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ENSC 201

ENSC 201. ENSC 411. Summary of Wednesday ’ s Class. Purpose of the course is to evaluate business strategies based on cashflows Cashflows occurring at different times cannot be compared directly There are factors of the form “ (X/Y,i,N) ” which can be used to move money through time.

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ENSC 201

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  1. ENSC 201 ENSC 411

  2. Summary of Wednesday’s Class • Purpose of the course is to evaluate business strategies based on cashflows • Cashflows occurring at different times cannot be compared directly • There are factors of the form “(X/Y,i,N)” which can be used to move money • through time

  3. Conversion Factors There are formulas, found in the back of the textbook, for evaluating the conversion factors. Warning! On no account should you remember these formulas! Write out the solutions to problems leaving the conversion factors unevaluated till the last stage. Then look them up in Appendix B. Sometimes you will find it useful to enter the formulas on spreadsheets.

  4. Some of the formulas from the back of the textbook.

  5. One page from Appendix B.

  6. Cash Flow Diagrams Receive $500 for the next 3 years Time Pay out $1000 now These are helpful in making sure we have taken all the important cash flows into account. They need not be exactly to scale, but it helps if they’re close.

  7. Present Value This is an application of the notion of equivalence: We compare a series of cash flows by bringing them all to the present and adding them up. The sum is called the present value of the series. If the series represents cash flows coming to us, we want the present value to be positive and the bigger the better.

  8. Present Value $500 $1000 For example, the present value of this series of cash flows is PV = -1000 + 500(P/F,i,1) +500(P/F,i,2) + 500(P/F,i,3)

  9. Annuities A The Present The pattern of a regular series of annual payments comes up often enough that we give it a special name: an annuity. By convention, an annuity starts one time period after the present and continues for N years. We can find its equivalent present value using another conversion factor: PV = A(P/A,i,N)

  10. Annuities A The Present The pattern of a regular series of annual payments comes up often enough that we give it a special name: an annuity. PV = A(P/A,i,N) As N increases, does (P/A,i,N) increase or decrease?

  11. Annuities A The Present The pattern of a regular series of annual payments comes up often enough that we give it a special name: an annuity. PV = A(P/A,i,N) As i increases, does (P/A,i,N) increase or decrease?

  12. Annuities A The Present The pattern of a regular series of annual payments comes up often enough that we give it a special name: an annuity. PV = A(P/A,i,N) As A increases, does (P/A,i,N) increase or decrease?

  13. Present Value $500 $1000 So a more concise expression for the present value of this series would be PV = -1000 + 500(P/A,i,3)

  14. Some Tips for the Assignments and Exams Say what you're doing. In the exams, you can get 25% credit for an answer if we can just tell what method it is you're using, and an additional 25-50% if it's the right method. You won't necessarily get exactly the numerical values we have on the model answer sheets -- in many questions there are several defensible ways of solving the problem. To make it easy for us to mark it right, say what the numbers you're writing down are supposed to be, e.g., ``Present worth of wages = A(P/A,i,N)'’ If we're just confronted by a page of anonymous calculations, there's not much we can do except glance through it and see if any of the numbers look anything like any of the numbers in the model answer.

  15. Use explicit conversion factors, i.e., expressions like `(P/A,i,N)'. Using an algebraic formula instead is more work, and there are many more opportunities to make a numerical slip. The only time you should use the formulas is when creating a spreadsheet. Even then, it's a good idea to write out what it is you're calculating in terms of the conversion factors -- this makes it easy for us to give credit even when there's a mistake in the spreadsheet (which can easily happen). If you don't have a copy of the text, you can find tables of conversion formulae on line, for example at: http://www.uic.edu/classes/ie/ie201/discretecompoundinteresttables.html

  16. Avoid excessive precision. If you're calculating the present value of a million-dollar investment, don't bother specifying it to the nearest thousandth of a cent. Three significant figures is usually adequate, and anything after the fifth significant figure is just imaginative fiction. Bad: “In 10 years, your investment will be worth $121,987.12531” Good: “In 10 years, your investment will be worth about $122,000”

  17. Bad When presenting a table of numbers, they should all be given to the same level of precision, and the decimal points should align vertically. Let the table entries be in thousand-dollar or million-dollar units, so there are only a few digits on either side of the decimal point. If you do have more than three digits to one side of the decimal point, separate them into groups of 3 by commas or spaces. Good

  18. Answer the question asked. If the question asks, `` which alternative is best? '', don't just calculate the value of each alternative and leave it to the reader to figure it out. Say it explicitly.

