1. Principles of Corporate Finance Session 10 Unit II: Time Value of Money

2. TIME INTEREST TIME allows you the opportunity to postpone consumption and earn INTEREST. Why TIME? TIME Why is TIME such an important element in your decision?

3. Types of Interest Compound InterestCompound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Simple Interest Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent).

4. Simple Interest Formula Formula FormulaSI = P 0 (i)(n) SI:Simple Interest P 0 :Deposit today (t=0) i:Interest Rate per Period n:Number of Time Periods

5. $140SI = P 0 (i)(n) = $1,000(0.07)(2) = $140 Simple Interest Example Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

6. FV $1,140 FV = P 0 + SI = $1,000 + $140 = $1,140 Future ValueFuture Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (FV) Future Value FVWhat is the Future Value (FV) of the deposit?

7. The Present Value is simply the $1,000 you originally deposited. That is the value today! Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (PV) Present Value PVWhat is the Present Value (PV) of the previous problem?

8. Why Compound Interest? Future Value (U.S. Dollars)

9. Simple Interest Year 1: 5% of $100=$5 + $100 = $105 Year 2: 5% of $100=$5 + $105 = $110 Year 3: 5% of $100=$5 + $110 = $115 Year 4: 5% of $100=$5 + $115 = $120 Year 5: 5% of $100=$5 + $120 = $125 With simple interest, you don’t earn interest on interest.

10. Compound Interest Year 1: 5% of $100.00= $5.00 + $100.00= $105.00 Year 2: 5% of $105.00= $5.25 + $105.00= $110.25 Year 3: 5% of $110.25 = $5.51+ $110.25= $115.76 Year 4: 5% of $115.76= $5.79 + $115.76= $121.55 Year 5: 5% of $121.55= $6.08 + $121.55= $127.63 With compound interest, a depositor earns interest on interest!

11. Principles of Corporate Finance Session 11 & 12 Unit II: Time Value of Money

12. The Role of Time Value in Finance Most financial decisions involve costs & benefits that are spread out over time. Time value of money allows comparison of cash flows from different periods. Question? Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in one year, or one that would return $500,000 after two years?

13. Answer! It depends on the interest rate! The Role of Time Value in Finance Most financial decisions involve costs & benefits that are spread out over time. Time value of money allows comparison of cash flows from different periods.

14. Basic Concepts Future Value: compounding or growth over time Present Value: discounting to today’s value Single cash flows & series of cash flows can be considered Time lines are used to illustrate these relationships

15. Computational Aids Use the Equations Use the Financial Tables Use Financial Calculators Use Spreadsheets

16. Computational Aids

20. Time Value Terms PV 0 =present value or beginning amount k= interest rate FV n =future value at end of “n” periods n=number of compounding periods A=an annuity (series of equal payments or receipts)

21. Four Basic Models FV n = PV 0 (1+k) n = PV(FVIF k,n ) PV 0 = FV n [1/(1+k) n ] = FV(PVIF k,n ) FVA n = A (1+k) n - 1= A(FVIFA k,n ) k PVA 0 = A 1 - [1/(1+k) n ] = A(PVIFA k,n ) k

22. Future Value Example You deposit $2,000 today at 6% interest. How much will you have in 5 years? $2,000 x (1.06) 5 = $2,000 x FVIF 6%,5 $2,000 x 1.3382 =$2,676.40 Algebraically and Using FVIF Tables

23. Future Value Example You deposit $2,000 today at 6% interest. How much will you have in 5 years? Using Excel Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,, 2000)

24. Future Value Example A Graphic View of Future Value

25. Compounding More Frequently than Annually Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently. As a result, the effective interest rate is greater than the nominal (annual) interest rate. Furthermore, the effective rate of interest will increase the more frequently interest is compounded.

