1、1,公 司 理 財,2,教 材,美 斯蒂芬 A.羅斯 (Stephen A. Ross) 倫道夫 W.韋斯特菲爾德(Randolph W.Westerfield) 布拉德福德 D.喬丹(Braford D.Jordan) 著,公司理財精要，第2版，張建平譯， 北京：人民郵電出版社，2003,3,斯蒂芬 A.羅斯,史蒂芬羅斯先生目前是麻省理工學(xué)院斯隆管理學(xué)院財務(wù)經(jīng)濟學(xué)教授。作為在財務(wù)和經(jīng)濟領(lǐng)域著述最為豐富的作者之一，羅斯教授以他在發(fā)展套利價格理論上所做的工作，以及通過研究信息折射理論、代理理論、利率期限結(jié)構(gòu)理論和其他諸多領(lǐng)域所做出的大量貢獻，成為備受稱道的著名學(xué)者。羅斯曾任美國金融協(xié)會主席，現(xiàn)在

2、擔(dān)任數(shù)家學(xué)術(shù)型和實戰(zhàn)型雜志的副主編。他還是CalTech的受托人，大學(xué)退休股權(quán)基金和GenRe公司的董事。此外，他還兼任勞爾 Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount 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 ?,89,Part 6,Annu

3、ities and Perpetuities,90,Annuities and Perpetuities,Annuity finite series of equal payments that occur at regular intervals If each payment 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 Perpetui

4、ty infinite series of equal payments,91,年金,年金：相等間隔期（通常為年，但 是也可為其他間隔期，如： 季 、月、每兩年，等）的 一系列 相同 金額的 收款 或 付款.,92,年金實例,學(xué)生貸款償還 汽車貸款償還 保險金 抵押貸款償還 養(yǎng)老儲蓄,93,年金例 解答見后,某人現(xiàn)年51歲，希望在60歲退休后從61歲初開始的9年內(nèi)每年年初能從銀行得到10,000元，那么他在從52歲初開始到60歲初的9年內(nèi)必須每年年初存入銀行多少錢才行 ？ 年利率6% 某人從銀行貸款100萬買房，年利率為6%，若在5年內(nèi)還清，那么他每個月必須還多少錢才行？,94,普通年金: 若

5、所求終值的時刻為最后一筆年金所在的時刻，or 所求現(xiàn)值的時刻為第一筆年金所在的時刻的前1期，則稱該年金為普通年金 - 求該年金的現(xiàn)值or終值可查普通年金現(xiàn)值or終值因子表。 先付年金: 若所求現(xiàn)值的時刻為第一筆年金所在的時刻，or 所求終值的時刻為最后一筆年金所在的時刻的后1期，則稱該年金為先付年金 。,年金分類,95,0 1 2 3年末,假定現(xiàn)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),相等現(xiàn)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

6、年年末 的普通年金）,若視第1年末為現(xiàn)值時刻，則紅年金為先付年金; 若視第2年末為終值時刻，則青年金為后付年金 !,假定終值：,96,對于任何年金，都可以 直接 查普通年金因子表 or 套普通年金公式，求其現(xiàn)值或終值，但須注意：求到的現(xiàn)值或終值 在時間軸上的位置，即，所在的時刻 求到的終值在最后一筆年金所在的時刻，求到的現(xiàn)值在第一筆年金所在時刻的前1時刻。,年金計算之要點,97,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

7、n年末的 終值 FVA(n),0 1 2 n,r,FVA n,R：每年現(xiàn)金流,年末,. . .,年末,?,98,FVA3 = $1,000 (1.07)2 + $1k (1.07)1 + $1k (1.07)0 = $1,145 + $1,070 + $1,000 = $3,215,普通年金終值 - FVA例,$1,000 $1,000 $1,000,0 1 2 3,$3,215 = FVA3,年末,7%,$1,070,$1,145,年末,99,FVA n = R (FVIFA r,n) FVA3 = $1,000 (FVIFA7%,3) = $1,000 (3.215) = $3,215,查

