金融数学引论答案第二章北京大学出版[1]

《金融数学引论》复习提纲第一章 利息的基本计算 第一节 利息基本函数一. 累积函数a(t)与总量函数A(t)某一度量期的实际利率是指该度量期内得到的利息金额与此度量期开始时投资的本金金额之比，通常用字母i 来表示。

利息金额I n =A(n)-A(n-1)对于实际利率保持不变的情形，i=I 1/A(0)； 对于实际利率变动的情形，则i n =I n /A(n-1)； 二．单利和复利考虑投资一单位本金，（1） 如果其在t 时刻的积累函数为 a(t)=1+i*t ，则称这样产生的利息为单利；实际利率 )()()()(1111-+=---=n i in a n a n a i n （2） 如果其在t 时刻的积累函数为a(t)=(1+i)t ，则称这样产生的利息为复利。

实际利率 i i n =例题：1.1三.. 贴现函数一个度量期的实际贴现率为该度量期内取得的利息金额与期末的投资可回收金额之比，通常用字母d 来表示实际贴现率。

等价的利率i 、贴现率d 和贴现因子（折现因子）v 之间关系如下：,(1),1111,,,1d i i d i i d d iv d d iv v i d idi=+==-+=-==-=+例题：1.2四．名利率与名贴现率用()m i 表示每一度量期支付m 次利息的名义利率，这里的m 可以不是整数也可以小于1。

所谓名义利率，是指每1/m 个度量期支付利息一次，而在每1/m 个度量期的实际利率为()/m i m 。

与()m i 等价的实际利率i 之间的关系：()1(1/)m m i i m +=+。

名义贴现率()m d ，()1(1/)m m d d m -=-。

名义利率与名义贴现率之间的关系：()()()()m m m m i d i d m m m m-=⋅。

例题：1.3五．连续利息计算定义利息强度（利息力）为()()()()t A t a t A t a t δ''==， 0()ts dsa t e δ⎰=一个常用的关系式如下：()()11[1]1(1)[1]m p m p i d i v d e m pδ---+=+==-=-=例题：1.4要求：δ,,,,)()(p m d i d i ，之间的计算。

金融数学引论答案第二版【篇一：北大版金融数学引论第二章答案】>第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。

如果它们前十年每年底存款1000元，后十年每年底存款1000+x 元，年利率7%。

计算x 。

解：s = 1000s?7%+xs?7%20p10p20px = 50000 ? 1000s?7% = 651.72s?p7%102．价值10,000元的新车。

购买者计划分期付款方式：每月底还250元，期限4年。

月结算名利率18%。

计算首次付款金额。

解：设首次付款为x ，则有10000 = x + 250a?p1.5%48解得x = 1489.3613．设有n年期期末年金，其中年金金额为n，实利率i =n解：p v = na?npi= 1nn+2 =(n + 1)nn2n4．已知：a?pn= x，a?p2n= y 。

试用x和y 表示d 。

解： a?p2n= a?pn+ a?p (1 ? d)则nny ? xd = 1 ? ( x ) n5．已知：a?p7= 5.58238, a?= 7.88687, a?= 10.82760。

计算i。

11p18p解：a?p = a?p + a?p v718711解得=i = 6.0%10?p +a∞?p6.证明： 11?v10s。

s10?p北京大学数学科学学院金融数学系第 1 页版权所有，翻版必究证明：s?p + a∞?p=s?10p10+101 = 107．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半年200元，然后减为每次100元。

解：p v = 100a?+ 100a20?8p3% p3% = 2189.7168．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。

然后，从65岁开始每年初领取一定的退休金，共计15年。

设前25年的年利率为8%，后15年的年利率7%。

计算每年的退休金。

解：设每年退休金为x，选择65岁年初为比较日=解得x = 8101.658。

第一章习题答案1.设总量函数为A(t) = t2 + 2/ + 3 o试计算累积函数a(t)和第n个吋段的利息【仇°解：把t =()代入得4(()) = 3于是：4(t) t? + 2t + 3啲=丽=3In = 4(北)一A(n一1)=(n2 + 2n + 3) — ((n — I)2 + 2(n — 1) + 3))= 2n+l2.对以下两种情况计算从t时刻到冗(£ < n)时刻的利息：(1)厶(0 < r < n);(2)/r =2r(0<r <n).解：(1)I = A(n) - A(t)—In + in-1+ • • • + A+l n(n + 1) t(t + 1)=2 2I = A(n) - A(t)n n=乞h = 土hk=t+l A：=t+13.已知累积函数的形式为：Q(t) = at2 +几若0时刻投入的100元累积到3吋刻为172元,试计算：5时刻投入的10()元在10时刻的终值。

