1. 1 ECGD3110 Systems Engineering & Economy Lecture 3 Interest and Equivalence

2. 2 Estimating Benefits For the most part, we can use exactly the same approach to estimate benefits as to estimate costs: –Fixed and variable benefits –Recurring and non-recurring benefits –Incremental benefits –Life-cycle benefits –Rough, semi-detailed, and detailed benefit estimates –Difficulties in estimation –Segmentation and index models Major differences between benefit estimation and cost estimation: –Costs are more likely to be underestimated –Benefits are most likely to be overestimated –Benefits tend to occur further in the future than costs

3. 3 Example Two summer Camps have the following data for a 12-week session: a. Develop the mathematical relationships for total cost and total revenue for camp A b. What is the total number of campers that will allow camp B to break even? c. What is the profit or loss for the 12-week session if camp A operates at 80% capacity? d. Determine the breakeven number of campers for the two camps to have equal total costs for a 12-week session. Camp A Charge per camper $120 per week Fixed costs $48,000 per session Variable cost per camper $80 per week Capacity 200 campers Camp B Charge per camper $100 per week Fixed costs $60,600 per session Variable cost per camper $50 per week Capacity 150 campers

4. 4 Time Value of Money Question: Would you prefer $100 today or $120 after 1 year? There is a time value of money. Money is a valuable asset, and people would pay to have money available for use. The charge for its use is called interest rate. Question: Why is the interest rate positive? Argument 1: Money is a valuable resource, which can be “ rented, ” similar to an apartment. Interest is a compensation for using money. Argument 1: Money is a valuable resource, which can be “ rented, ” similar to an apartment. Interest is a compensation for using money. Argument 2: Interest is compensation for uncertainties related to the future value of the money. Argument 2: Interest is compensation for uncertainties related to the future value of the money.

5. 5 Simple Interest Simple interest is interest that is computed on the original sum. If you loan an amount P for n years at a rate of i % a year, then after n years you will have: n years you will have: P + n  (i P) = P + n  i  P = P (1 + i  n). Note: Interest is usually compound interest, not simple interest. Example: You loan your friend $5000 for five years at a simple interest rate of 8% per year. Example: You loan your friend $5000 for five years at a simple interest rate of 8% per year. At the end of each year your friend pays you 0.08  5000 = $400 as an interest. At the end of five years your friend also repays the $5000 After five years your friend has paid you: 5000 + 5  400 = 5000 + 2000 = $7000 Note:The borrower has used the $400 for 4 years without paying interest on it.

6. 6 Compound Interest Compounded interest is interest that is charged on the original sum and un-paid interest. You put $500 in a bank for 3 years at 6% compound interest per year. At the end of year 1 you have (1.06)  500 = $530. At the end of year 1 you have (1.06)  500 = $530. At the end of year 2 you have (1.06)  530 = $561.80. At the end of year 2 you have (1.06)  530 = $561.80. At the end of year 3 you have (1.06)  $561.80 = $595.51. At the end of year 3 you have (1.06)  $561.80 = $595.51. Note: Note: $595.51 = (1.06)  561.80 = (1.06) (1.06) 530 = (1.06) (1.06) 530 = (1.06) (1.06) (1.06) 500 = 500 (1.06) 3 = (1.06) (1.06) (1.06) 500 = 500 (1.06) 3

7. 7 Single Payment Compound Formula If you put P in the bank now at an interest rate of i for n years, the future amount you will have after n years is given by: F = P (1+i) n F = P (1+i) n The term (1+i) n is called the single payment compound factor. The factor is used to compute F, given P, and given i and n. Handy Notation. (F/P,i,n) = (1+i) n (F/P,i,n) = (1+i) n F = P (1+i) n = P (F/P,i,n)

8. 8 Present Value Example If you want to have $800 in savings at the end of four years, and 5% interest is paid annually, how much do you need to put into the savings account today? We solve P (1+i) n = F for P with i = 0.05, n = 4, F = $800 P = F/(1+i) n = F(1+i) -n ( P = F (P/F,i,n) ) P = F/(1+i) n = F(1+i) -n ( P = F (P/F,i,n) ) = 800/(1.05) 4 = 800 (1.05) -4 = 800 (0.8227) = $658.16 Single Payment Present Worth Formula P = F/(1+i) n = F(1+i) -n

9. 9

10. 10 Present Value Example: You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back in 5 years. There are many ways the debt can be repaid. Plan A: At end of each year pay $1,000 principal plus interest due. Plan A: At end of each year pay $1,000 principal plus interest due. Plan B: Pay interest due at end of each year and principal at end of five years. Plan B: Pay interest due at end of each year and principal at end of five years. Plan C: Pay in five end-of-year payments. Plan C: Pay in five end-of-year payments. Plan D: Pay principal and interest in one payment at end of five years. Plan D: Pay principal and interest in one payment at end of five years.

