1. 8/25/04 Valerie Tardiff and Paul Jensen Operations Research Models and Methods Copyright 2004 - All rights reserved Time Value of Money Don’t put your money in your mattress

2. 2 Growth of Money

3. 3 $150$105 $140 $104 $130 $103 $120 $102 $110 $10 $100 1 Capital at EndInterest (10%) Capital at BeginningPeriod (Year) Simple Interest Interest = initial capital * interest rate

4. 4 $161.05$14.645 $146.41 $13.314 $133.10 $12.103 $121 $112 $110 $10$1001 Capital at EndInterest (10%) Capital at BeginningPeriod (Year) Compounded Interest Interest = accumulated capital * interest rate

5. 5 Impact of Interest Rate

6. 6 Impact of Time

7. 7 Equivalence When we are willing to trade one cash flow for another, the two cash flows are said to be economically equivalent. Equivalence depends on interest rate and time. In comparisons we will always assume compound interest.

8. 8 Simple Questions I have $100 now (at time 0). I invest it for 10 years at interest rate i. What will I have at time 10 if compound interest is used? $100(1+ i ) 10 At 10% interest $100(1.1) 10 = 100(2.594)=$259.4

9. 9 Simple Questions I expect to receive $100 at time 10. How much is the promise of that future amount worth to me today assuming compound interest? $100/(1+ i ) 10 At 10% interest $100/(1.1) 10 = 100/(2.594)=$38.55

10. 10 Conclusion –Time Value of Money A dollar today is worth more than a dollar in the future because of the interest (profit) it can earn. To compare alternatives with different cash flows at different times, you must consider the time value of money.

11. 11 Cash Flows A cash flow refers to a receipt or payment of an amount of money and is defined by 1) its dollar value and 2) the time of its occurrence Many analyst use cash flow diagrams to represent costs and revenues over time. They help maintain a consistent and time-equivalent view point.

12. 12 Cash Flow Diagrams Take the lender’s point of view

13. 13 Notation i = the interest rate per interest period. N = the total number of interest periods. P = a sum of money at a time chosen for purposes of analysis as time zero (present value or present worth). F = a future sum of money at the end of the analysis. A n = a discrete payment or receipt occurring at the end of some interest period. A = the special situation when A 1 = A 2 =... = A N. G = the gradient or amount by which payments increase or decrease linearly (e.g., A n = A 1 + (n-1)G ). g = the gradient or amount by which payments increase or decrease geometrically (e.g., A n = A 1 (1+g) n-1 )

14. 14 Simple Questions What is asked? What is i? What is N?