货币的时间价值.ppt

CHAPTER 4,THE TIME VALUE OF MONEY 资金的时间价值,COPYRIGHTZHULI,1,定义：,货币的时间价值就是指当前所持有的一定量货币比未来获得的等量货币具有更高的价值。从经济学的角度而言，现在的一单位货币与未来的一单位货币的购买力之所以不同，是因为要节省现在的一单位货币不消费而改在未来消费，则在未来消费时必须有大于一单位的货币可供消费，作为弥补延迟消费的贴水。,COPYRIGHTZHULI,2,本章导读,公司需要筹集资金（raise funds）以支付投资，并因此承担了在未来时点还款的责任；个人也可能通过借贷（student loan）获得大学教育所需资金，并计划在未来用工资收入偿债。 因此所有的财务决策（financial decision）都必须对不同时点的现金流量（cash flows）进行比较。 本章首先分析了按特定利率（given interest rate）进行投资的资金如何在投资期内增值，并给出了计算多笔现金流价值的一些公式，然后分析了通货膨胀（inflation）对财务计算的影响。,1-3,Content,Future values and compound interest终值与复利 Present values现值 Multiple cash flows多重现金流 Level cash flows: perpetuities and annuities 水平现金流：永续年金和年金 Inflation and the time value of money通胀与资金的时间价值 Effective annual interest rates有效年利率,COPYRIGHTZHULI,4,COPYRIGHTZHULI,5,Suppose you have $100 invested in a bank account, interest rate of 6% per year 假设银行账户中有100刀，年利率为6%,Interest in year 1 = .06 $100 = $6 Value of investment after 1 year = $100+$6 = $106,Interest in year 2 = .06 $106 = $6.36 Value of investment after 1 year = $106+$6.36 = $112.36,利率为r时，初始投资一年以后的价值等于初始金额乘以（1+r）,4.1 终值FV与复利COMPOUND INTEREST,COPYRIGHTZHULI,6,For an interest rate of r and a horizon of t years, 一笔时限为t年利率为r的初始金额为$100的投资的终值： Future value of $100 = $100(1+r)t,Future value: amount to which an investment will grow after earning interest 终值：投资的未来价值，即一定量的资金在将来某一时点的价值，表现为本利和. (P92.),COPYRIGHTZHULI,7,Compound interest: interest earned on interest 复利：利滚利（将所生利息计入本金再计利息）( P92).,Simple interest单利：the interest only on your original investment. 在贷款期间只就本金计算利息，利息不再加入本金计算利息。(P92.),COPYRIGHTZHULI,8,How an investment of $100 grows with compound interest at different interest rate 本金为$100的投资不同利率水平下的复利增长,Obviously, the higher the rate of interest, the faster your savings will grow with years.,COPYRIGHTZHULI,9,An example of a future value table, showing how an investment of $1 grows with compound interest,FV factor终值系数 = (1+r)t,COPYRIGHTZHULI,10,$100 invested for 1 year at 6% will grow to 本金$100，投资期1年，利息6%，终值(FV)为 $106= (1001.06),How much we need to invest now in order to produce $106 at the end of the year? 反之，若想在一年后获得$106，那么初始投资金额应为多少呢？ Present value = future value/l.06 = $100,4.2 PV现值,Present value (PV): value today of a future cash flow 现值：未来一定时间特定资金的现在价值。(P95.) To calculate present value, we discounted the future value at the interest rate r. 为了计算现值，我们按利率r将未来价值贴现。 The calculation is therefore termed a discounted cash flow (DCF贴现现金流) calculation, an the interest rate r is known as the discounted rate。利率r即是贴现率。 (P96.),COPYRIGHTZHULI,11,In general, for a future value or payment t periods away 公式化之后：现值即等于终值除以终值系数 Present value = future value after t periods/(1+r)t,COPYRIGHTZHULI,12,Present value of a future cash flow of $100,The longer, the less it is worth today 支付期限越长，现在所需的初始投资额越少。,COPYRIGHTZHULI,13,An example of present value table, showing the value today of $1 received in the future. 未来特定时间的1美元的现值：,Discount factor贴现系数 = 1/(1+r)t,4.3.1 FUTURE VALUE OF MULTIPLE CASH FLOWS 多笔现金流的终值,COPYRIGHTZHULI,14,Suppose you plan to save some amount of money each year to purchase a computer. 