8、普通年金終值表計算,100,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: 每年現(xiàn)金流,年初,r,. . .,年初,年末,0 1 n-1 n,年末,年末,年初,年末,現(xiàn)在：,101,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 $

9、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,現(xiàn)在：,102,FVAD n = R (FVIFA r,n)(1+r) FVAD3 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,440,1-查普通年金終值表算先付年金終值,103,FVAD n = R (FVIFA r,n +1 -1) FVAD3 = $1,000 (FVIFA7%,4 -1) = $1,000 (4.440 -1) = $3,440,2-查普通年金終值

10、表算先付年金終值,104,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,普通年金現(xiàn)值 - PVA,R R R,0 1 2 n,PVA n,R: 每年現(xiàn)金流,年末,r,. . .,年末,年末,年末,?,105,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,普通年金現(xiàn)值 - PVA例,0 1 2 3,$1,000

11、$1,000 $1,000,$2,624.32 = PVA3,年末,7%,$934.58 $873.44 $816.30,106,PVA n = R (PVIFA r,n) PVA3 = $1,000 (PVIFA7%,3) = $1,000 (2.624) = $2,624,查普通年金現(xiàn)值表計算,107,PVAD n = R/(1+r)0 + R/(1+r)1 + . + R/(1+r)n-1 = PVA n (1+r) = PVA n -1 + R,先付年金現(xiàn)值 - PVAD,R R R,1 2 n,PVAD n,R：每年現(xiàn)金流,年初,r,. . .,年初,年初,現(xiàn)在：,108,PVAD

12、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,$ 873.44,現(xiàn)在：,年初,年初,年初,109,PVAD n = R (PVIFA r,n)(1+r) PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808,1-查普通年金現(xiàn)值表 算先付年金現(xiàn)值,110,PVAD n = R (PV

13、IFA r,n -1 + 1) PVAD3 = $1,000 (PVIFA7%,2 + 1) = $1,000 (1.808 + 1) = $2,808,2 -查普通年金現(xiàn)值表 算先付年金現(xiàn)值,111,Annuities and Perpetuities,Perpetuity永續(xù)年金: PV = Constant / r Annuities:,112,解-1-年金例-1,某人51歲，希望在60歲退休后從61歲初開始的 9年內(nèi)每年年初能從銀行得到 1 萬元, 那么他必須在從52歲初開始的 9年內(nèi)每年年初存入銀行多少錢 ？ 年利率 6% 以60歲初為前后兩個年金流的比較時點: A(FV/A, 6%

14、, 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 5919,113,解-2-年金例-1,某人51歲，希望在60歲退休后從61歲初開始的 9年內(nèi)每年年初能從銀行得到 1 萬元，那么他必須在從52歲初開始的 9年內(nèi)每年年初存入銀行多少錢 ？ 年利率 6% 以69歲初為兩個年金流的比較時點: A (F/A, 6%, 9)(1+6%)9=1W (F/A, 6%, 9) A = 1W萬an /(1+ 6%)9 A 1Wan /1.6895 5919,114,

15、解-3-年金例-1,某人51歲，希望在60歲退休后從61歲初開始的 9年內(nèi)每年年初能從銀行得到1 萬元，那么他必須在從52歲初開始的9年內(nèi)每年年初存入銀行多少錢 ？ 年利率 6% 以51歲初為兩個年金流的比較時點: A(P/A,6%,9)=1W(P/A,6%,9)/(1+6%)9 A = 1W萬an /(1+ 6%)9 A 1Wan /1.6895 5919,115,解-1-年金例-1,某人55歲，希望在60歲退休后從61歲初開始的 9年內(nèi)每年年初能從銀行得到1 萬元，那么他必須在從56歲初開始的 5年內(nèi)每年年初存入銀行多少錢 ？ 年利率 6% 以60歲初為前后兩個年金流的比較時點: A(FV

