2. TIME VALUE OF MONEY 2.1 Theory of Equivalence 2.2 Equivalence Relationship 2.3 Nominal and Effective Interest Rates 2.4 Continuous Payments 2.5 Gradient Equations for Continuous Payment技术经济学英文版演示文稿C(8)课件2.1 Theory of Equivalence（等值等值）In any decision making process, we have to account for the benefits and the costs of a project. In a typical project, the costs occur at the beginning of the project and the benefits accur over a period of time. For example, installation of a waterflood project results in benefits in terms of additional oil recovery over several future years. 技术经济学英文版演示文稿C(8)课件However, to install waterflood, we may have to incur sufficient costs at the present time. This money has to come from internal capital of a corporation or from a lending institute. Either way, by using internal capital or by borrowing money, we will either lose the opportunity to invest the money somewhere else or we will have to pay interest to the lending institution. 技术经济学英文版演示文稿C(8)课件That is, instead of investing the money in a waterflooding project, we could have earned interest from a bank by investing that money in the bank; or, if we have borrowed the money to invest in this project, we have to pay interest on the borrowed amount. This lost opportunity (opportunity cost) or the interest payment has to be accounted for in our cost benefit analysis. One way to do this is through the understanding of the time value of money. 技术经济学英文版演示文稿C(8)课件Money is a valuable commodity. People will pay you to use your money. You will have to pay someone to use their money. The cost of money is established and measured by an interest rate. Interest rate is periodically applied and added to(额外的额外的) the amount (the principal) over a specific period of time. 技术经济学英文版演示文稿C(8)课件 For example, depositing $100 in the bank for one year may generate an interest of $6 at the end of the year. That is, the bank has paid you 6% interest to use your money. In the same vane, the bank will turn around and lend that money to another individual and charge 10% interest. The borrower will have to pay back $110 at the end of one year. We can also say that, for you, $100 today is equivalent to $106 one year from now. For the bank, $100 today is equivalent to $110 one year from now. This principal is called a theory of equivalence. 技术经济学英文版演示文稿C(8)课件This simple example clearly illustrates that money has an earning power. Just like any other commodity, money can be put to work and earn more money for its owner. Because of this, a dollar received today is worth more than a dollar received one year from now. Todays dollar can be invested to earn more than one dollar one year from now.技术经济学英文版演示文稿C(8)课件2. l Theory of EquivalenceOnce we know that money is a valuable commodity and it has an earning power, we need to establish a method to compare the monies collected at different times. For example, if you have the option of receiving $100 today or $110 one year from now, what would you prefer? If you can only invest $100 in a bank at 5% interest rate, you can only earn $5 one year from now. That will total up to $100 + $5 = $105 one year from now. Obviously, $105 is less than $110 you could receive one year from now. Therefore, you would prefer $110 one year from now.技术经济学英文版演示文稿C(8)课件The economic equivalence is also affected by the time. Consider the extension of the previous example. If some one offers you $100 today, $110 one year from now or $121 two years from now, which one would you prefer? As before, It depends on what you can do with the $100 you will have today. If you invest it at 5% interest rate, after one year you will have $100 + $100(.05) = $105After two years, you will have $105+ $105(.05) = $110.25Since $110.25 is smaller than $121, you would prefer $121 two years from today. On the other hand, if you can earn 25% interest rate, you will have, after one year $100 + $100(0.25) = $125and after two years $125 + $125(0.25) = $156.25技术经济学英文版演示文稿C(8)课件Think about why this principle is so important. When conducting the cost benefit analysis of any project, if the benefits are received in the future, we cannot directly compare the present costs to the future benefits unless we can convert the future benefits to equivalent present benefits; or, in the alternative, we will have to convert the present cost to equivalent future costs. In this chapter, we will learn how we can accomplish this task.