1. Business Finance Michael Dimond

2. Michael Dimond School of Business Administration The Time Value of Money (TVM)

3. Michael Dimond School of Business Administration Compounding Compounding is the growth of value resulting from some sort of return (such as interest payments) being added to the original amount. If you put $100 in the bank and receive 10% annual interest After 1 year: $100 x (1+10%) = $110 After 2 years: $110 x (1+10%) = $121 After 3 years: $121 x (1+10%) = $133.10 The three-year compounding could be rewritten like this: After 3 years: $100 x (1+10%) x (1+10%) x (1+10%) = $133.10 or $100 x (1+10%) 3 = $133.10 The general formula for compounding: PV x (1+i) n = FV where PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate

4. Michael Dimond School of Business Administration Discounting Discounting is the opposite of compounding. Instead of growing an amount by a specific rate, we are taking that expected growth out of a future total to find what the starting figure would be. Since compounding multiplies by (1+i) n, discounting will do the opposite: divide by (1+i) n. If you will need $133.10 at the end of three years, and you can receive 10% annual interest, how much would you need to deposit today? $133.10 ÷ (1+10%) 3 = $100 The general formula for discounting: FV ÷ (1+i) n = PV where PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate

5. Michael Dimond School of Business Administration Moving parts of compounding & discounting There are four “moving parts” in a compounding or discounting computation: PV (Present Value) FV (Future Value) n (Number of Periods) i (Rate of Return per Period) The general formula for compounding: PV x (1+i) n = FV The more periods something is compounded, the greater the future value is. The general formula for discounting: FV ÷ (1+i) n = PV The more periods something is discounted, the smaller the present value is.

6. Michael Dimond School of Business Administration What if compounding happens more frequently? APR means Annual Percentage Rate For example: 12% APR means 12% interest rate for the year. If interest compounds more frequently, divide that rate by the periods per year. 12 % APR compounded… Annually1 period/yr12% ÷ 1 = 12.00% interest/period Quarterly4 periods/yr12% ÷ 4 = 3.00% interest/period Monthly12 periods/yr12% ÷ 12 = 1.00% interest/period Daily360 periods/yr12% ÷ 360 = 0.03% interest/period Why do financiers use 360 days instead of 365? After 1 year, how much will $100 be at 12% APR… compounded at the end of the year?$100 x (1.1200) 1 = $112.00 compounded at the end of each quarter? $100 x (1.0300) 4 = $112.55 compounded at the end of each month ? $100 x (1.0100) 12 = $112.68 compounded at the end of each day ? $100 x (1.0003) 360 = $112.75 Remember to watch out for rounding errors: 12/360 = 0.03333…

7. Michael Dimond School of Business Administration Effective Annual Rate The Effective Annual Rate (EAR) is the APR adjusted for the value of compounding. EAR = (1+APR ÷ n) n - 1 12% APR compounded annually = (1.1200) 1 -1 = 12.00% EAR 12% APR compounded quarterly = (1.0300) 4 -1 = 12.55% EAR 12% APR compounded monthly = (1.0100) 12 -1 = 12.68% EAR 12% APR compounded daily = (1.0003) 360 -1= 12.75% EAR Sometimes this is called the APY (Annual Percent Yield)

8. Michael Dimond School of Business Administration Time vs Return: Basic TVM A dollar is worth more now than it will be at any time in the future. The concept is called the Time Value of Money (TVM). What makes money lose value over time? How long an investment takes to pay out will affect the price you would pay. If you require a 12% annual return, how much would you pay for $100 to be given to you in… 1 year? 3 years? 10 years? The further in the future a cash flow is, the less it is worth.

9. Michael Dimond School of Business Administration Understanding TVM problems Time Value of Money scenarios are examined with a timeline. Each tick mark on the timeline represents the end of one period. The first tick mark on the left is labeled 0 because zero periods have elapsed. It indicates the present, or the planned beginning of a project. The last tick mark indicates the end of the last period being analyzed. Payments and compounding happen at the end of each period. Consider our basic compounding example: 010123456789 0123 -100133.10 i = 10% PV = -100 i = 10% n = 3 FV = 133.10 You could use this diagram to analyze the future value or the present value

10. Michael Dimond School of Business Administration You could use this diagram to analyze the future value or the present value. Notice the cash outflow (money you invested) is shown with a minus sign. Financial calculators require this to give you the correct answer. This is called the sign convention. Understanding TVM problems 100 x (1+0.10) 3 = 133.10 :. FV = 133.10 0123 -100? i = 10% 0123 ?133.10 i = 10% 133.1 ÷ (1+0.10) 3 = 100 :. PV = -100

