Chapter 3 :Chapter 3 Understanding The Time Value of Money了解貨幣的時間價值
Time Value of Money貨幣的時間價值 :Prentice Hall inc. 2 Time Value of Money貨幣的時間價值 A dollar received today is worth more than a dollar received in the future.今天所收到的一塊錢，比未來所收到的一塊錢更值錢。
The sooner your money can earn interest, the faster the interest can earn interest.你的錢能夠越快投入賺取利息，則這些利息能夠越快再幫你賺取利息。
Interest and Compound Interest利息與複利 :Prentice Hall inc. 3 Interest and Compound Interest利息與複利 Interest – the return you receive for investing your money利息—你投資所賺取的報酬
Compound interest – the interest that your investment earns on the interest that your investment previously earned複利—藉由本金所賺取的利息再投資，產生以利息賺利息的效果。
Future Value Equation終值的公式 :Prentice Hall inc. 4 Future Value Equation終值的公式 FVn = PV(1 + i)n
FV = the future value of the investment at the end of n year投資在第 n年底的終值
i = the annual interest (or discount) rate年利率(或折現率)
PV = the present value, in today’s dollars, of a sum of money現值，即未來金額彙總的現在價值。
This equation is used to determine the value of an investment at some point in the future.這個公式用來決定你的投資在未來某個時點的價值。
Compounding Period複利期間 :Prentice Hall inc. 5 Compounding Period複利期間 Definition – the frequency that interest is applied to the investment 定義—該投資利息計算的次數
Examples – daily, monthly, or annually例如—每日，每月，每年
Reinvesting – How to Earn Interest on Interest再投資—如何以利滾利 :Prentice Hall inc. 6 Reinvesting – How to Earn Interest on Interest再投資—如何以利滾利 Future-value interest factor (FVIFi,n) is a value used as a multiplier to calculate an amount’s future value, and substitutes for the (1 + i)n part of the equation.終值的利息因子(FVIFi,n) ，亦即(1 + i)n 的數值，用來計算終值的金額。
FVIFs are found in Appendix A of the text. FVIFs在本書的附錄A中可找到。
The Future Value of a Wedding結婚成本的終值 :Prentice Hall inc. 7 The Future Value of a Wedding結婚成本的終值 In 1998, the average wedding cost $19,104. Assuming 4% inflation, what will it cost in 2028?在1998，結婚成本為$19,104. 假設通貨膨脹率為4% ，在2028年結婚成本為何？
FVn = PV (FVIFi,n)
FVn = PV (1 + i)n
FV30 = PV (1 + 0.04)30
FV30 = $19,104 (3.243)
FV30 = $61,954.27
The Rule of 72 72原則 :Prentice Hall inc. 8 The Rule of 72 72原則 Estimates how many years an investment will take to double in value慨算出投資要成長一倍，所需要的年數
Number of years to double = 成長一倍的年數=
72 / annual compound growth rate72/年複利率
Example – 72 / 8 = 9 therefore, it will take 9 years for an investment to double in value if it earns 8% annually例如– 72/8=9 每年複利率為8%時，必須有9年的時間，投資才可成長一倍。
Compound Interest With Nonannual Periods非每年計息之複利 :Prentice Hall inc. 9 Compound Interest With Nonannual Periods非每年計息之複利 The length of the compounding period and the effective annual interest rate are inversely related; therefore, the shorter the compounding period, the quicker the investment grows.複利計息期間越短，投資成長越快。
Compound Interest With Nonannual Periods (cont’d)非每年計息之複利(續) :Prentice Hall inc. 10 Compound Interest With Nonannual Periods (cont’d)非每年計息之複利(續) Effective annual interest rate = 實質年複利率=
amount of annual interest earned 每年所賺利息金額
amount of money invested投資的金額
Examples – daily, weekly, monthly, and semi-annually例如—每日，每週，每月，每半年。
The Time Value of a Financial Calculator使用財務計算機計算貨幣的時間價值 :Prentice Hall inc. 11 The Time Value of a Financial Calculator使用財務計算機計算貨幣的時間價值 The TI BAII Plus financial calculator keys: TI BAII Plus 財務計算機的按鍵：
N = stores the total number of payments輸入期數
I/Y = stores the interest or discount rate輸入利率或折現率
