【正文】 mpounding = Interest on Interest) 不僅借貸的本金要求支付利息，而且前期的利息 在下一期也要求支付利息 . ?單利 (Simple Interest) 只對(duì)借貸的 原始金額 或 本金 支付利息。 ?在上頁(yè)例子中，你 先 前 存的 $1,000 原始金額，就是 兩年后的 $1,140 的 單利現(xiàn)值 PV ,即 在 先 前 時(shí)點(diǎn) 的價(jià)值 ! 單利現(xiàn)值 SIPV 22 假設(shè)投資者按 7% 的 復(fù)利 把$1,000 存入銀行 2 年， 那么它的復(fù)利終值 是多少？ 復(fù)利終值 0 1 2 $1,000 FV2 7% 23 Basic Definitions ?Present Value earlier money on a time line（ 相對(duì)于 其后的 money 而言 ） ?Future Value later money on a time line（ 相對(duì)于 其前的 money 而言 ） ?PV 與 FV 是 相對(duì)而言 的 ！ ?Interest rate “exchange rate” between earlier money and later money 24 FV1 = P0 (1+r)1 = $1,000 () = $1,070 復(fù)利 在第一年年末你得了 $70的利息 . 這與 單利 利息相等 . 復(fù)利終值公式 —1/2 25 FV1 = P0 (1+r)1 = $1,000 () = $1,070 FV2 = FV1 (1+r)1 = P0 (1+r)(1+r) = $1,000()() = P0 (1+r)2 = $1,000()2 = $1,144 . 90 第 2 年，你比 單利 利息多得 $4 . 90 = $70 () 復(fù)利終值公式 —2/2 26 Future Values: General Formula ?FV = PV (1 + r)t ? FV = Future Value ? PV = Present Value ? r = period interest rate, expressed as a decimal ? t = number of periods ?Future Value Interest Factor = (1 + r)t 27 FV1 = P0(1+r)1 FV2 = P0(1+r)2 , etc. F V 公式 : FV n = P0 (1+r)n or FV n = P0 (FVIF r,n) – 可查表 (FVIF r,n)可直觀地記為 (FV/PV, r , n) 一般 復(fù)利 終值公式 28 FVIF r,n 可以查表如下所示： 復(fù)利 終值因子表： FVIF r,n 期限 6% 7% 8% 1 2 3 4 5 29 FV2 = $1,000 (FVIF7%,2) = $1,000 () = $1,145 [四舍五入 ] 查表 FVIF r,n計(jì)算 Pe rio d 6% 7% 8% 1 60 70 80 2 24 45 66 3 91 25 60 4 62 11 60 5 38 03 69 30 按 10% 的 復(fù)利 把 $1000存入銀行 ， 5年 后的終值是多少？ Example – 復(fù)利終值 0 1 2 3 4 5 $1000 FV5 10% 31 ?查表 : FV5 = $1,000 (FVIF10%, 5) = $1,000 () = $1,611 [四舍五入 ] 解： ?用一般公式 : FV n = P0 (1+r)n FV5 = $1,000 (1+ )5 = $1,610. 51 32 Effects of Compounding 1 ?Consider the previous example ?FV with simple interest = 1000 + 70 + 70 = 1140 ?FV with pound interest = ?The extra es from the interest of .07(70) = earned on the first interest payment 33 Effects of Compounding 2 ?Suppose you invest the $1000 from the previous example for 5 years. How much would you have? ?FV = 1000 ()5 = ?Simple interest would have a future value of $1350, for a difference of $ ?The effect of pounding is small for a small number of periods, but increases as the number of periods increases. 34 Figure : 終值、單利 vs. 復(fù)利 35 Effects of Compounding 3 ?Suppose you had a deposit $1k at 7% interest 30 years ago. How much would the investment be worth today ? ?FV = 1k ()30 ? 7,612 ?What is the effect of pounding ? ?Simple interest = 1k + 30(1k)(.07) = 3100 ?Compounding added $ 4,512 to the value of the investment 36 050001000015000202101 年 10 年 20 年 30 年 一筆 $1,000 存款的終值10% 單利7% 復(fù)利10% 復(fù)利單利 . 