1公公 司司 理理 财财2斯蒂芬 A.罗斯 史史蒂蒂芬芬罗斯斯先先生生目目前前是是麻麻省省理理工工学学院院斯斯隆隆管管理理学学院院财务经济学学教教授授。作作为在在财务和和经济领域域著著述述最最为丰丰富富的的作作者者之之一一，罗斯斯教教授授以以他他在在发展展套套利利价价格格理理论上上所所做做的的工工作作，以以及及通通过研研究究信信息息折折射射理理论、代代理理理理论、利利率率期期限限结构构理理论和和其其他他诸多多领域域所所做做出出的的大大量量贡献献，成成为备受受称称道道的的著著名名学学者者。罗斯斯曾曾任任美美国国金金融融协会会主主席席，现在在担担任任数数家家学学术型型和和实战型型杂志志的的副副主主编。他他还是是CalTech的的受受托托人人，大大学学退退休休股股权基基金金和和GenRe公公司司的的董董事事。此此外外，他他还兼兼任任劳尔&罗斯斯资产管理公司的管理公司的联席主席。席主席。3伦道夫 W.韦斯特菲尔德 伦道道夫夫韦斯斯特特菲菲尔德德先先生生是是南南加加州州大大学学马歇歇尔商商学学院的院院的院长。 从从1988到到1993年年，韦斯斯特特菲菲尔德德教教授授担担任任马歇歇尔商商学学院院财务和和商商业经济系系主主任任和和查尔斯斯桑桑顿财务教教授授。在在来来到到南南加加州州大大学学之之前前，他他曾曾是是宾夕夕法法尼尼亚大大学学沃沃顿商商学学院院财务系系主主任任，并并在在那那里里执教教20年年。他他还曾曾是是沃沃顿商商学学院院罗德德尼尼怀特特财务研研究究中中心心的的高高级研研究究员。他他所所擅擅长的的领域域包包括括：公公司司财务政政策策、投投资管管理理和和分分析析、兼兼并并和和收收购以以及及股票市股票市场价格行价格行为。 韦斯斯特特菲菲尔德德教教授授是是大大陆银行行信信托托委委员会会的的成成员，负责指指导信信托托部部门的的所所有有决决策策。他他还担担任任着着许多多公公司司和和机机构构的的顾问，包包括括：美美国国电报电话公公司司、莫莫比比尔石石油油和和太太平平洋洋公公司司以以及及联合合国国、美美国国司司法法和和劳工工部部以以及及加加利利福福尼尼亚州州州州政府。政府。 4布拉德福德 D.乔丹 布布拉拉德德福福德德乔丹丹先先生生是是肯肯塔塔基基大大学学财务教教授授和和加加藤藤研研究究员。长期期以以来来，他他对公公司司理理财的的理理论和和实务一一直直保保持持着着浓厚厚兴趣趣，而而且且在在公公司司理理财和和财务管管理理政政策策的的各各个个层面面上上拥有有丰丰富富的的经验。乔丹丹教教授授在在诸如如资本本成成本本、资本本结构构和和证券价格行券价格行为等等领域域发表了大量的表了大量的论文。文。 5作者写作该教材的基本理念v快速而广泛的变化，给财务课程的教学者设置了快速而广泛的变化，给财务课程的教学者设置了重重障碍。保持资料的及时性，变得越来越困难。重重障碍。保持资料的及时性，变得越来越困难。况且，还必须将持久性的内容与临时的东西区别况且，还必须将持久性的内容与临时的东西区别开来开来, ,以避免追随那些仅是昙花一现的时尚。以避免追随那些仅是昙花一现的时尚。我我们的方法是：强调财务的现代基础，并使所探讨们的方法是：强调财务的现代基础，并使所探讨的主题与当代实践紧密结合。的主题与当代实践紧密结合。v当时光流逝，细节将湮没，当时光流逝，细节将湮没，留下来的，将是对基留下来的，将是对基本原理的深刻把握本原理的深刻把握，如果我们的努力能够成功的，如果我们的努力能够成功的话。这就是为什么我们能自始至终，从本书第一话。这就是为什么我们能自始至终，从本书第一页到最后一页，页到最后一页，一直专注于财务决策的基本逻辑一直专注于财务决策的基本逻辑6未来现金流量的价值评估未来现金流量的价值评估财务管理的核心问题：财务管理的核心问题：未来的现金流量在今天价值多少未来的现金流量在今天价值多少 ？7货币的时间价值货币的时间价值& 现金流量的折现现金流量的折现8对于对于 今天的今天的$1000 和和 5 年后的年后的 $1000,你选择哪一个呢你选择哪一个呢 ？ 货币的时间价值货币的时间价值货币的时间价值货币的时间价值显然是显然是, 今天的今天的 $1000.你已经承认了你已经承认了 货币的时间价值货币的时间价值 !9为什么：为什么： 在决策中必须考虑在决策中必须考虑 货币的时间价值货币的时间价值 ? Why TIME ?Why TIME ?因为：因为： 如果眼前就能取得如果眼前就能取得 $1000，那么，那么我们我们 至少有一个至少有一个 用这笔钱去用这笔钱去 获获取取 利息利息收益收益 的的 投资机会投资机会.10实例 -货币的时间价值v19821982年年1212月月2 2日日, ,通用汽车的子公司通用汽车的子公司 Acceptance Acceptance 公司公司(GMAC)(GMAC)，公开发行了一些债券。在此债券，公开发行了一些债券。在此债券的条款中，的条款中，GMAC GMAC 承诺将在承诺将在20122012年年1212月月1 1日按照日按照每张每张 $10,000$10,000 的价格向该债券的所有者进行偿的价格向该债券的所有者进行偿付，但是投资者们在此日期之前不会有任何收付，但是投资者们在此日期之前不会有任何收入。投资者购入每一张债券支付给入。投资者购入每一张债券支付给 GMACGMAC $500$500，因此，他们在因此，他们在19821982年年1212月月2 2日日放弃放弃 $500$500, ,是为了是为了在在3030年后获得年后获得$10,000$10,000。这样的债券让你在今天。这样的债券让你在今天付出一笔钱，从而得到在将来的某日收到一大付出一笔钱，从而得到在将来的某日收到一大笔钱的承诺，它是所有可能性中最简单的形式笔钱的承诺，它是所有可能性中最简单的形式. .11续实例 -货币的时间价值v今天放弃今天放弃 $500$500 以在以在3030年年后获得后获得 $10,000$10,000 是不是不是一项好交易呢是一项好交易呢? ? 从正面来看，每从正面来看，每$1$1投入都投入都能得到能得到 $20$20 的回报的回报, ,它听起来很不错；但是从它听起来很不错；但是从负面来看，你必须等待负面来看，你必须等待3030年才能得到这笔钱。年才能得到这笔钱。如何来分析这种此消彼涨的关系如何来分析这种此消彼涨的关系 ？v这堂课的目标是介绍财务中最重要的一个原理：这堂课的目标是介绍财务中最重要的一个原理：货币的时间价值。货币的时间价值。如何如何确定未来的现金流入在确定未来的现金流入在今日的价值，是一项非常基础的商业技能，并今日的价值，是一项非常基础的商业技能，并且是对各种不同类型的投资和融资计划进行分且是对各种不同类型的投资和融资计划进行分析的理论基础。析的理论基础。v几乎所有的商业活动，都要几乎所有的商业活动，都要将今天的耗费与将将今天的耗费与将来的预计收益相比较。来的预计收益相比较。12OutlinevFuture Value and CompoundingvPresent Value and DiscountingvDiscount RatevThe Number of PeriodsvMultiple Cash FlowsvAnnuities and PerpetuitiesvAPR and EARvLoan Types & Loan Amortization13Part 1Future Value and Compounding14主要内容主要内容主要内容主要内容vv 单单利利利利vv 复利复利复利复利15利息利息利息利息v复利复利 (Compounding Compounding = = Interest on Interest on InterestInterest) 不仅借贷的本金要求支付利息，而且不仅借贷的本金要求支付利息，而且前期的利息前期的利息在下一期也要求支付利息在下一期也要求支付利息. .单利单利 (Simple InterestSimple Interest)只对借贷的只对借贷的原始金额原始金额或或本金本金支付利息。支付利息。16公式公式SI = P0 * *(r)* *(n)SI:单利利息利利息(Simple Interest)P0:原始金原始金额 (时间下下标 t = 0) r:利率利率n:期数期数单利公式单利公式单利公式单利公式17单利单利单利单利 ExampleExamplev假设投资者按假设投资者按 7% 的的单利单利把把 $1,000 存入存入银行银行 2年年. 第第 2 年年末的利息额是多少年年末的利息额是多少 ?vSI = P0 * (r) * (n) = $1,000(.07)(2) = $14018v单利单利 Future Value (SIFV) 是多少是多少?vSIFV = P0 + SI = $1,000 + $140= $1,140v终值终值FV 现在或将来的现在或将来的一笔钱一笔钱或或一系列一系列支付款按给定的利率计算，所得到的支付款按给定的利率计算，所得到的 在在某个某个更远的更远的未来时点的未来时点的 价值价值. .单利终值单利终值单利终值单利终值 SISIFVFV19v上页例子中，上页例子中，两年后的两年后的$1,140的的 单利单利现值现值 (SIPV) 是多少是多少?v现值现值PV 未来的未来的一笔钱或一系列支付款一笔钱或一系列支付款按给定的利率折算，所得到的按给定的利率折算，所得到的在在先先前时点前时点的价值的价值. . 这种折算称为这种折算称为折现折现。v在上页例子中，你在上页例子中，你先先前前存的存的 $1,000 原原始金额，就是始金额，就是两年后的两年后的$1,140 的的 单利单利现值现值PV ,即即在在先先前前时点时点的价值的价值 !