1 ECGD3110 Systems Engineering & Economy Lecture 3 Interest and Equivalence

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 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 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 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:  400 = = $7000 Note:The borrower has used the $400 for 4 years without paying interest on it.

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 = $ At the end of year 2 you have (1.06)  530 = $ At the end of year 3 you have (1.06)  $ = $ At the end of year 3 you have (1.06)  $ = $ Note: Note: $ = (1.06)  = (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 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 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) = $ Single Payment Present Worth Formula P = F/(1+i) n = F(1+i) -n

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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 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, ,4001,0001,400 24, ,3201,0001,320 33, ,2401,0001,240 42, ,1601,0001,160 51,000801,0801,0001,080 SUM15,0001,20016,2005,0006,200

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, , , , , , , , , ,4005,0005,400 SUM25,0002,00027,0005,0007,000

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, , ,252 24, , ,252 33, , ,252 42, ,4121,0741,252 51,160931,2521,1601,252 SUM15,7681,26117,0295,0006,261

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, , , , , , , , , ,3475,0007,347 SUM29,3332,34731,6805,0007,347

15 Example (cont'd)

16 Example (cont'd)

17 Example (cont'd)

18 Example (cont'd)

19 Example (cont'd)

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 = 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 ( )  $ 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 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) (0.5674) = 400 (0.7118) (0.5674) = = $ = = $ 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) (0.5504) = = $ Example

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 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