復(fù)利 ? Future Value (U.S. Dollars) 37 Effects of Compounding 4 ?Suppose you had a deposit $1 at 7% interest 200 years ago. How much would the investment be worth today ? ?FV = 1 ()200 ? 752,932 ?What is the effect of pounding ? ?Simple interest = 1 + 200(1)(.07) = 15 ?Compounding added $ 752,917 to the value of the investment 38 美國(guó)曼哈頓島值多少錢 ？ 1/4 ?為了闡述復(fù)利在 長(zhǎng)時(shí)期 中的作用，不妨看看彼得麥紐因特和印第安人買賣曼哈頓島的交易。這個(gè)價(jià)格聽(tīng)起來(lái)很便宜，但是印第安人也能從這個(gè)交易中獲得很不錯(cuò)的結(jié)果。 這項(xiàng)投資到今天 會(huì)值 多少錢呢 ? 請(qǐng)同學(xué)們猜一猜 ！ 39 美國(guó)曼哈頓島值多少錢 ？ 2/4 ?這項(xiàng)投資到今天 大約過(guò)了 377年。到底增長(zhǎng)到多少呢 ? ?該復(fù)利終值因子為： (1+ r)377 = ? ? 4 000 000 000 000 000 ?那也就是 4后面跟 15個(gè)零。 ?這么多錢可以買下整個(gè)美國(guó) ,乃至整個(gè)世界 .對(duì)此同學(xué)們有何感想 ？ 40 美國(guó)曼哈頓島值多少錢 ？ 3/4 ?這當(dāng)然是一個(gè) 夸張 的例子。 ?假設(shè)印第安人將賣島所得 $24 以 5％ 的利率進(jìn)行投資。終值因子帶來(lái)的結(jié)果是 $24x1 后面再加 8個(gè)零，也就是 $24億 。 這項(xiàng)投資到今天 會(huì)值 多少錢呢 ? ?該復(fù)利終值因子為： (1+ r)377 = ? 11,038 42 Effects of Compounding – 小結(jié) ?The effect of pounding is small for a low interest rate, and for a small number of periods. ?For a given interest rate – the longer the time period, the bigger the future value. ?For a given time period – the higher the interest rate, the bigger the future value. 43 Figure : $1于不同利率、多期后的復(fù)利終值 44 $1,000 按 12% 復(fù)利，需要多久成為 $2,000 (近似 ) ? 想使自己的財(cái)富倍增嗎 ! 快捷計(jì)算方法 : 72 法則 45 近似 . N = 72 / i% 72 / 12% = 6 年 [精確計(jì)算是 年 ] 快捷計(jì)算方法： 72法則 $1,000 按 12% 復(fù)利，需要多久成為 $2,000 (近似 ) ? 46 Quick Quiz: Part 1 ?What is the difference between simple interest and pound interest ? ?Suppose 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 pound interest ? 47 Part 2 Present Value and Discounting 48 Present Values ?How much do I have to invest today to have some amount in the future? ?FV = PV(1 + r)t ?Rearrange to solve for PV = FV / (1 + r)t ?When we talk about discounting, we mean finding the present value of some future amount. ?When we talk about the “value” of someth., we are talking about the present value unless we specifically indicate that we want the future value. 49 假設(shè) 2 年 后你需要 $1,000. 那么按 7%復(fù)利，你現(xiàn)在要存多少錢 ？ 0 1 2 $1,000 7% PV1 PV0 復(fù)利現(xiàn)值 50 PV0 = FV2 / (1+r)2 = $1,000 / ()2 = FV2 / (1+r)2 = $ 復(fù)利現(xiàn)值公式 0 1 2 $1,000 7% PV0 51 PV0 = FV1 / (1+r)1 = FV1(1+r)1 PV0 = FV2 / (1+r)2 = FV2(1+r)2 P V 公式 : PV0 = FV n / (1+r)n = FV n (1+r)n or PV0 = FV n (PVIF r,n) 見(jiàn)下頁(yè)表格 (PVIF r,n) 可