1Chapter 16Interest 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 alternative Capital 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 timeDeposit funds in a bank today• bank agrees to pay you interest rate, i = 4%• one year from now, bank pays for every dollar you loan it: $1.04 = 1 + iCompounding• “interest on interest”• “accumulation of interest”2Example• place $100 in bank account that pays 4% a year • you can take out interest payment of $4 each year and leave your $100 in bank: earn a flow of $4-a-year payments forever• if you leave the interest rate in back, in second year bank owes you:• interest of $4 on your original deposit of $100 • interest of $4 G 0.04 = $0.16 on your first-year interest• total interest is $4.16Compounding over timeassets at the end of year 1: $104.00 = $100 G 1.04 = $100 G 1.041year 2: $108.16 = $104 G 1.04 = $100 G 1.042year 3: $112.49 G $108.16 G 1.04 = $100 G 1.043General compounding formulaFor every $1 you loan the bank, it owes you:year 1: $(1 + i)1year 2: $(1 + i)2= $(1 + i)q(1 + i) year 3: $(1 + i)3 = $(1 + i)q(1 + i)q(1 + i) ……year t: $(1 + i)tFrequency of compounding• for a given i, more frequent compounding, greater payment at end of a year• annual interest rate is i = 4%• if bank pays interest 2 times a year, • half a year's interest, i/2 = 2%, after six month:$(1 + i/2) = $1.02• at end of year, bank owes:$(1 + i/2)´ (1 + i/2) = $(1 + i/2)2=$(1.02)2 = $1.0404U.S. Truth-in-Lending Actrequires lenders to tell borrowers equivalent noncompounded annual percentage rate (APR) of interest3Interest 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 q (1 + i)tPower of compounding: Manhattan Island• Dutch allegedly bought Manhattan in 1626 for about $24 worth of beads and trinkets• if Native Americans had invested in tax-free bonds 7% APR bonds, it would now be worth over $2.0 trillion > assessed value of ManhattanAlaska• if US had invested $7.2 million it paid Russia in 1867 in tax-free 7% APR bonds• money worth only $50.9 billion < Alaska's current valuePresent 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)tExample• general formulaPV = FV/(1 + i)t• FV = $100 at end of year• i = 4%PV = $100/1.04 = $96.154When 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 forever• PV in a bank account earning i produces a flow of f (at end of each year) of f = i´ PV•to receive f each year forever need to investPV = f / i• to get $10 a year invest$200 = $10/0.05 at i = 5%$100 = $10/0.10 at i = 10%$50 = $10/0.20 at i = 20Stream 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 iéù=+++êú++ +ëûFigure 16.1 Present Value of a Dollar in the FuturePresent value,PV, of $12010405060708090$1t , Years0 102030405060708090100i = 0%i = 5%i = 10%i = 20%30Future 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]5Starting Early• it pays to start saving early (take advantage of compounding)• two approaches to savings• early bird: you save $3,000 a year for first 15 years of your working life and then let your savings accumulate interest until you retire• late bloomer: after not savingfor first 15 years, you save $3,000 a year for next 33 yearsuntil retirementEarly Bird• save $3,000 a year for first 15 years then let it accumulate• after 15 years, early bird has $3,000[1+1.071+1.072 +...+1.0714] = $75,387• interest compounds for next 33 years, so fund grows 9.3 times to$75,387.07 q 1.0733= $703,010Late Bloomer• no investments for 15 years, then invests $3,000 a year until retirement so funds at retirement are$3,000[1+1.071+1.072+...+1.0732] = $356,800• thus, late bloomer • contributes to account more than twice as long as the early bird • but saves only about half as much by retirement• to have same amount at retirement, late bloomer has to save nearly $6,000 a year for 33 yearsInflation and discounting• we've been assuming inflation rate = 0%• suppose general inflation occurs: nominal prices rise at a constant rate g over time• by adjusting for rate of inflation, we convert nominal prices to real pricesAdjusting for inflation• nominal amount you pay next year is • future debt in today's dollars is•if g = 10%, a nominal payment of next year is in today’s (real) dollars°/(1 )ffγ=+°f°°/1.1 0.909fff==°fNominal 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 + inext year• with inflation rate of g, $1 today is worth(1 + i)(1 + g) nominal dollars tomorrow•if i = 5% and g = 10%, $1 today is worth 1.05 q1.1 = 1.155 nominal dollars next year6Nominal vs. real interest rates• banks pay a nominal interest rate, • if real discount rate is i, banks' nominal interest rate is such that dollar today pays (1 + i)(1 +
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