Lecture 20Key issuesCapital and durable goodsInterest ratesGeneral compounding formulaFrequency of compoundingInterest rates connect present and futurePresent valueExampleWhen is future money nearly worthless? Stream of payments foreverStream of payments for t yearsPresent Value of a Dollar in the FutureFuture value of payments over timeInflation and discountingAdjusting for inflationNominal and real rates of interestNominal vs. real interest ratesReal interest rateReal present valueInvestment decisionNet present value approachNPV rulesInternal rate of return approachIRR for flowHuman capitalRetire at 70Compare earning streamsExhaustible resourcesIntuitionMany periodsInterpretationGap over timePrice of an Exhaustible Resource1. Comparing money today to money in future2. Choices over timeLecture 20Interest Rates, Investments, and Capital MarketsKey issues1. comparing money today to money in the future: interest rates 2. choices over time: invest in a project if return from investment > return on best alternativeCapital and durable goods• durable goods: products that are usable for years • if durable good or capital is rented, rent up to the point where the marginal benefit = MC• if bought or built rather than rented, firm compares current cost of capital to future higher profits it will make from using capitalInterest rates• assume no inflation: consuming $1 worth of candy today is better than consuming $1 worth in 10 years• how much more you must pay in future to repay a loan today is specified by an interest rate: percentage more that must be repaid to borrow money for a fixed period of timeGeneral compounding formulaFrequency of compounding• for a given i, more frequent compounding, greater payment at end of a year• annual interest rate is i = 4%• if bank pays interest 2 times a year, • half a year's interest, i/2 = 2%, after six month:$(1 + i/2) = $1.02• at end of year, bank owes:$(1 + i/2) × (1 + i/2) = $(1 + i/2)2=$(1.02)2 = $1.0404Interest rates connect present and future• future value (FV) depends on the present value (PV), the interest rate, and the number of years• put PV dollars in bank today and allow interest to compound for t years:FV = PV × (1 + i)tPresent value• 2 equivalent questions:• how much is $1 in the future worth today?• how much money, PV, must we put in bank today at i to get a specific FV at some future time?•answer:PV = FV/(1 + i)tExampleWhen is future money nearly worthless? • at high interest rates, money in future is virtually worthless today • $1 paid to you in 25 years is worth only 1¢today at a 20% interest rateStream of payments foreverStream of payments for t years•What’s PV of payments per period of fmade every year?• you agree to pay $10 at end of each year for 3 years to repay a debt (i = 10%)PV = $10/1.11+ $10/1.12+ $10/1.13 ≈ $24.87• generally:1211 1...(1 ) (1 ) (1 )tPV fii i⎡⎤=+++⎢⎥++ +⎣⎦Present Value of a Dollar in the FutureFuture value of payments over time•What’s FV after t years if you save f each year?• year 1: put f in account• year 2: add a second f, so you have first year's payment + accumulated interest of f (1 + i)1or f [1 + (1 + i)1] in total• year 3: total is f [1 + (1 + i) + (1 + i)2]•after t years:FV = f [1 + (1+i)1 + (1+i)2+…+(1+i)t-1]Inflation and discounting• we've been assuming inflation rate = 0%• suppose general inflation occurs: nominal prices rise at a constant rate γ over time• by adjusting for rate of inflation, we convert nominal prices to real pricesAdjusting for inflationNominal and real rates of interest• to calculate PV of this future real payment, we discount using real interest rate• without inflation, $1 today is worth 1 + i next year• with inflation rate of γ, $1 today is worth(1 + i)(1 + γ) nominal dollars tomorrow•if i = 5% and γ = 10%, $1 today is worth 1.05x1.1 = 1.155 nominal dollars next yearNominal vs. real interest ratesReal interest rate• depends on inflation and nominal rate• if inflation rate is small , then we can closely approximate the real rate by • if nominal rate is 15.5%, γ = 10%, real rate is (15.5%-10%)/1.1=5% and approximation is 5.5%1iiγγ−=+iiγ=−0γ≈Real present valueInvestment decision1. net present value approach2. internal rate of return approachNet present value approachdepends on PV of revenues, R, and cost, C1201212012...(1 ) (1 ) (1 )...(1 ) (1 ) (1 )TTTTNPV R CRR RRii iCC CCii i=−⎡⎤=+ + ++⎢⎥++ +⎣⎦⎡⎤−+ + ++⎢⎥++ +⎣⎦11 2 2001212012...(1 ) (1 ) (1 )...(1 ) (1 ) (1 )TTTTTRC RC R CNPV R Cii iii iππ ππ⎡⎤−− −=−+ + ++⎢⎥++ +⎣⎦⎡⎤=+ + ++⎢⎥++ +⎣⎦NPV rules• invest if NPV > 0 • it isn’t necessary for cash flow in each year, πt(loosely, annual profit), to be positiveInternal rate of return approach• at what discount rate (rate of return) is firm indifferent between making investment and not?• internal rate of return (irr) is discount rate where NPV = 0• replacing i with irr and setting NPV = 0, find irrby solving for iir12012... 0(1 ) (1 ) (1 )TTNPVirr irr irrππ ππ⎡⎤=++++ =⎢⎥++ +⎣⎦IRR for flowHuman capital• individuals decide whether to invest in their own human capital• does going to college increase your lifetime earnings?• graduate high school at 18 years old and either go to work or go to collegeRetire at 70• suppose you• graduate from college in 4 years• do not work when in college• pay $10,000 a year for school expenses: tuition, books, fees• opportunity cost of college: tuition payments plus 4 years of foregone earnings (at HS grad wage)• at age 22 • typical college grad earns $29K ($1995)• HS grad earns $18KAnnual Earnings of High School and College GraduatesAnnual earnings,Thousands of 1995 dollars–1010020304018 22 30 40 50 60 70High school graduateCollege graduateBenefitCostAge, YearsCompare earning streams• earnings peak• for college grad at 40 years of age at $39K• for HS grad at 43 years at $34K• decide whether invest in college by comparing PV at age 18 of the two earnings streamsExhaustible resources• discounting determines how fast we consume exhaustible resources• exhaustible resources:• nonrenewable natural assets that cannot be increased, only depleted• examples: oil, gold, copper, uraniumIntuition• storing coal in the ground is like keeping money in the bank• if you sell a pound today, net p1– m,
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