技术经济学英文版演示文稿C(8)课件2.2 Equivalence RelationshipsTo understand the theory of equivalence in a more rigorous way, we need to establish certain relationships starting from some basic parameters. We can define these parameters as: P = present sum of money F = future sum of money A = end of period cash payment or receipt i = interest rate per period n = number of periods 技术经济学英文版演示文稿C(8)课件In addition to defining the necessary parameters, we will also define the cash flow diagrams we will be using in this book. The cash flow diagram represents the cash flow pro the life of the project. For example, after investing $10,000 at the beginning of the project, if we receive $3,000 each year in benefits for five years, we can draw the cash diagram as in Figure 2.1. 技术经济学英文版演示文稿C(8)课件 Figure 2.1: Cash Flow diagramP=$10,000A=$3,000 01 2 3 4 5=n技术经济学英文版演示文稿C(8)课件In this cash flow diagram, the horizontal line represents the time scale; the arrows signify cash flows and are placed at the end of period. Upward arrows represent cash receipts (benefits), whereas the downward arrows represent expenses (costs). Note that cash flow diagram is a function of whose point of view it represents. 技术经济学英文版演示文稿C(8)课件 0 1 2 3 4 5P=$10,000A=$3,000Figure 2.2 :Cash Flow Diagram-Borrowers Perspective技术经济学英文版演示文稿C(8)课件2.2.1 Relationship Between P and FThe cash flow diagram for establishing the relationship between the present sum and the future sum can be drawn as in Figure 2.4. 0 1 2 3 n-1 nPFFigure 2.4: Relationship Between P and F技术经济学英文版演示文稿C(8)课件(1) P, i, n-F=?after one period P+Pi=P(1+i)If we continue to invest the new principal, p(l + i), for another period ,after n periods, (2.1)技术经济学英文版演示文稿C(8)课件(2) F, i, n- P =? 0 1 2 3 n-1 nPFFigure Relationship Between P and F技术经济学英文版演示文稿C(8)课件Example 2.2 If you need $10,000 after 5 years, how much should you invest today at an interest rate of 10%?SolutionGiven: F = $10.000, n = 5 years, i = 10%Find: PEq. 2. I can also be written as, 技术经济学英文版演示文稿C(8)课件Using Eq. 2.2,You need to invest $6,209 today.技术经济学英文版演示文稿C(8)课件Example 2.3 If you invest $4,000 in a bank at an interest rate 6.25%,the money would you have at the end of three years?SolutionGiven: P = $4,000, n = 3 years, i = 6.25%, Find: F Using Eq. 2.1You will have $4,798 after three years.技术经济学英文版演示文稿C(8)课件Example 2.4 If you want to invest $3,000 at an interest rate of 7%, how long will it take to double the initial investment?SolutionGiven: P = $3,000, F = $6,000, i = 7%Find: nUsing Eq. 2.1,Taking log on both sidesThe amount will double in 10.2 years.技术经济学英文版演示文稿C(8)课件Thumb of rule (rule of 72)技术经济学英文版演示文稿C(8)课件2.2.2 Relationship Between A and FLet us extend the previous relationships to a case where a payment is made at the end of each period. We would like to calculate the future value of these payments at the end of the total period. Cash flow diagram for this arrangement is shown in Fig. 2.5.Figure 2.5: Periodic PaymentsFn0 1 2 3 n-1 A A A A A技术经济学英文版演示文稿C(8)课件(3) A, i, n-F=?Considering that for the first payment, we earned interest for (n - l) periods (see Fig. 2.6), and for the last payment we earned no interest, using Eq. 2.l, we can write, (2 3)multiplying Eq. 2.3 by (l + i), (2.4)subtracting Eq. 2.3 from Eq. 2.4 and rearranging, we obtain, 技术经济学英文版演示文稿C(8)课件Therefore, (2.5) (4) F, i, n-A=?Figure Periodic PaymentsFn 0 1 2 3 n-1 A A A A A技术经济学英文版演示文稿C(8)课件 (4) F, i, n-A=?We can rewrite q. 2.5 as, (2.6)Eq. 2.5 and Eq. 2.6 establish the relationships between A and F.