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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §5.5 Int Apps Biz & Econ. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 5.4. Review §. Any QUESTIONS About §5.4 → Applying the Definite Integral Any QUESTIONS About HomeWork §5.4 → HW-25. §5.5 Learning Goals.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §5.5 Int AppsBiz & Econ Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 5.4 Review § • Any QUESTIONS About • §5.4 → Applying the Definite Integral • Any QUESTIONS About HomeWork • §5.4 → HW-25

  3. §5.5 Learning Goals • Use integration to compute the future and present value of an income flow • Define consumer willingness to spend as a definiteintegral, and use it to explore consumers’ surplus and producers’ surplus

  4. Time Value of Money • We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return • In other words, “a dollar received today is worth more than a dollar to be received tomorrow” • That is because today’s dollar can be invested so that we have more than one dollar tomorrow

  5. Terminology of Time Value • Present Value → An amount of money today, or the current value of a future cash flow • Future Value → An amount of money at some future time period • Period → A length of time (often a year, but can be a month, week, day, hour, etc.) • Interest Rate → The compensation paid to a lender (or saver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate)

  6. Time Value Abreviations • PV → Present value • FV → Future value • Pmt → Per period payment $-amount • N → Either the • total number of cash flows or • the number of a Payment periods • i→ The interest rate per period • Usually %/Yr expressed as a fraction

  7. 0 1 2 3 4 5 TimeLines • A timeline is a graphical device used to clarify the timing of the cash flows for an investment • Each tick represents one time period PV FV Today 5 TimePeriods Later

  8. 0 1 2 3 4 5 Finding Future Value by Arith • Consider $100 ($1 cNote) invested today at an interest rate of 10% per year or 0.10/yr as decimal • Then the $-Value expected in 1 year $100 ?

  9. 0 1 2 3 4 5 Finding Future Value by Arith • Now Extend the Investment for another Year (“Let it Ride”) • Then the $-Value expected after the 2nd year with no additional investment $110 ?

  10. Recognize Future Value Pattern • Engaging in the EXTREMELY VALUABLE Practice of PATTERN RECOGNITION surmise The Pattern that is developing for FV in year-N • If the $1c investment were extended for a 3rd Year then the FV

  11. Present Value by Arithmetic • What is the $Amount TODAY (the Present Value, PV) needed to realize a FV $Goal after N-years invested at interest rate i per year? • Solve the FV equation for PV

  12. Example  Present Value • Tadesuz has a 5-year old daughter for which he now plans for college $-expenses. • Tadesuz lives in San Leandro, and he develops this college plan for his daughter • She can live at home until she is 22 • Attends Chabot and takes the Lower-Division Courses needed for University Transfer • Transfers to UCBerkeley (GoBears!) where she earns her Bachelors Degree

  13. Example  Present Value • Tadesuz estimates that he will need about $30k on her 18th birthday to pay for her bacaluarate Education. • If he can earn 8% per year on his ONE-Time Initial investment, then how much must he invest today to achieve the $30k Future-Value Goal? • SOLUTION: After 18-5 periods the PV:

  14. 0 1 2 3 4 5 Annuities • An annuity is a series of nominally equal $-payments equally spaced in time • Annuities are very common: Bldg Leases, Mortgage payments, Car payments, Pension income • The timeline shows an example of a 5-year, $100 ($1 cNote) annuity 100 100 100 100 100

  15. Principle of Value Additivity • To find the value (PV or FV) of an annuity first consider principle of value additivity: • The value of any stream of cash flows is equal to the sum of the values of the components • Thus can move the cash flows to the same time period, and then simply add them all together to get the total value

  16. PRESENT Value of an Annuity • Use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components:

  17. 0 1 2 3 4 5 Present Value of an Annuity • Using the example, and assuming a discount rate (a.k.a., interest rate) of 10% per year, find that the PVA as: 62.09 68.30 75.13 82.64 90.91 100 100 100 100 100 379.08

  18. Present Value of an Annuity • Actually, there is no need to take the present value of each cash flow separately • Using Convergent Series Analysis find a closed-form of the PVA equation instead:

  19. Present Value of an Annuity • Using the PVA equation in the $1c example • Thus a 5yr constant yearly income Annuity of $100/yr, discounted by 10% has PV of $379 • The PVA equation works for all regular annuities, regardless of the number of payments