解：由题意得。

(0) = 1, «(3) = = L72=> a = 0.0& 6=14(5) = 100>1(10) = 4(0) • «(10) = 4⑸• W = 100 x 3 = 300.a(5)4.分别对以下两种总量函数计算订和讪:(1) A(t) = 100 + 5t; (2) A(t) = 100(1 + 0.1尸・解：(1)_ 4(5) - 4(4)5 _ 4(4)5二面-.17% . 4(10)-4(9)210 =—4(9)—5=—^ 3.45%145⑵_ 4(5) - 4(4)5 - 4⑷_ 100(1 + 0.1)5 - 100(1 + 0.1)4 = 100(1+ 0.1)4=10%. 4(10) —4(9)皿=_ 100(1+ O.1)10-100(1+ 0.1)9 = 100(1 + 0.1)9=10%5•设4(4) = 1000, i n = O.Oln.试计算4(7)。

第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。

如果它们前十年每年底存 款1000元，后十年每年底存款1000+X 元，年利率7%。

计算X 。

解：20|7%10|7%50000100020|7%10|7% 1000 651.72s s s S s X X -=+==2．价值10,000元的新车。

购买者计划分期付款方式：每月底还250元，期限4年。

月结算名利率18%。

计算首次付款金额。

解： 设首次付款为X ，则有48|1.5%1000250X a =+解得X = 1489.363．设有n 年期期末年金，其中年金金额为n ，实利率i = 1。

试计算该年金的现值。

解：22|1( 1)1( 1)n n n n i nv n n n PV na n n n+-+-===+ 4．解： ]]]2(1)nn n n a a a d =+-则1 1()n Y X d X -=- 5．已知：]]]71118 5.58238, 7.88687, 10.82760a a a ===。

计算i 。

解：]]]718711a a a v =+解得i = 6.0%6.证明：]]]10101 110s a v s ∞+=- 证明：]]]1010101010(1)111(1)11i s a i i i s v i∞+-++==+-- 7．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半 年200元，然后减为每次100元。

解：8p]3%20]3%100100 2189.716a a PV =+=8．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。

然后，从65岁开始每年初领取一定的退休金，共计15年。

设前25年的年利率为8%，后15年的年利率7%。

计算每年的退休金。

解： 设每年退休金为X ，选择65岁年初为比较日15]7%100025]8%a s X =￢解得X = 8101.659．已知贴现率为10%，计算8]a 。

第一章习题答案1.设总量函数为A(t) = t2 + 2/ + 3 o试计算累积函数a(t)和第n个吋段的利息【仇°解：把t =()代入得4(()) = 3于是：4(t) t? + 2t + 3啲=丽=3In = 4(北)一A(n一1)=(n2 + 2n + 3) — ((n — I)2 + 2(n — 1) + 3))= 2n+l2.对以下两种情况计算从t时刻到冗(£ < n)时刻的利息：(1)厶(0 < r < n);(2)/r =2r(0<r <n).解：(1)I = A(n) - A(t)—In + in-1+ • • • + A+l n(n + 1) t(t + 1)=2 2I = A(n) - A(t)n n=乞h = 土hk=t+l A：=t+13.已知累积函数的形式为：Q(t) = at2 +几若0时刻投入的100元累积到3吋刻为172元,试计算：5时刻投入的10()元在10时刻的终值。

