Chapter 3 Measuring Wealth: Time Value of Money

Future Value and Present Value These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package Formula solution FV = PV (1+i)n Formula solution PV = FV/ (1+i)n

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 Example – 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? Enter 115000 / 76900, yx, .1, –, 1, =, .0411 or 4.11%

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) Example – 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? 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

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).

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

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

Present Value of an Uneven Stream of Year-end Cash Flows Example – You can invest in an athletic endorsement that will increase net cash flows to your firm by: $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

Rate of Return on an Uneven Stream of Year-end Cash Flows Example – you can invest in an athletic endorsement that will increase net cash flows to your firm by: $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

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

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

Valuing Perpetuities Value perpetual no-grow cash flows Formula Present value = cash flow / discount rate Value perpetual growing cash flows Present value = cash flow /(discount rate - growth rate)