假设你想每年存一点钱来买一台电脑。 You might be able to put $1200 in the bank, a second deposit of $1400 in 1 year in 2 years. 两年内存了三笔，第一笔$1200，第二笔$1400，第三笔$1000，假设都是年初存入。 If you earn an 8% rate of interest. How much be available to spend 2 years from now? 若年利率为8%，复利计算，那么三年后可用资金额是多少？,COPYRIGHTZHULI,15,To find the value at some future date of a stream of cash flows, calculate what each flow will be worth at that future date, and then add up these future values. 计算多笔现金流的终值，只需分别计算单笔现金流的终值，然后加总即可。,4.3.2 PRESENT VALUE OF MULTIPLE CASH FLOWS 多笔现金流的现值,COPYRIGHTZHULI,16,Suppose your auto dealer gives you a choice between paying $15500 for a new car or entering into an installment plan where you pay $8000 down today and make payments of $4000 in each of the next 2 years. Assume that the interest rate you can earn on safe investments is 8%. 假设你有一个购车计划，可以选择一次性支付$15500，或者分期付款首付$8000，之后的两年每年支付$4000，无风险投资收益率为8%。你会如何选择？ Which is the better deal?,COPYRIGHTZHULI,17,$15133.06 15500 The installment plan is preferable,The present value of a stream of future cash flows is the amount you need to invest today to generate that stream. To find the PV, you just calculate the PV of each flow and then add them. 计算多笔未来现金流量的现值，只需分别计算出每笔现金流的现值，然后加总即可。,4.4 level cash flows水平现金流,Annuity: equally spaced level stream of cash flows, with a finite maturity 年金：在相等的间隔期，连续地分批支付或收入相等金额的款项。(P105.) Perpetuity: stream of level cash payments that never ends 永续年金：无限期定额支付的年金。 (P105.) Annuity due: level stream of cash flows starting immediately 期初年金：指在一定时期内，以相同的时间间隔在各期期初收入或支出的等额的款项。,COPYRIGHTZHULI,18,4.4.1 HOW TO VALUE PERPETUITIES如何计算永续年金,COPYRIGHTZHULI,19,Suppose that you could invest $100 at an interest rate of 10%. You would earn annual interest of .10 $100 = $10 per year and could withdraw this amount from your investment account each year without ever running down your balance. 初始投资为$100年利率为10%的投资可使投资者每年支取$10的利息而不会减少账户资金。 In other words, a $100 investment could provide a perpetuity of $10 per year. In general,Cash payment from perpetuity = interest rate present value 永续年金可获得的现金收入是固定的：C = r PV (P105.),Two warnings about the formula关于永续年金的计算，有两点需要注意： Do not confuse the formula with the present value of a single cash payment 1/(1+r)不能与单一现金流的计算公式混淆。 The value of a regular stream of payments starting one period from now。永续年金的计算公式给出的是从第一年年末开始的连续等额的现金支付的现值。,COPYRIGHTZHULI,20,反之，永续年金的现值=现金支付除以利率 PV of perpetuity = C/r = cash payment/interest rate,Delayed perpetuity 递延永续年金or后付年金or延期年金,指在最初若干期没有收付款项的情况下，后面若干期等额的系列收付款项。 如某慈善机构打算在四年后每年捐款$100。即在第三年年末，这笔捐款变成了正常的永续年金价值:$100/r。但是现在的价值并不是$100/r，而是要乘以贴现系数： PV of delayed perpetuity=$100 1/r1/(1+r)3,4.4.2 HOW TO VALUE ANNUITIES 年金的计算方式,COPYRIGHTZHULI,22,方法一：The value of an annuity is equal to the difference between the value of two perpetuities普通年金的现值等于永续年金与递延永续年金现值之差。