技术经济学英文版演示文稿C(8)课件Example 2.5 If you deposit $10,000 at the end of each year, how much would you accumulate at the end of five years at an interest rate of 6%?SolutionGiven: A = $10,000, n = 5 years, i = 6%Find: FUsing Eq. 2.5,You would have $56,371 at the end of 5 years.技术经济学英文版演示文稿C(8)课件Example 2.6 If you need $100,000 at the end of 10 years for a college education, how much should you invest at the end of each year at an interest rate of 8%?SolutionGiven: F = $100,000, n = 10 years, i = 8%Find: AUsing Eq. 2.6, You should invest $6,903 at the end of each year to receive $100,000 at the end of 10 years.技术经济学英文版演示文稿C(8)课件Example 2.7 You intend to invest $1,000 per year in mutual funds. If the average annual yield from this fund is expected to be 12%, how long will it take before you would have accumulated $15,000 in your account?SolutionGiven: A=$1,000, F = $15,000, i = 12%Find: nUsing Eq. 2.5,Substituting,Therefore, n = 9.1 yearsIt will take approximately 9. 1years before $15,000 would be accumulated.技术经济学英文版演示文稿C(8)课件Example 2.8 After graduating from college, Betty desires to buy a house worth $150,000 with a 20% down payment after 5 years. Bettys father gives her $10,000 as a graduation gift. If Betty invests that money at 6% interest rate, how much additional annual savings will she have to invest at the same interest rate to accumulate the desired 20% down payment at the end of 5 years?SolutionGiven: F = 20% of $150,000, P = $10,000, i = 6%, n = 5 yearsFind: A技术经济学英文版演示文稿C(8)课件In this example, Betty is investing $10,000 at the beginning of year l plus additional annual investments to get $30,000 at the end of 5 years. The cash flow diagram can be drawn as shown in Fig. 2.7.A A A A AFigure 2.7: Cash Flow Diagram for Example 2.8F=$30,000$10,0000 1 2 3 4 5技术经济学英文版演示文稿C(8)课件As a first step, we can calculate the future value (F1 ) of $10,000 after 5 years.Using Eq. 2. l,The remaining future value has to be the result of annual investments. We can calculate the remaining future value Using Eq. 2.6. That is, Betty will have to invest $2,948 at the end of each year.技术经济学英文版演示文稿C(8)课件2.2.3 Relationship Between A and PLet us extend the relationship one step further by relating the present value to the periodic payments. As shown in Fig. 2.8, we want to calculate the present value of future periodic payments. 0 1 2 3 n-1 n A A A A AP=?Figure 2.8: Relationship Between P and A技术经济学英文版演示文稿C(8)课件(5) A, i, n-P=?From Eq. 2.l, we know that From Eq. 2.5, we know thatSubstituting Eq. 2. l in Eq. 2.5, we can writeSimplifying, (2.7) 技术经济学英文版演示文稿C(8)课件(6) P, i, n-A=?Eq.2.7 can also be written as (2.8)Eq. 2.7 and Eq. 2.8 establish the relationship between the periodic payment (A) and the present worth ( P) .技术经济学英文版演示文稿C(8)课件Example 2.9 If you take a home improvement loan of $10,000 to be paid over a five year period, what would be the yearly payment if the interest rate is 12% per year?SolutionGiven: P = $10,000, n = 5 years, i = 12%Find: AUsing Eq. 2.8, The yearly payment would be $2,774.技术经济学英文版演示文稿C(8)课件Example 2.10 If you want to invest sufficient money in the bank such that at an interest rate of 8%, you will receive $20,000 per year for the next 10 years, how much should you invest in the bank?SolutionGiven: A= $20,000, n = 10 years, i = 8%Find: PUsing Eq. 2.7, You will have to invest $ 134,202 today to earn $20,000 per year for the next ten years.技术经济学英文版演示文稿C(8)课件Example 2.11 Able borrows $1,000 from a loan shark. In return, the loan shark demands that Able pay $100 per month for a one year period. What is the monthly interest rate the loan shark is charging?SolutionGiven: P = $l.000, A = $100 per month, n = 12 monthsFind: iNotice that the periodic payment and the number of periods are given in terms of monthly units.Using Eq. 2.7. Substituting 技术经济学英文版演示文稿C(8)课件There is no explicit solution for i. By trial and error,ForTherefore, the interest rate charged is 2.9% per month.