解：由题意得。

(0) = 1, «(3) = = L72=> a = 0.0& 6=14(5) = 100>1(10) = 4(0) • «(10) = 4⑸• W = 100 x 3 = 300.a(5)4.分别对以下两种总量函数计算订和讪:(1) A(t) = 100 + 5t; (2) A(t) = 100(1 + 0.1尸・解：(1)_ 4(5) - 4(4)5 _ 4(4)5二面-.17% . 4(10)-4(9)210 =—4(9)—5=—^ 3.45%145⑵_ 4(5) - 4(4)5 - 4⑷_ 100(1 + 0.1)5 - 100(1 + 0.1)4 = 100(1+ 0.1)4=10%. 4(10) —4(9)皿=_ 100(1+ O.1)10-100(1+ 0.1)9 = 100(1 + 0.1)9=10%5•设4(4) = 1000, i n = O.Oln.试计算4(7)。

北大版金融数学引论第二章答案版权所有，翻版必究第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。

如果它们前十年每年底存 款1000元，后十年每年底存款1000+X 元，年利率7%。

计算X 。

解：S = 1000s 20¬p7%+Xs 10¬p7%X =50000 −1000s 20¬p 7%s 10¬p7%= 651.722．价值10,000元的新车。

购买者计划分期付款方式：每月底还250元，期限4年。

月结算名利率18%。

计算首次付款金额。

解： 设首次付款为X ，则有10000 = X + 250a 48¬p1.5%解得X = 1489.363．设有n 年期期末年金，其中年金金额为n ，实利率i =1。

试计算该年金的现值。

解：P V = na¬n pi1 − v nn= n 1n=(n + 1)nn 2− n n+2 (n + 1)n4．已知：a¬n p= X ，a 2¬n p= Y。

试用X 和Y 表示d 。

解： a 2¬n p= a¬n p+ a¬np (1 − d )n则Y − X1d = 1 − ( X ) n5．已知：a¬7p= 5.58238, a 11¬p= 7.88687, a 18¬p= 10.82760。

计算i 。

解：a 18¬p = a¬7p + a 11¬p v 7解得6.证明： 11−v10=i = 6.0%s。

10¬p +a∞¬ps10¬p北京大学数学科学学院金融数学系第 1 页版权所有，翻版必究证明：s 10¬p + a ∞¬p(1+i)10−1+11 s 10¬p=i(1+i)10−1ii= 1 − v 107．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半 年200元，然后减为每次100元。

一、单选1.已知风险资产收益为u，损失为d，无风险资产收益为r，则二叉树模型无套利机会的充要条件是（D）A．r d B.u r C.d r u D.d<r<u2.对于不支付红利的股票，假设欧式看涨期权价格EC看跌期权价格E P的施权价都是X，施权日都是T，若()rTEE XeSPC-->-0，问该如何套利（A）A．以价格EP买入一份看跌期权，以价格EC卖出一份看涨期权B. 以价格EP买入一份看涨期权，以价格EC卖出一份看跌期权C. 以价格EP卖出一份看跌期权，以价格EC买入一份看涨期权D. 以价格EP卖出一份看涨期权，以价格EC买入一份看跌期权3.已知u=0.2，d=-0.1,r=0.1,则风险中性的概率p*为（ C ）A.0.1B.1/2C.2/3D.1/34.关于期权，下列说法错误的是（C）A．欧式期权是一种合约，它给予持有者在未来某个指定的时间以预先固定的价格买卖一种资产的权利B．美式期权可以再到期日之前包括到期日的任意时间施权C．期权费不是期权的价格D．期权的购买者的收益是由期权支付的期权费修改的回报5.以利率10%按半年复合，跟以利率9%按月复合相比，那种更好？（A）A．以利率10%按半年复合更好B．以利率9%按月复合更好C．两种方法一样好D．无法比较二、填空1、是一种合约，它给予其持有者以施权价在未来某个指定的时间T购买表的资产的权利。

（欧式看涨期权）2、揭示了有效组合的收益和风险之间的均衡关系，而没有给出任何证券或组合的收益风险关系。

（资本市场线）3、1952年，哈里∙马科威兹建立了的基本框架，提出了解决投资决策中最优化投资配置问题。

（均值方差证券组合模型）4. (远期合约)是在将来固定的日期<成为（交割日）>,以预先指定的价格<称为（远期价格）>买入或卖出一种资产的协议。

（3分）5. （欧式看涨期权）是一种合约，它给予其持有者以预先固定的价格（称为 施权价）在未来某个指定的时间T （称为施权日或到期日）购买一种资产（称为 标的资产）的权利。

