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The Time Value of Money A core concept in financial management

The Time Value of Money A core concept in financial management. Lesson Objectives . To introduce the time value concept Calculate present and future values of any set of expected future cash flows. Time Value ???. Rs.1000 you received today or Rs.1000 will be received tomorrow.

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The Time Value of Money A core concept in financial management

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  1. The Time Value of Money A core concept in financial management

  2. Lesson Objectives To introduce the time value concept Calculate present and future values of any set of expected future cash flows.

  3. Time Value ??? Rs.1000 you received today or Rs.1000 will be received tomorrow. What do you prefer? Simple reason is your time preference for money. Therefore, you may expect to get an extra cash amount as compensation for delaying. It is called the interest. This Interest for the Time Preference for Money is called the Time Value of Money

  4. Why we need a premium for future cash flows? Alternative uses of money. (Investment opportunities) Individual’s preference for early consumption (time preference theory ) The risk associated with future cash flows The interest rate for the time value of money can be regarded as the opportunity cost

  5. Comparison of CF at different Time intervals • A rupee received today is more valuable than a rupee received tomorrow. • Thus, cash flows in different time periods cannot be compared as they are. • There are two ways • Future value • Present value

  6. Translate Rs.1 today into its equivalent in the future (COMPOUNDING). Today Future ? Today Future ? • Translate Rs.1 in the future into its equivalent today (DISCOUNTING). ?

  7. ASSUMPTIONS A point in time is denoted by the letter “t”. Unless otherwise stated, t=0 represents today (the decision point). Unless otherwise stated, cash flows occur at the end of a time interval. Cash inflows are treated as positive amounts, while cash outflows are treated as negative amounts. Compounding frequency is the same as the cash flow frequency.

  8. The Time Line Beginning of the fourth year End of the third year Today t=1 t=2 t=3 t=4 t=0

  9. Cash Flows Single cash flow Annuity Multiple/ uneven cash flows

  10. Future Value and Compounding Process Future value Is the total of the principle amount and the interest accumulated on the principle for a given period. Is the sum which an initial amount of principle (or present value (PV)) is expected to grow over a given (n) period at a given interest rate.

  11. Example 1: Suppose you place Rs.100 in a savings account that earns 6% interest compounded annually. How much can you get at the end of each period?

  12. Future Value Formula Let PV = Present Value FVn= Future Value at time n r = interest rate (ordiscount rate) per period.

  13. Future Value Interest Formula

  14. Future Value Factor r = 15% r = 10% r = 5% r = 0%

  15. More Frequent Compounding Interest may be compounded more than once a year. The Nominal Rate (Annual Percentage Rate (APR)) is the periodic rate times the number of periods per year. The Effective rate (Annual Percentage Yield (APY)) is the “true” annually compounded interest rate.

  16. Effect of Compounding Frequency on Future Value Find the future value at the end of one year if the present value is Rs.20,000 and the interest rate is 16%. Use the following compounding frequencies: • Annual Compounding • Semiannual Compounding • Quarterly Compounding • Monthly Compounding • Daily Compounding • Continuous Compounding

  17. Annual Compounding - Once a year The periodic rate is 16%. APY = APR = 16% Compounding m-times in a year

  18. Semi - Annual Compounding Since m = 2, the periodic rate is 8%. APY > APR

  19. Quarterly Compounding Since m = 4, the periodic rate is 4%.

  20. Effect of Compounding Frequency on Future Value

  21. Continuous Compounding With continuous compounding, m becomes very large. • As m approaches infinity, the value of (1+r/m)mn goes to er n. Thus, • Then the effective rate = er - 1, where e = 2.71828. • Thus, effective rate = (2.71828)0.16 - 1 = 0.17351 or 17.351%. • FV = Rs.20,000 (1.17351)1 = Rs.23,469.39

  22. Present Value Future value Thus the Present Value When we get the present value, the interest rate is referred as the Discount Rate and this process is called as Discounting

  23. Present Value Interest Formula

  24. 1. If you will receive Rs.100 one year from now, what is the PV of that Rs.100 if your opportunity cost is 6%2. If you receive Rs.100 5 years from now, what is the PV of that Rs.100, if your opportunity cost is 6%?3. What is the PV of Rs.1,000 to be received 15 years from now if your opportunity cost is 7%? Present Value - single sums

  25. Present Value Factor and Time r = 0% r = 5% r = 10% r = 15%

  26. 1. If you sold land for Rs.11,933 that you bought 5 years ago for Rs.5,000, what is your annual rate of return?2. Suppose you placed Rs.100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to Rs.500? Present Value - single sums Ex.

