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1. INTRODUCTION TO ENGINEERINGFUNDAMENTALS OF ENGINEERING PROJECT ECONOMICS - 2 Prepared by Prof T.M.Lewis

2. Location of Notes – to be updated • http://www.eng.uwi.tt/depts/faculty/ugrad/courses/engr0300/engr0300.html

3. Effect of Time – Interest Rates • Interest is the amount that has to be paid for the use of borrowed money, and it is a rate (usually a percentage) for a period of time (usually a year), and so interest is a measure of the time value of money. • Within the level of accuracy normally required for engineering projects, interest rates can be assumed proportional to the length of time period: • 12% per annum = 1% per month • 1% per month = (1/30)% per day

4. Time Value of Money • Cash inflows to a project are normally treated as positive (+) whilst cash outflows are negative (-). • The net cash flow during any time period is simply the difference between the cash inflows and the outflows, and it may be either positive or negative • For an engineering project, receipts (benefits) and expenditures (costs) are generally compared at the start of the project - this is ‘the end of year zero' or ‘the start of year 1’ – so they are all usually converted to their equivalent Present Value.

5. Time Value of Money • Which is better? • How can you tell which gives the best return? • By converting them all to equivalent values.

6. Present Value • All cash sums can be converted to an equivalent value by working out what sum would have to be 'invested' now (start of year 1 = end year 0) at the going interest rate to give that specific sum at that specific time in the future. • The equivalent sums at the start of year 1, are Present Values (P) • The process of converting an amount to its equivalent at an earlier date is known as discounting, and the interest rate involved is called the discount rate.

7. SIMPLE INTEREST • Simple Interest is where the interest on some capital (present) sum or principal is not reinvested to earn more interest on itself. • Where: i = interest rate; I = Interest earned; P = Present sum; F = Future sum • Interest earned in one year I1 = P.i • Interest earned in 2 years I2 = Pi + Pi = P.2i • Interest earned in n years In = P.ni

8. SIMPLE INTEREST • At end of n years the future sumwill be: • F = P + In = P + P.ni = P(1+ni) • F = P(1+ni) • Alternatively, by rearranging the equation, we can get an expression for the present value given the future sum. • P = F/(1+ni) • Simple interest is rarely used in practice • Normally it is compound interest that is required. Now Forget About Simple Interest

9. COMPOUND INTEREST • Where the interest each year is reinvested to earn more interest on itself. • Again where: i = interest rate; P = Principal (or Present Sum); F = Future sum • Year Amount at Interest earned Amount at end start of year during year of year (a) (b) (a+b) • 1 P Pi P(1+i) • 2 P(1+i) P(1+i).i P(1+i)2 • 3 P(1+i)2P(1+i)2.i P(1+i)3 • n P(1+i)n-1P(1+i)n-1.i P(1+i)n + = + = + = + =

10. Present Values • Thus, the future value \$F, of the principal \$P, after n years at compound interest rate i, is given by the last entry in this table – this is the equation we arrived at by simple logic last week: F = P(1+i)n • Rearranging the equation also tells us that: P = F (1 + i)n

11. Simple or Compound Interest? • Why do we rarely use simple interest? • In 1626 the Government of the Dutch settlement in America – New Amsterdam – bought Manhattan Island from the Indians for beads and trinkets worth \$24. If that money had been invested to earn an average of 8% annually – what would it be worth today? • Simple Interest – F = 24(1+380(0.08)) = \$753.60 • Compound Interest – F = 24(1+.08)380 = \$120,569,740,656,495

12. Present & Future Value Factors • If you have to do a number of calculations, it is possible to take the money out of the equation and simply calculate a PV Factor for any period and any interest rate. When you have done it once you can just use the factor afterwards : Hence the Future Value factors can be calculated for all values of I and n so that the PV Factor = 1/(1+i)n And the FV Factor = (1 + i)n

13. Present Value Factors • PV Factor = 1/(1+i)n After 1 year, with 10% per annum interest rate PV Factor = 1 = 0.9091 (1.1) After 2 years, with 10% per annum interest rate PV Factor = 1 = 0.8264 (1.1)2 After 3 years, with 10% per annum interest rate PV Factor = 1 = 0.7513 (1.1) 3

14. Going back to the table earlier • Which is best? • Option 1 requires lowest investment - so risk is low • Option 2 gives the highest cash value. • Option 3 gives earliest payback

15. To Convert to Present Values at an Interest Rate of 10% per year apply Factors (P=F(1/(1+i)n)

16. Present values • Which one has the highest net present value? • Project 3. • But Project 2 gives the highest return per \$ invested. • Different choices may be appropriate at different times, but usually people want to know the highest returns

17. Uniform Cash Flows - Annuities Often cash flows are uniform – with the same amount being paid or received during each time period – as in a mortgage, pension or insurance premium. These can be discounted individually but doing the 240 calculations for a monthly payment on a 20 year mortgage is tiresome so they are usually discounted as an ‘annuity’

18. Annuities An annuity is a periodic payment of a fixed amount, over a given period of time, whose accumulated value, whether present or future, is required (the payments do not have to be yearly to be called an annuity). If a sum of \$A is invested at the end of each year for n years, the total final amount will obviously be the sum of the compounded amounts for the individual investments. Remember that the Future Value of a Present Sum is given by the expression F = P(1+i)n

19. ANNUITIES Payment \$A Payment \$A Payment \$A Payment \$A Payment \$A Payment \$A Payment \$A End of Year 0 = now End of Year (n-2) End of Year (n-1) End of Year 1 End of Year 2 End of Year 3 End of Year 4 End of Year n

