Engineering Economic Analysis 9th Edition. Chapter 3 INTEREST AND EQUIVALENCE

Transcription

1 Engineering Economic Analysis 9th Edition Chapter 3 INTEREST AND EQUIVALENCE

2 Economic Decision Components Where economic decisions are immediate we need to consider: amount of expenditure taxes Where economic decisions occur over a considerable period of time we also need to consider: interest inflation 2

3 Computing Cash Flows Cash flows have: Costs (disbursements) > a negative number Benefits (receipts) > a positive number Example 3- End of Year Cash flow 0 $ (,000.00) $ $

4 Time Value of Money Money has value Money can be leased or rented The payment is called interest If you put $00 in a bank at 9% interest for one time period you will receive back your original $00 plus $9 Original amount to be returned = $00 Interest to be returned = $00 x.09 = $9 4

5 Simple Interest Interest that is computed only on the original sum or principal Total interest earned = I = P x i x n Where P present sum of money i interest rate n number of periods (years) I = $00 x.09/period x 2 periods = $8 5

6 Future Value of a Loan with Simple Interest Amount of money due at the end of a loan F = P + P i n or F = P ( + i n ) Where F = future value F = $00 ( +.09 x 2) = $8 Would you accept payment with simple interest terms? Would a bank? 6

7 Compound Interest Interest that is computed on the original unpaid debt and the unpaid interest Total interest earned = I n = P (+i) n -P Where P present sum of money i interest rate n number of periods (years) I 2 = $00 x (+.09) 2 - $00 = $8.8 7

8 Future Value of a Loan with Compound Interest Amount of money due at the end of a loan F = P(+i) (+i) 2..(+i) n or F = P ( + i) n Where F = future value F = $00 ( +.09) 2 = $8.8 Would you be more likely to accept payment with compound interest terms? Would a bank? 8

9 Comparison of Simple and Compound Interest Over Time Simple and compound interest Single payment If you loaned a friend money for short period of time the difference between simple and compound interest is negligible. If you loaned a friend money for a long period of time the difference between simple and compound interest may amount to a considerable difference. Check the table to see the difference over time. Short or long? When is the $ difference significant? You pick the time period. Principal = Interest = 9.00% Period Simple amount factor Compound amount factor n Find Fs Given P Fs/P Find F Given P F/P

10 Four Ways to Repay a Debt Plan Repay Principal Equal annual installments End of loan Repay Interest Interest on unpaid balance Interest on unpaid balance Equal annual installments End of loan Compound and pay at end of loan Interest Earned Declines Constant Declines at increasing rate Compounds at increasing rate until end of loan 0

11 Loan Repayment Four Options This calculator is partially complete. If you complete the calculator you can earn 0 bonus points for your team. Loan Repayment Option Calculator Years Amount owed at the beginning of the year $5,000 Principal 0.00% Interest rate (enter as. for 0%) 0 Years Plan Enter through 4 Principal payment Equal annual installments Interest payment EOY on unpaid principal Interest owed for that year Total owed at the end of year Principal payment Total end of year payment 5, , , , , , , , , , , , , , , ,500 50, ,000 00, ,750 5,000 7,750 5, , , , , , , , , ,500 5,000 5,500 2,500 5,000 7,500 5, , , , , , , , , , ,000,95 4,069 5, , , , , , , , , ,053 7,32 8,053 3,053 5,000 8,053

12 Equivalence When an organization is indifferent as to whether it has a present sum of money now or the assurance of some other sum of money (or series of sums of money) in the future, we say that the present sum of money is equivalent to the future sum or series of sums. Each of the plans on the previous slide is equivalent because each repays $5000 at the same 0% interest rate. 2

13 Given the choice of these two plans which would you choose? Year Total Plan $ $6200 Plan 2 $ $7000 To make a choice the cash flows must be altered so a comparison may be made. 3

14 Technique of Equivalence Determine a single equivalent value at a point in time for plan. Determine a single equivalent value at a point in time for plan 2. Both at the same interest rate. Judge the relative attractiveness of the two alternatives from the comparable equivalent values. 4

