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Time Value of Money

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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

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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 $1,000 today $1,000 in 5 years? Which would you rather have -- $1,000 today or $1,000 in 5 years? $1,000 today Obviously, $1,000 today. TIME VALUE OF MONEY 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 INTEREST TIME allows one the opportunity to postpone consumption and earn INTEREST.

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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) FV n = PV 0 × (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

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$2,000 today in an account that pays 6 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? Future Value(Graphic) 0 1 2 $2,000 FV 6%

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FV 1 PV FV 1 = PV (1+r) n $2,000 = $2,000 (1.06) 2 $2,247.20 = $2,247.20 Future Value (Formula) 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 y x) 2 x 2000 =

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$5,000 5 years 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. Future Value (Example) 5 0 1 2 3 4 5 $5,000 FV 5 8%

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

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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) PV 0 = FV n / (1 + r) n PVIF (r,n) Examples

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Assume that you need to have exactly $4,000 saved 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? 5 0 5 10 $4,000 6% PV 0 Present Value (Graphic)

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

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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 todays deposit will grow at a compound rate of 4% annually. Present Value Example 5 0 1 2 3 4 5 $2,500 PV 0 4%

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PV 0 FV n PV 0 $2,500/(1.04) 5 Calculation based on general formula: PV 0 = FV n / (1+r) n PV 0 = $2,500/(1.04) 5 = $2,054.81 Calculator keystrokes: 1.04 2nd y x 5 = 2 nd 1/x X 2500 = Present Value Solution

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General Formula: PV 0 FV n = PV 0 (1 + [r/m]) mn n: Number of Years m: Compounding Periods per Year r: Annual Interest Rate FV n,m : FV at the end of Year n PV 0 PV 0 : PV of the Cash Flow today Frequency of Compounding

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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

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

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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

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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

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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

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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

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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

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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

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Present Value of an Annuity – Examples Computing Cash Flow per period in annuity – Examples

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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

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Effective Annual Rate Compounding other than annual

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Multiple Cash Flows –Future Value Example 6.1

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