PV = stores the present value輸入現值
FV = stores the future value輸入終值
PMT = stores the dollar amount of each annuity payment輸入每年必須的支付額
CPT = is the compute key計算按鍵
The Time Value of a Financial Calculator (cont’d)使用財務計算機計算貨幣的時間價值(續) :Prentice Hall inc. 12 The Time Value of a Financial Calculator (cont’d)使用財務計算機計算貨幣的時間價值(續) Step 1 – input the values of the known variables步驟1 –輸入已知變數的金額
unknown variable未知變數
Note: always set your calculator to “end of year” and “one payment per year” modes unless otherwise directed. 注意：除非有其他狀況，將計算機設定在“end of year” ，以及“one payment per year” 。
Tables Versus Calculator查表與計算機的比較 :Prentice Hall inc. 13 Tables Versus Calculator查表與計算機的比較 REMEMBER – The tables have a discrepancy due to rounding error; therefore, the calculator is more accurate.記住–查表由於小數點的問題存在，所以計算機的答案比較正確。
Compounding and the Power of Time複利機制和時間的力量 :Prentice Hall inc. 14 Compounding and the Power of Time複利機制和時間的力量 In the long run, money saved now is much more valuable than money saved later.長期來看，現在存的一塊錢永遠比以後存的一塊錢還有價值。
Don’t ignore the bottom line, but also consider the average annual return.不要戶略底線，但是要同時考慮平均的年報酬率。
The Power of Time in Compounding Over 35 Years35年複利所顯示的時間的力量 :Prentice Hall inc. 15 The Power of Time in Compounding Over 35 Years35年複利所顯示的時間的力量 Selma contributed $2,000 per year in years 1 – 10, or 10 years. Selma在第1年到第10年每年提撥$2,000 ，共提撥10年。
Patty contributed $2,000 per year in years 11 – 35, or 25 years. Patty 在第11年到第35年每年提撥$2,000 ，共提撥25年。
Both earned 8% average annual return.兩者的年報酬率均為8%。
The Importance of the Interest Rate in Compounding在複利機制下利率的重要性 :Prentice Hall inc. 16 The Importance of the Interest Rate in Compounding在複利機制下利率的重要性 From 1926-2001 the compound growth rate of large company stocks was approximately 10.7%, whereas long-term government bonds returned only 5.3%.在1926-2001年間，美國大公司股票的年平均報酬率為10.7%，同期間長期政府公債的報酬率只有5.3%。
The “Daily Double” – states that you are earning a 100% return compounded on a daily basis.”每天都加倍”—顯示每天的報酬率為100%。
Present Value現值 :Prentice Hall inc. 17 Present Value現值 Is also known as the discount rate, or the interest rate used to bring future dollars back to the present.折現率，或者是將未來的價值折回現在價值的利率。
Present-value interest factor (PVIFi,n) is a value used to calculate the present value of a given amount.現值利率因子(PVIFi,n) 是[1/(1 + i)n ]的數值，用來計算現值的金額。
PVIFs are found in Appendix B of the text. PVIFs可以在本書的附錄B找到。
Present Value Equation現值公式 :Prentice Hall inc. 18 Present Value Equation現值公式 PV = FVn (PVIFi,n)
PV = the present value, in today’s dollars, of a sum of money現值，即未來金額彙總目前的價值。
FVn = the future value of the investment at the end of n years投資在第n年底的終值。
PVIFi,n = the present value interest factor現值利率因子
This equation is used to determine today’s value of some future sum of money.這個公式用來決定未來金額彙總目前的價值。
Calculating Present Value for the “Prodigal Son”計算現值”揮霍男”的例子 :Prentice Hall inc. 19 Calculating Present Value for the “Prodigal Son”計算現值”揮霍男”的例子 If promised $500,000 in 40 years, assuming 6% interest, what is the value today?如果在40年後給他$500,000 ，假設年利率6%，目前的價值如何？
PV = FVn (PVIFi,n)
PV = $500,000 (PVIF6%, 40 yr)
PV = $500,000 (0.097)
PV = $48,500
Annuities年金 :Prentice Hall inc. 20 Annuities年金 Definition – a series of equal dollar payments coming at the end of a certain time period for a specified number of time periods定義–在特定期間內，每期的期末均支付一系列相同的金額。
