1. CTC 475 Review Cost Estimates Job Quotes (distributing overhead) – Rate per Direct Labor Hour – Percentage of Direct Labor Cost – Percentage of Prime (Labor+Matl) Cost Present Economy Problems – No capital investment – Long-term costs are same – Alternatives have identical results

2. CTC 475 Interest and Single Sums of Money

3. Objectives Know the difference between simple and compound interest Know how to find the future worth of a single sum Know how to find the present worth of a single sum Know how to solve for i or n

4. Time Value of Money Value of a given sum of money depends on when the money is received

5. Which would you prefer? EOYCash Flow 0(100,000) 170,000 250,000 330,000 410,000 Total +160,000 EOYCash Flow 0(100,000) 110,000 230,000 350,000 470,000 Total +160,000

6. Which Would you Prefer? EOYCash Flow 0(5,000) 11,500 2 3 4 50 Total +6,000 EOYCash Flow 0(5,000) 10 21,500 3 4 5 Total +6,000

7. Money Has a Time Value Money at different time intervals is worth different amounts Time (or year at which cash flow occurs) must be taken into account

8. Simple vs Compound Interest If $1,000 is deposited in a bank account, how much is the account worth after 5 years, if the bank pays 3% per year ---simple interest? 3% per year ---compound interest?

9. Simple vs Compound Interest EOYCash Flow- Simple 0$1,000 1$1,030 2$1,060 3$1,090 4$1,120 5$1,150 EOYCash Flow- Compound 0$1,000.00 1$1,030.00 2$1,060.90 3$1,092,73 4$1,125, 51 5$1,159.27

10. Simple Interest Equation Simple—every year you earn 3% ($30) on the original $1000 deposited in the account at year 0 F n =P(1+i*n) Where: F=Future amount at year n P=Present amount deposited at year 0 i=interest rate

11. Compound Interest Equation Compound—every year you earn 3% on whatever is in the account at the end of the previous year F n =P(1+i) n Where: F=Future amount at year n P=Present amount deposited at year 0 i=interest rate

12. Example-Simple vs Compound An individual borrows $1,000. The principal plus interest is to be repaid after 2 years. An interest rate of 7% per year is agreed on. How much should be repaid using simple and compound interest? Simple: F=P(1+i*n)=1000(1+.07*2)=$1,140 Compound: F=P(1+i) n =1000(1.07) 2 =$1,144.90

13. Simple or Compound? In practice, banks usually pay compound interest Unless otherwise stated assume compound interest is used

14. Factor Form Previous slide shows equation form for compound interest The factor form is a shortcut used to find answers faster from tables in the book

15. Factor Form F=P(F/P i,n ) Find the future worth (F) given the present worth (P) at interest rate (i) at number of interest periods (n) Future worth=Present worth * factor Note that the factor=(1+i) n

16. Example of Find F given P problem- Equation vs Factor An individual borrows $1,000 at 6% per year compounded annually. If the loan is to be repaid after 5 years, how much will be owed? Equation: F=P(1+i) n =1000(1.06) 5 =$1,338.20 Factor: F= P(F/P 6,5 )=1000(1.3382)=$1,338.20 Note that the factor comes from Appendix C, Table C-9, from your book. Also note that the factor = (1.06) 5 =1.3382

17. Find P given F Can rewrite F=P(1+i) n equation to find P given F: Equation Form: P=F/(1+i) n =F*(1+i) -n OR Factor Form: P=F(P/F i,n )

18. Example of Find P given F problem- Equation vs Factor What single sum of money does an investor need to put away today to have $10,000 5 years from now if the investor can earn 6% per year compounded yearly? Equation: P=F*(1+i) -n =10,000(1.06) -5 =$7,473 Factor: P=F(P/F i,n )=1000(0.7473)=$7,473 Note that the factor comes from appendix C out of your book. Also note that the factor = (1.06) -5 =0.7473. Also note that the F/P factor is the reciprocal of the P/F factor

19. Example of Find P given F If you wish to accumulate $2,000 in a savings account in 2 years and the account pays interest at a rate of 6% per year compounded annually, how much must be deposited today? F=$2,000 P=? i=6% per year compounded yearly n=2 years Answer: $1,780

20. Relationship between P and F F occurs n periods after P P occurs n periods before F

21. Find i given P/F/n Can rewrite F=P(1+i) n equation and solve for i 15 years ago a textbook costs $25.00. Today it costs $50.00. What is the inflation rate per year compounded yearly? Answer: 4.73%

22. Find n given P/F/i Can rewrite F=P(1+i) n equation and solve for n How long (to the nearest year) does it take to double your money at 7% per year compounded yearly? Answer: 10 years

23. Solve for N Method 1-Solve directly F=P(1+i) n 2D=D(1.07) n 2=1.07 n log 2 = n*log(1.07) n=10.2 years

24. Solve for n Method 2-Trial & Error n 2=1.07 n nValue 11.07 51.40 101.97 152.76

25. Solve for N Method 3-Use factors in back of book F/P=2 @ n=10; F/P=1.9727 @ n=11; F/P=2.1049 To the nearest year; n=10 Interpolate to get n=10.2

26. Series of single sum cash flows How much must be deposited at year 0 to withdraw the following cash amounts? (i=2% per year compounded yearly) EOYCash Flow 0P=? 1$1,000 2$3,000 3$2,000 4$3,000

27. Cash Flow Series (Present Worth) P(at year 0)=: 1000(P/F 2,1 )+ 3000(P/F 2,2 )+ 2000(P/F 2,3 )+ 3000(P/F 2,4 ) EOYCash Flow 0P 1$1,000 2$3,000 3$2,000 4$3,000

28. Series of single sum cash flows How much would an account be worth if the following cash flows were deposited? (i=2% per year compounded yearly) EOYCash Flow 00 1$1,000 2$3,000 3$2,000 4$3,000

29. Cash Flow Series (Future worth) F(at year 4)=: 1000(F/P 2,3 )+ 3000(F/P 2,2 )+ 2000(F/P 2,1 )+ 3000(F/P 2,0 ) EOYCash Flow 00 1$1,000 2$3,000 3$2,000 4$3,000

30. Next lecture Uniform Series