  20. FUTURE Value of an Annuity • Use the principle of value additivity to find the Future Value of an annuity, by simply summing the future values of each of the components

  21. 0 1 2 3 4 5 Future Value of an Annuity • Again consider a $1c annuity, and assume a discount rate of 10% per year, find that the future value: } 146.41 133.10 = 610.51 at year 5 121.00 110.00 100 100 100 100 100

  22. Future Value of an Annuity • As was done for the PVA equation, use series convergence to find a closed-form of the FVA equation: • As with The PVA, the FVAeqn works for all regular annuities, regardless of the number of payments

  23. PV of an Income Stream • Now assume the Pmt is divided into k payments per year (say 12) and then the discount is also applied k times a yr • Call this an Income Stream as the payments are NO Longer made annually • Then the FVA → FVISeqn • Note that Pmtktmay VARY in time; e.g., Pmt73≠ Pmt74

  24. PV of an Income Stream • The discounts Occur infinitely often, and the Pmtkt becomes continuously variable in time, then the PVIS equation becomes • Or textbook notation

  25. Example  PV of Income Stream • Yasiel’s grandfather promises to contribute continuously at a rate of $10,000 per year to a trust fund earning 3% interest as long as the boy maintains a 3.0 GPA in school. • If Yasiel maintains the required grades for 8 years during high school and college, what is the value of the trust at the end of that period?

  26. Example  PV of Income Stream • The trust’s value is the future value of the annuity into which Yasiel’s grandfather is paying. Since the money accrues at a rate of $10,000 per year, and is simultaneously invested, its future value is

  27. Example  PV of Income Stream • Running the Numbers • Thus the fund is worth about $90,416 at the end of eight years.

  28. Present Value of an Income Stream • From the PV discussion, taking payment infinitely often, and the payments to becomes continuously variable in time, then the FVIS equation becomes

  29. Example  Present Value • Instead of investing at a continuous rate of $10,000 per year over the eight years, Yasiel’s grandfather in could have invested for eight years a lump sum of

  30. Value of Ticket to Potential Demanders Peter $200 Paul $150 Mary $100 Jack $50 Jill $50 Value of Ticket to Potential Suppliers: Professor V $50 Professor W $50 Professor X $100 Professor Y $150 Professor Z $200 Price Peter Z 200 Paul Y 150 100 X Jack and Jill 50 V and W Tickets CALFootball Tickets Mary 0 1 2 3 4 5

  31. Equilibrium Price = $100 Peter, Paul and Mary buy tickets from Professors V, W and X. If they all Buy & Sell at the equilibrium price, does it matter who buys from whom? → No Gains: Peter = $200 - $100 = $100 Paul = $150 - $100 = $50 Mary = $100 - $100 = $0 V = $100 - $50 = $50 W = $100 - $50 = $50 X = $100 - $100 = $0 Total Gain: $250 Price Consumer Surplus Peter Z 200 Paul Y 150 100 X Jack and Jill 50 V and W Tickets Producer Surplus CALFootball Tickets Mary 0 1 2 3 4 5

  32. Price($/unit) Consumer Surplus Maximum Willingness to Pay (or Spend) for Qo What is Actually Paid D(Q) Quantity (Units) Consumers’ Surplus • By Supply & Demand the Price settles at P0, but SOME Consumers are willing to pay MUCH MORE, thus these consumers save $-Money Po Qo

  33. Consumers’ Surplus • Thus the Surplus Total $-Funds kept by the consumers: • With Reference to the Areas shown on the Supply-n-Demand Graph CS = [Amount Willing to Spend] − [Amount Actually Paid]

  34. Price($/Unit) Producer Surplus $-Amount Actually paid Minimum $-Amount Needed to Supply Qo Quantity (units) Producers’ Surplus • By Supply & Demand the Price settles at P0, but SOME Producrs are willing to accept a LOWER Price, thus these Producers make extra $-Money S(Q) Po Qo

  35. Price($/Unit) Quantity (Units) Consumer & Producer Surplus • The Two Surpluses usually exist simultaneously Consumer Surplus S(Q) EquilibriumPrice-Point Po Producer Surplus D(Q) Qo

  36. WhiteBoard Work • Problems From §5.5 • P18 → Supply & Demand • P26 → Construction Decision • P42 → InventionProfit

  37. All Done for Today TreeNPV

  38. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

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