (Continued) NI/YP/YPVPMTFVMODE 161-1000106 Calculator Solution (TI BA II PLUS) NNumber of periods I/YInterest per Year P/YPayment per Year C/YCompounding per Year PVPresent Value PMTPayMenT (Periodic and Fixed) FVFuture Value MODEEND for ending and BGN for beginning

6. Slide 6 Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: FV n = $1 x (1 + i) n FV n = $100 x (1 + 0.06) 5 $133.82 FV n = $133.82 NI/YP/YPVPMTFVMODE 561-1000133.82

7. Slide 7 Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 year? Mathematical Solution: FV n = $1 x (1 + i) n FV n = $100 x (1 + 0.06/4) 5x4 $134.69 FV n = $134.69 NI/YP/YPVPMTFVMODE 2064-1000134.69

8. Slide 8 Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 year? Mathematical Solution: FV n = $1 x (1 + i) n FV n = $100 x (1 + 0.06/12) 5x12 $134.89 FV n = $134.89 NI/YP/YPVPMTFVMODE 60612-1000134.89

9. Slide 9 Future Value – Single Sums (Continued) If you deposit $1,000 in an account earning 8% with daily compounding, how much would you have in the account after 100 year? Mathematical Solution: FV n = $1 x (1 + i) n FV n = $1,000 x (1 + 0.08/365) 100x365 $2,978,346.07 FV n = $2,978,346.07 NI/YP/YPVPMTFVMODE 36,5008365-100002,978,346.07

10. Slide 10 The Present Value Present Value equation:

11. Slide 11 Present Value – Single Sums (Continued) If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? Mathematical Solution: PV 0 = $1 / (1 + i) n PV 0 = $100 / (1 + 0.06) 1 -$94.34 PV 0 = -$94.34 NI/YP/YPVPMTFVMODE 161-94.340100

12. Slide 12 Present Value – Single Sums (Continued) If you receive $100 five year from now, what is the PV of that $100 if your opportunity cost is 6%? Mathematical Solution: PV 0 = $1 / (1 + i) n PV 0 = $100 / (1 + 0.06) 5 -$74.73 PV 0 = -$74.73 NI/YP/YPVPMTFVMODE 561-74.730100

13. Slide 13 Present Value – Single Sums (Continued) If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? Mathematical Solution: NI/YP/YPVPMTFVMODE 5191-5,000011,933

14. Slide 14 Present Value – Single Sums (Continued) Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: NI/YP/YPVPMTFVMODE 2029.612-1000500

15. Slide 15 Hint for Single Sum Problems In every single sum future value and present value problem, there are 4 variables: FV, PV, i, and n When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable Keeping this in mind makes “time value” problems much easier!

16. Slide 16 Compounding and Discounting Cash Flow Streams Annuity: a sequence of equal cash flows, occurring at the end of each period 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

17. Slide 17 Future Value – Annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? Mathematical Solution: NI/YP/YPVPMTFVMODE 3810-10003,246.40

18. Slide 18 Present Value – Annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? Mathematical Solution: NI/YP/YPVPMTFVMODE 3812,577.10-10000

19. Slide 19 Perpetuities Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity You can think of a perpetuity as an annuity that goes on

20. Slide 20 Perpetuities (Continued)

21. Slide 21 Perpetuities (Continued) What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? PV = $10,000 / 0.08 = $125,000

22. Slide 22 Future Value – Annuity Due Annuity Due: The cash flows occur at the beginning of each year, rather than at the end of each year If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: NI/YP/YPVPMTFVMODE 3810-10003,506.11BEGIN

23. Slide 23 Present Value – Annuity Due Annuity Due: The cash flows occur at the beginning of each year, rather than at the end of each year What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%? Mathematical Solution: NI/YP/YPVPMTFVMODE 3812,783.26-10000BEGIN

24. Slide 24 Uneven Cash Flows How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate) PeriodCFPVCF 0-10,000-10,000.00 12,0001,818.15 24,0003,305.79 36,0004,507.89 47,0004,781.09 Total4,412.95

25. Slide 25 Uneven Cash Flows

26. Slide 26 Annual Percentage Yield (APY) or Effective Annual Rate (EAR) Which is the better loan: 8.00% compounded annually, or 7.85% compounded quarterly? We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year! We need to calculate the APY

27. Slide 27 Annual Percentage Yield (APY) or Effective Annual Rate (EAR) (Continued) Find the APY for the quarterly (m = 4) loan: The quarterly loan is more expensive than the 8% loan with annual compounding!

28. Slide 28 Annual Percentage Yield (APY) or Effective Annual Rate (EAR) (Continued) 2nd ICONV NOM7.85ENTER(EFF) C/Y4ENTER(EFF) CPT8.08