11. 11 Example Plan A: At end of each year pay $1,000 principal plus interest due. abcdef Year Amnt.Owed Int. Owed Total Owed Princip.PaymentTotalPayment int*bb+c 15,0004005,4001,0001,400 24,0003204,3201,0001,320 33,0002403,2401,0001,240 42,0001602,1601,0001,160 51,000801,0801,0001,080 SUM15,0001,20016,2005,0006,200

12. 12 Example (cont'd) Plan B: Pay interest due at end of each year and principal at end of five years. abcdef Year Amnt.Owed Int. Owed Total Owed Princip.PaymentTotalPayment int*bb+c 15,0004005,4000400 25,0004005,4000400 35,0004005,4000400 45,0004005,4000400 55,0004005,4005,0005,400 SUM25,0002,00027,0005,0007,000

13. 13 Example (cont'd) Plan C: Pay in five end-of-year payments. abcdef Year Amnt.Owed Int. Owed Total Owed Princip.PaymentTotalPayment int*bb+c 15,0004005,4008521,252 24,1483324,4809201,252 33,2272583,4859941,252 42,2331792,4121,0741,252 51,160931,2521,1601,252 SUM15,7681,26117,0295,0006,261

14. 14 Example (cont'd) Plan D: Pay principal and interest in one payment at end of five years. abcdef Year Amnt.Owed Int. Owed Total Owed Princip.PaymentTotalPayment int*bb+c 15,0004005,40000 25,4004325,83200 35,8324676,29900 46,2995046,80200 56,8025447,3475,0007,347 SUM29,3332,34731,6805,0007,347

15. 15 Example (cont'd)

16. 16 Example (cont'd)

17. 17 Example (cont'd)

18. 18 Example (cont'd)

19. 19 Example (cont'd)

20. 20 Quarterly Compounded Interest Rates Example. You put $500 in a bank for 3 years at 6% compound interest per year. Interest is compounded quarterly. Example. You put $500 in a bank for 3 years at 6% compound interest per year. Interest is compounded quarterly. The bank pays you i = 0.06/4 = 0.015 every 3 months; 1.5% for 12 periods (4 periods per year  3 years). 1.5% for 12 periods (4 periods per year  3 years). At the end of three years you have: F = P (1+i) n = 500 (1.015) 12 F = P (1+i) n = 500 (1.015) 12 = 500 (1.19562)  $597.81 Note. Usually the stated interest is for a 1-year period. If it is compounded quarterly then an interest period is 3 months long. If the interest is i per year, each quarter the interest paid is i/4 since there are four 3-month periods a year. Note. Usually the stated interest is for a 1-year period. If it is compounded quarterly then an interest period is 3 months long. If the interest is i per year, each quarter the interest paid is i/4 since there are four 3-month periods a year.

21. 21 Example. In 3 years, you need $400 to pay a debt. In two more years, you need $600 more to pay a second debt. How much should you put in the bank today to meet these two needs if the bank pays 12% per year? Example. In 3 years, you need $400 to pay a debt. In two more years, you need $600 more to pay a second debt. How much should you put in the bank today to meet these two needs if the bank pays 12% per year? Interest is compounded yearly P = 400(P/F,12%,3) + 600(P/F,12%,5) P = 400(P/F,12%,3) + 600(P/F,12%,5) = 400 (0.7118) + 600 (0.5674) = 400 (0.7118) + 600 (0.5674) = 284.72 + 340.44 = $625.16 = 284.72 + 340.44 = $625.16 Interest is compounded monthly P = 400(P/F,12%/12,3*12) + 600(P/F,12%/12,5*12) P = 400(P/F,12%/12,3*12) + 600(P/F,12%/12,5*12) = 400(P/F,1%,36) + 600(P/F,1%,60) = 400 (0.6989) + 600 (0.5504) = 279.56 + 330.24 = $609.80 Example

22. 22 Borrower point of view:You borrow money from the bank to start a business. Investors point of view:You invest your money in a bank and buy a bond. Example: Points of view

23. 23 The Blue Pages in the text book tabulate: Compound Amount Factor (F/P,i,n) = (1+i) n (F/P,i,n) = (1+i) n Present Worth Factor (P/F,i,n) = (1+i) -n These terms are in columns 2 and 3 in the illustrated factor sheet, identified as These terms are in columns 2 and 3 in the illustrated factor sheet, identified as Compound Amount Factor: “ Find F Given P: F/P ” Compound Amount Factor: “ Find F Given P: F/P ” Present Worth Factor: “ Find P Given F: P/F ” Concluding Remarks