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Time Value of Money

Time Value of Money. Outline. Meaning of Time Value Concept of Future Value and Compounding (FV) Concept of Present Value and Discounting (PV) Frequency of Compounding Present Value versus Future Value Determining the Interest rate (r) Determining the Time Period (n)

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Time Value of Money

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  1. Time Value of Money

  2. Outline • Meaning of Time Value • Concept of Future Value and Compounding (FV) • Concept of Present Value and Discounting (PV) • Frequency of Compounding • Present Value versus Future Value • Determining the Interest rate (r) • Determining the Time Period (n) • Future Value and Present Value of Multiple Cash Flows • Annuities and Perpetuities

  3. Time Value of Money • Basic Problem: • How to determine value today of cash flows that are expected in the future? • Time value of money refers to the fact that a dollar in hand today is worth more than a dollar promised at some time in the future • Which would you rather have -- $1,000 today or $1,000 in 5 years? • Obviously, $1,000 today. • Money received sooner rather than later allows one to use the funds for investment or consumption purposes. This concept is referred to as the TIME VALUE OF MONEY!! • TIME allows one the opportunity to postpone consumption and earn INTEREST.

  4. Future Value and Compounding • Future value refers to the amount of money an investment will grow to over some length of time at some given interest rate • To determine the future value of a single cash flows, we need: • present value of the cash flow (PV) • interest rate (r), and • time period (n) • FVn = PV0× (1 + r)n • Future Value Interest Factor at ‘r’ rate of interest for ‘n’ time periods • Examples on computation of future value of a single cash flow

  5. Future Value (Graphic) If you invested $2,000 today in an account that pays 6% interest, with interest compounded annually, how much will be in the account at the end of two years if there are no withdrawals? 0 1 2 6% $2,000 FV

  6. Future Value (Formula) FV1 = PV (1+r)n = $2,000(1.06)2 = $2,247.20 FV = future value, a value at some future point in time PV = present value, a value today which is usually designated as time 0 r = rate of interest per compounding period n = number of compounding periods Calculator Keystrokes: 1.06 (2nd yx) 2 x 2000 =

  7. Future Value (Example) • John wants to know how large his $5,000 deposit will become at an annual compound interest rate of 8% at the end of 5 years. 0 1 2 3 4 5 8% $5,000 FV5

  8. Future Value Solution • Calculation based on general formula: FVn = PV (1+r)n • FV5 = $5,000 (1+ 0.08)5 • = $7,346.64 • Calculator keystrokes: 1.08 2nd yx x 5000 =

  9. Present Value and Discounting • The current value of future cash flows discounted at the appropriate discount rate over some length of time period • Discounting is the process of translating a future value or a set of future cash flows into a present value. • To compute present value of a single cash flow, we need: • Future value of the cash flow (FV) • Interest rate (r) and • Time Period (n) • PV0 = FVn / (1 + r)n • PVIF (r,n) • Examples

  10. Present Value (Graphic) Assume that you need to have exactly $4,000saved 10 years from now. How much must you deposit today in an account that pays 6% interest, compounded annually, so that you reach your goal of $4,000? 0 5 10 6% $4,000 PV0

  11. Present Value (Formula) PV0 = FV / (1+r)10 = $4,000/ (1.06)10 = $2,233.58 0 5 10 6% $4,000 PV0

  12. Present Value Example Joann needs to know how large of a deposit to make today so that the money will grow to $2,500 in 5 years. Assume today’s deposit will grow at a compound rate of 4% annually. 0 1 2 3 4 5 4% $2,500 PV0

  13. Present Value Solution • Calculation based on general formula: PV0 = FVn / (1+r)nPV0= $2,500/(1.04)5 = $2,054.81 • Calculator keystrokes: 1.04 2nd yx 5 = 2nd 1/x X 2500 =

  14. Frequency of Compounding General Formula: FVn = PV0(1 + [r/m])mn n: Number of Years m: Compounding Periods per Year r: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today

  15. Frequency of Compounding Example • Suppose you deposit $1,000 in an account that pays 12% interest, compounded quarterly. How much will be in the account after eight years if there are no withdrawals? PV = $1,000 r = 12%/4 = 3% per quarter n = 8 x 4 = 32 quarters

  16. Solution based on formula: FV= PV (1 + r)n = 1,000(1.03)32 = 2,575.10 Calculator Keystrokes: 1.03 2nd yx 32 X 1000 =

  17. Present Value versus Future Value • Present value factors are reciprocals of future value factors • Interest rates and future value are positively related • Interest rates and present value are negatively related • Time period and future value are positively related • Time period and present value are negatively related

  18. Determining the Interest Rate (r) • At what rate of interest should we invest our money today to get a desired amount of money after a certain number of years? • Essentially, we are trying to determine the interest rate given present value (PV), future value (FV), and time period (n) • Examples • The rate which money can be doubled/tripled

  19. Determining the Time Period (n) • For how long should we invest money today to get a desired amount of money in the future at a given rate of interest • Determining the time period (n) for which a current amount (PV) needs to be invested to get a certain future value (FV) given a rate of interest (r). • Examples • The time period needed to double/triple our current investment

  20. Future Value of Multiple Uneven Cash Flows • Compute the future value of each single cash flow using future value formula and add them up over all the cash flows • Example

  21. Present Value of Multiple Uneven Cash Flows • Compute the present value of each single cash flow using present value formula and add them over all the cash flows • Examples

  22. Annuities • A series of level/even/equal sized cash flows that occur at the end of each time period for a fixed time period • Examples of Annuities: • Car Loans • House Mortgages • Insurance Policies • Some Lotteries • Retirement Money

  23. Present Value of an Annuity • Examples • Computing Cash Flow per period in annuity • Examples

  24. Perpetuities • A series of level/even/equal sized cash flows that occur at the end of each period for an infinite time period • Examples of Perpetuities: • Consoles issued by British Government • Preferred Stock • Present Value of a Perpetuity

  25. Effective Annual Rate • Compounding other than annual

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