  27. Multiple Cash Flows PV of multiple cash flows = the sum of the present values of the individual cash flows. FV of multiple cash flows at a common point in time = the sum of the future values of the individual cash flows at that point in time.

  28. How do we find the FV/PV of a cash flow stream when cash flows are different? (Use a 10% interest rate). Uneven Cash Flows FV -10,000 2,000 4,000 6,000 7,000 0 1 2 3 4

  29. Annuities An annuity is a series of identical cash flows that are expected to occur each period for a specified number of periods. Thus, CF1 = CF2 = CF3 = Cf4 = ... = CFn Examples of annuities: Installment loans (car loans, mortgages). Coupon payment on corporate bonds. Rent payment on your apartment.

  30. Types of Annuities Ordinary Annuity: An annuity with End-of-Period cash flows, beginning one period from today. Annuity Due: An annuity with Beginning-of-Period cash flows. Deferred Annuity: An annuity that begins more than one period from today.

  31. Future Value of an Annuity 1,000 1,000 1,000 1,000 0 1 2 3 4 When it has n periods, the equation is

  32. Simplification When you substitute above variables and simplify the equation, You can arrive at

  33. If you invest Rs.1,000 at the end of each next 3 years, at 8%, how much would you have after 3 years? Future Value - annuity Future Value Interest Factor for Annuity (FVIFA) Tables can be used to get the answer

  34. FVIFA Table for 8%

  35. Present Value Of an Annuity 1,000 1,000 1,000 1,000 0 1 2 3 4 When it has n periods, the equation is

  36. Simplification When you substitute above variables and simplify the equation, You can arrive at

  37. What is the PV of Rs.1,000 CF at the end of each of the next 3 years, if the opportunity cost is 8%? Present Value Of an Annuity Present Value Interest Factor for Annuity (PVIFA) Tables can be used to get the answer

  38. PVIFA Table for 8%

  39. Using an interest rate of 8%, we find that: The Future Value (at 3) is Rs.3,246.40. The Present Value (at 0) is Rs.2,577.10. 0 1 2 3 Earlier, we examined this “ordinary” annuity: 1000 1000 1000

  40. Same 3-year time line, Same 3 Rs.1000 cash flows, but The cash flows occur at the beginning of each year, rather than at the end of each year. This is an “annuity due.” 0 1 2 3 Annuity Due 1000 1000 1000

  41. If you invest Rs.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? Equation and the Solution is Future Value - annuity due

  42. What is the PV of Rs.1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%? 0 1 2 3 Present Value - Annuity due 1000 1000 1000 • Equation and the solution is

  43. Deferred Annuity The first cash flow in a deferred annuity is expected to occur later than t=1. The PV of the deferred annuity can be computed as the difference in the PVs of two annuities.

  44. Deferred Annuity An annuity’s first cash flow is expected to occur 3 years from today. There are 4 cash flows in this annuity, with each cash flow being Rs.500. At an interest rate of 10% per year, find the annuity’s present value. 0 1 2 3 4 5 6 Rs.500 Rs.500 Rs.500 Rs.500

  45. Example Cash flows from an investment are expected to be Rs.40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows? After graduation, you plan to invest Rs. 400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years? If you borrow Rs.100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?

  46. Upon retirement, your goal is to spend 5 years travelling around the world. To travel in style, it will require Rs.250,000 per year at the beginning of each year. If you plan to retire in 30 years, what are the equal monthly (end of month) payments necessary to achieve this goal? The funds in your retirement account will compound at 10% per annum on monthly basis. Team Assignment

  47. Present Value of Your Bank Loan Cynthia Smart agrees to repay her loan in 24 monthly installments of Rs.250 each. If the interest rate on the loan is 0.75% per month, what is the present value of her loan payments? You wish to retire 25 years from today with Rs.2,000,000 in the bank. If the bank pays 10% interest per year, how much should you save each year to reach your goal? Rob Steinberg borrows Rs.10,000 to be repaid in four equal annual installments, beginning one year from today. What is Rob’s annual payment on this loan if the bank charges him 14% interest per year?

  48. Loan Amortization Schedule It shows how a loan is paid off over time. It breaks down each payment into the interest component and the principal component. We will illustrate this using Rob Steinberg’s 4-year Rs.10,000 loan which calls for annual payments of Rs.3,432.05. Recall that the interest rate on this loan is 14% per year.

  49. Loan Amortization Schedule

  50. Loan Amortization Schedule

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