  19. Some Minor Details

  20. Simple Interest and Compound Interest In the case of simple interest, we just charge interest on the principal amount. In the case of compound interest, we add the interest to the principal at regular intervals – the compounding interval – and charge interest on the sum.

  21. For example, suppose we borrow $100 at 10% interest. After N years, the amount we owe is:

  22. No-one ever uses simple interest, and we will never speak of it again. Any compound interest rate can be described as i % per time_period1, compounded every time_period2 For example, a bank may charge 12% interest per year, compounded every month.

  23. If time_period1 is the same as time_period2, then the interest rate is an effective interest rate. Otherwise it is a nominal interest rate. The appendices and formulas at the back of the book all assume that we are talking about effective interest rates. So if we are quoted a nominal interest rate, we have to convert it to an effective interest rate before doing any calculations. For example, if a bank charges 12% interest per year, compounded every month, we can convert this to 1% interest per month, compounded every month. This is now an effective interest rate, and we can do calculations with it.

  24. An interest rate of 36% per year, compounded monthly, is the same thing as an interest rate of 9% per quarter, compounded monthly, which is the same as an interest rate of 3% per month, compounded monthly. The last of these is an effective interest rate, since the two time periods match, and we can look up its conversion factors in the back of the book.

  25. I put $100 in the bank for one year. Which interest rate gives me more money at the end of the year: 12% interest per year, compounded every year 1% interest per month, compounded every month?

  26. 12% interest per year, compounded every year gives me $100(F/P,12%,1) = $112.00 1% interest per month, compounded every month gives me $100(F/P,1%,12) = $112.68

  27. Suppose I have an effective interest rate – 10% per month, compounded monthly, say – and I want to transform it to an equivalent effective annual rate. How do I do this? The two rates are equivalent if they give me the same amount of money after the same period of time. So if the effective monthly rate is i and the equivalent effective yearly rate is j, we must have (1+j) = (1+i)12

  28. Exercise: what effective biennial interest rate is equivalent to an effective annual rate of 10%? (Where `biennial’ means `every two years’)

  29. Answer: Let the effective biennial interest rate be j. Then: (1+j) = (1+0.1)2 =1.21 So j = 21%.

  30. Reassuring note: Almost every interest rate you come across in real life will be an effective annual rate.

  31. Continuous compounding: Suppose we keep the nominal yearly interest rate constant -- r, say -- and decrease the compounding interval towards zero, what happens to the effective interest rate, j? Decrease to months: j = (1 + r/12)12 – 1 Decrease to weeks: j = (1 + r/52)52 – 1 Decrease to days: j = (1 + r/365)365 – 1 Decrease forever: j = er – 1

  32. Further reassuring note: continuous compounding rarely shows up in real life. When it does, there are special interest tables for looking it up.

  33. Final minor detail: If we compound monthly or annually, we can sum up weekly and daily cash flows and treat them as occurring at the end of the month or year. But if we’re compounding continuously, we are better off treating these cash flows as continuous. There are other special interest tables for this.

  34. Example: The SFU library charges you $1/day for an overdue book and continuously compounds the amount you owe at a nominal rate of 10% per year. How much do you owe after two years?