CHAPTER 2THE FINANCIAL S YS TEMObjectives∙To provide a conceptual framework for understanding how the financial system works and how it changes over time.∙To understand the meaning and determinants of rates of return on different classes of assets.Outline2.1 What Is the Financial System?2.2 The Flow of Funds2.3 The Functional Perspective2.4 Financial Innovation and the “Invisible Hand”2.5 Financial Markets2.6 Financial Market Rates2.7 Financial Intermediaries2.8 Financial Infrastructure and Regulation2.9 Governmental and Quasi-Governmental OrganizationsSummary∙The financial system is the set of markets and intermediaries used by households, firms, and governments to implement their financial decisions. It includes the markets for stocks, bonds, and other securities, as well as financial intermediaries such as banks and insurance companies.∙Funds flow through the financial system from entities that have a surplus of funds to those that have a deficit.Often these fund flows take place through a financial intermediary.∙There are six core functions performed by the financial system:1.To provide ways to transfer economic resources through time, across borders, and among industries.2.To provide ways of managing risk.3.To provide ways of clearing and settling payments to facilitate trade.4.To provide a mechanism for the pooling of resources and for the subdividing of shares in variousenterprises.5.To provide price information to help coordinate decentralized decision-making in various sectors of theeconomy.6.To provide ways of dealing with the incentive problems created when one party to a transaction hasinformation that the other party does not or when one party acts as agent for another.∙The fundamental economic force behind financial innovation is competition, which generally leads to improvements in the way financial functions are performed. The basic types of financial assets traded in markets are debt, equity, and derivatives.∙Debt instruments are issued by anyone who borrows money—firms, governments, and households.∙Equity is the claim of the owners of a firm. Equity securities issued by corporations are called common stocks.∙Derivatives are financial instruments such as options and futures contracts that derive their value from the prices of one or more other assets.∙An interest rate is a promised rate of return, and there are as many different interest rates as there are distinct kinds of borrowing and lending. Interest rates vary depending on the unit of account, the maturity, and the default risk of the credit instrument. The nominal interest rate is the promised amount of money you receive per unit you lend.∙The real rate of return is defined as the nominal interest rate you earn corrected for the change in the purchasing power of money. For example, if you earn a nominal interest rate of 8% per year and the rate of price inflation is also 8% per year, then the real rate of return is zero.∙There are four main factors that determine rates of return in a market economy:∙the productivity of capital goods—expected rates of return on mines, dams, roads, bridges, factories, machinery, and inventories,∙the degree of uncertainty regarding the productivity of capital goods,∙time preferences of people—the preference of people for consumption now versus consumption in the future, and∙risk aversion—the amount people are willing to give up in order to reduce their exposure to risk.∙Indexing is an investment strategy that seeks to match the returns of a specified stock market index.∙Financial intermediaries are firms whose primary business is to provide customers with financial products that cannot be obtained more efficiently by transacting directly in securities markets. A mong the main types of intermediaries are banks, investment companies, and insurance companies. Their products include checking accounts, loans, mortgages, mutual funds, and a wide range of insurance contracts.Solutions to Problems at End of Chapter1. Do you agree with Adam Smith’s view that society can rely more on the “invisible hand” than on government to promote economic pros perity?Student answers will vary of course.SAMPLE ANSWER:The communist system is the exact opposite of Adam Smith’s invisible hand. And of course we have recently seen the downfall of many of the communist countries around the world. In the communist world, it was believed that government could make better decisions promoting economic prosperity than individuals could. Clearly this system failed to promote economic prosperity. It seems that Adam Smith’s view was that competitive market systems as a whole (rather than government) could best allocate resources to promote economic prosperity. However, a completely unfettered capitalist society such as in the late 1800s in the Western world may n ot have been the perfect system either as the invisible hand helped the “rich get richer” while the poor and needy had no formal assistance. This outraged the moral fabric of society and government programs were eventually set up to formally address thisi ssue of general welfare and “fairness”.2. How does the financial system contribute to economic security and prosperity in a capitalist society?In a capitalist society, it is the price system which helps make capital resource decisions. Capital flows to those operations which can employ it to earn the highest rate of return. This therefore allocates capital to its most productive use, thereby enhancing society’s economic prosperity. In addition, the financial system has markets and intermediaries which transfer risks from those who are least willing to bear it to those who are most willing to bear it. This benefits society