Engineering Economics ECIV 5245 Chapter 3 Interest and Equivalence
Cash Flow Diagrams (CFD) Used to model the positive and negative cash flows. At each time at which cash flow will occur, a vertical arrow is added, point down for costs and up for revenues. Cash flow are drawn to relative scale Rent and insurance are beginning-of-period cash flows; i.e. just put an arrow in where it occurs. O&M, salvages, and revenues are assumed to be endof-period cash flows. 2
Example 3-1 Purchase a new $30,000 mixing machine. The machine may be paid for in one of two ways A. Pay the full price now minus a 3% discount B. Pay $5000 now, $8000 at the end of 1 st yr, and $6000 at end of each following year List the alternatives in the form of a table of cash flows 3
Continue Example 3-1 Cash flow table: End of year 0 (now) 1 2 3 4 5 Pay in Full Now Pay over 5 Yrs -$29,100 -$5000 0 -$8000 0 -$6000 0 -$6000 0 -$6000 0 -$6000 4
Example 3-2 A man borrowed $1000 from a bank at 8% interest. At the end of 1 st yr: Pay half of the $1000 principal amount plus the interest. At the end of 2 nd yr: Pay the remaining half of the principal amount plus the interest for the second year. Compute the borrower s cash flow End of Year Cash Flows 0 (Now) +$1000 1-580 2-540 5
Time Value of Money If monetary consequences occur in a short period of time Simply add the various sums of money What if time span is greater? $100 cash today vs. $100 cash a year from now? Money is rented. The rent is called the interest If you put $100 in the bank today, and interest rate is 9% $109 a year from now 6
Interest Simple Interest Compound interest 7
Simple Interest Interest that is computed only on the original sum and not on accrued interest. e.g. if you loaned someone the amount of P at a simple interest rate of i for a period of n years: Total interest earned = P i n = P i n The amount of money due after n years: F = P + P i n Or F = P(1+ i n) 8
Example 3-3 You loaned a friend $5000 for 5 years at a simple interest rate of 8% per year. How much interest you receive from the loan? How much will your friend pay you at the end of 5 yrs. Total interest earned = P i n = (5000)(0.08)(5) = $2000 Amount due at the end of loan = P + P i n = 5000 + 2000 = $7000 9
Compound Interest This is the interest normally used in real life Interest on top of interest Next year s interest is calculated based on the unpaid balance due, which includes the unpaid interest from the preceding period. 10
Example 3-4 11
Compound Interest Compound 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 2 you have (1.06) 530 = $561.80. At the end of year 3 you have (1.06) $561.80 = $595.51. Note: $595.51 = (1.06) 561.80 = (1.06) (1.06) 530 = (1.06) (1.06) (1.06) 500 = 500 (1.06) 3 12
Single Payment Compound Amount 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 i = interest rate per interest period (stated as decimal) n = number of interest periods P = a present sum of money F = a future sum of money The term (1+i) n is called the single payment compound factor. F = P (1+i) n = P (F/P,i,n) Also P = F (1+i) -n = F (P/F,i,n) The factor (F/P,i,n) is used to compute F, given P, and given i and n. The factor (P/F,i,n) is used to compute P, given F, and given i and n. 13
Present Value Example 3-6 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 F = P (1+i) n for P with i = 0.05, n = 4, F = $800. P = F/(1+i) n = F(1+i) -n P = 800/(1.05) 4 = 800 (1.05) -4 = 800 (0.8227) = $658.16. Alternate Solution Single Payment Present Worth Formula P = F/(1+i) n = F(1+i) -n P =? P = F (P/F,i,n), i = 5% and n = 4 periods From tables in Appendix B, (P/F,i,n) = 0.8227 P = 800 x 0.8227 = $658.16 F = 800 14
Factors in the Book (page 573 in 9-th edition) 15
Present Value Example: You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back in 5 years. The debt can be repaid in many ways. 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 C: Pay in five end-of-year payments. Plan D: Pay principal and interest in one payment at end of five years. 16