Examples – life insurance benefits, lottery payments, retirement payments例如–人壽保險給付，樂透彩給付
Compound Annuities複利年金 :Prentice Hall inc. 21 Compound Annuities複利年金 Definition – depositing an equal sum of money at the end of each time period for a certain number of periods and allowing the money to grow定義–當你在確定的期間，每期期末均存入相同的金額，而且在這段期間內，原來的儲蓄或投資仍然會複利成長，稱之為複利年金。
Example – saving $50 a month to buy a new stereo two years in the future例如–每個月存$50存兩年去買身歷聲音響。
By allowing the money to gain interest and compound interest, the first $50, at the end of two years, is worth $50 (1 + 0.08)2 = $58.32.藉由金額可以賺取利息以及複利，第一個$50，在兩年終了後，的價值為$50 (1 + 0.08)2 = $58.32。
Future Value of an Annuity Equation年金終值的公式 :Prentice Hall inc. 22 Future Value of an Annuity Equation年金終值的公式 FVn = PMT (FVIFAi,n)
FVn = the future value, in today’s dollars, of a sum of money終值，用現在的金額，彙總的金額。
PMT = the payment made at the end of each time period每期期末所支付的金額
FVIFAi,n = the future-value interest factor for an annuity年金終值的利息因子
Future Value of an Annuity Equation (cont’d)年金終值的公式(續) :Prentice Hall inc. 23 Future Value of an Annuity Equation (cont’d)年金終值的公式(續) This equation is used to determine the future value of a stream of payments invested in the present, such as the value of your 401(k) contributions.用來決定目前投入一系列支付金額(提撥)的未來價值，例如你的401(k)退休計畫的提撥。
FVIFAs are found in Appendix C of the text. FVIFAs可以在本書的附錄C找到。
Calculating the Future Value of an Annuity: An IRA計算你的複利終值： IRA退休計畫 :Prentice Hall inc. 24 Calculating the Future Value of an Annuity: An IRA計算你的複利終值： IRA退休計畫 Assuming $2000 annual contributions with 9% return, how much will an IRA be worth in 30 years?假設每年提撥$2000 ，年報酬率為9%，30年後你的IRA的價值為何？
FVn = PMT (FVIFA i, n)
FV30 = $2000 (FVIFA 9%,30 yr)
FV30 = $2000 (136.305)
FV30 = $272,610
Present Value of an Annuity Equation年金現值的公式 :Prentice Hall inc. 25 Present Value of an Annuity Equation年金現值的公式 PV = PMT (PVIFAi,n)
PV = the present value, in today’s dollars, of a sum of money現值，即未來金額彙總的現在價值
PMT = the payment to be made at the end of each time period每期期末所支付的金額
PVIFAi,n = the present-value interest factor for an annuity 年金現值利率因子
Present Value of an Annuity Equation (cont’d)年金現值的公式(續) :Prentice Hall inc. 26 Present Value of an Annuity Equation (cont’d)年金現值的公式(續) This equation is used to determine the present value of a future stream of payments, such as your pension fund or insurance benefits.用來決定未來一系列支付金額(提撥)的現在價值，例如你的退休基金或者保險給付。
PVIFAs are found in Appendix D of the text. PVIFAs 可以在本書附錄D找到。
Calculating Present Value of an Annuity: Now or Wait?計算年金現值：現在或者再等一下？ :Prentice Hall inc. 27 Calculating Present Value of an Annuity: Now or Wait?計算年金現值：現在或者再等一下？ What is the present value of the 25 annual payments of $50,000 offered to the soon-to-be ex-wife, assuming a 5% discount rate?一個離婚的配偶，在未來25年，每年年底可以得到$50,000 的給付，如果每年折現率為5%，目前的現值為何？
PV = PMT (PVIFA i,n)
PV = $50,000 (PVIFA 5%, 25)
PV = $50,000 (14.094)
PV = $704,700
Amortized Loans分期攤還借款 :Prentice Hall inc. 28 Amortized Loans分期攤還借款 Definition – loans that are repaid in equal periodic installments定義每期償還固定金額分期償還的借款
With an amortized loan, the interest payment declines as your outstanding principal declines; therefore, with each payment you will be paying an increasing amount towards the principal of the loan在分期攤還借款下，利息的支付逐期降低，因為未清償本金也是逐期降低，因此，你每期償還的本金，將會逐期增加。.