26. Compounding More Frequently than Annually For example, what would be the difference in future value if I deposit $100 for 5 years and earn 12% annual interest compounded (a) annually, (b) semiannually, (c) quarterly, an (d) monthly? Annually: 100 x (1 +.12) 5 =$176.23 Semiannually:100 x (1 +.06) 10 = $179.09 Quarterly:100 x (1 +.03) 20 =$180.61 Monthly:100 x (1 +.01) 60 =$181.67

27. Compounding More Frequently than Annually

28. Continuous Compounding With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes: FV n (continuous compounding) = PV x (e kxn ) where “e” has a value of 2.7183. Continuing with the previous example, find the Future value of the $100 deposit after 5 years if interest is compounded continuously.

29. Continuous Compounding With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes: FV n (continuous compounding) = PV x (e kxn ) where “e” has a value of 2.7183. FVn = 100 x (2.7183).12x5= $182.22

30. Principles of Corporate Finance Session 12 & 13 Unit II: Time Value of Money

31. Nominal & Effective Rates The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. The effective interest rate is the rate actually paid or earned. In general, the effective rate > nominal rate whenever compounding occurs more than once per year EAR = (1 + k/m) m -1

32. Nominal & Effective Rates For example, what is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly? EAR = (1 +.18/12) 12 -1 EAR= 19.56%

33. “Rule-of-72”. We will use the “Rule-of-72”. Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?

34. 72 Approx. Years to Double = 72 / i% 726 Years 72 / 12% = 6 Years [Actual Time is 6.12 Years] The “Rule-of-72” Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?

35. Present Value Present value is the current dollar value of a future amount of money. It is based on the idea that a dollar today is worth more than a dollar tomorrow. It is the amount today that must be invested at a given rate to reach a future amount. It is also known as discounting, the reverse of compounding. The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, and the cost of capital.

36. Present Value Example How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit? $2,000 x [1/(1.06) 5 ] =$2,000 x PVIF 6%,5 $2,000 x 0.74758 = $1,494.52 Algebraically and Using PVIF Tables

37. Present Value Example How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit? Excel Function =PV (interest, periods, pmt, FV) =PV (.06, 5,, 2000) Using Excel

38. Present Value Example A Graphic View of Present Value

39. Annuities Annuities are equally-spaced cash flows of equal size. Annuities can be either inflows or outflows. An ordinary (deferred) annuity has cash flows that occur at the end of each period. An annuity due has cash flows that occur at the beginning of each period. An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

40. Annuities

41. Future Value of an Ordinary Annuity Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years. FVA = 100(FVIFA,5%,3) = $315.25 Year 1 $100 deposited at end of year=$100.00 Year 2 $100 x.05 = $5.00 + $100 + $100 =$205.00 Year 3 $205 x.05 = $10.25 + $205 + $100=$315.25 Using the FVIFA Tables

42. Future Value of an Ordinary Annuity Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years. Using Excel Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, )

43. Future Value of an Annuity Due Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years. FVA = 100(FVIFA,5%,3)(1+k) = $330.96 Using the FVIFA Tables FVA = 100(3.152)(1.05) = $330.96

44. Future Value of an Annuity Due Annuity = Equal Annual Series of Cash Flows Example: How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years. Using Excel Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, ) =315.25*(1.05)

45. Present Value of an Ordinary Annuity Annuity = Equal Annual Series of Cash Flows Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest? PVA = 2,000(PVIFA,10%,3) = $4,973.70 Using PVIFA Tables

46. Present Value of an Ordinary Annuity Annuity = Equal Annual Series of Cash Flows Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest? Using Excel Excel Function =PV (interest, periods, pmt, FV) =PV (.10, 3, 2000, )

47. Present Value of a Mixed Stream A mixed stream of cash flows reflects no particular pattern Find the present value of the following mixed stream assuming a required return of 9%. Using Tables

48. Present Value of a Mixed Stream A mixed stream of cash flows reflects no particular pattern Find the present value of the following mixed stream assuming a required return of 9%. Using EXCEL Excel Function =NPV (interest, cells containing CFs) =NPV (.09,B3:B7)

49. Present Value of a Perpetuity A perpetuity is a special kind of annuity. With a perpetuity, the periodic annuity or cash flow stream continues forever. PV = Annuity/k For example, how much would I have to deposit today in order to withdraw $1,000 each year forever if I can earn 8% on my deposit? PV = $1,000/.08 = $12,500