第五章 評價：貨幣的時間價值 5.1 未來值(future value, FV)和複利 (compounding) 第五章 評價：貨幣的時間價值 5.1 未來值(future value, FV)和複利 (compounding) 未來值：在已知利率下，一筆金額投資一段期間所能成長到達的數量 一期投資：在r利率下投資一期，所投資的每$1，將收到$(1+r) 假設存$100到一個年息10%的儲蓄帳戶中，一年後將有$100*(1+10%)=$110 Ch5

多期投資： 若第一年年底時，繼續把$110留在銀行，則第二年年底時為$110*(1+10%)=$121 把本金(principal)和累積的利息留在投資中超過一期，因而再投資的過程就叫做複利(compound-ing) 複利是賺取利上利(interest on interest)，這部份的利息稱為複利利息(compound interest) 如果是單利(single interest)，利息就不再投資，每一期只賺得本金的利息 [範例5.1] Ch5

[範例5.1] Suppose you locate a two-year investment that pays 14% per year. If you invest $325, how much will you have at the end of the two years? How much of this is simple interest? How much is compound interest? The end of 1st year : $325 * ( 1 + 0.14 ) = $370.50 You reinvest this entire amount, and compound the interest, The end of second year : $370.50 * 1.14 = $422.37 The total interest you earn : $422.37 - 325 = $97.37 Original principal earns : $325 * 0.14 = $45.50 in interest each year, for a two - year total of $91 in simple interest. The remaining $97.37 - 91 = $6.37 result from compounding.. Ch5

未來值利率因子(future value interest factor, FVIF) FVIF(r, t) = (1+r)t [表5.1] 投資三年之情形： 1.$110 = $100*(1+10%) 2.$121 = $110*(1+10%) = $100*(1+10%)2 3.$133.1 = $121*(1+10%) = $100*(1+10%)3 =>FVt = $1*(1+r)t 未來值利率因子(future value interest factor, FVIF) FVIF(r, t) = (1+r)t [表5.1] Ch5

[表5.1] Beginning Simple Compound Total Ending Year Amount Interest Interest Interest Amount Earned 1 $100.00 $10 $0.00 $10.00 $110.00 2 110.00 10 1.00 11.00 121.00 3 121.00 10 2.10 12.10 133.10 4 133.10 10 3.31 13.31 146.41 5 146.41 10 4.64 14.64 161.05 Total $50 Total $11.05 Total $61.05 simple compound interest interest interest Ch5

[圖5.1] Future value ($) $161.05 160 $146.41 150 140 $133.10 130 $121 120 $110 110 100 $0 Time (years) 1 2 3 4 5 Ch5

[圖5.2] Future value of $1 ($) Time (years) 6 5 20% 4 15% 3 10% 2 5% 1 7 8 9 10 Ch5

[表5.2] 利率與未來值成正向關係 期間與未來值成正向關係 Interest Rate Number of Periods 5% 10% 15% 20% 1 1.0500 1.1000 1.1500 1.2000 2 1.1025 1.2100 1.3225 1.4400 3 1.1576 1.3310 1.5209 1.7280 4 1.2155 1.4641 1.7490 2.0736 5 1.2763 1.6105 2.0114 2.4883 利率與未來值成正向關係 期間與未來值成正向關係 Ch5

[範例5.2] You’ve located an investment that pays 12%. That rate sounds good to you, so you invest $400. How much will you have in three years? How much will you have in seven years? At the end of seven years, how much interest will you have earned? How much of that interest results from compounding? Future value factor ( 1+ r )t = 1.123 = 1.4049 Your $400 grows to $400 * 1.4049 = $561.97 After 7 years, you will have $400 * 1.127 = $400 * 2.2107 = $884.27 The interest you earn $884.27 - 400 = $484.27 If you invest in simple interest $400 * 0.12 = $48/per year Over 7 years, you will have 7 * $48 = $336 The interest results from compounding is $484.27 - 336 = $148.27 Ch5

[範例5.3] In 1626, Minuit bought all of Manhattan Island for about $24 in goods and trinkets. This sounds cheap, but the Indians may have gotten the better end of the deal. To see why, suppose the Indians had sold the goods and invested the $24 at 10%. How much would it be worth today? Years : 377(=2003-1626) The future value factor ( 1 + r )t = 1.1377  4,000,000,000,000,000 (4千兆) The future value $24 * 4*1015 = $96 quadrillion (9.6萬兆) Ch5

如果把錢存在一個計息帳戶中，假設不提走任何錢，則該帳戶的利率就是錢的成長率 複成長： 如果把錢存在一個計息帳戶中，假設不提走任何錢，則該帳戶的利率就是錢的成長率 [範例5.4] The TICO Corporation currently pays a cash dividend of $5 per share. You believe the dividend will be increased by 4% each year indefinitely. How big will the dividend be in 8 years? Future value = $5 * 1.048 = $5 * 1.3686 = $6.84 The dividend will grow by $1.84 over that period. Ch5

5.2 現值(present value, PV)和折現(discount) 一期現值 假設在年利率10%下，今天我們必須投資多少，才能在一年後拿到$1呢？ 現值*1.1=$1 現值=$1/1.1=$0.909 Ch5

Present value = $400 * ( 1 / 1.07 ) = $373.83 [範例5.5] Suppose you need EU€400 to buy textbooks next year. You can earn 7% your money. How much do you have to put up today? Present value * 1.07 = $400 Present value = $400 * ( 1 / 1.07 ) = $373.83 Ch5

