1. 1 Increasing Speed of Exponential Functions The story of “A Grain of Rice” F(t) = 2 t-1

2. 2 Money-Time Relationships One dollar today is more valuable than one dollar tomorrow

3. 3 Capital and Interest You deposit $1,000 to the bank, and after one year, the bank will give you $1,050 The principal, or capital, is the initial amount of money in a transaction –The principal is $1,000 Interest is the “cost of using money” –Interest is $50 per year –Interest rate is 50/1000=5% per year How much is the interest if –the principal is $2,000? –after two years?

4. 4 Simple Interest The interest earned is linearly proportional to the principal, the interest rate, and the number of interest periods I=P*N*i –I: total interest –P: principal –N: number of interests periods –i: interest rate per interest period I borrow $1,000 at simple interest rate 10% per year. How much do I need to repay after 3 years? –I = 1000*3*10%=$300 –Total payment = 1000+300 = $1300

5. 5 Compound Interest Interest can also generate interest –For each period, the interest is calculated based on the principal and the accumulated interest at the beginning of that period I borrow $1,000 for 3 years at a compound interest rate of 10% Yearvalue at the beginning Interest generated value at the end 110001001100 2 1101210 3 1211331

6. 6 Comparison 1000 1100 1200 1300 1210 1331 Year 1 Year 2Year 3 Compound interest is assumed in this course 1000(1+0.1t) 1000(1+0.1) t : exponential function

7. 7 Facts about Compound Interest In 1626, the Manhattan Island in NY was bought for $24 from the Indians. Now we are in 2007. –At 8% simple interest, this $24 becomes 24(1+8%*381)=$756 –At 8% compound interest, this $24 becomes $1.3*10 14 –It is estimated that the value of Manhattan Island is about $1*10 12 in 2000 When can the money be doubled? –At 7% interest rate, about 10 years –At 10% interest rate, about 7 years –For interest rate i below 20%, the doubling time is approximately 72/i. Becoming a millionaire –Deposit $1 each day at the annual rate of 10%. After 60 years, the total money becomes $1.1 million.

8. 8 The $555,000 Student-Loan Burden The Wall Street Journal, Feb. 16, 2010 When Michelle Bisutti, a 41-year-old family practitioner in Columbus, Ohio, finished medical school in 2003, her student-loan debt amounted to roughly $250,000. Since then, it has ballooned to $555,000. It is the result of her deferring loan payments while she completed her residency, default charges and relentlessly compounding interest rates. Loan terms, including interest rates, are disclosed "multiple times and in multiple ways," says Martha Holler, a spokeswoman for Sallie Mae,

9. 9 Equivalence I borrow $8000 for 4 years at annual interest rate of 10% –There are different ways to repay the principal and interests. –They are all equivalent provided that the interest rate does not change Plan 1. Pay nothing until the end of year 4 –Interests incurred at different time: 800; 880; 968; 1065 –The amount of debt at different time:8800; 9680; 10648; 11713 –Final payment: 11713 Plan 2. Pay the interest generated at the end of each year –Interests incurred (and paid) at different time: 800; 800; 800; 800 –The amount of debt at different time :8000; 8000; 8000; 8000 –Total payment: 8000+3200=11200 (Meaning?) Plan 3. Pay in four Equal End-of-Year payment –Each year pay $2524 –End of Year 1 balance: 8000*1.1-2524 = 6276 –End of Year 2 balance: 6276*1.1-2524 = 4380 –End of Year 3 balance: 4380*1.1-2524 = 2294 –End of Year 4 balance: 2294*1.1-2524 = 0

10. 10 Cash Flow Diagram 8000 11713 1 2 3 4 8000 8000+800 1 2 3 4 800 8000 1 2 3 4 2524 Plan 1 Plan 2 Plan 3

11. 11 Cash Flow Diagram Time is the end-of-period time point A downward arrow represents cash outflow, and an upward arrow represents cash inflow. If needed, the viewpoint should be indicated –Previous examples are from the lender’s point of view –In this book, we often use the company’s (investor’s) point of view

12. 12 Present and Future Equivalence Notation i: interest rate per interest period N: number of compounding periods P: present sum of money; the equivalent value of a cash flow at a time reference point called present F: future sum of money; the equivalent value of a cash flow at a time reference point called future A: end-of-period cash flow in a uniform series cash flow, also called annuity

13. 13 Given P, to find F Formula: F=P(1+i) N The term (1+i) N, denoted by (F/P, i%, N), is called single payment compound amount factor Values of (F/P, i%, N) can be found in the appendix of the book P is given F to be found 1 2 … N

14. 14 Example Suppose I borrow $8,000 at the annual interest rate of 10%, promising to repay at the end of the 4 th year. How much do I need to repay? P=8000 From the table, we get (F/P,10%,4)=1.4641 –F=P(F/P,10%,4)=$11713 Or, we can calculate directly –F=8000(1+0.1) 4 =$11713

15. 15

16. 16 Given F, to find P Formula: P=F(1+i) -N The term (1+i) -N, denoted by (P/F, i%, N), is called single payment present worth factor Values of (P/F, i%, N) can be found in the appendix of the book P to be found F is given 1 2 … N

17. 17 Example An investor has an option to buy a tract of land that will be worth $10,000 in 6 years. If the value of the land increases at 8% each year, how much should the investor be willing to pay now for this property? P=F(P/F, i%, N) P=10000(P/F, 8%, 6)=10000*0.6302=$6302