as a whole without costing it anything. In addition, by allowing individuals to reduce or eliminate risks, it fosters an atmosphere of undertaking business ventures which also benefits society.3. Give an example of how each of the six functions of the financial system are performed more efficiently today than they were in the time of Adam Smith (1776).Clearing and settling payments:In Adam Smith’s day, just as today there was paper and coin currency. However, due to technological innovations (primarily the computer) today there are many additional forms of payment settlement such as personal checks, credit cards, debit cards and electronic transfer of funds. In addition, certain credit cards and traveler’s checks are accepted everywhere in the world making currency exchange a relic of the past. Pooling resources and subdividing shares:In Adam Smith’s day, most businesses were s mall and were financed by sole proprietorships. Therefore the need to pool resources to finance large investments was not as prevalent or as important as it is today. Again, the technological revolution of computers and telephones allow for global capital marke ts to efficiently finance today’s much larger businesses. Today these companies can access huge pools of money around the world and find the cheapest source of financing for large scale projects.Transfer economic resources: Today there is a worldwide financial system which facilitates the transfer of resources and risk from one individual to another and from one point in time to another. In Adam Smith’s day, although there were financial markets which played a limited role, they were localized, small and much less efficient and innovative than they are today.Managing risk: Of course during Adam Smith’s day individuals and businesses faced many of the same risks they do today (risk of property damage, risk of financial loss, risk of crop failure etc.) Ho wever, there were limited means to offset this risk. There were some insurance companies in place at that time, however, they concentrated on managing business risk rather than personal risk and certainly there was not the same type of insurance. A good ex ample is that in Adam Smith’s day, there was no unemployment insurance. In Adam Smith’s day, there was very little a farmer could do about reducing his risk of crop failure or lower crop prices. Today there are a vast number of markets and securities which can be used to offset individual and business risk as well as a huge network of insurance companies whose role is to transfer risk from those who want to reduce risk to those who want to take on more risk.Price information: During Adam Smith’s day, info rmation traveled slowly. Of course, there were no phones, televisions or radios. News traveled by newspaper and by the mail. Today, information travels around the worldinstantaneously. Due primarily to the growth and innovation in computer and telephone t echnology, information about security prices and performance is known at virtually the same time everywhere around the world.Incentive problems: As discussed above, today’s financial system is large, innovative and global. In Adam Smith’s day, while there were problems of moral hazard and adverse selection (but less of a principal-agent problem) there was not the same financial system and sophistication to deal with these problems as there is today.4. How does a competitive stock market accomplish the result that Adam Smith describes? Should the stock market be regulated? How and why?Student answers will vary.SAMPLE ANSWER:Adam Smith talked about free and competitive markets as a system which allocates capital to its most productive use and greatest value. In a competitive stock market, prices are set through supply and demand. Those companies returning the highest return will be rewarded with the highest prices (or cheapest source of financing). Those companies which are under performing will not be allocated as much capital because they are not as productive. Because the universe of possible investments is huge and because it is at times difficult for investors to discern which companies are the most productive employers of capital, regulation shou ld be required to make sure relevant and standardized information is disseminated to potential investors. This would include regulation on disclosure and also insider trading and stock manipulation. However other forms of market regulation are perhaps not so important from a market efficiency point of view and may even impede society’s overall financial welfare.5. Would you be able to get a student loan without someone else offering to guarantee it?Since most students do not have any earning power (yet) or source of savings or other capital, it is doubtful any intermediary would take that credit risk at any reasonable interest rate.6. Give an example of a new business that would not be able to get financing if insurance against risk were not available.EXAMPLES:∙Chemical company∙Child safety products company∙Airline∙Bank∙Hospital∙Environmental consulting∙Hazardous waste disposal7. Suppose you invest in a real-estate development deal. The total investment is $100,000. You invest $20,000 of your own money and borrow the other $80,000 from the bank. Who bears the risk of this venture and why?The $20,000 of my own money is considered the equity capital and the $80,000 is debt financing. In general it is the equity investors who absorb the primary risk of business failure. This is because if the business goes bankrupt, I will unlikely get any or my money back as the debt holders get paid back before I do. However, the debt holder also faces some risk that it will not even get