15 Repayment Plans Establish the Interest Rate. Principal outstanding over time 2. Amount repaid over time As an example: If F = P ( + i) n Then i=(f/p) /n - Year s $5, % 5 Plan Principal payment Equal annual Interest payment EOY on unpai Interest owed for that year Total owed at the end of year Amount owed at the beginning of the year 5, , , , , , , ,60 5,000 80,080 Totals,200 Year s Equivalence Calculator Interest paid over time,200 Total owed over time 5,000 $4, % 5 Plan Principal payment Equal annual Interest payment EOY on unpai Interest owed for that year = 8.00% Total owed at the end of year Amount owed at the beginning of the year 4, ,36 2 3, , , ,89 4, , ,063 Totals,37 5

16 Application of Equivalence Calculations Pick an alternative. Which would you choose? Change the interest rate. What happens at 8%,5%,3%? Comparing alternatives Interest rate 0.00% Alternative Year A B C D 0 $600 -$600 -$850 $850 $5 -$5 -$80 $80 2 $5 -$5 -$80 $80 3 $5 -$5 -$80 $80 4 $5 -$5 -$80 $80 5 $5 -$5 -$80 $80 6 $5 -$5 -$80 $80 7 $5 -$5 -$80 $80 8 $5 -$5 -$80 $80 9 $5 -$5 -$80 $80 0 $5 -$5 -$80 $80 P $, ($,306.63) ($,34.57) $,34.57 A $22.65 ($22.65) ($28.33) $28.33 F $3, ($3,389.05) ($3,479.68) $3, Present worth Annual worth Future worth 6

17 Interest Formulas To understand equivalence, the underlying interest formulas must be analyzed. Notation: I = Interest rate per interest period n = Number of interest periods P = Present sum of money (Present worth) F = Future sum of money (Future worth) 7

18 Single Payment Compound Interest Year Beginning balance Interest for period Ending balance P ip P(+i) 2 P(+i) ip(+i) P(+i) 2 3 P(+i) 2 ip(+i) 2 P(+i) 3 n P(+i) n- ip(+i) n- P(+i) n P at time 0 increases to P(+i) n at the end of time n. Or a Future sum = present sum (+i) n 8

19 Notation for Calculating a Future Value Formula: F=P(+i) n is the single payment compound amount factor. Functional notation: F=P(F/P,i,n) F=5000(F/P,6%,0) F =P(F/P) which is dimensionally correct. 9

20 Notation for Calculating a Present Value P=F(/+i) n =F(+i) -n is the single payment present worth factor. Functional notation: P=F(P/F,i,n) P=5000(P/F,6%,0) 2 0

21 Examples F=P(F,i,n) P=F(F,i,n) F=$5000 i=0.0 n=5 P=? F=P(+i) n =$5000(+0.0) 5 =$5000(.6)=$8055 F=P(F/P,0,5)=$5000(.6) =$8055 P=F(P/F,0,5)=$8055(.62092) =$ % Compound Interest Factors Compound Amount Factor Single Amount Factor Present Worth Factor n F/P P/F 0 $.00 $5, $.00 $8, $5, $7, $6, $6, $6, $6, $7, $5, $8, $5, $8, $4, $9, $4, $0, $3, $, $3, $2, $3, $4, $2, $5, $2, $7, $2, $8, $2, $20, $, $22, $, $25, $, $27, $, $30, $, $33, $, $54, $ $87, $ $226, $ $586, $ $,522, $ $4,777, $ , $68,903, $0.58 2

22 8% Compounded Monthly 8% interest: Assume a yearly rate if not stated Compounded monthly: Indicates 2 periods/year [8%/year] / [2months/year] =.5% / month Effective vs Nominal Interest Comparator Nominal Interest rate 365 Periods/year Effective Interest rate 9.42% per year Number of years.00 Single Amount Factor Compound Amount Factor Present Worth Factor F/P P/F i n $.00 $ $.00 $ % $ $ % $ $