1. Chapter 3-- Measuring Wealth: Time Value of Money u Why must future dollars be put on a common basis before adding? u Cash is a limited and controlled resource. u Those controlling the resource can charge for its use. u The longer the period of use the higher the interest (or rental fee) for the use of the cash. u Therefore, you cannot add a 1 year dollar with a two year dollar.

2. Future Dollar Equivalent (Future Value) of a Present Amount u These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package u Formula solution FV = PV (1+i) n u Example -- How much will $1000 grow to at 12% in 15 years? u Enter 1.12, y x, 15, times 1000 = $5473.56

3. Present Dollar Equivalent (Present Value) of a Future Amount u These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package u Formula solution PV = FV/ (1+i) n u Example -- how much do you need today to have $1,000,000 in 40 years if your money is earning 12%? u Enter 1.12, y x, 40, =, 1/x, times 1000000 = $10,747

4. Finding the Rate Between Two Single Amounts u These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package u Formula solution -- i = (FV/PV) 1/n -1 u 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? u Enter 115000 / 76900, y x,.1, –, 1, =,.0411 or 4.11%

5. Finding the Number of Periods Needed Between Two Amounts u These can be solved using formulas, tables, a financial calculator or a computer spreadsheet package u Formula solution -- n = LN(FV/PV)/LN(1+i) u 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? u Enter – (1000000 / 120000),=, ln -- this gives you 2.1203 u Enter – (1.08), ln -- this gives you.0770 u Divide the two results to get 27.55 years

6. Different Types of Annuities. u Ordinary annuities -- dollars are received or paid at the end of the period and grow until the end of the period. u All annuity formulas to be discussed will work for ordinary annuities with no adjustments. u Annuities due -- dollars are received or paid at the beginning of the period and grow until the end of the period. u All annuity formulas to be discussed will need adjustment (for the extra year’s worth of interest).

7. Future Value of an Ordinary Annuity and an Annuity Due u 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?) u Formula solution ordinary annuity – u FV = [((1+i) r –1) / r ] payment u Enter – (1.12,y x, 35, =, –1, = ) /.12 times 10000 u The answer is $4,316,635 u To solve for an annuity due just remove the –1 from the formula above – the answer is then $4,834,631

8. Present Value of an Ordinary Annuity and an Annuity Due u 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? u Formula solution ordinary annuity – u FV = [[1-(1/(1+i) r )] / r ] payment u Enter – 1.12,y x, 35, =, 1/x, –,1, +/-, = ) /.12 times 10000 u this will give you the answer of $81,755 u 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

9. Present Value of an Uneven Stream of Year-end Cash Flows u Example – You can invest in an athletic endorsement that will increase net cash flows to your firm by: u $800,000 at the end of year 1 u $600,000 at the end of year 2 u $400,000 at the end of year 3 u 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? u Formula solution discount each future cash flow to present by dividing by (1+i) n and then add up these results u Answer -- $1,572,679

10. Rate of Return on an Uneven Stream of Year-end Cash Flows u Example – you can invest in an athletic endorsement that will increase net cash flows to your firm by: u $800,000 at the end of year 1 u $600,000 at the end of year 2 u $400,000 at the end of year 3 u 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? u Calculator solution – 2 nd, CLR WRK, CF 1000000, +/-, enter, 800000, enter, 600000, enter, 400000, enter, IRR, CPT u The answer 42.06% appears

11. Adjusting for Compounding More Than Once a Year u 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 u 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) u You Enter --.10/12, =, +1, =, y x,120 times 5000 = $13,535

12. Adjusting for Compounding More Than Once a Year u To adjust an APR or nominal rate to an effective rate use the following formula: u Effective rate = [(1+ nominal rate / # of comp.) n times # of comp ]-1

13. Adjusting for When Cash Flows Are Received Daily u A close approximation for level daily cash flows is the use of mid-year cash flows. u 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. u 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).

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