技术经济学英文版演示文稿C(8)课件Example 2.12 Betty buys a new computer at a price of $10,000. Betty expects that the use of the computer should result in an annual income of $2,500. If Betty wants to earn at least a 15% return on her investment at what price would the computer have to be sold after 4 years?SolutionGiven: P = $10,000, A = $2,500, i = 15%, n = 4yearsFind: Salvage value (resale price) of the computerWe can draw a cash flow diagram for this example as shown in Fig. 2.9. A A A 0 1 2 3 4A=$2500P=$10,000F=?AFigure 2.9: Cash Flow Diagram for Example 2.12技术经济学英文版演示文稿C(8)课件Using Eq. 2.7, we can calculate the present value of periodic payments asSubstituting The remaining present investment will have to be recovered by the future resale price.The remaining present investment is 10,000-7,137=$2,863Using Eq. 2.l, The computer would have to be sold at a price of $5,007 at the end of 4 years.技术经济学英文版演示文稿C(8)课件In solving these and the other examples and problems, please remember that the units of the periodic payment, the interest rate per period and the number of periods have to be consistent. For example, if the payment is paid per month, then the number of periods have to be in months and the interest rate has to be defined per month. Similarly, if the payment is per year, the interest rate has to be defined per year and the number of periods have to be in years.技术经济学英文版演示文稿C(8)课件2.2.4 Gradient Equations（梯度方程）（梯度方程）Arithmetic GradientThe arithmetic gradient series is graphically represented in Fig. 2. 10. In this series, the initial payment is A at the end of period 1. The payment changes by a constant sum during each period. An example would be maintenance cost of a computer. The maintenance cost is, say, $500 in the first year. It increases by $50 each year. This can be represented by arithmetic gradient series where A is $500 and G is $50.A+(n-2)G0 1 2 3 n-1FnAA+GA+2GA+(n-1)GFigure 2.10: Arithmetic Gradient Series技术经济学英文版演示文稿C(8)课件技术经济学英文版演示文稿C(8)课件0 1 20 1 2AA+GA+(n-1)GFnFAeq Aeq Aeq=Figure 2.111 Equivalent Constant Value Payment for Arithmetic Gradient Seriesn技术经济学英文版演示文稿C(8)课件技术经济学英文版演示文稿C(8)课件Example 2.13 If you invest $1,000 in the bank in the first year followed by a steady increase of $50 per year over the next ten years, how much money will you get back at the end of the ten year period if the interest rate is 8%?SolutionGiven: A = $l,000, G = $50, i = 8%, n= 10yearsFind: FUsing Eq. 2.13,You will get $17,291 at the end of ten years.技术经济学英文版演示文稿C(8)课件Example 2.14 A company buys a new computer. The maintenance agreement requires that the company pay $2,000 in the first year for the maintenance cost to be increased at a rate of $200 per year. If the company intends to keep the computer for six years, how much money should it set aside to cover the maintenance costs if the money is invested at a 6% interest rate?What would be an equivalent constant annual maintenance cost for the above cost schedule?SolutionGiven: A = $2,000, G = $200, n = 6 years, i= 6%Find: P, AeqUsing Eq. 2.14.技术经济学英文版演示文稿C(8)课件A total sum of $ 12,127 has to be set aside to cover the maintenance costs.To calculate an equivalent annual cost, we use Eq. 2. 15,The equivalent, constant annual maintenance cost is $2,466.技术经济学英文版演示文稿C(8)课件Example 2.15 Able intends to invest $200 in the first month followed by an increase of $3 per month in a mutual fund account for his daughters education. His daughter is 8 years old and he intends to have $100,000 at the time she is 18 years old. What interest rate does he need to earn on his investment? SolutionGiven: A = $200, G = $3, n = 120 months, F = $100,000Find: iUsing Eq. 2.13,技术经济学英文版演示文稿C(8)课件Substituting,There is no explicit solution. We will have to solve for i through trial and error.For i =1%/month, right hand side of above equation = 79,019For i =l.5%/month, right hand side of above equation = 108,515For i = 1.37%/month, right hand side of above equation = 99,680Therefore, Able will have to earn at least l.37% per month interest to get $100,000after ten years.