单利现值单利现值单利现值单利现值 SISIPVPV20 假假设投投资者按者按 7% 的的 复利复利 把把$1,000 存入存入银行行 2 年，年，那么它的那么它的复利复利终值 是多少？是多少？复利终值复利终值复利终值复利终值 0 1 2$1,000FV27%21Basic DefinitionsvPresent Value - earlier money on a time line（相对于相对于其后的其后的 money 而言而言） vFuture Value - later money on a time line（相对于相对于其前的其前的 money 而言而言） vPV 与与 FV 是是 相对而言相对而言的的 ！vInterest rate - “exchange rate” between earlier money and later money22 FV1 = P0 (1+r)1 = $1,000 (1.07) = $1,070 复利复利 在第一年年末你得了在第一年年末你得了$70的利息的利息.这与这与单利单利利息相等利息相等. .复利终值公式复利终值公式复利终值公式复利终值公式1 1/2/223 FV1= P0 (1+r)1 = $1,000 (1.07) = $1,070 FV2 = FV1 (1+r)1 = P0 (1+r)(1+r) = $1,000(1.07)(1.07)= P0 (1+r)2= $1,000(1.07)2= $1,144 . 90第第 2 年，你比年，你比单利单利利息多得利息多得 $4 . 90 = $70 (0.07) 复利终值公式复利终值公式复利终值公式复利终值公式2 2/2/224Future Values: General FormulavFV = PV (1 + r)t FV = Future Value PV = Present Value r = period interest rate, expressed as a decimal t = number of periodsvFuture Value Interest Factor = (1 + r)t25 FV1 = P0(1+r)1FV2 = P0(1+r)2 , etc. F V 公式公式:FV n = P0 (1+r)n or FV n = P0 (FVIF r,n) 可查表可查表(FVIF r,n)可直观地记为可直观地记为(FV/PV, r , n)一般一般一般一般复利复利复利复利终值公式终值公式终值公式终值公式26FVIF r,n 可以查表如下所示：可以查表如下所示：复利复利复利复利终值因子表：终值因子表：终值因子表：终值因子表： FVIFFVIF rr,nn27FV2 = $1,000 (FVIF7%,2)= $1,000 (1.145)= $1,145 四舍五入四舍五入查表查表查表查表FVIFFVIF rr,nn计算计算计算计算28 按按 10% 的的复利复利把把$1000存入银行存入银行， 5年年后的终值是多少？后的终值是多少？Example Example 复利终值复利终值复利终值复利终值 0 1 2 3 4 5$1000FV510%29v查表查表 : FV5 = $1,000 (FVIF10%, 5)= $1,000 (1.611)= $1,611 四舍五入四舍五入解：解：解：解：用一般公式用一般公式: : FV n = P0 (1+r)n FV5 = $1,000 (1+ 0.10)5= $1,610. 5130Effects of Compounding - 1vConsider the previous exampleFV with simple interest = 1000 + 70 + 70 = 1140FV with compound interest = 1144.90The extra 4.90 comes from the interest of .07(70) = 4.90 earned on the first interest payment31Effects of Compounding - 2vSuppose you invest the $1000 from the previous example for 5 years. How much would you have?FV = 1000 (1.07)5 = 1402.55vSimple interest would have a future value of $1350, for a difference of $52.55vThe effect of compounding is small for a small number of periods, but increases as the number of periods increases. 32Figure 4.1: 终值、单利 vs. 复利33Effects of Compounding - 3vSuppose you had a deposit $1k at 7% interest 30 years ago. How much would the investment be worth today ?FV = 1k (1.07)30 7,612vWhat is the effect of compounding ?Simple interest = 1k + 30(1k)(.07) = 3100Compounding added $ 4,512 to the value of the investment34单利单利单利单利 v.s.v.s. 复利复利复利复利 ? ?Future Value (U.S. Dollars)35Effects of Compounding - 4vSuppose you had a deposit $1 at 7% interest 200 years ago. How much would the investment be worth today ?FV = 1 (1.07)200 752,932vWhat is the effect of compounding ?Simple interest = 1 + 200(1)(.07) = 15Compounding added $ 752,917 to the value of the investment36美国曼哈顿岛值多少钱美国曼哈顿岛值多少钱 ？-1/4v为了阐述复利在为了阐述复利在长时期长时期中的作用，不妨看看彼中的作用，不妨看看彼得麦纽因特和印第安人买卖曼哈顿岛的交易。得麦纽因特和印第安人买卖曼哈顿岛的交易。v16261626年，麦纽因特以价值仅仅年，麦纽因特以价值仅仅$24$24的商品和小的商品和小饰品，购买了整个曼哈顿岛。这个价格听起来饰品，购买了整个曼哈顿岛。这个价格听起来很便宜，但是印第安人也能从这个交易中获得很便宜，但是印第安人也能从这个交易中获得很不错的结果。很不错的结果。v为了弄明白其中原由，假设印第安人将卖岛所为了弄明白其中原由，假设印第安人将卖岛所得得 $24$24 以以1010的利率的利率进行进行投资投资。 这项投资到这项投资到今天今天 会值会值 多少钱呢多少钱呢 ? ? 请同学们猜一猜请同学们猜一猜 ！ 37美国曼哈顿岛值多少钱美国曼哈顿岛值多少钱 ？ -2/4v这项投资到今天这项投资到今天大约过了大约过了377377年。年。 当利率为当利率为1010时，时，$24$24能够在这段时期内大幅增长。到能够在这段时期内大幅增长。到底增长到多少呢底增长到多少呢? ? v该复利终值因子为：该复利终值因子为：(1(1+ + r r) )377377 = = 1.11.1377377 4 000 000 000 000 0004 000 000 000 000 000v那也就是那也就是4 4后面跟后面跟1515个零。终值因子带来的结个零。终值因子带来的结果是果是 $24$24x x4 4 后面再加后面再加1515个零，也就是个零，也就是 $96 $96 后后面加面加1515个零（个零（9.69.6亿亿亿亿）。）。v这么多钱可以买下整个美国这么多钱可以买下整个美国, ,乃至整个世界乃至整个世界. .对此同学们有何感想对此同学们有何感想 ？38美国曼哈顿岛值多少钱美国曼哈顿岛值多少钱 ？ -3/4v这当然是一个这当然是一个夸张夸张的例子。在的例子。在16261626年，要想进年，要想进行一笔利率为行一笔利率为1010且且377377年都不减息年都不减息的投资可的投资可不是件容易的事。不是件容易的事。v假设印第安人将卖岛所得假设印第安人将卖岛所得$24 $24 以以5 5的利率进的利率进行投资。行投资。 这项投资到今天这项投资到今天 会值会值 多少钱呢多少钱呢 ? ?v该复利终值因子为：该复利终值因子为：(1(1+ + r r) )377377 = = 1 1. .0505377377 1 00 000 0001 00 000 000v那也就是那也就是1 1后面跟后面跟 8 8个零个零。终值因子带来的结。终值因子带来的结果是果是 $24$24x x1 1 后面再加后面再加 8 8个零，也就是个零，也就是$24$24亿亿。39美国曼哈顿岛值多少钱美国曼哈顿岛值多少钱 ？ -4/4v假设印第安人将卖岛所得假设印第安人将卖岛所得$24$24 以以2 2. .5 5的的利率进行投资。