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Chabot Mathematics §5.5 Int Apps Biz & Econ Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot College Mathematics 1 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Review § 5.4 Any QUESTIONS About • §5.4 → Applying the Definite Integral Any QUESTIONS About HomeWork • §5.4 → HW-25 Chabot College Mathematics 2 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx §5.5 Learning Goals Use integration to compute the future and present value of an income ﬂow Deﬁne consumer willingness to spend as a deﬁnite integral, and use it to explore consumers’ surplus and producers’ surplus Chabot College Mathematics 3 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 Chabot College Mathematics 4 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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) Chabot College Mathematics 5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 Chabot College Mathematics 6 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 0 Today Chabot College Mathematics 7 FV 1 2 3 4 5 5 TimePeriods Later Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 ? 0 1 2 3 4 5 FV1 1001 010 . 110 Chabot College Mathematics 8 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 ? 1 2 0 3 4 5 FV2 1001 010 . 1 0.10 121 or 2 FV2 1001 010 . 121 Chabot College Mathematics 9 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Recognize Future Value Pattern Engaging in the EXTREMELY VALUABLE Practice of PATTERN RECOGNITION surmise The Pattern that is developing for FV in year-N FVN PV1 i N If the $1c investment were extended for a 3rd Year then the FV FV3 1001 010 . 13310 . 3 Chabot College Mathematics 10 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 FVN PV 1 i N Divide Both Sides by 1 i FVN PV N 1 i N Chabot College Mathematics 11 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 (Go Bears!) where she earns her Bachelors Degree Chabot College Mathematics 12 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 ONETime Initial investment, then how much must he invest today to achieve the $30k Future-Value Goal? SOLUTION: After 18-5 periods the PV: Chabot College Mathematics 13 $30k PV $11.031k 13 1 .0.08 $30k PV $11.031k Bruce Mayer, PE 13 BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 5year, $100 ($1 cNote) annuity 0 Chabot College Mathematics 14 100 100 100 100 100 1 2 3 4 5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 Chabot College Mathematics 15 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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: N PVA 1 i Pmt t t 1 Chabot College Mathematics 16 t Pmt 1 1 i 1 Pmt 2 1 i 2 Pmt N 1 i Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx N 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: 100 PVA . 110 1 100 . 110 2 100 . 110 3 100 . 110 4 100 . 110 5 379.08 62.09 68.30 75.13 82.64 90.91 379.08 100 100 100 100 100 1 2 3 4 5 0 Chabot College Mathematics 17 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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: N PVA 1 i t 1 Chabot College Mathematics 18 Pmt t t 1 1 N 1 i Pmt i N Pmtt PVA t 1 i Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Present Value of an Annuity Using the PVA equation in the $1c 1 1 example 5 PVA Pmt . 110 010 . 379.08 • 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 Chabot College Mathematics 19 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 N FVA Pmt 1 i t Nt Pmt 1 1 i N 1 Pmt 2 1 i N 2 Pmt N t 1 Chabot College Mathematics 20 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Future Value of an Annuity Again consider a $1c annuity, and assume a discount rate of 10% per year, find that the future value: FVA 100110 . 100110 . 100110 . 100110 . 100 610.51 4 3 2 1 146.41 133.10 121.00 110.00 0 100 100 100 100 100 1 2 3 4 5 Chabot College Mathematics 21 } = 610.51 at year 5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Future Value of an Annuity As was done for the PVA equation, use series convergence to find a closedform of the FVA equation: N FVA Pmt 1 i t t 1 Nt 1 i N 1 Pmt i As with The PVA, the FVA eqn works for all regular annuities, regardless of the number of payments Chabot College Mathematics 22 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 N t k Then the Nk i FVA → FVIS FVIS Pmt kt 1 k kt 1 eqn • Note that Pmtkt may VARY in time; e.g., Pmt73≠ Pmt74 Chabot College Mathematics 23 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx PV of an Income Stream The discounts Occur infinitely often, and the Pmtkt becomes continuously variable in time, then the PVIS equation becomes N FVIS e i N 0 f t e dt it Or textbook notation FV e Chabot College