技术经济学英文版演示文稿C(8)课件A(1+g)n-1A(1+g)n-2Geometric GradientGeometric gradient represents a series where the periodic payment changes by a constant factor. The relationship between the changing payment and the future sum is shown in Fig. 2.12. A0 1 2 n-1FnA(1+g)Figure 2.121 Geometric Gradient Series技术经济学英文版演示文稿C(8)课件This equation is applicable only if . If , Eq. 2.16 simplifies to,技术经济学英文版演示文稿C(8)课件If , an equation for Aeq can be written asIf , an equation for Aeq can be written as, 技术经济学英文版演示文稿C(8)课件Example 2.16 If a person invests $10,000 in the first year with a 10% increase in each subsequent year, how much money would be accumulated at the end of ten years at an interest rate of 8%?SolutionGiven: A =$10,000, g =10%, i=8%, n=10yearsFind: FUsing Eq. 2.18,After ten years, the person would collect $217,410.技术经济学英文版演示文稿C(8)课件Example 2.17 If as an employer, you guarantee one of your employees an initial salary of $20,000 per year and an annual increase of at least 6% for the next five years. What is the minimum amount of money you need to set aside at an interest rate of 7% to cover the cost of the employees salary? If the interest rate is 6%, how much more needs to be set aside?SolutionGiven: A= $20,000, g=6%, i=7%, n=5 yearsFind: P技术经济学英文版演示文稿C(8)课件Using Eq. 2.20,At 7% interest rate, $91,727 has to be set aside to cover the cost of the employees salaries. If the interest rate is 6%, i=g . Therefore, using Eq. 2.21,Therefore, at an interest rate of 6%, an additional $2,621 (94340-91,719) has to be set aside. 技术经济学英文版演示文稿C(8)课件2. 3 Nominal and Effective Interest Rates名义利率名义利率 实际利率实际利率Example 2.19 $1000 is invested in a bank at a rate of 12% per year. The bank statement states that“the interest is compounded every month based on average daily balance during that month.” How much money would you accumulate at the end of one year?技术经济学英文版演示文稿C(8)课件SolutionThe interest rate is equal to 12% a year. Since the interest is compounded each month, we need to calculate the monthly interest rate.技术经济学英文版演示文稿C(8)课件In this example, P = $1,000, i = 1%/month, n = 1 year= 12 months.Substituting,That is, we would have accumulated $1,126.80 at the end of the one year period.Instead of accumulating the interest every month, if the interest is accumulated at the end of the year, we would receive,技术经济学英文版演示文稿C(8)课件The two amounts are not equal. By compounding the interest more frequently, we have accumulated more money. We can define an equivalent yearly rate which will give us the same amount of money as the monthly compounding. For example, if we assume that our interest rate is 12.68%, and the interest is compounded at the end of each year, at the end of one year, we would accumulate,This is exactly the same amount we would receive if the interest is 12% per year, but is compounded each month.12.68% in this example, therefore, is called the effective interest rate, and 12% is called the nominal interest rate.技术经济学英文版演示文稿C(8)课件Let us consider the development of the relationship between the nominal and the effective interest rate. If we define, j = nominal interest rate per period M=number of compounding sub-periods.Using Eq. 2.25, we can calculate the nominal interest rate per sub-period as, If we invest principal P for one period ( M sub-periods), we can calculate the future value of the principal as, using Eq. 2. l, (2.26)技术经济学英文版演示文稿C(8)课件 (2.28)Note that if compounding sub-periods, M, is equal to one, then, 技术经济学英文版演示文稿C(8)课件Example 2.20 A bank advertises in a newspaper: “Invest a minimum of $5,000 today in a CD (Certificate of Deposit) account at an interest rate of 8.5% per year and receive an effective yield of 8.87% per year by compounding the interest daily.” Do you believe that this statement is accurate?SolutionGiven: j = 8.5% per year, M = 365 (365 sub-periods in one year)Find: iUsing Eq. 2.28, The effective interest rate is 8.87%. Therefore, the statement is accurate.技术经济学英文版演示文稿C(8)课件Example 2.21 A credit card agreement states that“ The finance charge will be calculated on a monthly basis based on the Average Daily Balance.” Further it states that the finance charge is“ l.75% which is an ANNUAL PERCENTAGE RATE OF 21%.”Is this an accurate statement? SolutionThe finance charge is calculated on a monthly basis. If we assume that the nominal interest rate per year is 21%, then the monthly nominal interest rate is, using Eq.2.25.