利率进行投资。 这项投资到今天这项投资到今天 会值会值 多少钱呢多少钱呢 ? ?v该复利终值因子为：该复利终值因子为：(1(1+ + r) )377 377 = = 1 1. .025025377377 1111, ,03803840Effects of Compounding 小结vThe effect of compounding is small for a low interest rate, and for a small number of periods.vFor a given interest rate the longer the time period, the bigger the future value.vFor a given time period the higher the interest rate, the bigger the future value.41Figure 4.2: $1于不同利率、多期后的复利终值42 $1,000 按按 12% 复利，需要复利，需要多久成为多久成为$2,000 (近似近似) ?想使自己的财富倍增吗想使自己的财富倍增吗想使自己的财富倍增吗想使自己的财富倍增吗 ! !快捷计算方法快捷计算方法: : 72 法则法则43近似近似. N = 72 / i% 72 / 12% = 6 年年 精确计算是精确计算是 6.12 年年快捷计算方法：快捷计算方法：7272法则法则法则法则 $1,000 按按 12% 复利，需要复利，需要多久成为多久成为$2,000 (近似近似) ?44Quick Quiz: Part 1vWhat is the difference between simple interest and compound interest ?vSuppose you have $500 to invest and you believe that you can earn 6% per year over the next 12 years.How much would you have at the end of 12 years using simple interest ?How much would you have at the end of 12 years with compound interest ?45Part 2Present Value and Discounting46Present ValuesvHow much do I have to invest today to have some amount in the future?FV = PV(1 + r)tRearrange to solve for PV = FV / (1 + r)tvWhen we talk about discounting, we mean finding the present value of some future amount.vWhen we talk about the “value” of someth., we are talking about the present value unless we specifically indicate that we want the future value.47 假假设 2 年年后你需要后你需要$1,000. 那么那么按按 7%复利，你复利，你现在要存多少在要存多少钱 ？ 0 1 2$1,0007%PV1PV0复利现值复利现值复利现值复利现值48 PV0 = FV2 / (1+r)2 = $1,000 / (1.07)2 = FV2 / (1+r)2 = $873.44复利现值公式复利现值公式复利现值公式复利现值公式 0 1 2$1,0007%PV049 PV0 = FV1 / (1+r)1 = FV1(1+r)-1PV0 = FV2 / (1+r)2 = FV2(1+r)-2 P V 公式公式:PV0= FV n / (1+r)n = FV n (1+r)-nor PV0 = FV n (PVIF r,n) - 见下下页表格表格(PVIF r,n) 可直可直观地地记为 (PV/FV, r , n)一般复利现值公式一般复利现值公式一般复利现值公式一般复利现值公式etc.50PVIF r,n表如下所示：表如下所示：复利现值因子表：复利现值因子表：复利现值因子表：复利现值因子表：PVPVIFIF rr,nn51PV2 = $1,000 (PVIF7%,2)= $1,000 (.873)= $873 四舍五入四舍五入查复利现值因子表查复利现值因子表查复利现值因子表查复利现值因子表52 按按10% 的复利折算，的复利折算，5 年年后的后的 $1,000 的的现值是多少？是多少？Example -Example -复利现值复利现值复利现值复利现值 0 1 2 3 4 5$1,000PV010%53v用公式用公式:PV0 = FV n / (1+r)n PV0 = $1,000 / (1+ 0.10)5= $620. 92v查表表: PV0 = $1,000 (PVIF10%, 5)= $1,000 (.621)= $621. 00 四舍五入四舍五入解：解：解：解：54欺骗性广告欺骗性广告 1/2v最近，一些商家都这样宣称最近，一些商家都这样宣称 “来试来试一下我们的产品。如果你试了，我一下我们的产品。如果你试了，我们将为你的光顾支付们将为你的光顾支付 $100$100 ！”v你会发现他们给你的是一个在你会发现他们给你的是一个在2525年年之后之后支付给你支付给你 $100$100 的存款证书。的存款证书。v如果该存款的年利率是如果该存款的年利率是 1010的话，的话，今天他们真正给你多少钱呢今天他们真正给你多少钱呢 ? ?55欺骗性广告欺骗性广告 2/2v你真正得到的是在你真正得到的是在2525年后才能得到的年后才能得到的$100 $100 在今日的现值。在今日的现值。v假如折现率是每年假如折现率是每年1010，那么折现因子应，那么折现因子应是：是：1 1/ /1 1. .1 12525 = = 1 1/ /1010. .83478347 = = 0 0. .09230923 v这告诉你在折现率为这告诉你在折现率为1010时，时，2525年后的年后的 $1$1在在今天今天仅值仅值9 9美分美分多一点。多一点。 v由此可见，该促销仅能给你由此可见，该促销仅能给你0 0. .09230923x x$l00$l00 = = $9$9. .2323。可能这。可能这$9$9. .2323 已经足以吸引顾客，已经足以吸引顾客，但它的确不是但它的确不是 $l00$l00 。56PV One Period ExamplevSuppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today ?vPV = 10,000 / (1.07)1 = 9345.7957Present Values Example 2vYou want to begin saving for your daughters college education and you estimate that she will need $150,000 in 17 years. vIf you feel confident that you can earn 7% per year, how much do you need to invest today?PV = 150,000 / (1.07)17 = 47,486.1658Present Values Example 3vYour parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest ?vPV = 19,671.51 / (1.07)10 = 10,00059PV Important Relationship IvFor a given interest rate the longer the time period, the lower the present valuevWhat is the present value of $100 to be received in 5 years ? 