16、/A, 6%, 5) = 1萬 (PV/A, 6%, 9) A = 1萬(PV/A, 6%, 9) / (FV/A, 6%, 5) A = 1W萬an (6.8017) / 5.6371 A 12066,116,解-2-年金例-1,某人55歲，希望在60歲退休后從61歲初開始的 9年內(nèi)每年年初能從銀行得到1 萬元，那么他必須在從56歲初開始的 5年內(nèi)每年年初存入銀行多少錢 ？ 年利率 6% 以69歲初為兩個年金流的比較時點: A(F/A, 6%, 5)(1+6%)9 = 1W萬 (F/A, 6%, 9) A=1W (FV/A, 6%, 9)(1+6%)- 9/(F/A, 6%, 5) A =1

17、萬 (PV/A, 6%, 9) / (FV/A, 6%, 5) A = 1W萬an (6.8017) / 5.6371 A 12066,117,解-3-年金例-1,某人55歲，希望在60歲退休后從61歲初開始的 9年內(nèi)每年年初能從銀行得到1 萬元，那么他必須在從56歲初開始的 5年內(nèi)每年年初存入銀行多少錢 ？ 年利率 6% 以55歲初為兩個年金流的比較時點: A(P/A, 6%, 5) = 1萬 (PV/A, 6%, 9) (1+6%)- 5 A = 1萬(P/A, 6%, 9) / (PV/A, 6%, 5)(1+6%)5 A = 1萬(P/A, 6%, 9) / (FV/A, 6%, 5)

18、 A 1萬(6.8017) / 5.6371 12066,118,年金例解-2,某人從銀行貸款100萬買房，年利率為6% (.5% = .005 per month)，若在 5年內(nèi)還清，那么他每個月須還多少錢 ？ 100萬 = A (P/A, .005, 60) 100萬 = A 1 1/1.00560 / .005 5000 = A 1 1/1.00560 5000 = A 1 1/1.34885 A = 19332.80,119,Buying a House,You are ready to buy a house and you have $20,000 for a down payme

19、nt 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

20、month) for a 30-year fixed rate loan. How much money will the bank loan you ? How much can you offer for the house ?,120,Buying a House - Continued,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

21、(140,105) = 5,604 Down payment = 20,000 5604 = 14,396 Total Price = 140,105 + 14,396 = 154,501,121,1.全面閱讀問題 2.決定是PV,還是FV 3.畫一條時間軸 4.將現(xiàn)金流的箭頭標(biāo)示在時間軸上 5.決定問題是單個的現(xiàn)金流、年金或混合現(xiàn)金流 6.解決問題,解決資金時間價值問題的步驟,122,如下現(xiàn)金流，按10%折現(xiàn)的 PV 是多少 ？,混合現(xiàn)金流 Example,0 1 2 3 4 5,$600 $600 $400 $400 $100,PV0,10%,年末,123,現(xiàn)金流逐個折算法,0 1 2 3

22、 4 5,$600 $600 $400 $400 $100,10%,$545.45 $495.87 $300.53 $273.21 $ 62.09,$1677.15 = PV0,124,分組年金 (#1),0 1 2 3 4 5,$600 $600 $400 $400 $100,10%,$1,041.60 $ 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

23、(PVIF10%,5) = $100 (0.621) = $62.10,125,分組年金 (#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,$62.10,+,+,126,例：,某企業(yè)購買一大型設(shè)備，若貨款 現(xiàn)在(0年末) 一次性付清需100萬元；也可采用分期付款，從第二年年末到第四年年末每年付款40萬元。假設(shè)資金利率為10%，問該企業(yè)應(yīng)選擇何種付款方式？,127,方法 1：選 0年末為比較的時點,分期付款好于一次付款,128,方法 2：選 1

24、年末為比較的時點,分期付款好于一次付款,129,方法 3：選 4年末為比較的時點,分期付款好于一次付款,130,方法 4：比較等價年金 “A”,分期付款好于一次付款,131,Part 7,APR and EAR,132,公式: FV n,m = PV0 (1 + r/m)m n n : 年頭數(shù) m: 每年的復(fù)利次數(shù) r : 名義年利率,復(fù)利頻率,133,按年利率12%將 $1,000 投資 2 Years： 計息期是1年 FV2 = 1,000(1+ .12/1)(1)(2) = 1,254.40 計息期是半年FV2 = 1,000(1+ .12/2)(2)(2) = 1,262.48,復(fù)利頻