Example (cont d) You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back in 5 years. Plan A: At end of each year pay $1,000 principal plus interest due. a b c d e f Year Amnt. Owed Int. Owed int*b Total Owed b+c Princip. Payment Total Payment 1 5,000 400 5,400 1,000 1,400 2 4,000 320 4,320 1,000 1,320 3 3,000 240 3,240 1,000 1,240 4 2,000 160 2,160 1,000 1,160 5 1,000 80 1,080 1,000 1,080 SUM 1,200 5,000 6,200 17
Example (cont'd) You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back in 5 years. Plan B: Pay interest due at end of each year and principal at end of five years. a b c d e f Year Amnt. Owed Int. Owed int*b Total Owed b+c Princip. Payment Total Payment 1 5,000 400 5,400 0 400 2 5,000 400 5,400 0 400 3 5,000 400 5,400 0 400 4 5,000 400 5,400 0 400 5 5,000 400 5,400 5,000 5,400 SUM 2,000 5,000 7,000 18
Example (cont'd) You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back in 5 years. Plan C: Pay in five equal end-of-year payments. a b c d e f Year Amnt. Owed Int. Owed int*b Total Owed b+c Princip. Payment Total Payment 1 5,000 400 5,400 852 1,252 2 4,148 332 4,480 920 1,252 3 3,227 258 3,485 994 1,252 4 2,233 179 2,412 1,074 1,252 5 1,160 93 1,252 1,160 1,252 SUM 1,261 5,000 6,261 19
Example (cont'd) You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back in 5 years. Plan D: Pay principal and interest in one payment at end of five years. a b c d e f Year Amnt. Owed Int. Owed int*b Total Owed b+c Princip. Payment Total Payment 1 5,000 400 5,400 0 0 2 5,400 432 5,832 0 0 3 5,832 467 6,299 0 0 4 6,299 504 6,802 0 0 5 6,802 544 7,347 5,000 7,347 SUM 2,347 5,000 7,347 20
The four plans were Year Plan 1 Plan 2 Plan 3 Plan 4 1 $1400 $400 $1252 0 2 1320 400 1252 0 3 1240 400 1252 0 4 1160 400 1252 0 5 1080 5400 1252 7347 Total $6200 $7000 $6260 $7347 How do we know whether these plans are equivalent or not? We won t be able to know by simply looking at the cash flows, therefore some effort should be made. 21
Equivalence In the previous example, four payment plans were described. The four plans were used to accomplish the task of repaying a debt of $5000 with interest at 8%. All four plans are equivalent to $5000 now. i.e. all four plans are said to be equivalent to each other and to $5000 now. 22
Present Value Example 3-8 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 F = P (1+i) n for P with i = 0.05, n = 4, F = $800. P = F/(1+i) n = F(1+i) -n P = 800/(1.05) 4 = 800 (1.05) -4 = 800 (0.8227) = $658.16. Alternate Solution Single Payment Present Worth Formula P = F/(1+i) n = F(1+i) -n P = F (P/F,i,n), i = 5% and n = 4 periods From tables in Appendix B, (P/F,i,n) = 0.8227 P = 800 x 0.8227 = $658.16 F = 800 P =? 23
Example 3-8 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? P 0 1 2 3 4 5 Interest is compounded yearly P = 400(P/F,12%,3) + 600(P/F,12%,5) = 400 (0.7118) + 600 (0.5674) = 284.72 + 340.44 = $625.16 $400 $600 Alternate Solution P = F(1+i) -n P = 400(1+0.12) -3 + 600(1+0.12) -5 P = $625.17 24
Example 3-8 (Interest Compounded monthly) 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% compounded monthly? P 0 1 2 3 4 5 Interest is compounded yearly P = 400(P/F,12%,3) + 600(P/F,12%,5) = 400 (0.7118) + 600 (0.5674) = 284.72 + 340.44 = $625.16 $400 $600 Interest is compounded monthly 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 25
Points of view Borrower point of view:you borrow money from the bank to start a business. Year Cash flow 0 - P 1 0 2 0 3 +400 4 0 5 +600 Year Cash flow 0 + P 1 0 2 0 3-400 4 0 5-600 Investors point of view:you invest your money in a bank and buy a bond. 26
Concluding Remarks Appendix B in the text book tabulate: Compound Amount Factor (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, identified as Compound Amount Factor: Find F Given P: F/P Present Worth Factor: Find P Given F: P/F 27