技术经济学英文版演示文稿C(8)课件Therefore, the statement in the credit card is accurate to the extent that 21% is the Nominal annual percentage rate.However, the interest is compound monthly. Therefore, M is equal to 12. Using Eq. 2.28,The effective interest rate is 23.14%, which is much greater than 21% as advertised.We, therefore, can say that the statement in the agreement is misleading(令人误解的令人误解的) because it does not specify whether the interest is nominal or effective.技术经济学英文版演示文稿C(8)课件In defining the time value of money, we should use the effective interest rate rather than the nominal interest rate in our computations. The effective interest rate is the actual interest rate charged to the principal. In applying any of the equations we learned in the previous section, it should be remembered that the units for many of the terms should be consistent. The determining factor in most instances is the periodic payment. 技术经济学英文版演示文稿C(8)课件The interest rate used in any calculation should be the effective interest rate per period (period defined based on the periodic payment) and the number of periods should be defined in a consistent unit as the periodic payment. For example, if the periodic payment is per month, then the effective interest rate should be per month, and the number of periods should be in months.技术经济学英文版演示文稿C(8)课件Example 2.22 Able borrows $50,000 for a lending institution for building a house. The lending institution charges 8% per year nominal interest rate compounded daily. If Able intends to pay the loan back in 10 years, what would be his monthly payment?SolutionGiven: P = $50,000, j = 8% per year compounded daily, n = 10 yearsFind: A per monthIn this example, the payment needs to be calculated per month. Therefore, we need to define the effective interest rate per month as well as the number of periods in months. 技术经济学英文版演示文稿C(8)课件Using Eq. 2.25,Number of sub-periods per month = 30.4 days/month. Using Eq. 2.28,Number of periods = 10 years = 10 12 = 120 months. Using Eq. 2.8,The monthly payment would be $607.30.技术经济学英文版演示文稿C(8)课件Example 2.23 A department store advertises on September 1st. “Buy now and dont make the first payment till January 31st at 0% interest.” At the bottom of the advertisement, in small print, the advertisement states that “After January 31st, the interest will be charged at 18% annual rate compounded daily.”If you buy a computer for $2,500 on September 1st, and pay it off on January 31st,what is the effective interest rate you have paid?Instead of paying on January 31st, you start making the first payment on February 28. You make twelve constant monthly payments, what would be your monthly payment? What is the effective interest rate charged to you?技术经济学英文版演示文稿C(8)课件SolutionPart I: Payment on January 31stSince the interest charged till January 31st is 0%, the payment will be $2,500 on January 31st. The payment and the borrowed amount is the same (by buying the computer and not paying for it, you are borrowing $2,500 from the store!), therefore, the effective interest rate is zero.Part II: Constant monthly payments starting at the end of February.In this case, the interest will not start accumulating till the first day of February. In other words, whether we bought the computer on January 31st or September 1st,the loan amount will be the same - $2,500. We, therefore, can calculate the monthly payment on $2,500 by using the appropriate monthly effective interest rate. 技术经济学英文版演示文稿C(8)课件Using Eq. 2.25,For daily compounding, M = 30.4 days. Using Eq. 2.28,Using Eq. 2.8. for n=12,You will have to make a monthly payment of $229.21 for 12 months to pay off the $2,500 loan.