  35. Solution: You will owe $F, where F = A(F/A,r,2) r is 10% per year. A is $365/year (note that we have to express A in terms of the same time unit as r.). We look the conversion factor up in the appropriate table. So you will owe F = 365(2.2140) = $808.11

  36. The mid-period convention: An alternative way of representing continuous cash flows is to suppose that they arrive in one lump in the middle of the period you’re considering. On this approximate model, you consider that you owe the library two years’s fines, which is $730, from halfway through the period, that is, after one year. So the approximate solution would be: F = 730(F/P,0.1,1), or slightly better, F = 730(F/P,e0.1,1) And looking these up in Appendix B gives $803 and $806.77 respectively.

  37. How much does it matter? Compare the values for (F/A,i,N):

  38. End of Digression into Minor Details

  39. Recap of the Important Parts Cash flows can only be usefully compared if they are converted to equivalent cash flows at the same time period. This is accomplished through the use of conversion factors. Among the conversion factors we have met so far are:

  40. Some Important Conversion Factors Present worth of a future cashflow: (P/F,i,N) Future worth of a present cashflow: (F/P,i,N) Present worth of an annuity: (P/A,i,N) Future worth of an annuity: (F/A,i,N)

  41. General Approach: I want to know X I only know Y So I write down an equation of the general form X = Y(X/Y,i,N) Then I look up (X/Y,i,N) in the back of the book.

  42. More Important Conversion Factors `Capital Recovery Factor’: (A/P,i,N) (This is how you calculate mortgage payments, for example.) `Sinking Fund Factor’: (A/F,i,N) (`I want to have a million dollars by the time I retire’.) `Capitalised Cost’: P = A/i (Present worth of an infinite annuity.)

  43. The purpose of all the conversion factors is to help us make choices. For example:

  44. A drafting company employs 10 drafters at $800/week each. The CEO considers • three alternatives: • Buy 8 low-end workstations at $2 000 each. Give two drafters 1 year’s notice. • At the end of the year they each get $5 000 severance pay. Train the • remaining eight in AutoCAD. The first training course is available in a year, • and costs $2 000 per participant. After completing the course, each drafter • gets a $100/week raise. • Buy 5 high-end workstations at $5 000 each. All the drafters get a year’s notice and $5 000 severance pay at the end of the year. Five new graduates are hired at $1 200 per week. They are trained in Pro-Engineer; to keep current, they will each need a $5 000 retraining session every six months. • Do nothing. • Any of these options would allow the company to service its current customers. Any money saved can be invested at 10%, which is also the • cost of borrowing money. What should they do?

  45. A drafting company employs 10 drafters at $800/week each. The CEO considers three alternatives: • Buy 8 low-end workstations at $2 000 each. Give two drafters 1 year’s notice. At the end of the year they each get $5 000 severance pay. Train the remaining eight in AutoCAD. The first training course is available in a year, and costs $2 000 per participant. After completing the course, each drafter gets a $100/week raise. • 2. Buy 5 high-end workstations at $5 000 each. All the drafters get a year’s notice and $5 000 severance pay at the end of the year. Five new graduates are hired at $1 200 per week. They are trained in Pro-Engineer; to keep current, they will need a $5 000 retraining session every six months. • 3. Do nothing. • Any of these options would allow the company to service its current customers. Any money saved can be invested at 10%, which is also the cost of borrowing money. What should they do? What time frame should we use? How do we represent `Do nothing’? Sketch the cash-flow diagrams. What non-monetary factors would matter?

  46. Option 1 $40,000 $16 000 $16000 $10 000 PW = -16,000 – (16,000+10,000)(P/F,0.1,1) + 40,000(P/A,0.1,N)(P/F,0.1,1)

  47. $40,000 P=F(P/F,i,1) $16 000 $10 000 PW = -16,000 - ….

  48. $40,000 26,000(P/F,i,1) PW = -16,000 – 26,000(P/F,i,1)…

  49. P’=40,000(P/A,i,4) $40,000 P’ PW = -16,000 – 26,000(P/F,i,1) + …

  50. P=F(P/F,i,1) P’=40,000(P/A,i,4) PW = -16,000 – 26,000(P/F,i,1) + …

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