bài giảng kinh tế vi mô tiếng anh ch16 capital market

1 Chapter 16 Interest Rates, Investments, and Capital Markets Key issues 1. 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 capital Interest 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 time Deposit 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 + i Compounding • “interest on interest” • “accumulation of interest” 2 Example • 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 $4.16 Compounding over time assets at the end of year 1: $104.00 = $100 G 1.04 = $100 G 1.04 1 year 2: $108.16 = $104 G 1.04 = $100 G 1.04 2 year 3: $112.49 G $108.16 G 1.04 = $100 G 1.04 3 General compounding formula For every $1 you loan the bank, it owes you: year 1: $(1 + i) 1 year 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) t Frequency 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 = $1.0404 U.S. Truth-in-Lending Act requires lenders to tell borrowers equivalent noncompounded annual percentage rate (APR) of interest 3 Interest 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) t Power 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 Manhattan Alaska • 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 value Present 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) t Example • general formula PV = FV/(1 + i) t • FV = $100 at end of year • i = 4% PV = $100/1.04 = $96.15 4 When 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 rate Stream 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 invest PV = 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 = 20 Stream of payments for t years •What’s PV of payments per period of f made every year? • you agree to pay $10 at end of each year for 3 years to repay a debt i = 10% PV = $10/1.1 1 + $10/1.1 2 + $10/1.1 3 » $24.87 • generally: 12 11 1 (1 ) (1 ) (1 ) t PV f ii i éù =+++ êú ++ + ëû Figure 16.1 Present Value of a Dollar in the Future Present value, PV, of $1 20 10 40 50 60 70 80 90 $1 t , Years 0 102030405060708090100 i = 0% i = 5% i = 10% i = 20% 30 Future 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) 1 or 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 ] 5 Starting 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 saving for first 15 years, you save $3,000 a year for next 33 years until retirement Early Bird • save $3,000 a year for first 15 years then let it accumulate • after 15 years, early bird has $3,000[1+1.07 1 +1.07 2 + +1.07 14 ] = $75,387 • interest compounds for next 33 years, so fund grows 9.3 times to $75,387.07 q 1.07 33 = $703,010 Late Bloomer • no investments for 15 years, then invests $3,000 a year until retirement so funds at retirement are $3,000[1+1.07 1 +1.07 2 + +1.07 32 ] = $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 years Inflation 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 prices Adjusting 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 )ff γ =+ ° f ° ° /1.1 0.909 f ff== ° f Nominal 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 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 year 6 Nominal 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 + g) in next year’s dollars • because • nominal interest rate is i  1(1)(1)1ii ii γγγ += + + =++ +  iii γγ =+ +  Real 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%, g = 10%, real rate is (15.5%-10%)/1.1=5% and approximation is 5.5% 1 i i γ γ − = +  ii γ =−  0 γ ≈ Real present value • real present value of a nominal payment one year from now is • you agree to pay $100 next year for a tape recorder you get today, g = 10%, i = 5%, then PV = $100/(1.1 q 1.05) = $86.58 ° 1(1)(1) ff PV ii γ == +++ Winning the lottery lottery: a tax on people who are bad at math • several states boast (lie) that their lottery pays a winner one million dollars • winner gets $50,000 a year for 20 years • total nominal payments are $1 million •if C = 5% and i = 4% (nominal rate of interest is 9.2%) •real PV of 20 payments is $491,396 • without inflation real PV = $706,697 Comparing 2 contracts • professional basketball player is offered a choice of 2 contracts • one contract pays $1 million today • other contract pays $500,000 today and $2 million 10 years from now • both contracts guaranteed: payments will be made even if he’s injured Player’s choice • assume there is no inflation and that our pro wants to maximize his PV • PV of first contract is $1 million • to calculate PV of second contract, he uses market i • PV = $500,000 + $2,000,000/(1 + i) 10 • PV depends on interest rate • PV = $1,727,827 at i = 5% • PV = $823,011 at i = 20% • choose second contract if i = 5% but not at 20% • break-even interest rate is 14.87% 7 Investment decision 1. net present