多期現值 假設兩年後需要$1,000，利率為7%，現在必須投資多少，兩年後才有$1,000 $1,000=PV*1.07*1.07 Ch5

現值利率因子(present value interest factor, PVIF) PVIF(r, t) = 1/(1+r)t [範例5.6] You would like to buy a new automobile. You have ¥500,000 or so, but the car costs ¥ 685,000. If you can earn 9%, how much do you have to invest today to buy the car in two years? Do you have enough? Assume the price will stay the same. PV = ¥ 685,000 / 1.092 = $685,000 / 1.1881 = ¥ 576,550.80 You will still about ¥ 108,449 short. PV=$1*[1/(1+r)t] 現值利率因子(present value interest factor, PVIF) PVIF(r, t) = 1/(1+r)t Ch5

折現現金流量評價(discounted cash flow, DCF, valuation) [表5.3] r = 折現率(discount rate) 折現現金流量評價(discounted cash flow, DCF, valuation) [表5.3] Interest Rate Number of Periods 5% 10% 15% 20% 1 0.9524 0.9091 0.8696 0.8333 2 0.9070 0.8264 0.7561 0.6944 3 0.8638 0.7513 0.6575 0.5787 4 0.8227 0.6830 0.5718 0.4823 5 0.7835 0.6209 0.4972 0.4019 Ch5

[範例5.7] 折現率與現值成反向關係 期間與現值成反向關係 “Come try our product. If you do, we‘ll give you $100 just for coming by!” If you read the fine print, what you find out is that they will give you a savings certificate that will pay you $100 in 25 years or so. If the going interest rate on such certificate is 10% per year, how much are they really giving you today? The discount factor 1 / 1.125 = 1 / 10.8347 = 0.0923 The promotion is actually paying you about 0.0923 * $100 = $9.23 折現率與現值成反向關係 期間與現值成反向關係 Ch5

[圖5.3] Ch5

5.3 現值和未來值 現值和未來值 現值因子就是未來值因子的倒數 未來值因子=(1+r)t 現值因子=1/(1+r)t 5.3 現值和未來值 現值和未來值 現值因子就是未來值因子的倒數 未來值因子=(1+r)t 現值因子=1/(1+r)t 基本現值等式(basic present value equation) PV* (1+r)t=FVt PV= FVt/(1+r)t= FVt*[1/(1+r)t] 基本現值等式包含PV、 FVt、r 和 t 四個部份，只要知道其中任何三個，就可以求得第四個 Ch5

Because the proposed investment only pays out $400, it is not as [範例5.8] Your company proposes to buy an asset for $335. This investment is very safe. You would sell off the asset in 3 years for $400. You know you could invest the $335 elsewhere at 10% with very little risk. What do you think of the proposed investment? This is not a good investment. Why not? Future value If we invest $335 at 10%, after 3 years it will grow to $335 * (1+r)t = $335 * 1.13 = $445.8 Because the proposed investment only pays out $400, it is not as good as other alternatives we have. Another way ---- Present value of $400 in 3 years at 10% $400 * [1/( 1+r )t ] = $400/1.13 = $300.53 < $335 Ch5

決定折現率 [範例5.9] If you put up Au$1,250 for one year, you will get back Au$1,350. What rate is this investment paying? Au$1,350 - Au$1,250 = Au$100 The implicit rate is Au$100/Au$1,250 = 8% From the basic present value Au$1,250 = Au$1,350 / ( 1 + r )1 1 + r = Au$1,350/Au$1,250 = 1.08 r = 8% Ch5

r = 折現率 = 報酬率(rate of return，或return) 72法則(rule of 72)：72 / r% = t [範例5.10] In 2001, when Mark McGwire was chasing baseball’s single-season home run record, there was much speculation as to what might be the value of the baseball he hit to break the record. One “expert” on such collectibles said, “No matter what it’s worth today, I’m sure it will double in value over the next 10 years.”So, would the record-breaking home run ball have been a good investment? By the Rule of 72, you already know that it would earn about 72/10 = 7.2% per year, which is only so-so. Ch5

[範例5.11] You estimate that you will need about $80,000(pesos) to send your child to college in 8 years. You have about $35,000 (pesos) now. If you can earn 20% per year, will you make it? At what rate will you just reach your goal? FV = $35,000 * 1.28 =$35,000 * 4.2998 = $150,493.59 So you make it easily. The minimum rate is FV = $35,000 * (1+r)8 = $80,000 (1+r)8 = $80,000/35,000 = 2.2857 查表--11% 1+r = 2.2857(1/8) =1.1089 r = 10.89% Ch5

[範例5.12] You would like to retire in 50 years as a millionaire. If you have $10,000 today, what rate of return do you need to earn to achieve your goal? FV = $1,000,000 PV = $10,000 Year = 50 $10,000 = $1,000,000 / (1 + r )50 ( 1 + r )50 = 100 So the implicit rate is 9.65% Ch5

找出期數 [範例5.13] You have been saving up to buy the Godot Company. The total cost will be $10 million. You currently have about $2.3 million. If you can earn 5% on your money, how long will you have to wait? At 16%, how long must you wait? At 5% $2.3 million = $10 million / 1.05t 1.05t = 4.35 t = 30 years At 16%, t = 10 years. Ch5

[表5.4] I. Symbols: PV = Present value, what future cash flows are worth today FVt = Future value, what cash flows are worth in the future r = Interest rate, rate of return, or discount rate per period t = number of periods C = cash amount II. Future value of C invested at r percent per period for t periods: FVt = C  (1 + r )t The term (1 + r )t is called the future value factor. Ch5

[表5.4] III. Present value of C to be received in t periods at r percent per period: PV = C/(1 + r )t The term 1/(1 + r )t is called the present value factor. IV. The basic present value equation giving the relationship between present and future value is: PV = FVt/(1 + r )t Ch5