Econ 100AChapter 16Interest Rates, Investments, and Capital Markets*1.Comparing Money Today to Money in the Future: Interest rates2.Choices over Time: Invest in a project if return from investment > return on bestalternativeCapitaland otherdurable goods: products that are usable for years.If capital is rented,• rent up to the point where the marginal benefit = marginal cost• Ex: Firm rents trucks untilcurrentmarginal rental cost =currentmarginal benefit(marginal revenue product of the trucks)If capital must be bought or built rather than rented,• firm must compare thecurrentcost of the capital to thefuturehigher profits it will makefrom using the plant.• Such comparisons involvestocksandflows•stock: quantity or value that is measured independently of time• A firm owns "an apartment buildingthisyear" (not "an apartment buildingperyear")•flow: quantity or value that is measured per unit of time (hotdogs per week)Interest RatesAssume no inflation: Consuming a $1 worth of candy today is better than consuming $1 worthin 10 yearsHow much more you must pay in the future to repay a loan today is specified by aninterestrate:the percentage more that must be repaid to borrow money for a fixed period oftime• Deposit funds in a bank today• Bank agrees to pay you• an interest rate,i=4%• $1.04 (= 1 +i) one year from now for every dollar you loan itan individual’s personal "interest" rate that person’sdiscount rate:a rate reflecting the relative value an individual places on future consumptioncompared to current consumptionA person’s willingness to borrow or lend depends on whether his or her discount rate isgreater or less than the market interest rateCompounding:• "interest on interest"• "accumulation of interest"You place $100 in a bank account that pays 4%, at the end of a year• Remove interest payment each year:*Based on Jeffrey M. Perloff,Microeconomics(© Addison Wesley Longman, 1999). Thesenotes are © Jeffrey M. Perloff.• you can take out the interest payment of $4 each year and leave your $100 inthe bank to earn more interest in the future, or• you have convert your $100 stock into a flow of $4-a-year payments forever• Let interest accumulate:• must pay you interest on $104 at end of the second year• bank owes you interest of $4 on your original deposit of $100 and interest of $4× 0.04 = $0.16 on your interest from the first year, for a total of $4.16.• End of Year 1, your account contains$104.00 = $100 × 1.04 = $100 × 1.041.• End of the Year 2, you have$108.16 = $104 × 1.04 = $100 × 1.042.• End of Year 3, your account has$112.49 ≈ $108.16 × 1.04 = $100 × 1.043.• End of yeart, you have$100 × 1.04tGenerally (with compounding):For every $1 you loan the bank, it owes you:1+idollars after 1 year(1 +i) × (1 +i)=(1+i)2dollars after 2 years(1 +i) × (1 +i) × (1 +i)=(1+i)3after three years...(1 +i)tdollars at the end oftyears.Frequency of CompoundingThe more frequently interest is compounded (holding the interest rate constant), the greaterthe payment at the end of a year.Let a bank’s annual interest rate bei=4%• It pays interest two times a year, get half a year’s interest,i/2 = 2%, after six month:(1 +i/2) = 1.02 per dollars• At the end of the year, it owes:(1 +i/2) × (1 +i/2)=(1+i/2)2= (1.02)2= $1.0404U.S. Truth-in-Lending Act requires lenders to tell borrowers what is the equivalentnoncompounded annual percentage rate (APR) of interestSee Table 16.1Interest Rates Connect the Present and Futurefuture value(FV)=present value(PV) + interestPutPVdollars in the bank today and allow interest to compound fortyears:Table 16.2(16.1)FV PV× (1i)t.Application: Power of CompoundingManhattan:2• Dutch allegedly bought it in 1626 for about $24 worth of beads and trinkets• If Native Americans had invested in tax-free bonds with an APR of 7%, the bond wouldnow be worth over $2.0 trillion, which is much more than the assessed value ofManhattan IslandAlaska:• If US had invested the $7.2 million it paid Russia in 1867 in the same type of bonds forthe next 127 years, that money would now be worth only $50.9 billion < Alaska’scurrent valuePresent ValueHow much is a $1 in the future worth today?Or, much money,PV, must we put in the bank today at an interest rateito get a specificamount of money,FV, in the future?• WantFV= $100 at the end of a year•i=4%• From Equation 16.1:PV× 1.04 = $100• Dividing both sides of this expression by 1.04:PV= $100/1.04 = $96.15General formula:(16.2)PVFV(1i)t.At high interest rates, money in the future is virtually worthless today:A dollar paid to you in 25 years is worth only 1¢ today at a 20% interest rate.Stream of PaymentsPayments per period (flow measure) may be used instead ofPVorFV(stock measures)PVof payments for afinite numberof yearsExample:• you agree to pay $10 at the end of each year for 3 years to repay a debt•i= 10%PV= $10/1.1 + $10/1.12+ $10/1.13≈ $24.87.Generally:future paymentoffper year fortyears at an interest rate ofi, the present value(stock) of this flow of payments is(16.3)PV f1(1i)11(1i)21(1i)t.Table 16.4 shows that the present value of a payment off= $10 a year for 5 years is $43 at5%, $38 at 10%, and $30 at 20% annual interest.3Payments ForeverPVdollars in a bank account earningiproduces a flow off=i×PVat the end of yearDividing both sides byi:(16.4)PVfiThus, you’d have to deposit $10/iin the bank to ensure a future payment off= $10 forever.PVof $10 a year forever:• $200 at 5%• $100 at 10%• $50 at 20%.Future Value of Payments over TimeHow much will you have in your savings account,FV, at some future time if you savefeachyear?• Year 1: Putfin the account• Year 2: Add a secondf, so you have the first year’s payment plus its accumulatedinterest,f(1 +i)1,orf[1+(1+i)1] in total.• Year 3: total isf[1+(1+i)+(1+i)2]• General (Equation 16.5):FV f1 (1i)1(1i)2(1i)tApplication: Saving for RetirementIt pays to start saving early (take adv. of compounding):Two approaches to savings:•Early bird: You save $3,000 a year for the first 15 years of your working life and thenlet your savings accumulate interest until you retire.•Late bloomer: After not saving for the first 15 years, you save $3,000 a year for thenext 33 years until retirement.Early BirdSave $3,000 a year for the first 15 years then let it accumulate:Using Equation 16.5, at the end of 15 years, the early bird has$3,000[1 + 1.071+ 1.072+ ... + 1.0714] = $75,387Interest
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