10 years ? The discount rate is 10% 5 years: PV = 100 / (1.1)5 = 62.0910 years: PV = 100 / (1.1)10 = 38.5560PV Important Relationship IIv For a given time period the higher the interest rate, the smaller the present value.vWhat is the present value of $100 received in 5 years if the interest rate is 10%? Or 20%?Rate = 10%: PV = 100 / (1.1)5 = 62.09Rate = 20%: PV = 100 / (1.2)5 = 40.1961Figure 4.3 : $1于不同利率、折现期的现值62Quick Quiz: Part 2vWhat is the relationship between present value and future value?vSuppose you need $7,000 in 12 years. If you can earn 6% annually, how much do you need to invest today ?vIf you could invest the money at 12%, would you have to invest more or less than at 6% ?63Part 3Discount Rate64The Basic PV Equation - RefreshervPV = FV / (1 + r)tvThere are four parts to this equationPV, FV, r and tIf we know any three, we can solve for the fourth65Discount RatevOften we will want to know what the implied interest rate is in an investmentvRearrange the basic PV equation and solve for r FV = PV (1 + r)t r = (FV / PV)1/t 166Discount Rate Example 1vYou are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest ?v r = (1200 / 1000)1/5 1 = .03714 = 3.714%67Discount Rate Example 2vSuppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest ?v r = (20,000 / 10,000)1/6 1 = .122462 = 12.25%68Discount Rate Example 3vSuppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. vYou currently have $5000 to invest. What interest rate must you earn to have the $75,000 when you need it ?v r = (75,000 / 5,000)1/17 1 = .172688 = 17.27%69Quick Quiz : Part 3vWhat are some situations where you might want to compute the implied interest rate ?vSuppose you are offered the following two choices of investment. Which is better ?You can invest $1000 today with low risk and receive $2000 in 12 years. Or you can invest the $1000 in a bank account paying 6% for 12 years.What is the implied interest rate for the first choice and which investment should you choose?70Part 4Finding the Number of Periods71Finding the Number of PeriodsvStart with basic equation and solve for t (remember your logs)FV = PV (1 + r)t t = l n (FV / PV) / l n (1 + r)72Number of Periods Example 1vYou want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car ?v t = l n (20,000 / 15,000) / l n (1.1) = 3.02 years73Number of Periods Example 2vSuppose you want to buy a new house. You currently have $18,000 and you figure you need to have a 10% down payment plus an additional 5% of the loan value in closing costs. vIf the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs ?74Example 2 Continuedv How much do you need to have in the future?Down payment = .1(150,000) = 15,000Bank Loan = 150,000 15,000 = 135,000 Closing costs = .05 (135,000) = 6,750Total needed = 15,000 + 6,750 = 21,750v t = l n (21,750 / 18,000) / l n (1.075) = 2.62 years75Work the Web Examplevfinancial calculators are available online such as w3.cigna.com/consumer/tools/present_value.htmlvYou may go to Cignas web site and work the following example:You need $40,000 in 15 years. If you can earn 9.8% interest, how much do you need to invest today ?You should get $9,84176Quick Quiz: Part 4vWhen might you want to compute the number of periods?vSuppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $1000. If you can earn 12%, how long will you have to wait if you dont add any additional money ?77Part 5Multiple Cash Flows78Multiple Cash Flows FV Example 1vSuppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.9779Example 1 - Timeline510001234300349.92136.05485.9780Multiple Cash Flows PV Example 5.3vYou are offered an investment that will pay you $200 in one year, $400 in two years, $600 in three years, and $800 in four years. You can earn 12% on very similar investments. vWhat is the most you should pay for this one ?81Multiple Cash Flows PV Example 5.3vFind the PV of each cash flow and add them PV1 of Year 1 CF1: 200 / (1.12)1 = 178.57PV2 of Year 2 CF2: 400 / (1.12)2 = 318.88PV3 of Year 3 CF3: 600 / (1.12)3 = 427.07PV4 of Year 4 CF4: 800 / (1.12)4 = 508.41vTotal PV+ = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93vIf you can earn 12% on your money, this is the most you should be willing to pay.82Example 5.3 Timeline01234200400600800178.57318.88427.07508.411432.9383Decisions IvYour broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment ?vPV = 91.49, No - the broker is charging more than you would be willing to pay.84Saving For RetirementvYou