20. Annuities The present sum of money \$A invested at the end of the first year will earn interest for (n-1) years, so it will mount to F = A(1+i)n-1. The second year's payment of \$A will be compounded up to A(1+i)n-2, and so on until the (n-1)thyear, when that payment will earn interest for 1 year The last payment on the last day will earn no interest. The total future amount \$F is, thus, given by the expression: F = A[1 + (1+i) + (1+i)2.....(1+i)n-1]

21. Annuities F = A[1 + (1+i) + (1+i)2.....(1+i)n-1] This is not a very convenient expression but it can be put into a simpler form by multiplying both sides by (1+i) to get F(1+i) = A[(1+i) + (1+i)2.......(1+i)n-1 + (1+i)n] Now subtract the first equation from the second, and you will get: Fi = A[(1+i)n – 1]

22. Annuities F = A{[(1+i)n - 1]/i} Thus, A = F[ i ] (1+i)n - 1 Previous calculations showed that F = P(1+i)n, so P can be obtained in terms of A, i and n from these equations by substitution, i.e. P(1+i)n = A{[(1+i)n - 1]/i} giving P = A [ (1+i)n - 1 ] i(1+i)n and from this it is easy to see that: A = P [ i(1+i)n ] (1+i)n - 1

23. The Formulae To Find the Future Value given the Present Value F = P(1+i)n (1) To Find the Present Value given the Future Value P = F (2) (1 + i)n

24. The Formulae To Find the Future Value given the Annual Value F = A[(1+i)n - 1] (3) i To Find the Annual Value given the Future Value A = F[ i ] (4) (1+i)n – 1)

25. The Formulae To Find the Present Value given the Annual Value P = A[(1+i)n - 1] (5) i(1+i)n To Find the Annual Value given the Present Value A = P[ i(1+i)n ] (6) ((1+i)n – 1)

26. Uniform Periodic Cash Flows When it is often necessary to consider payments over a significant period of time in equal amounts. This is the way that mortgages on properties are normally amortised. Likewise, in providing capital for future use (e.g. pensions), payments are often made monthly or annually into a fund which is built up over a period of time while accumulating interest.

27. Tables • The factors from these equations can be used to create look-up tables that can be used to calculate the equivalent values of cash flows. • These tables are widely published (occasionally in diaries) and are built into spreadsheet packages like MS Excel • Typical tables are shown below

28. To Find a Future Value given a Present Value

29. To Find a Present Value given a Future Value

30. To Find a Future Value given a Periodic (Annual) Value

31. To Find a Present Value given aPeriodic (Annual) Value

32. Example • If you set aside a sum of \$500 eachmonth for 50 months (just over 4 years) at an interest rate of 12% per annum, how much will you have at the end of that period? • Use table "To Find a Future Value given a Periodic (Annual) Value” using the 1% interest rate (per month) and 50 periods, you get a factor 64.463. Multiply by the sum of \$500. • And the answer is \$32,231

33. Internal Rate of Return • When it is not clear what interest rate should be used to discount the cash flows on a project. • The problem becomes one of determining what interest rate the cash flows from the project are actually earning - it is the interest rate that would make the discounted costs and benefits exactly equal. • In other words, it involves finding which discount rate would result in a zero NPV. • This interest rate is referred to as the Internal Rate of Return.

34. Example 7 • Find the internal rate of return for the investment with a capital outlay of \$100,000 now, which earns receipts of \$40,000, \$50,000, and \$40,000 at the end of the first, second and third years

35. Solution • In order to find the internal rate of return, it is necessary to try a discount rate and see what net present value (NPV) it gives. • If the resulting NPV is positive then a higher interest rate must be used for the discounting (to reduce the value of the discounted future flows). • If the NPV is negative, a lower interest rate must be used. • Hence, for this example, using the expression • NPV = -100,000 + 40,000 + 50,000 + 40,000 • 1+i (1+i)2 (1+i)3 • Solving for i = 10% NPV = +\$7,738.54 • i = 15% NPV = -\$1,109.56

36. Internal Rate of Return • To find the value at which the NPV is zero, we can interpolate by using the rule of simple proportions, i.e. the internal rate of return i is • i = {10 + 5(7,738.54/8,848.10)} = 14.3% • The criterion usually applied under this method is that a project is acceptable if it produces an internal rate of return greater than the minimum return required on capital. • Thus, in this example, if the cost of capital is 10%, the project should be accepted, whereas if it is 15%, it should be rejected.

37. Using Excel

38. Excel Functions Description

39. Excel Functions

40. Cell ID Result Formula

41. IRR Function

42. Problem THE ANNUAL GROSS DOMESTIC PRODUCT PER CAPITA OF TRINIDAD AND TOBAGONIANS IS US\$9,000 AND IS EXPECTED TO GROW AT 4% PER ANNUM FOR THE NEXT 15 YEARS (UNTIL 2026), WHAT WILL IT BE THEN?

43. Solution F = P (1+i)n F = 9,000 (1.04)15 F = \$16,208

44. Problem DUE TO IMPROVEMENTS IN RICE SEED VARIETIES, FERTILISATION AND IRRIGATION, THE PRODUCTIVITY OF RICE PADDY HAS INCREASED BY A STEADY 3% PER YEAR SINCE 1960. WHAT WAS THE PRODUCTIVITY OF PADDY IN 2010 COMPARED WITH 1960?

45. Solution PRODUCTION IN 1960 = X TONS/HECTARE N = 50 YEARS F = X(1.03)50 F = 4.4X HENCE RICE PADDY PRODUCTIVITY HAS GROWN BY 440%