value approach 2. internal rate of return approach Net present value approach depends on PV of revenues, R, and cost, C 12 0 12 12 0 12 (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) T T T T NPV R C RR R R ii i CC C C ii i =− é ù =+ + ++ ê ú ++ + ë û é ù −+ + ++ ê ú ++ + ë û 11 2 2 00 12 12 0 12 (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) TT T T T R CRC RC NPV R C ii i ii i ππ π π é ù −− − =−+ + ++ ê ú ++ + ë û éù =+ + ++ êú ++ + ë û NPV rules • invest if NPV > 0 • it isn’t necessary for cash flow in each year, Q t (loosely, annual profit), to be positive Solved problem • Snyder, Zuckerman, and Drasner bought Washington Redskins football team and its home stadium for $800 million in 1999 • estimated 1999 net income was f = $32.8 million • if new owners believe they’ll earn this annual profit (adjusting for inflation) forever, was this investment more lucrative than putting $800 million in a savings account that pays i = 4%? Answer • NPV is positive if PV of expected returns, $32.8 million/0.04 = $820 million, minus PV of cost, $800 million, is positive: NPV = $820 million - $800 million = $20 million > 0 • thus, they buy team if best alternative is 4% rate of return Internal 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 irr by solving for iir 12 0 12 0 (1 ) (1 ) (1 ) T T NPV irr irr irr ππ π π é ù =+ + ++ = ê ú ++ + ë û 8 IRR for flow • if investment produces a steady stream of profit, f, then irr = f / PV • make investment if irr > i, if i is next best alternative Solved problem • group of investors can buy Redskins football team for PV = $800 million • they expect an annual real flow of payments (profits) of f = $32.8 million forever • if real interest rate is 4%, do they buy team? Answer • because this rate of return, 4.1%, is greater than interest rate, 4%, they buy the team $32.8 4.1% 4% $800 f million irr PV million == ≈ > Buying vs. renting a phone • choose between • renting a telephone at $10 per year (flow payment) • buy phone (lifetime warranty) for $100 (stock payment) • plan to use phone for 50 years; no inflation • 2 ways to decide • calculate PV of flow at a given interest rate and compare that to the cost of buying (NPV) • determine irr and compare to interest rate PV approach • PV of renting (f = $10/year) for 50 years using Table 16.3 shows PV payment of $10 a year for 50 years is • $183 at 5% • $99 at 10% • $50 at 20% • thus, if i U 10%, it is better to rent rather than buy 9 Approximation • approximation of PV of a stream of payments for 50 years: assume payments go on forever: • PV = $10 / i • PV = $200 at 5% (vs. $183 for 50 years) • PV = $100 at 10% (vs. $99) • PV = $50 at 20% (vs. $50) • thus, approximation is better, higher interest rate Calculate irr • put $100 you'd spend on the phone into an account that pays you interest of $100 q irr per year • that is, $100 q irr = $10 º irr = 10% • rent if irr < i; otherwise buy Human 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 college Retire 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 $18K Figure 16.2 Annual Earnings of High School and College Graduates Annual earnings, Thousands of 1995 dollars –10 10 0 20 30 40 Age, Years 18 22 30 40 50 60 70 High school graduate College graduate Benefit Cost 10 Compare 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 streams Exhaustible resources • discounting determines how fast we consume exhaustible resources • exhaustible resources: • nonrenewable natural assets that cannot be increased, only depleted • examples: oil, gold, copper, uranium When to sell • if own a coal mine and want to maximize PV, when do you • mine the coal • sell the coal • for simplicity, suppose • can only sell this year or next in a competitive market • interest rate is i • cost of mining a pound of coal, m, is constant over time When to mine • present value of cost of mining •this year: m • next year: m/(1 + i) • thus, if you are not selling until next year, mine at the last possible moment (lower PV of cost) When to sell • answer depends on how price changes over time • suppose price increases from p 1 this year to p 2 next year • compare PV of selling today to PV of selling next year •this year: PV 1 = p 1 – m • next year: PV 2 = (p 2 - m)/(1 + i) . 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. irr = $10 º irr = 10% • rent if irr < i; otherwise buy Human capital • individuals decide whether to invest in their own human capital • does going to college increase your lifetime earnings? •. bought or built rather than rented, firm compares current cost of capital to future higher profits it will make from using capital Interest rates • assume no inflation: consuming $1 worth of