are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12% ?vPV = 1084 .7185Saving For Retirement - Timeline0 1 2 39 40 41 42 43 44 0 0 0 0 25K 25K 25K 25K 25KNotice the year 0 cash flow = 0 (CF0 = 0)The cash flows years 1 39 are 0The cash flows years 40 44 are 25,00086Quick Quiz: Part 5vSuppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7%vWhat is the value of the cash flows at year 5?vWhat is the value of the cash flows today?vWhat is the value of the cash flows at year 3 ?87Part 6Annuities and Perpetuities88Annuities and PerpetuitiesvAnnuity finite series of equal payments that occur at regular intervalsIf each payment occurs at the end of each period, it is called an ordinary annuityIf each payment occurs at the beginning of each period, it is called an annuity duevPerpetuity infinite series of equal payments89年金年金年金年金年金：年金：相等间隔期相等间隔期（通常为年，但（通常为年，但是也可为其他间隔期，如：是也可为其他间隔期，如：季季、月、每两年，等）、月、每两年，等）的的 一系列一系列 相同相同 金额的金额的 收款收款 或或 付款付款. .90年金实例年金实例v 学生学生贷款款偿还v 汽汽车贷款款偿还v 保保险金金v 抵押抵押贷款款偿还v 养老养老储蓄蓄91年金例 解答见后v某人某人现年年51岁，希望在，希望在60岁退休后从退休后从61岁初初开始的开始的9年内每年年初能从年内每年年初能从银行得到行得到10,000元，那么他在从元，那么他在从52岁初初开始到开始到60岁初初的的9年内必年内必须每年年初每年年初存入存入银行多少行多少钱才行才行 ？ 年利率年利率6%v某人从某人从银行行贷款款100万万买房，年利率房，年利率为6%，若在，若在5年内年内还清，那么他每个清，那么他每个月必月必须还多少多少钱才行？才行？92v普通年金普通年金: : 若所求终值的时刻为若所求终值的时刻为最后一最后一笔年金笔年金所在的时刻，所在的时刻，oror 所求现值的时刻所求现值的时刻为为第一笔年金第一笔年金所在的时刻的所在的时刻的前前1 1期期，则称，则称该年金为该年金为普通年金普通年金 - 求求该年金的现值该年金的现值oror终值可查终值可查普通年金普通年金现值现值oror终值终值因子表因子表。v先付年金先付年金: : 若所求现值的时刻为若所求现值的时刻为第一笔第一笔年金年金所在的时刻，所在的时刻，or or 所求终值的时刻为所求终值的时刻为最后一笔年金最后一笔年金所在的时刻的所在的时刻的后后1 1期期，则称，则称该年金为该年金为先付年金先付年金 。年金分类年金分类年金分类年金分类930 1 2 3年末年末假定现值：假定现值：Parts of an AnnuityParts of an Annuity年末年末普通年金：普通年金： $100 $100 $100(第第1年年末年年末的的普通年金）普通年金）(第第1年年年初年初年初年初的的先付年金先付年金)相等相等现金流现金流 (第第2年年年初年初年初年初的的先付年金先付年金)(第第3年年年初年初年初年初的的先付年金先付年金)(第第2年年末年年末的的普通年金）普通年金）(第第3年年末年年末的的普通年金）普通年金） 若若视视第第1 1年末年末为为现值时刻，则现值时刻，则红年金红年金为先付为先付年金年金; ; 若若视视第第2 2年末年末为为终值时刻，则终值时刻，则青年金青年金为后付为后付年金年金 ! !假定终值：假定终值：94 对于对于任何任何年金，都可以年金，都可以 直接直接 查查普通年金普通年金因子表因子表 oror 套套普通年金公式普通年金公式，求其现值或终值，但须注意：求到的求其现值或终值，但须注意：求到的现值或终值现值或终值 在时间轴上的位置，即，在时间轴上的位置，即，所在的时刻所在的时刻 求到的终值在求到的终值在最后最后一笔年金一笔年金所在的时刻，求到的现值在所在的时刻，求到的现值在第一笔年金第一笔年金所在时刻所在时刻的前的前1 1时刻时刻。年金计算之要点年金计算之要点年金计算之要点年金计算之要点95FVA n = R(1+r)n-1 + R(1+r)n-2 + . . . + R(1+r)1 + R(1+r)0= R(1+r)n 1/r = RFVIFA r,n = RFVIF r,n 1/r普通年金普通年金普通年金普通年金 于第于第于第于第 n n年末的年末的年末的年末的终值终值终值终值 FVA FVA( (n n) )0 1 2 n n r R R RFVAFVA nR：每年现金流每年现金流年末年末. . . 年末年末?96FVAFVA3 3 = $1,000 (1.07)2 + $1k (1.07)1 + $1k (1.07)0 = $1,145 + $1,070 + $1,000 = $3,215$3,215普通年金普通年金普通年金普通年金终值终值终值终值 - FVA- FVA例例例例$1,000 $1,000 $1,0000 1 2 3 3$3,215 $3,215 = = FVA FVA3年末年末7%$1,070$1,145年末年末97FVA n = R (FVIFA r,n) FVA3 = $1,000 (FVIFA7%,3)= $1,000 (3.215) = $3,215查普通年金终值表计算查普通年金终值表计算查普通年金终值表计算查普通年金终值表计算 98FVAD n = R(1+r)n + R(1+r)n-1 + . + R(1+r)2 + R(1+r)1= FVA n (1+r) = FVA n+1 - R 先付年金先付年金先付年金先付年金 FVA FVADD（DDueue） R R R 1 2 n n FVADFVAD nR: 每年现金流年年初初r. . .年年初初年末年末 0 1 n-1n-1 n年末年末年末年末年年初初年末年末现在：现在：99 FVADFVAD3 3 = $1,000 (1.07)3 + $1k (1.07)2 + $1k (1.07)1 = $1,225 + $1,145 + $1,070 = $3,440$3,440先付年金先付年金先付年金先付年金 - FVAD- FVAD例例例例$1,000 $1,000 $1,000 $1,0700 1 2 2 3FVADFVAD3 3 = = $3,440 $3,440年末年末7%$1,225$1,145 1 2 3 3年初年初年初年初年初年初年末年末年末年末年末年末现在：现在：100FVAD n = R (FVIFA r,n)(1+r) FVAD3 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,44011-查查查查普通普通普通普通年金终值表年金终值表年金终值表年金终值表算算算算先付先付先付先付年金终值年金终值年金终值年金终值101FVAD n = R (FVIFA r,n +1 -1) FVAD3 = $1,000 (FVIFA7%,4 -1) = $1,000 (4.440 -1) = $3,44022-查查查查普通普通普通普通年金终值表年金终值表年金终值表年金终值表算算算算先付先付先付先付年金终值年金终值年金终值年金终值102PVA n = R/(1+r)1 + R/(1+r)2 + . + R/(1+r)n = R 1 (1+r)- n /r = R PVIFA r,n = R 1 PVIF r,n/r普通年金现值普通年金现值普通年金现值普通年金现值 - PVA- PVA R R R0 1 2 n nPVAPVA n nR: 每年现金流每年现金流年末年末r. . .年末年末年末年末年末年末?103 PVAPVA3 3 = $1,000/(1.07)1 + $1,000/(1.07)2 + $1,000/(1.07)3 = $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32普通年金现值普通年金现值普通年金现值普通年金现值 - PVA- PVA例例例例 0 1 2 3 3$1,000 $1,000 $1,000$2,624.32 $2,624.32 = = PVA PVA3年末年末7%$934.58$873.44 $816.30104PVA n = R (PVIFA r,n)PVA3 = $1,000 (PVIFA7%,3)= $1,000 (2.624) = $2,624查查查查普通年金现值表普通年金现值表普通年金现值表普通年金现值表计算计算计算计算105PVAD n = R/(1+r)0 + R/(1+r)1 + . + R/(1+r)n-1 = PVA n (1+r) = PVA n -1 + R先付年金现值先付年金现值先付年金现值先付年金现值 - PVAD- PVAD R R R1 2 n nPVADPVAD nR：每年现金流每年现金流年初年初r. . .