11. Michael Dimond School of Business Administration A TVM problem has one more “moving part” than a simple compounding or discounting problem. PV (Present Value) FV (Future Value) n (Number of Periods) i (Rate of Return per Period) PMT (Payment) There may be payments which happen between the beginning and end of the timeline. Each payment is discounted separately. The PV of the stream of cash flows is the sum of the individual PVs. Moving parts of TVM

12. Michael Dimond School of Business Administration If you require a 12% annual return, what would you pay for… …$100 to be delivered in 1 year? ($89.2857) …$100 to be delivered in 2 years? ($79.7194) …$100 to be delivered in 3 years? ($71.1780) …all of the above (i.e. $100 to be paid at the end of each of the next three years)? By adding together the present values, you find the value of all the cash flows in the stream. Discounting payments 0123 ?100 i = 12% 100 ÷ (1+0.12) 3 100 100 ÷ (1+0.12) 2 100 100 ÷ (1+0.12) 1 89.2857 79.7194 + 71.1780 240.1831

13. Michael Dimond School of Business Administration Remember the magic machine? $100 per month for 5 years. What if you require a 12% annual return? Discounting a stream of cash flows 060123456575859 ?100 i = 1% monthly (12% APR) 100 100 ÷ (1+0.01) 60 100 ÷ (1+0.01) 1 Each payment has its own present value. Adding up those PVs gives the total value of the stream of cash flows. 100 ÷ (1+0.01) 2 100 ÷ (1+0.01) 3 100 ÷ (1+0.01) 4 100 ÷ (1+0.01) 56 100 ÷ (1+0.01) 57 100 ÷ (1+0.01) 58 100 ÷ (1+0.01) 59 99.01 57.28 98.03 97.06 96.10...... 56.71 56.15 55.60 55.04

14. Michael Dimond School of Business Administration Timelines & PMTs i and n are always in the same increment. Monthly periods → monthly rate. Annual periods → annual rate. What happens to PV as n increases? As n increases, PV becomes smaller 100 ÷ 1.01 2 = 98.03 100 ÷ 1.02 60 = 55.04 Value = Sum of PVs So if you demand a 12% rate of return, the value of the machine’s monthly payments is: There is also an easier way to compute that value…

15. Michael Dimond School of Business Administration Ordinary Annuity: FV & PV A stream of cash flows where all payments are equal is called an Annuity. In an Ordinary Annuity, each payment happens at the end of the period. Your financial calculator can solve these easily and quickly. Find PV given n, i, and PMT Find FV given n, i, and PMT For the magic machine, the inputs would be: PV = ? (This is what we’re solving for) n = 60 (monthly payments) i = 12/12 (12% ÷ 12 months) PMT = 100 (per month) FV = 0 (This has no value once the final payment is delivered) Notice that these three items must always be in the same timeframe: monthly annually, daily… whatever is in the scenario

16. Michael Dimond School of Business Administration An Annuity Due has payments which happen at the beginning of each period instead of the end. Typically used in real estate… Timelines… Difference in value… Calculator setting… Comments about Annuity Due 0123 100 i = 12% 100 ÷ (1+0.12) 3 100 100 ÷ (1+0.12) 2 100 100 ÷ (1+0.12) 1 89.2857 79.7194 + 71.1780 240.1831 0123 100 0 i = 12% 100 100 ÷ (1+0.12) 2 100 100 ÷ (1+0.12) 1 100.0000 89.2857 + 79.7194 269.0051 Ordinary Annuity Annuity Due

17. Michael Dimond School of Business Administration Ordinary Annuity with an additional payout What happens if there is a stream of payments, and also a lump sum being paid at the end of the timeline? Timeline… Find PV given n, i, PMT & FV

18. Michael Dimond School of Business Administration Uneven cash flows Covered later in the quarter “Part 2” of How Do I Use This Financial Calculator explains how to use the calculator for uneven cash flows

19. Michael Dimond School of Business Administration

20. If you require a 12% annual return, what would you pay for… …$90 to be delivered in 1 year? ($80.3571) …$95 to be delivered in 2 years? ($75.7334) …$99 to be delivered in 3 years? ($70.4662) …all of the above? By adding together the present values, you find the value of all the cash flows in the stream. Discounting unequal payments 0123 ? 99 i = 12% 99 ÷ (1+0.12) 3 95 95 ÷ (1+0.12) 2 90 90 ÷ (1+0.12) 1 80.3571 75.7334 + 70.4662 226.5567

21. Michael Dimond School of Business Administration Using the calculator (NPV function)

22. Michael Dimond School of Business Administration Using the calculator (NPV function)