(P107.),Present value of a 3-year $1 annuity = 1/r 1/r(1+r)3,COPYRIGHTZHULI,23,期限t年的普通年金的现值=现金支付*年金系数 Present value of t-year annuity = C 1/r 1/r(1+r)t Present value of t-year annuity =cash payment annuity factor,An example of an annuity table, showing the present value today of $1 a year received for each of t years,COPYRIGHTZHULI,24,方法二：To find the value of an annuity, you can calculate the value of each cash flow. 分别计算每笔现金流量的现值，然后加总。 It is usually quicker to use the annuity formula.,FUTURE VALUE OF AN ANNUITY普通年金的终值,COPYRIGHTZHULI,25,Suppose you are setting aside $3,000 at the end of every year in order to buy a car. If your savings earn interest of 8 percent a year, how much will they be worth at the end of 4 years? 假设为了购车于4年内，每年年末存入$3000，储蓄利息为8%, 4年后收益是多少？,PV = $3,000 4-year annuity factor,4.4.3 ANNUITIES DUE期初年金 （P114.),COPYRIGHTZHULI,26,Present value of an annuity due = PV of an ordinary annuity (1+r) 期初年金的现值等于普通年金的现值乘以（1+r) Future value of annuity due = FV of ordinary annuity (1+r) 期初年金的终值等于普通年金的终值乘以（1+r),definition：a level stream of payments starting immediately 每期期初支付的年金,4.5 inflation the time value of money,通货膨胀与货币的时间价值,4.5.1 Real versus nominal cash flows实际与名义现金流的对比,An overall general rise in price is known as inflation 物价的普遍上涨称其为通货膨胀。(P119.) Economists track the general level of prices using several different price indexes. The best known is the consumer price index (CPI) 消费者价格指数它反映了一篮子代表性消费商品和服务项目不同时期的价格水平。 Current or nominal dollars refer to the actual number of dollars of the day 流动美元或名义美元指的是美元的实际金额。 Constant or real dollars refer to the amount of purchasing power 不变美元或实际美元则指美元的购买力大小。,COPYRIGHTZHULI,28,例题：,参照表格4.7 1950年，一户普通家庭每周的日常支出是$250，那么到了1980年，这些消费品需要花费多少美金？ 如果1980年，年收入是$30,000，那么在1950年这笔收入的实际价值是多少？,1-29,4.5.2 Inflation and interest rates通货膨胀与利率,Nominal interest rate: rate at which money invested grows 名义利率：投资金额的增长率 Real interest rate: rate at which the purchasing power of an investment increases 实际利率：投资金额所代表的购买力的增长 (P120.) 它们之间的差异是由通货膨胀导致的。,COPYRIGHTZHULI,30,通货膨胀率与名义利率和实际利率的关系式,精确算法： 1+real interest rate = (1+nominal interest rate)/(1+inflation rate) 近似算法：r R - i Real interest rate nominal interest rate inflation rate 注意：当通货膨胀率与实际利率都非常小时，近似算法才是可行的。,1-31,4.5.3 Valuing real cash payments对实际支付的现金进行估值,COPYRIGHTZHULI,32,Suppose that the nominal interest rate is 10%. How much do you need to invest now to produce $100 in a years time?,名义利率计算的终值PV = $100/1.10 = $90.91,Assume that you expect inflation of 7% over the next year. The real value of that $100 is therefore only $100/1.07 = $93.46. Also with a 7% inflation rate, the real rate of interest is 2.8%,实际利率计算的终值PV = $93.46/1.028 = $90.91,The two should give the same answer!,COPYRIGHTZHULI,33,结论：Current dollar cash flows must be discounted by the nominal interest rate; real cash flows must be discounted by the real interest rate. 用名义利率来贴现名以现金流，用实际利率来贴现实际现金流，且二者计算结果相同。(P121.),例题：,詹姆斯先生年过六旬，已经积攒了一笔$3,000,000的财富，正考虑提前退休。 假设詹姆斯先生可以活90岁，他打算把这笔财富在接下来的30年都享受完。假定他每年的实际支出是固定的。名义利率是10%，通货膨胀率为5%。 那么詹姆斯先生每年可以支出多少钱？,1-34,4.6 effective annual interest rates实际年利率,利率和利息计算期要保持一致。如果给定的利率是月利率，那么必须以月份数作为终值计算的利息计算期。,1-35,COPYRIGHTZHULI,36,Consider your credit card. Suppose you have to pay