技术经济学英文版演示文稿C(8)课件 To calculate the effective rate, note that for five months (September 1 - January 31st), you did not pay any interest. After that over a 12 month period, you paid an interest rate of l.51% per month, the effective rate should be somewhere between. If we define the effective interest rate, i, per month, we can write that the principal loan $2,500 should have accumulated at that rate. Therefore, on January 31st, the future value of the loan should be, using Eq. 2.l,技术经济学英文版演示文稿C(8)课件This will be the new principal which should be paid off in twelve months. So. we can write, using Eq. 2.8, our monthly payment as,But we already know that our monthly payment is $229,21. Therefore, we need to find a value of such that the right hand side of the above equation is equal to $229,21. By trial and error,For That is, the effective interest rate is 0.85% per month. This is smaller than the nominal interest rate.技术经济学英文版演示文稿C(8)课件In some instance, we can simplify our relationship between the nominal and the effective interest rate if the compounding is continuous. Recall that, (2.28)represents the relationship between the effective and the nominal interest rate. If M is very large (continuous compounding), we can write,where i=the effective interest rate per period and j is the nominal interest rate per period. This equation represents the relationship between the nominal and the effective interest rate for continuous compounding.技术经济学英文版演示文稿C(8)课件Example 2.24 If the nominal interest rate is 16% per year, calculate the effective interest rate if.a. The interest is compounded quarterly.b. The interest is compounded monthly.c. The interest is compounded daily.d. The interest is compounded continuously.Solutiona.Quarterly compounding,b. Monthly compounding,c. Daily compounding,D.Continuous compounding, using Eq. 2.30,技术经济学英文版演示文稿C(8)课件When the interest is compounded continuously, instead of using the conventional relationships between F, P and, A, we can use the alternate expressions. We know that, (2.30)for continuous compounding. We can write Eq. 2.30 as, (2.31)Knowing the relationship between F and P as, (2.1)we can substitute (l + i) with Eq. 2.31. We obtain, (2.32)技术经济学英文版演示文稿C(8)课件技术经济学英文版演示文稿C(8)课件To generalize, if we want to calculate the remaining principal after k payments, we can first calculate the remaining balance after k payments based on the remaining payments. The remaining payments are equal to (n-k), where n is the total number of payments. If i is the effective interest rate per month, we can write the balance of the principal after k payments as, using Eq. 2.8,技术经济学英文版演示文稿C(8)课件where is the balance of the principal after k payments.Therefore, the interest paid for (k + l) payment, (2.37)where is the interest payment in (k + l) payment.Therefore, part of the payment towards principal, (2.38)where is the part of the payment which goes towards the principal.技术经济学英文版演示文稿C(8)课件Example: A company decides to borrow $100,000 of the required capital from a bank, the five year loan is borrowed at an interest rate of 10%. How much the payment of each year as well as the interest and the principal payment of each year ?For 1 year, interest payment as: orPrincipal payment as: 技术经济学英文版演示文稿C(8)课件Or yearInterest Payment Principal payment Total Payment110,00016,38026,38028,36218,01826,38036,56019,81826,38044,57821,80126,38052,39823,88226,380技术经济学英文版演示文稿C(8)课件Summary 1、基本复利公式、基本复利公式 公公 式式 名名 称称 已已 知知 求求 解解 计计 算算 公公 式式一次复利终值公式一次复利终值公式 P，i，n F一次复利现值公式一次复利现值公式 F，i，n P 年金终值公式年金终值公式 A，i，n F资金储存公式资金储存公式 F，i，n A年金现值公式年金现值公式 A，i，n P资金恢复公式资金恢复公式 P，i，n A技术经济学英文版演示文稿C(8)课件 以上各公式系数之间的关系是：以上各公式系数之间的关系是： （1） 倒数关系倒数关系 （2） 乘积关系乘积关系 （F/A，i，n）（）（P/A，i，n）（F/P，i，n） （F/P，i，n）（）（A/P，i，n）（F/A，i，n） （P/A，i，n）（）（P/F，i，n）（）（F/A，i，n） （3） 加减关系加减关系 （A/P，i，n）（A/F，i，n）i （F/A，i，n）1+（F/P，i，1）+（F/P，i，2）+（F/P，i，n-1） （P/A，i，n）（）（P/F，i，1）+（P/F，i，2）+（P/F，i，n）技术经济学英文版演示文稿C(8)课件 2、基本复利公式极限值、基本复利公式极限值公公 式式 名名 称称 计计 算算 公公 式式 n i0 一次复利终值公式一次复利终值公式 FP1一次复利现值公式一次复利现值公式 0 PF1年金终值公式年金终值公式 FAn资金储存公式资金储存公式 0 AF年金现值公式年金现值公式 PA PAn资金恢复公式资金恢复公式 APi AP技术经济学英文版演示文稿C(8)课件Summary技术经济学英文版演示文稿C(8)课件Su技术经济学英文版演示文稿C(8)课件