Transcript 下載/瀏覽
Chapter 3-- Measuring Wealth: Time Value of Money Why must future dollars be put on a common basis before adding? Cash is a limited and controlled resource. Those controlling the resource can charge for its use. The longer the period of use the higher the interest (or rental fee) for the use of the cash. Therefore, you cannot add a 1 year dollar with a two year dollar. 1 Future Dollar Equivalent (Future Value) of a Present Amount These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package solution FV = PV (1+i)n Example -- How much will $1000 grow to at 12% in 15 years? Formula Enter 1.12, yx, 15, times 1000 = $5473.56 2 Present Dollar Equivalent (Present Value) of a Future Amount These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package Formula solution PV = FV/ (1+i)n Example -- how much do you need today to have $1,000,000 in 40 years if your money is earning 12%? Enter 1.12, yx, 40, =, 1/x, times 1000000 = $10,747 3 Finding the Rate Between Two Single Amounts These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package Formula solution -- i = (FV/PV)1/n -1 – you purchased your house for $76,900 in 1994. Your neighbor’s house of similar value sold for $115,000 in 2004 ( 10 years later). What rate of return are you earning on your house? Example Enter 115000 / 76900, yx, .1, –, 1, =, .0411 or 4.11% 4 Finding the Number of Periods Needed Between Two Amounts These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package Formula solution -- n = LN(FV/PV)/LN(1+i) – you inherit $120,000 from your great aunt and invest it to earn 8% interest. How long will it take for this to grow to $1,000,000? Example Enter – (1000000 / 120000) ,=, ln -- this gives you 2.1203 Enter – (1.08), ln -- this gives you .0770 Divide the two results to get 27.55 years 5 Different Types of Annuities. Ordinary annuities -- dollars are received or paid at the end of the period and grow until the end of the period. All annuity formulas to be discussed will work for ordinary annuities with no adjustments. Annuities due -- dollars are received or paid at the beginning of the period and grow until the end of the period. All annuity formulas to be discussed will need adjustment (for the extra year’s worth of interest). 6 Future Value of an Ordinary Annuity and an Annuity Due Example -- How much will you have at the end of 35 years if can earn 12% on your money and place $10,000 per year in you 401k account at the beginning of the year? (at the end of the year?) Formula solution ordinary annuity – FV = [((1+i)r –1) / r ] payment Enter – (1.12,yx, 35, =, –1, = ) / .12 times 10000 The answer is $4,316,635 solve for an annuity due just remove the –1 from the formula above – the answer is then $4,834,631 To 7 Present Value of an Ordinary Annuity and an Annuity Due Example -- How much is a trust fund worth today that promises to pay you $10,000 at the end (or beginning) of each year for 35 years if can earn 12% on your money? Formula solution ordinary annuity – FV = [[1-(1/(1+i)r)] / r ] payment Enter – 1.12,yx, 35, =, 1/x, –,1, +/-, = ) / .12 times 10000 this will give you the answer of $81,755 To solve for an annuity due, change the 35 to 34 in the formula above then add an additional 10000 payment to the answer of $81,566 to get $91,566 8 Present Value of an Uneven Stream of Year-end Cash Flows – You can invest in an athletic endorsement that will increase net cash flows to your firm by: Example $800,000 at the end of year 1 $600,000 at the end of year 2 $400,000 at the end of year 3 After that, you do not expect any additional benefit from her endorsement. What is the present value of this endorsement if the firm has a cost of funds of 8 percent? Formula solution discount each future cash flow to present by dividing by (1+i)n and then add up these results Answer -- $1,572,679 9 Rate of Return on an Uneven Stream of Year-end Cash Flows – you can invest in an athletic endorsement that will increase net cash flows to your firm by: Example $800,000 at the end of year 1 $600,000 at the end of year 2 $400,000 at the end of year 3 After that, you do not expect any additional benefit from her endorsement. If this endorsement cost the firm $1,000,000 today, what is the rate of return of this endorsement? Calculator solution – 2nd , CLR WRK, CF 1000000, +/-, enter, 800000, enter, 600000, enter, 400000, enter, IRR, CPT The answer 42.06% appears 10 Adjusting for Compounding More Than Once a Year In the formula, you divide the interest rate by the number of compoundings and multiple the n by the number of compoundings to account for monthly, quarterly or semi-annual compounding Excel Example -- What will $5,000 dollars invested today grow to at the end of 10 years if your account promises a 10% APR compounded monthly? You Enter -- for the monthly answer -=FV(.10/12,10*12,0,-5000,0) You Enter -- .10/12, =, +1, =, yx ,120 times 5000 = $13,535 11 Adjusting for Compounding More Than Once a Year To adjust an APR or nominal rate to an effective rate use the following formula: Effective rate = [(1+ nominal rate / # of comp.)n times # of comp]-1 12 Adjusting for When Cash Flows Are Received Daily A close approximation for level daily cash flows is the use of mid-year cash flows. When using a computer package with both mid year and year-end cash flows it is easiest to use the PV function to discount each period’s cash flow back to present individually. When looking for the internal rate of return of daily cash flows the problem must be worked as a goal seek (solving for the interest rate). 13 Valuing Perpetuities Value perpetual no-grow cash flows Formula Present value = cash flow / discount rate Value perpetual growing cash flows Formula Present value = cash flow /(discount rate - growth rate) 14