# MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

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BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §5.5 Int Apps Biz & Econ

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §5.4 → Applying the Definite Integral  Any QUESTIONS About HomeWork §5.4 → HW-25 5.4

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 5 Bruce Mayer, PE Chabot College Mathematics 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)

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 6 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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 012345 PVFV Today5 TimePeriods Later

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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 012345 \$100?

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 9 Bruce Mayer, PE Chabot College Mathematics Finding Future Value by Arith  Now Extend the Investment for another Year (“Let it Ride”)  Then the \$-Value expected after the 2 nd year with no additional investment 012345 \$110?

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 10 Bruce Mayer, PE Chabot College Mathematics 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 3 rd Year then the FV

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 13 Bruce Mayer, PE Chabot College Mathematics 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:

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 14 Bruce Mayer, PE Chabot College Mathematics 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 012345 100

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 15 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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:

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 17 Bruce Mayer, PE Chabot College Mathematics 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 PV A as: 012345 100 62.09 68.30 75.13 82.64 90.91 379.08

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 18 Bruce Mayer, PE Chabot College Mathematics 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 PV A equation instead:

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 19 Bruce Mayer, PE Chabot College Mathematics Present Value of an Annuity  Using the PV A equation in the \$1c example Thus a 5yr constant yearly income Annuity of \$100/yr, discounted by 10% has PV of \$379  The PV A equation works for all regular annuities, regardless of the number of payments

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 21 Bruce Mayer, PE Chabot College Mathematics Future Value of an Annuity  Again consider a \$1c annuity, and assume a discount rate of 10% per year, find that the future value: 100 012345 146.41 133.10 121.00 110.00 } = 610.51 at year 5

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 22 Bruce Mayer, PE Chabot College Mathematics Future Value of an Annuity  As was done for the PV A equation, use series convergence to find a closed- form of the FV A equation:  As with The PV A, the FV A eqn works for all regular annuities, regardless of the number of payments

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 23 Bruce Mayer, PE Chabot College Mathematics 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 FV A → FV IS eqn Note that Pmt kt may VARY in time; e.g., Pmt 73 ≠ Pmt 74

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 24 Bruce Mayer, PE Chabot College Mathematics PV of an Income Stream  The discounts Occur infinitely often, and the Pmt kt becomes continuously variable in time, then the PV IS equation becomes  Or textbook notation

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 25 Bruce Mayer, PE Chabot College Mathematics 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?

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 26 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  PV of Income Stream  Running the Numbers  Thus the fund is worth about \$90,416 at the end of eight years.

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 28 Bruce Mayer, PE Chabot College Mathematics 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 FV IS equation becomes

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 29 Bruce Mayer, PE Chabot College Mathematics 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

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 30 Bruce Mayer, PE Chabot College Mathematics CAL Football Tickets 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\$200 200 150 100 50 Price Tickets 0 1 2 3 4 5 Mary Peter Paul Jack and JillV and W X Y Z

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 31 Bruce Mayer, PE Chabot College Mathematics 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 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: \$250 200 150 100 50 Price Tickets 0 1 2 3 4 5 Mary Peter Paul Jack and JillV and W X Y Z Consumer Surplus Producer Surplus

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 32 Bruce Mayer, PE Chabot College Mathematics Consumers’ Surplus By Supply & Demand the Price settles at P 0, but SOME Consumers are willing to pay MUCH MORE, thus these consumers save \$-Money Price (\$/unit) Quantity (Units) D(Q) PoPo QoQo Maximum Willingness to Pay (or Spend) for Q o What is Actually Paid Consumer Surplus

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 33 Bruce Mayer, PE Chabot College Mathematics 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]

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 34 Bruce Mayer, PE Chabot College Mathematics Producers’ Surplus By Supply & Demand the Price settles at P 0, but SOME Producrs are willing to accept a LOWER Price, thus these Producers make extra \$-Money Minimum \$-Amount Needed to Supply Q o Price (\$/Unit) Quantity (units) PoPo QoQo \$-Amount Actually paid Producer Surplus S(Q)

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 35 Bruce Mayer, PE Chabot College Mathematics Consumer & Producer Surplus  The Two Surpluses usually exist simultaneously Price (\$/Unit) Quantity (Units) PoPo QoQo S(Q) Producer Surplus Consumer Surplus D(Q) Equilibrium Price-Point

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 36 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §5.5 P18 → Supply & Demand P26 → Construction Decision P42 → Invention Profit

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 37 Bruce Mayer, PE Chabot College Mathematics All Done for Today Tree NPV

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 38 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 39 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 40 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 41 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 42 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 43 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 44 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 45 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 46 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 47 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 48 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 49 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 50 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 51 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 52 Bruce Mayer, PE Chabot College Mathematics P55_42_AirPurifiers_1307.mn

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 53 Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 54 Bruce Mayer, PE Chabot College Mathematics