【正文】 f periods? ?Suppose 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 don’t add any additional money ? 79 Part 5 Multiple Cash Flows 80 Multiple Cash Flows – FV Example 1 ?Suppose 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()4 + 300()2 = + = 81 Example 1 Timeline 5 100 0 1 2 3 4 300 82 Multiple Cash Flows – PV Example ?You 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. ?What is the most you should pay for this one ? 83 Multiple Cash Flows – PV Example ?Find the PV of each cash flow and add them ?PV1 of Year 1 CF1: 200 / ()1 = ?PV2 of Year 2 CF2: 400 / ()2 = ?PV3 of Year 3 CF3: 600 / ()3 = ?PV4 of Year 4 CF4: 800 / ()4 = ?Total PV+ = + + + = ?If you can earn 12% on your money, this is the most you should be willing to pay. 84 Example Timeline 0 1 2 3 4 200 400 600 800 85 Decisions I ?Your 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 ? ?PV = , No the broker is charging more than you would be willing to pay. 86 Saving For Retirement ?You 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% ? ?PV = 1084 .71 87 Saving For Retirement Timeline 0 1 2 … 39 40 41 42 43 44 0 0 0 … 0 25K 25K 25K 25K 25K Notice the year 0 cash flow = 0 (CF0 = 0) The cash flows years 1 – 39 are 0 The cash flows years 40 – 44 are 25,000 88 Quick Quiz: Part 5 ?Suppose you are looking at the following possible cash flows: Year 1 CF = $100。 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 Annuities 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 ?Perpetuity – 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(shí)例 ? 學(xué)生貸款償還 ? 汽車貸款償還 ? 保險(xiǎn)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 。 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 。 年金計(jì)算之要點(diǎn) 97 FVA n = R(1+r)n1 + R(1+r)n2 + ... ... ... + R(1+r)1 + R(1+r)0 = R{[(1+r)n – 1]/r} = R{FVIFA r,n} = R{[FVIF r,n– 1]/r} 普通年金 于第 n年末的 終值 – FVA(n) 0 1 2 n r R R R FVA n R： 每年現(xiàn)金流 年末 . . . 年末 ? 98 FVA3 = $1,000 ()2 + $1k ()1 + $1k ()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 查普通年金終值表計(jì)算 Pe rio d 6% 7% 8% 1 00 00 00 2 60 70 80 3 84 15 46 4 75 40 06 5 37 51 67 100 FVAD n = R(1+r)n + R(1+r)n1 + ... + 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 n1 n 年末 年末 年 初 年末 現(xiàn)在： 101 FVAD3 = $1,000 ()3 + $1k ()2 + $1k ()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 現(xiàn)在： 102 FVAD n = R (FVIFA r,n)(1+r) FVAD3 = $1,000 (FVIFA7%,3)() = $1,000 ()() = $3,440 1查 普通 年金終值表 算 先付 年金終值 Pe rio d 6% 7% 8% 1 00 00 00 2 60 70 80 3 84 15 46 4 75 40 06 5 37 51 67 103 FVAD n = R (FVIFA r,n +1 1) FVAD3 = $1,000 (FVIFA7%,4 1) = $1,000 ( 1) = $3,440 2查