TOPIC 14: TIME VALUE OF MONEY CONCEPTS AND CALCULATIONS

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1 Present value (PV) A Present value is determined by taking the future value of a sum of money and calculating what it is worth today, using a discount rate The formula is PV = FV/(1 + I)N (1) Example: Calculate the present value of $10,000 to be received in ve years, using an annual interest rate of 10 percent (2) Solution: 10,000 FV, 10 I, 5 N, calculate PV $6,20921 (ignore the sign) B The more frequent the compounding, the smaller the present value 2 Future value (FV) is the future amount of a sum invested today that will grow over time when it is compounding interest The formula for nding the future value of a single cash ow is FV = PV(1 + I)N A Example: Calculate the future value of $10,000 invested for ve years, using an annual interest rate of 10 percent B Solution: 10,000 PV, 10 I, 5 N, calculate FV $16,10510 3 Ordinary annuity and annuity due A An annuity is a series of equal cash ows that occur at equal intervals over a period of time For example, the receipt of $1,000 at the end of each year for the next 10 years is an annuity (1) An ordinary annuity is one in which cash ows begin at the end of each year (2) An annuity due is one in which cash ows begin on the same day as the initial investment B Example 1: Finding the future value of an ordinary annuity (1) Calculate the future value of an ordinary annuity that will pay $1,000 per year for each of the next 10 years while earning a 12 percent rate of return (2) Solution: 10 N, 12 I, 1,000 PMT, compute (CPT) FV $17,54873 C Example 2: Finding the present value of an annuity due (1) Calculate the present value of an annuity of $2,000 received annually, beginning today and continuing for 15 years, earning a 10 percent rate of return (2) First, put your calculator in the begin mode (3) Solution: 15 N, 10 I, 2,000 payment (PMT), CPT PV $16,73338 D Example 3: Finding the annual payment in an ordinary annuity (1) Calculate the annual payments required to fund your retirement plan in order to have $25,000 at the end of 10 years while earning a 12 percent rate of return (2) Solution: 10 N, 12 I, FV 25,000, CPT PMT $1,42461

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36 - General Principles of Financial Planning E Example 4: Finding the monthly payment in an annuity due (1) Calculate the annual payments received at the beginning of each month for 10 years from an investment of $50,000 earning an annual return of 7 percent, compounded monthly (2) First, put your calculator in the begin mode (3) Solution: 120 N (10 12), 5833 (7/12) I, 50,000 PV, CPT PMT $57717 4 Net present value (NPV) A Net present value is the amount of cash ow (in present value terms) that a project generates after repaying the invested capital and required rate of return on that capital B If the project generates a positive NPV, then shareholder wealth increases In contrast, a negative NPV will decrease shareholder wealth C NPV is considered better than internal rate of return (IRR) because it measures pro tability in dollars added to shareholder value In contrast, IRR measures pro tability as a rate of return D NPV assumes that the reinvestment rate of cash ows is the cost of capital, whereas IRR assumes that the reinvestment rate is the IRR E When the IRR is equal to the cost of capital, the NPV will be zero If the IRR is less than the cost of capital, the result is a negative NPV F Example: Calculate the NPV of a project with an initial cost of $2,000 that produces the following cash ows (CF): year (1) +1,000; year (2) +500; year (3) +700; year (4) 500; year (5) +300 The cost of capital is 5 percent Solution: 2,000[CF0]; 1,000 CFj; 500 CFj; 700 [CFj]; 500 [CFj]; 300 [CFj]; 5 [I]; [NPV] $16571 5 Internal rate of return (IRR) A The IRR calculates the rate of return at which the present value of a series of cash in ows will equal the present value of the project s cost B It is also de ned as the rate of return in which the net present value of a project is zero It assumes that all cash ows are reinvested at the IRR C The IRR is equivalent to the yield to maturity (YTM), the geometric average return, and the compounded average rate of return D If IRR is less than the cost of capital, reject the project If IRR is greater than the cost of capital, accept the project E Example: Calculate the IRR of a project that has an initial out ow of 5,000 and will generate the following cash ows: year (1) 3,000; year (2) 500; year (3) 2,500; year (4) 500; year (5) 1,500 Solution: 5,000[CF0]; 3,000 CFj; 500 CFj; 2,500 [CFj]; 500 [CFj]; 1,500 [CFj]; [IRR] 1409 percent 6 Irregular cash ow A It is common for the stream of cash ows to change from year to year for projects or investments, so it is not an annuity The uneven cash ow is simply just a stream of (annual) single cash ows B To determine the FV/PV of irregular cash ows, you need to nd the FV/PV of each cash ow and then add them up The PV of an uneven cash ow stream is also calculated using the NPV function on your calculator C Example 1: Calculate the present value of an uneven cash ow series using a 10 percent discount rate and PV1 though PV5 Assume cash ows are:

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Topic 15: Characteristics and Consequences of Types of Entities - 37 0 1 2 3 4 5

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PV1: enter FV = 1,000; I/Y = 10; N = 1; CPT PV1 = 90909 PV2: enter FV = 700; I/Y = 10; N = 2; CPT PV2 = 57851 0; I/Y = 10; N = 3; CPT PV3 = 000 PV3: enter FV = PV4: enter FV = 5,000; I/Y = 10; N = 4; CPT PV4 = 3,41507 PV5: enter FV = 2,500; I/Y = 10; N = 5; CPT PV5 = 1,55230 Add up the PVs 3,47977 D Regarding Example 2, what you are really doing is nding the NPV of a series of cash ows: Solution: 0[CF0]; 1,000 CFj; 700 CFj; 0 [CFj]; 5,000 [CFj]; 2,500 [CFj]; 10 [I]; [NPV] $3,47977 7 In ation-adjusted earnings rate A Nominal rate of return investors require is: nominal risk-free rate = (1 + real risk-free rate) (1 + in ation rate) 1 B Real risk-free rate = [(1 + nominal risk-free rate)/(1 + in ation rate)] 1 C Calculate the nominal risk-free rate if the real rate is 5 percent and the expected in ation rate is 3 percent: (105) (103) 1 = 815 percent 8 Serial payments A A serial payment is a payment that increases at some constant rate on an annual basis; the constant rate is usually in ation B The last serial payment will have the same purchasing power as the initial serial payment C Serial payments are not xed payments like annuities; the rst serial payment will be less than an annuity payment, but the last serial payment will be more than an annuity payment D Example 1: Assume Jeff wants to start a business in ve years He needs to have $250,000 (today s dollars) in ve years to nance his business In ation is expected to average 3 percent, and Jeff can earn an 8 percent annual compounded rate on his investments What serial payment should Jeff invest at the end of the rst year Solution: 250,000 FV; 5 N; 0 PV; [(108/103 1)] 100[I]; [PMT] 103 [ ] $46,73678 E Example 2: What is Jeff s payment in the second year Solution: $46,73678 [ ] 103 $48,13888

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