Loan payments are found in Appendix E of the text.分期攤還借款可以在本書附錄E找到。
Examples – car loans or home mortgages例如—汽車借款，或者房屋借款。
Buying a Car With Four Easy Annual Installments每年償還，四年到期的汽車借款 :Prentice Hall inc. 29 Buying a Car With Four Easy Annual Installments每年償還，四年到期的汽車借款 What are the annual payments to repay $6,000 at 15% interest?如果汽車借款為$6,000 ，年利率為15%，則所借的汽車借款每年支付金額為何？
PV = PMT (PVIFA i%,n yr)
$6,000 = PMT (PVIFA 15%, 4 yr)
$6,000 = PMT (2.855)
$2,101.58 = PMT
Total Payment = $2,101.58 x 4 = $8,406.32
Buying a Car With Forty-Eight Easy Monthly Installments每月償還，48期的汽車借款 :Prentice Hall inc. 30 Buying a Car With Forty-Eight Easy Monthly Installments每月償還，48期的汽車借款 What are the monthly payments to repay $6,000 at 15% interest?如果汽車借款為$6,000 ，年利率為15%，則所借的汽車借款每月支付金額為何？
PV = PMT (PVIFA i%,n yr)
$6,000 = PMT (PVIFA 15%/12, 4 yr x 12)
$6,000 = PMT (35.932)
$166.98 = PMT
Total Payments = $166.98 x 48 = $8,015.04
Perpetuities永續年金 :Prentice Hall inc. 31 Perpetuities永續年金 Definition – an annuity that lasts forever年金的年限會持續到永遠。
PV = PP / i
PV = the present value of the perpetuity永續年金的現值
PP = the annual dollar amount provided by the perpetuity永續年金每年必須支付(提撥)的金額
i = the annual interest (or discount) rate年利率(或折現率)
Summary摘要 :Prentice Hall inc. 32 Summary摘要 Future value – the value, in the future, of a current investment終值–目前投資的未來價值
Rule of 72 – estimates how long your investment will take to double at a given rate of return72原則–在既定的報酬率下，要投資多少才會增長一倍
Present value – today’s value of an investment received in the future現值–未來投資的現在價值
Summary (cont’d)摘要(續) :Prentice Hall inc. 33 Summary (cont’d)摘要(續) Annuity – a periodic series of equal payments for a specific length of time年金–在特定的期間內，每期期末均支付一定金額
Future value of an annuity – the value, in the future, of a current stream of investments年金終值–目前一系列投資固定金額的未來價值
Present value of an annuity – today’s value of a stream of investments received in the future年金現值–未來投資一系列固定金額的現在價值
Summary (cont’d)摘要(續) :Prentice Hall inc. 34 Summary (cont’d)摘要(續) Amortized loans – loans paid in equal periodic installments for a specific length of time分期攤還借款–在未來一定的期間內，每期均償還固定金額的借款。
Perpetuities – annuities that continue forever永續年金–年金的年限會持續到永遠。