25、率的影響,134,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,復(fù)利頻率的影響,135,10%簡單年利率下計息次數(shù) 與 有效年利率EAR之間的關(guān)系,136,設(shè)一年中復(fù)利次數(shù)為m, 名義年利率APR 為 r ，則 有效年利率EAR 為： (1 + r / m )m - 1 ,有效年利率,er - 1,137,某公司在銀行 有 $1,000 CD (Certificates of De

26、posit)，名義年利率是 6%，一個季度計息一次，問： EAR = ? EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14% !,例：有效年利率,138,某公司在銀行有 $1,000(PV) CD，名義年利率是6%，一個季度計息一次,問： EAR = ? FV1 = PV ( 1 + 6% / 4 )4 FV1 = PV ( 1 + EAR )1 1 + EAR = ( 1 + 6% / 4 )4 EAR = ( 1 + 6% / 4 )4 - 1,例證：有效年利率,139,設(shè)一年中復(fù)利次數(shù)為 m， 名義年利率為 r ，問： EAR

27、= ? FV1 = PV ( 1 + r / m )m FV1 = PV ( 1 + EAR )1 1 + EAR = ( 1 + r / m )m EAR = ( 1 + r / m )m - 1,公式證明：有效年利率 EAR,140,Effective Annual Rate (EAR),This is the actual rate paid (or received) after accounting for compounding that occurs during the year If you want to compare two alternative investment

28、s with different compounding periods you need to compute the EAR and use that for comparison.,141,Annual Percentage Rate,This is the annual rate that is quoted By definition APR = Period Rate times the Number of Periods Per Year Consequently, to get the period rate we rearrange the APR equation: Per

29、iod 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,142,Computing 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

30、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.,143,Things to Remember,You ALWAYS need to make sure that the interest rate and the time period

31、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 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 t

32、han monthly,144,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 effectively earning? FV1 = 1 (1.01)12 = 1.1268 EAR = (1.1268 1) / 1 = .1268 = 12.68%,145,Computing EARs Example 2,Suppose if you put it in another ac

33、count, 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) / 1 = .1255 = 12.55%,146,EAR - Formula,Remember the APR is the quoted rate,147,Decisions II,You are looking at two savings accounts. One pays 5.25%, wi

34、th 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% Which account should you choose ?,148,Decisions II Continued,Lets verify the choice. Suppose you invest $100 in each account. How

35、 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 Account: Semiannual rate = .053 / 2 = .0265 FV 1 = 100(1.0265)2 = 105.37 You have more money in the first account.,149,Computing APRs from EAR s,If you

36、 have an effective rate, how can you compute the APR ? Rearrange the EAR equation and you get:,150,APR - Example,Suppose 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 ?,151,Computing Payments with APRs,Suppose yo

37、u 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 Nu

38、mber of months = 2 (12) = 24 4000 = A 1 1 / 1.01524 / .015 A = 4000 (A/P, .015, 24) = 199.70,152,Future Values with Monthly Compounding,Suppose 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 ra

39、te = .12 / 12 = .01 Number of months = 10(12) = 120 FV = 501.01120 1 / .01 = 11,502,153,Present Value with Daily Compounding,You 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? Da

40、ily rate = .055 / 365 = .00015068493 Number of days = 3(365) = 1095 PV = 15,000 / (1.00015068493)1095 = 12,718.56,154,Quick Quiz: Part 7,What is the definition of an APR ? What is the effective annual rate(EAR) ? Which rate should you use to compare alternative investments or loans ? Which rate do y

41、ou need to use in the time value of money calculations ?,155,Part 8,Loan Types & Loan Amortization,156,Pure Discount Loans Example 5.11,Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. If a T-bil

42、l 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 = 9345.79,157,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.

43、 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 is similar to the cash flows on corporate bonds and we will talk about them in greater detail later.,158,1. 計算 每期償還額. 2. 計算第 t 期償還的 利息. = (第 t-1 期的 貸款余額) x (APR/m期數(shù)/年) 3. 計算第 t 期 償還的本金. = (