年初年初年初年初现在：现在：106PVAD n = $1,000/(1.07)2 + $1,000/(1.07)1 + $1,000/(1.07)0 = $2,808.02先付年金先付年金先付年金先付年金 - PVAD- PVAD例例例例$1,000.00 $1,000 $1,0001 2 3 3 4PVADPVAD n n = $2,808.02$2,808.02年初年初7%$ 934.58$ 873.44现在：现在：年初年初年初年初年初年初107PVAD n = R (PVIFA r,n)(1+r) PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,80811-查查查查普通年金现值表普通年金现值表普通年金现值表普通年金现值表算算算算先付年金现值先付年金现值先付年金现值先付年金现值 108PVAD n = R (PVIFA r,n -1 + 1) PVAD3 = $1,000 (PVIFA7%,2 + 1)= $1,000 (1.808 + 1) = $2,80822 - -查查查查普通年金现值表普通年金现值表普通年金现值表普通年金现值表算算算算先付年金现值先付年金现值先付年金现值先付年金现值 109Annuities and Perpetuities vPerpetuity永续年金永续年金: PV = Constant / rvAnnuities:110解-1-年金例-1v某人某人51岁，希望在，希望在60岁退休后从退休后从61岁初初开开始的始的 9年年内每年年初能从内每年年初能从银行得到行得到 1 万万元元, 那么他必那么他必须在从在从52岁初初开始的开始的 9年年内每年内每年年初存入年初存入银行多少行多少钱 ？ 年利率年利率 6%v以以60岁初初为前后两个年金流的比前后两个年金流的比较时点点: A(FV/A, 6%, 9) = 10000 (PV/A, 6%, 9) (FV/A, 6%, 9) = (1+6%)9 (PV/A, 6%, 9) A = 10000/(1+ 6%)9 A 10000/1.6895 5919111解-2-年金例-1v某人某人51岁，希望在，希望在60岁退休后从退休后从61岁初初开开始的始的 9年年内每年年初能从内每年年初能从银行得到行得到 1 万万元，元，那么他必那么他必须在从在从52岁初初开始的开始的 9年年内每年内每年年初存入年初存入银行多少行多少钱 ？ 年利率年利率 6%v以以69岁初初为两个年金流的比两个年金流的比较时点点: A (F/A, 6%, 9)(1+6%)9=1W (F/A, 6%, 9) A = 1W万万an /(1+ 6%)9 A 1Wan /1.6895 5919112解-3-年金例-1v某人某人51岁，希望在，希望在60岁退休后从退休后从61岁初初开开始的始的 9年年内每年年初能从内每年年初能从银行得到行得到1 万万元，元，那么他必那么他必须在从在从52岁初初开始的开始的9年年内每年年内每年年初存入初存入银行多少行多少钱 ？ 年利率年利率 6%v以以51岁初初为两个年金流的比两个年金流的比较时点点: A(P/A,6%,9)=1W(P/A,6%,9)/(1+6%)9 A = 1W万万an /(1+ 6%)9 A 1Wan /1.6895 5919113解-1-年金例-1v某人某人55岁，希望在，希望在60岁退休后从退休后从61岁初初开开始的始的 9年年内每年年初能从内每年年初能从银行得到行得到1 万万元，元，那么他必那么他必须在从在从56岁初初开始的开始的 5年年内每年内每年年初存入年初存入银行多少行多少钱 ？ 年利率年利率 6% v以以60岁初初为前后两个年金流的比前后两个年金流的比较时点点: A(FV/A, 6%, 5) = 1万万 (PV/A, 6%, 9) A = 1万万(PV/A, 6%, 9) / (FV/A, 6%, 5) A = 1W万万an (6.8017) / 5.6371 A 12066114解-2-年金例-1v某人某人55岁，希望在，希望在60岁退休后从退休后从61岁初初开开始的始的 9年年内每年年初能从内每年年初能从银行得到行得到1 万万元，元，那么他必那么他必须在从在从56岁初初开始的开始的 5年内年内每年每年年初存入年初存入银行多少行多少钱 ？ 年利率年利率 6% v以以69岁初初为两个年金流的比两个年金流的比较时点点: A(F/A, 6%, 5)(1+6%)9 = 1W万万 (F/A, 6%, 9) A=1W (FV/A, 6%, 9)(1+6%)- 9/(F/A, 6%, 5) A =1万万 (PV/A, 6%, 9) / (FV/A, 6%, 5) A = 1W万万an (6.8017) / 5.6371 A 12066115解-3-年金例-1v某人某人55岁，希望在，希望在60岁退休后从退休后从61岁初初开开始的始的 9年年内每年年初能从内每年年初能从银行得到行得到1 万万元，元，那么他必那么他必须在从在从56岁初初开始的开始的 5年内年内每年每年年初存入年初存入银行多少行多少钱 ？ 年利率年利率 6%v以以55岁初初为两个年金流的比两个年金流的比较时点点: A(P/A, 6%, 5) = 1万万 (PV/A, 6%, 9) (1+6%)- 5 A = 1万万(P/A, 6%, 9) / (PV/A, 6%, 5)(1+6%)5 A = 1万万(P/A, 6%, 9) / (FV/A, 6%, 5) A 1万万(6.8017) / 5.6371 12066116年金例解-2v某人从某人从银行行贷款款100万万买房，房，年利率年利率为6% (.5% = .005 per month)，若在，若在 5年年内内还清，那么他每个月清，那么他每个月须还多少多少钱 ？v100万万 = A (P/A, .005, 60)v100万万 = A 1 1/1.00560 / .005 5000 = A 1 1/1.00560 5000 = A 1 1/1.34885 A = 19332.80117Buying a HousevYou are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. vHow much money will the bank loan you ? How much can you offer for the house ?118Buying a House - ContinuedvBank loanMonthly income = 36,000 / 12 = 3,000Maximum payment = .28(3,000) = 840PV = 8401 1/1.005360 / .005 = 140,105vTotal PriceClosing costs = .04 (140,105) = 5,604Down payment = 20,000 5604 = 14,396Total Price = 140,105 + 14,396 = 154,5011191.1.全面阅读问题全面阅读问题2.2.决定是决定是PV,PV,还是还是FVFV3.3.画一条时间轴画一条时间轴4.4.将现金流的箭头标示在时间轴上将现金流的箭头标示在时间轴上5.5.决定问题是单个的现金流、年金决定问题是单个的现金流、年金或混合现金流或混合现金流6.6.解决问题解决问题解决资金时间价值问题的步骤解决资金时间价值问题的步骤解决资金时间价值问题的步骤解决资金时间价值问题的步骤120如下现金流，按如下现金流，按10%折折现的的 PV 是多少是多少 ？混合现金流混合现金流混合现金流混合现金流 ExampleExample 0 1 2 3 4 5 $600 $600 $400 $400 $100PV010%年末年末121现金流现金流现金流现金流逐个逐个逐个逐个折算法折算法折算法折算法 0 1 2 3 4 5 $600 $600 $400 $400 $10010%$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09$1677.15 = PV0 122分组年金分组年金分组年金分组年金 (#1)(#1) 0 1 2 3 4 5 $600 $600 $400 $400 $100$600 $600 $400 $400 $10010%$1,041.60$1,041.60$ 573.57$ 573.57$ 62.10$ 62.10$1,677.27$1,677.27 = = PVPV0 0 查表如下：查表如下：查表如下：查表如下： $600(PVIFA10%,2) = $600(1.736) = $1,041.60$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57$100 (PVIF10%,5) = $100 (0.621) = $62.10123分组年金分组年金分组年金分组年金 (#2)(#2) 0 1 2 3 4 $400 $400 $400 $400$400 $400 $400 $400PV0 =$1677.30. 0 1 2 $200 $200$200 $200 0 1 2 3 4 5 $100$100$1,268.00$1,268.00$347.20$347.20$62.10$62.10+124例：v某企某企业购买一大型一大型设备，若，若货款款 现在在(0年末年末) 一次性付清需一次性付清需100万万元；也可采用分期付款，元；也可采用分期付款，从从第二年年末到第四年年末每年付款第二年年末到第四年年末每年付款40万万元。元。假假设资金利率金利率为10%，问该企企业应选择何种付何种付款方式？款方式？125方法 1：选 0年末为比较的时点分期付款好于一次付款分期付款好于一次付款126方法 2：选 1年末为比较的时点分期付款好于一次付款分期付款好于一次付款127方法 3：选 4年末为比较的时点分期付款好于一次付款分期付款好于一次付款128方法 4：比较等价年金 “A”分期付款好于一次付款分期付款好于一次付款129Part 7APR and EAR130公式公式: FV n,m = PV0 (1 + r/m)m nn : 年年头数数 m: 每年的复利次数每年的复利次数r : 名名义年利率年利率复利频率复利频率复利频率复利频率131 按年利率按年利率12%将将 $1,000 投资投资 2 Years：计息期是计息期是1 1年年 FV2 = 1,000(1+ .12/1)(1)(2) = 1,254.40计息期是计息期是半年半年FV2 = 1,000(1+ .12/2)(2)(2) = 1,262.48复利频率的影响复利频率的影响复利频率的影响复利频率的影响132季度季度：FV2 = 1,000(1+ .12/4)(4)(2) = 1,266.77月月： FV2 = 1,000(1+ .12/12)(12)(2) = 1,269.73天天：FV2 = 1,000(1+.12/365)(365)(2) = 1,271.20复利频率的影响复利频率的影响复利频率的影响复利频率的影响133 10%简单年利率下计息次数简单年利率下计息次数 与与 有效年利率有效年利率EAR之间的关系之间的关系134 设一年中复利次数一年中复利次数为m, 名名义年利率年利率APR 为 r ，则有效年利率有效年利率EAR 为：(1 + r / m )m - 1 有效年利率有效年利率有效年利率有效年利率er - 1135 某公司在银行某公司在银行 有有 $1,000 CD (Certificates of Deposit)，名义年利率是名义年利率是 6%，一个季度计一个季度计息一次息一次，问，问： EAR = ?EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14% ! 例：有效年利率例：有效年利率例：有效年利率例：有效年利率136 某公司在银行有某公司在银行有 $1,000(PV) CD，名义年利率是名义年利率是6%，一个季度计息一次一个季度计息一次, ,问：问： EAR = ?FV1 = PV ( 1 + 6% / 4 )4 FV1 = PV ( 1 + EAR )1 1 + EAR = ( 1 + 6% / 4 )4EAR = ( 1 + 6% / 4 )4 - 1 例证：有效年利率例证：有效年利率例证：有效年利率例证：有效年利率137设一年中复利次数为设一年中复利次数为 m， 名义年利率为名义年利率为 r ，问： EAR = ?FV1 = PV ( 1 + r / m )m FV1 = PV ( 1 + EAR )1 1 + EAR = ( 1 + r / m )mEAR = ( 1 + r / m )m - 1 公式证明：有效年利率公式证明：有效年利率公式证明：有效年利率公式证明：有效年利率 EAREAR138Effective Annual Rate (EAR)vThis is the actual rate paid (or received) after accounting for compounding that occurs during the yearvIf you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.139Annual Percentage RatevThis is the annual rate that is quotedvBy definition APR = Period Rate times the Number of Periods Per YearvConsequently, to get the period rate we rearrange the APR equation:Period Rate = APR / number of periods per yearvYou should NEVER divide EAR by the number of periods per year - it will NOT give you the Period Rate140Computing APRsvWhat is the APR if the monthly rate is .5%?.5 (12) = 6%vWhat is the APR if the semiannual rate is .5%?.5 (2) = 1%vWhat is the monthly rate if the APR is 12% with monthly compounding?12 / 12 = 1%Can you divide the above APR by 2 to get the semiannual rate? NO! You need an APR based on semiannual compounding to find the semiannual rate.141Things to RemembervYou ALWAYS need to make sure that the interest rate and the time period match.If you are looking at annual periods, you need an annual rate.If you are looking at monthly periods, you need a monthly rate.vIf you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly142Computing EARs Example 1vSuppose you can earn 1% per month on $1 invested today.vWhat is the APR ? 1%(12 Month) = 12%vHow much are you effectively earning? FV1 = 1 (1.01)12 = 1.1268vEAR = (1.1268 1) / 1 = .1268 = 12.68%143Computing EARs Example 2vSuppose if you put it in another account, you earn 3% per quarter.vWhat is the APR ? 3%(4 Quarter) = 12%vHow much are you effectively earning? FV1 = 1(1.03)4 = 1.1255vEAR = (1.1255 1) / 1 = .1255 = 12.55%144EAR - FormulaRemember the APR is the quoted rate145Decisions IIvYou are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use ?EAR 1 = (1 + .0525/365)365 1 = 5.39%EAR 2 = (1 + .053/2)2 1 = 5.37%vWhich account should you choose ?146Decisions II ContinuedvLets verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?First Account:Daily rate = .0525 / 365 = .00014383562FV 1 = 100(1.00014383562)365 = 105.39Second Account:Semiannual rate = .053 / 2 = .0265FV 1 = 100(1.0265)2 = 105.37vYou have more money in the first account.147Computing APRs from EAR s vIf you have an effective rate, how can you compute the APR ? vRearrange the EAR equation and you get:148APR - ExamplevSuppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay ?149Computing Payments with APRsvSuppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments. The entire computer system costs $4000. The loan period is for 2 years and the interest rate is 18% with monthly compounding. What is your monthly payment? Monthly rate = .18 / 12 = .015 Number of months = 2 (12) = 24 4000 = A 1 1 / 1.01524 / .015 A = 4000 (A/P, .015, 24) = 199.70150Future Values with Monthly CompoundingvSuppose you deposit $50 a month into an account that has an APR of 12%, based on monthly compounding. How much will you have in the account in 10 years ?Monthly rate = .12 / 12 = .01Number of months = 10(12) = 120FV = 501.01120 1 / .01 = 11,502151Present Value with Daily CompoundingvYou need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?Daily rate = .055 / 365 = .00015068493Number of days = 3(365) = 1095PV = 15,000 / (1.00015068493)1095 = 12,718.56152Quick Quiz: Part 7vWhat is the definition of an APR ?vWhat is the effective annual rate(EAR) ?vWhich rate should you use to compare alternative investments or loans ?vWhich rate do you need to use in the time value of money calculations ?153Part 8Loan Types & Loan Amortization154Pure Discount Loans Example 5.11vTreasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.vIf a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7%, how much will the bill sell for in the market ?PV = 10,000 / 1.07 = 9345.79155Interest Only Loan - ExamplevA 5-year, interest only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.vWhat would the stream of cash flows be ?Years 1 4: Interest payments of .07(10,000) = 700Year 5: Interest + principal = 10,700vThis cash flow stream is similar to the cash flows on corporate bonds and we will talk about them in greater detail later.156 1.计算计算 每期偿还额每期偿还额. 2.计算第计算第 t t 期偿还的期偿还的 利息利息. = (第第 t-1t-1 期的期的 贷款余额贷款余额) x (APR/m期数期数/年年) 3.计算第计算第 t t 期期 偿还的偿还的本金本金. = (每期偿还额每期偿还额 - - 第第 2 步的步的利息利息) 4.计算第计算第 t t 期的期的 贷款余额贷款余额. = (第第t-1t-1期的贷款余额期的贷款余额 - - 第第 3步的步的本金偿还本金偿还) 5.从第从第 2 步起循环步起循环.计算步骤：分期偿还贷款计算步骤：分期偿还贷款计算步骤：分期偿还贷款计算步骤：分期偿还贷款157 银行贷款银行贷款 $10,000，年利率年利率12%. 分分 5年年 等额偿还等额偿还.Step 1:每年偿还额每年偿还额 R PV0 = R (P/A, r, n)$10,000 = R (PVIFA12%,5)$10,000 = R (3.605)R = $10,000 / 3.605 = $2,774例：分期偿还贷款例：分期偿还贷款例：分期偿还贷款例：分期偿还贷款158例：分期偿还贷款例：分期偿还贷款例：分期偿还贷款例：分期偿还贷款- -年利率年利率 12%12%Last Payment Slightly Higher Due to Rounding159分期偿还的好处分期偿还的好处2.未未偿还债务 - The quantity of outstanding debt may be used in day-to-day activities of the firm.1. 利息费用利息费用 - 利息费用可以减少利息费用可以减少公司的应税收入公司的应税收入. .160Amortized Loan with Fixed Payment - ExamplevEach payment covers the interest expense plus reduces principalvConsider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5000.vWhat is the annual payment?v5000 = A(PV/A,8%,4) = A 1- 1/1.084 /.08vA = 5000 (A/PV, 8%, 4) = 1509.60161Amortization Table for ExampleYearBeg. BalanceTotal PaymentInterest PaidPrincipal PaidEnd. Balance15,000.001509.60400.001109.603890.4023890.401509.60311.231198.372692.0332692.031509.60215.361294.241397.7941397.791509.60111.821397.78 .01Total6038.401038.414999.99162Example: Work the WebvSeveral web sites have calculators that will prepare amortization tables quicklyvOne such site is www.LendingTree.comvYou may go to the site and enter the following information:Loan amount = 20,000Term = 10 yearsInterest rate = 7.625%What is the monthly payment?Using the calculator you will get $238.71Clicking “View Report” will give you the amortization table163Quick Quiz: Part 8vWhat is a pure discount loan? What is a good example of a pure discount loan ?vWhat is an interest only loan? What is a good example of an interest only loan ?vWhat is an amortized loan? What is a good example of an amortized loan ?164教 材美美 斯蒂芬斯蒂芬 A.罗斯斯 (Stephen A. Ross)伦道夫道夫 W.韦斯特菲斯特菲尔德德(Randolph W.Westerfield)布拉德福德布拉德福德 D.乔丹丹(Braford D.Jordan)著著,公司理公司理财精要精要，第，第2版，版，张建平建平译，北京：人民北京：人民邮电出版社，出版社，2003165参考书 v英英理理查德德 A.布雷利布雷利(Richard A. Brealey) and 美美斯斯图尔特特 C.迈尔斯斯 (Stewart C. Myers) 著著, 公司公司财务原理原理，第，第7版，方曙版，方曙红等等译，北京：北京：机械工机械工业出版社，出版社，2004v加加 西安大略大学毅西安大略大学毅伟管理学院管理学院 韦纳礼礼 ( Larry Wynant ) 编，国，国际通用通用MBA教材配套案例教材配套案例: 公公司司财务案例案例, 张梵等梵等译, 北京北京: 机械工机械工业出版社出版社, 1999