Mathematics 24 rT T 0 f t e dt rt Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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? Chabot College Mathematics 25 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 T future value is rT -rt FV = e ò 10000e dt 0 e Chabot College Mathematics 26 0.03(8) 8 10000e 0.03t 0 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx dt Example PV of Income Stream Running the Numbers 8 FV e 0.24 e 1 0.03t 10000 e 0.03 0 10000 0.03(8) 0.03( 0) e e 0.03 0.24 90,416.38 Thus the fund is worth about $90,416 at the end of eight years. Chabot College Mathematics 27 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 PV f t e dt T rt 0 Chabot College Mathematics 28 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 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 8 0.03t PV 10000e dt 0 10000 é -0.03(8) -0.03(0) ù = -e ëe û -0.03 = $71,124. Chabot College Mathematics 29 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx CAL Football Tickets Value of Ticket to Potential Demanders Peter $200 Paul Mary Jack Jill $150 $100 $50 $50 Price Peter Value of Ticket to Potential Suppliers: Professor V $50 Professor W Professor X Professor Y Professor Z $50 $100 $150 $200 Z 200 Paul Y 150 Mary 100 X Jack and Jill 50 V and W 0 1 2 3 4 5 Tickets Chabot College Mathematics 30 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx CAL Football Tickets 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 Price Consumer Surplus Peter Gains: Peter Paul Mary V W X Z 200 = $200 - $100 = $100 = $150 - $100 = $50 = $100 - $100 = $0 = $100 - $50 = $50 = $100 - $50 = $50 = $100 - $100 = $0 Paul Y 150 Mary 100 X Jack and Jill 50 V and W Total Gain: $250 0 1 2 3 4 Tickets Producer Surplus Chabot College Mathematics 31 5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Consumers’ Surplus Price ($/unit) Consumer Surplus Maximum Willingness to Pay (or Spend) for Qo WS Q Q0 Q 0 Po Q Q dQ What is Actually Paid AP Q Q0 Q 0 P0 dQ P0 Q0 D(Q) Qo Quantity (Units) • By Supply & Demand the Price settles at P0, but SOME Consumers are willing to pay MUCH MORE, thus these consumers save $-Money Chabot College Mathematics 32 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Consumers’ Surplus Thus the Surplus Total $-Funds kept by the consumers: CS = [Amount Willing to Spend] − [Amount Actually Paid] With Reference to the Areas shown on the Supply-n-Demand Graph CS WS AP Q Q0 Q 0 Chabot College Mathematics 33 QQ dQ P0 Q0 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Producers’ Surplus Price ($/Unit) Producer Surplus S(Q) $-Amount Actually paid Po AP Q Q0 Q 0 P0 dQ P0 Q0 Minimum $-Amount Needed to Supply Qo AN Q Q0 Q 0 Qo S Q dQ Quantity (units) • By Supply & Demand the Price settles at P0, but SOME Producrs are willing to accept a LOWER Price, thus these Producers make extra $-Money Chabot College Mathematics 34 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Consumer & Producer Surplus The Two Surpluses usually exist simultaneously Price ($/Unit) Consumer Surplus S(Q) Equilibrium Price-Point Po Producer Surplus D(Q) Qo Chabot College Mathematics 35 Quantity (Units) Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx WhiteBoard Work Problems From §5.5 • P18 → Supply & Demand • P26 → Construction Decision • P42 → Invention Profit Chabot College Mathematics 36 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx All Done for Today Tree NPV Chabot College Mathematics 37 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot Mathematics Appendix r s r s r s 2 2 Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu – Chabot College Mathematics 38 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 39 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 40 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 41 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 42 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 43 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 44 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 45 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 46 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 47 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 48 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 49 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 50 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 51 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx P55_42_AirPurifiers_1307.mn Chabot College Mathematics 52 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 53 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx Chabot College Mathematics 54 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx