公司理财-未来现金流量的价值评估

1、1,公 司 理 财,2,斯蒂芬 A.罗斯,史蒂芬罗斯先生目前是麻省理工学院斯隆管理学院财务经济学教授。作为在财务和经济领域著述最为丰富的作者之一，罗斯教授以他在发展套利价格理论上所做的工作，以及通过研究信息折射理论、代理理论、利率期限结构理论和其他诸多领域所做出的大量贡献，成为备受称道的著名学者。罗斯曾任美国金融协会主席，现在担任数家学术型和实战型杂志的副主编。他还是CalTech的受托人，大学退休股权基金和GenRe公司的董事。此外，他还兼任劳尔 Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount

2、 rate is 7% What is the value of the cash flows at year 5? What is the value of the cash flows today? What is the value of the cash flows at year 3 ?,87,Part 6,Annuities and Perpetuities,88,Annuities and Perpetuities,Annuity finite series of equal payments that occur at regular intervals If each pay

3、ment occurs at the end of each period, it is called an ordinary annuity If each payment occurs at the beginning of each period, it is called an annuity due Perpetuity infinite series of equal payments,89,年金,年金：相等间隔期（通常为年，但是也可为其他间隔期，如：季、月、每两年，等）的 一系列 相同 金额的 收款 或 付款.,90,年金实例,学生贷款偿还 汽车贷款偿还 保险金 抵押贷款偿还 养

4、老储蓄,91,年金例 解答见后,某人现年51岁，希望在60岁退休后从61岁初开始的9年内每年年初能从银行得到10,000元，那么他在从52岁初开始到60岁初的9年内必须每年年初存入银行多少钱才行 ？ 年利率6% 某人从银行贷款100万买房，年利率为6%，若在5年内还清，那么他每个月必须还多少钱才行？,92,普通年金: 若所求终值的时刻为最后一笔年金所在的时刻，or 所求现值的时刻为第一笔年金所在的时刻的前1期，则称该年金为普通年金 - 求该年金的现值or终值可查普通年金现值or终值因子表。 先付年金: 若所求现值的时刻为第一笔年金所在的时刻，or 所求终值的时刻为最后一笔年金所在的时刻的后1期

5、，则称该年金为先付年金 。,年金分类,93,0 1 2 3年末,假定现值：,Parts of an Annuity,年末,普通年金： $100 $100 $100,(第1年年末 的普通年金）,(第1年年初 的先付年金),相等现金流,(第2年年初 的先付年金),(第3年年初 的先付年金),(第2年年末 的普通年金）,(第3年年末 的普通年金）,若视第1年末为现值时刻，则红年金为先付年金; 若视第2年末为终值时刻，则青年金为后付年金 !,假定终值：,94,对于任何年金，都可以 直接 查普通年金因子表 or 套普通年金公式，求其现值或终值，但须注意：求到的现值或终值 在时间轴上的位置，即，所在的时刻

6、 求到的终值在最后一笔年金所在的时刻，求到的现值在第一笔年金所在时刻的前1时刻。,年金计算之要点,95,FVA 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 于第 n年末的终值 FVA(n),0 1 2 n,r,FVA n,R：每年现金流,年末,. . .,年末,?,96,FVA3 = $1,000 (1.07)2 + $1k (1.07)1 + $1k (1.07)0 = $1,145 + $1,070 + $1,000 = $3,21

7、5,普通年金终值 - FVA例,$1,000 $1,000 $1,000,0 1 2 3,$3,215 = FVA3,年末,7%,$1,070,$1,145,年末,97,FVA 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,98,FVAD 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 FVAD（Due）,R R R,1 2 n,FVAD n,R: 每年现金流,年初,r

8、,. . .,年初,年末,0 1 n-1 n,年末,年末,年初,年末,现在：,99,FVAD3 = $1,000 (1.07)3 + $1k (1.07)2 + $1k (1.07)1 = $1,225 + $1,145 + $1,070 = $3,440,先付年金 - FVAD例,$1,000 $1,000 $1,000 $1,070,0 1 2 3,FVAD3 = $3,440,年末,7%,$1,225,$1,145,1 2 3,年初,年初,年初,年末,年末,年末,现在：,100,FVAD n = R (FVIFA r,n)(1+r) FVAD3 = $1,000 (FVIFA7%,3)(

9、1.07) = $1,000 (3.215)(1.07) = $3,440,1-查普通年金终值表算先付年金终值,101,FVAD n = R (FVIFA r,n +1 -1) FVAD3 = $1,000 (FVIFA7%,4 -1) = $1,000 (4.440 -1) = $3,440,2-查普通年金终值表算先付年金终值,102,PVA 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 - PVA,R R R,0 1 2 n,PVA n,R: 每年

10、现金流,年末,r,. . .,年末,年末,年末,?,103,PVA3 = $1,000/(1.07)1 + $1,000/(1.07)2 + $1,000/(1.07)3 = $934.58 + $873.44 + $816.30 = $2,624.32,普通年金现值 - PVA例,0 1 2 3,$1,000 $1,000 $1,000,$2,624.32 = PVA3,年末,7%,$934.58 $873.44 $816.30,104,PVA n = R (PVIFA r,n) PVA3 = $1,000 (PVIFA7%,3)= $1,000 (2.624) = $2,624,查普通年金

11、现值表计算,105,PVAD 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 - PVAD,R R R,1 2 n,PVAD n,R：每年现金流,年初,r,. . .,年初,年初,现在：,106,PVAD n = $1,000/(1.07)2 + $1,000/(1.07)1 + $1,000/(1.07)0 = $2,808.02,先付年金 - PVAD例,$1,000.00 $1,000 $1,000,1 2 3 4,PVAD n = $2,808.02,年初,7%,$ 934.58,

12、$ 873.44,现在：,年初,年初,年初,107,PVAD n = R (PVIFA r,n)(1+r) PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808,1-查普通年金现值表算先付年金现值,108,PVAD n = R (PVIFA r,n -1 + 1) PVAD3 = $1,000 (PVIFA7%,2 + 1)= $1,000 (1.808 + 1) = $2,808,2 -查普通年金现值表算先付年金现值,109,Annuities and Perpetuities,Perpetuity永续年金: PV

13、= Constant / r Annuities:,110,解-1-年金例-1,某人51岁，希望在60岁退休后从61岁初开始的 9年内每年年初能从银行得到 1 万元, 那么他必须在从52岁初开始的 9年内每年年初存入银行多少钱 ？ 年利率 6% 以60岁初为前后两个年金流的比较时点: 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 5919,111,解-2-年金例-1,某人51岁，希望在60岁退休后从61岁初开始的 9年内每年

14、年初能从银行得到 1 万元，那么他必须在从52岁初开始的 9年内每年年初存入银行多少钱 ？ 年利率 6% 以69岁初为两个年金流的比较时点: A (F/A, 6%, 9)(1+6%)9=1W (F/A, 6%, 9) A = 1W万an /(1+ 6%)9 A 1Wan /1.6895 5919,112,解-3-年金例-1,某人51岁，希望在60岁退休后从61岁初开始的 9年内每年年初能从银行得到1 万元，那么他必须在从52岁初开始的9年内每年年初存入银行多少钱 ？ 年利率 6% 以51岁初为两个年金流的比较时点: A(P/A,6%,9)=1W(P/A,6%,9)/(1+6%)9 A = 1W

15、万an /(1+ 6%)9 A 1Wan /1.6895 5919,113,解-1-年金例-1,某人55岁，希望在60岁退休后从61岁初开始的 9年内每年年初能从银行得到1 万元，那么他必须在从56岁初开始的 5年内每年年初存入银行多少钱 ？ 年利率 6% 以60岁初为前后两个年金流的比较时点: A(FV/A, 6%, 5) = 1万 (PV/A, 6%, 9) A = 1万(PV/A, 6%, 9) / (FV/A, 6%, 5) A = 1W万an (6.8017) / 5.6371 A 12066,114,解-2-年金例-1,某人55岁，希望在60岁退休后从61岁初开始的 9年内每年年初

16、能从银行得到1 万元，那么他必须在从56岁初开始的 5年内每年年初存入银行多少钱 ？ 年利率 6% 以69岁初为两个年金流的比较时点: A(F/A, 6%, 5)(1+6%)9 = 1W万 (F/A, 6%, 9) A=1W (FV/A, 6%, 9)(1+6%)- 9/(F/A, 6%, 5) A =1万 (PV/A, 6%, 9) / (FV/A, 6%, 5) A = 1W万an (6.8017) / 5.6371 A 12066,115,解-3-年金例-1,某人55岁，希望在60岁退休后从61岁初开始的 9年内每年年初能从银行得到1 万元，那么他必须在从56岁初开始的 5年内每年年初存

17、入银行多少钱 ？ 年利率 6% 以55岁初为两个年金流的比较时点: A(P/A, 6%, 5) = 1万 (PV/A, 6%, 9) (1+6%)- 5 A = 1万(P/A, 6%, 9) / (PV/A, 6%, 5)(1+6%)5 A = 1万(P/A, 6%, 9) / (FV/A, 6%, 5) A 1万(6.8017) / 5.6371 12066,116,年金例解-2,某人从银行贷款100万买房，年利率为6% (.5% = .005 per month)，若在 5年内还清，那么他每个月须还多少钱 ？ 100万 = A (P/A, .005, 60) 100万 = A 1 1/1.

18、00560 / .005 5000 = A 1 1/1.00560 5000 = A 1 1/1.34885 A = 19332.80,117,Buying a House,You 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

19、 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. How much money will the bank loan you ? How much can you offer for the house ?,118,Buying a House - Continued

20、,Bank loan Monthly income = 36,000 / 12 = 3,000 Maximum payment = .28(3,000) = 840 PV = 8401 1/1.005360 / .005 = 140,105 Total Price Closing costs = .04 (140,105) = 5,604 Down payment = 20,000 5604 = 14,396 Total Price = 140,105 + 14,396 = 154,501,119,1.全面阅读问题 2.决定是PV,还是FV 3.画一条时间轴 4.将现金流的箭头标示在时间轴上

21、5.决定问题是单个的现金流、年金或混合现金流 6.解决问题,解决资金时间价值问题的步骤,120,如下现金流，按10%折现的 PV 是多少 ？,混合现金流 Example,0 1 2 3 4 5,$600 $600 $400 $400 $100,PV0,10%,年末,121,现金流逐个折算法,0 1 2 3 4 5,$600 $600 $400 $400 $100,10%,$545.45 $495.87 $300.53 $273.21 $ 62.09,$1677.15 = PV0,122,分组年金 (#1),0 1 2 3 4 5,$600 $600 $400 $400 $100,10%,$1,

22、041.60 $ 573.57 $ 62.10,$1,677.27 = PV0 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) = $62.10,123,分组年金 (#2),0 1 2 3 4,$400 $400 $400 $400,PV0 = $1677.30.,0 1 2,$200 $200,0 1 2 3 4 5,$100,$1,268.00,$347.20,

23、$62.10,+,+,124,例：,某企业购买一大型设备，若货款 现在(0年末) 一次性付清需100万元；也可采用分期付款，从第二年年末到第四年年末每年付款40万元。假设资金利率为10%，问该企业应选择何种付款方式？,125,方法 1：选 0年末为比较的时点,分期付款好于一次付款,126,方法 2：选 1年末为比较的时点,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,128,方法 4：比较等价年金 “A”,分期付款好于一次付款,129,Part 7,APR and EAR,130,公式: FV n,m = PV0 (1 + r/m)m n n : 年头数

24、 m: 每年的复利次数 r : 名义年利率,复利频率,131,按年利率12%将 $1,000 投资 2 Years： 计息期是1年 FV2 = 1,000(1+ .12/1)(1)(2) = 1,254.40 计息期是半年FV2 = 1,000(1+ .12/2)(2)(2) = 1,262.48,复利频率的影响,132,季度：FV2 = 1,000(1+ .12/4)(4)(2) = 1,266.77 月： FV2 = 1,000(1+ .12/12)(12)(2) = 1,269.73 天：FV2 = 1,000(1+.12/365)(365)(2) = 1,271.20,复利频率的影响,

25、133,10%简单年利率下计息次数 与 有效年利率EAR之间的关系,134,设一年中复利次数为m, 名义年利率APR 为 r ，则 有效年利率EAR 为： (1 + r / m )m - 1 ,有效年利率,er - 1,135,某公司在银行 有 $1,000 CD (Certificates of Deposit)，名义年利率是 6%，一个季度计息一次，问： EAR = ? EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14% !,例：有效年利率,136,某公司在银行有 $1,000(PV) CD，名义年利率是6%，一个季度计息一次,问

26、： EAR = ? FV1 = PV ( 1 + 6% / 4 )4 FV1 = PV ( 1 + EAR )1 1 + EAR= ( 1 + 6% / 4 )4 EAR= ( 1 + 6% / 4 )4 - 1,例证：有效年利率,137,设一年中复利次数为 m， 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 )m EAR= ( 1 + r / m )m - 1,公式证明：有效年利率 EAR,138,Effective Annual Rate (EAR),This

27、 is the actual rate paid (or received) after accounting for compounding that occurs during the year If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.,139,Annual Percentage Rate,This is the annual rate that i

28、s quoted By definition APR = Period Rate times the Number of Periods Per Year Consequently, to get the period rate we rearrange the APR equation: Period Rate = APR / number of periods per year You should NEVER divide EAR by the number of periods per year - it will NOT give you the Period Rate,140,Co

29、mputing APRs,What is the APR if the monthly rate is .5%? .5 (12) = 6% What is the APR if the semiannual rate is .5%? .5 (2) = 1% What 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 o

30、n semiannual compounding to find the semiannual rate.,141,Things to Remember,You 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. If you have an AP

31、R 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 monthly,142,Computing EARs Example 1,Suppose you can earn 1% per month on $1 invested today. What is the APR ? 1%(12 Month) = 12% How much are you

32、effectively earning? FV1 = 1 (1.01)12 = 1.1268 EAR = (1.1268 1) / 1 = .1268 = 12.68%,143,Computing EARs Example 2,Suppose if you put it in another account, you earn 3% per quarter. What is the APR ? 3%(4 Quarter) = 12% How much are you effectively earning? FV1 = 1(1.03)4 = 1.1255 EAR = (1.1255 1) /

33、1 = .1255 = 12.55%,144,EAR - Formula,Remember the APR is the quoted rate,145,Decisions II,You 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 +

34、 .053/2)2 1 = 5.37% Which account should you choose ?,146,Decisions II Continued,Lets 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 = .00014383562 FV 1 = 100(1.00014383562)365 = 105.39 Second Ac

35、count: Semiannual rate = .053 / 2 = .0265 FV 1 = 100(1.0265)2 = 105.37 You have more money in the first account.,147,Computing APRs from EAR s,If you have an effective rate, how can you compute the APR ? Rearrange the EAR equation and you get:,148,APR - Example,Suppose you want to earn an effective

36、rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay ?,149,Computing Payments with APRs,Suppose 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.

37、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) = 199.70,150,Future Values with Monthly Compounding,Suppose you dep

38、osit $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 = .01 Number of months = 10(12) = 120 FV = 501.01120 1 / .01 = 11,502,151,Present Value with Daily Compounding,You need $15,000 in 3 year

39、s 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 = .00015068493 Number of days = 3(365) = 1095 PV = 15,000 / (1.00015068493)1095 = 12,718.56,152,Quick Quiz: Part 7,What is the de

40、finition of an APR ? What is the effective annual rate(EAR) ? Which rate should you use to compare alternative investments or loans ? Which rate do you need to use in the time value of money calculations ?,153,Part 8,Loan Types & Loan Amortization,154,Pure Discount Loans Example 5.11,Treasury bills

41、are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. If 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.

42、79,155,Interest Only Loan - Example,A 5-year, interest only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually. What would the stream of cash flows be ? Years 1 4: Interest payments of .07(10,000) = 700 Year 5: Interest + principal = 10,700 This cash flow stream

43、 is similar to the cash flows on corporate bonds and we will talk about them in greater detail later.,156,1.计算 每期偿还额. 2.计算第 t 期偿还的 利息. = (第 t-1 期的 贷款余额) x (APR/m期数/年) 3.计算第 t 期 偿还的本金. = (每期偿还额 - 第 2 步的利息) 4.计算第 t 期的 贷款余额. = (第t-1期的贷款余额 - 第 3步的本金偿还) 5.从第 2 步起循环.,计算步骤：分期偿还贷款,157,银行贷款 $10,000，年利率12%. 分 5年 等额偿还. Step 1: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,158,例：分期偿还贷款-年利率 12%,Last Payment Slightly Higher Due to Rounding,159