back all its principal and interest. So lenders do share some of the business risk along with the equity investors.8. You are living in the United States and are thinking of traveling to Germany 6 months from now. You can purchase an option to buy marks now at a fixed rate of $0.75 per mark 6 months from now. How is the option like an insurance policy?An option means you have a choice. In this example you can choose to buy the marks at $0.75 in 6 months but you do not have to. You will only buy the marks at this price if it is cheaper for you to do so (if the spot market at that time is higher). Therefore, like an insurance policy you are protected against a potential loss. You know that the maximum price you will have to pay is $0.75 per mark and that you are protected against any higher price. Presumably you will have to pay something for the price of that option and that can be equated to an insurance premium.9. Give an example of how the problem of moral hazard might prevent you from getting financing for something you want to do. Can you think of a way of overcoming this problem?SAMPLE ANSWER:Suppose I want to start a biotechnology business and I need a lot of financing. The trouble is, I do not want to disclose my technology secrets to potential equity and debt investors. I will have great difficulty raising financing. But I could do the following: At a minimum, I could require all potential lenders and investors to sign agreements saying they will not disclose any of my secrets. Secondly, I could share some of my equity with potential lenders (equity-kickers) and investors (stock and stock options). At least that way they will not be motivated to disclose my secrets to others. Finally, if I decided I did not want to share secrets, I could give collateral in my new plant to the debt lenders and that might make them more comfortable with the issue of moral hazard.10. Give an example of how the problem of adverse selection might prevent you from getting financing for something you want to do. Can you think of a way of overcoming this problem?SAMPLE ANSWER:Suppose I want to start a car leasing business. Initially my plan was to purchase several automobiles and lease them out at attractive annual rates. However, potential lenders were worried that my business would attract individuals who drive great distances each year. Rather than buy their own car and lose significant value, they would lease my cars and take a new one each year. I would not be able to obtain financing for this business until I instituted annual mileage restrictions. This alteration in the business plan was enough to make the lenders comfortable with the potential problem of adverse selection.11. Give an example of how the principal-agent problem might prevent you from getting financing for something you want to do. Can you think of a way of overcoming this problem?SAMPLE ANSWER:Suppose you want to start a personal care products company. However, you have the idea for the business, but you do not want to actually run the business. To do that you have hired an executive from a competitor. He will own no equity in the business but will be paid a salary of $100,000 to start up the business.Trouble with this example is that the executive you have hired has little incentive to make the business really work other than his salary (which presumably he could earn at many different companies). What if this executive is really a spy? It may be difficult to get financing for this venture. The way to solve the problem is if you the owner decide to run the business (you certainly are motivated for it do well) or at a minimum, grant your new employee stock or stock options in the business.12. Why is it that a country’s postage stamps are not as good a medium of exchange as its paper currency? Postage stamps would be much easier to copy (to counterfeit) than paper currency which has intricate designs and is made of special fibers (not easily duplicated). Secondly, postage stamps would not be as durable as paper currency and because of their other use, could easily stick to other items! Finally, because postage stamps are used for another purpose, one might run out of them and have to make a special trip to the post office to get more. Of course, the post office is not as convenient as an ATM machine for getting a new supply of currency.13. Who is hurt if I issue counterfeit U.S. dollars and use them to purchase valuable goods and services?If this were done in great size, everyone would be hurt through the inflation that would result in the increased money supply. However, if done in a s mall amount, the individuals accepting the currency are taking on the risk (without knowing it) that the dollars will not be accepted by others as a medium of exchange.14. Some say the only criterion to use in predicting what will serve as money in the future is the real resource cost of producing it, including the transaction costs of verifying its authenticity. According to this criterion what do you think will be the money of the future?SAMPLE ANSWER:Payments via electronic transfer may become the medium of choice. It is a very cheap way to create currency. The biggest challenge will be to create security systems that do not allow for tampering and fraud. Once this is done and once most individuals and retail establishments have access to the system (through bank accounts and linking computer systems) then this should become the “currency” of choice.15. Should all governments issue debt that is indexed to their domestic price level? Is there a moral hazard problem that citizens face with regard to their public officials when government debt is fixed in units of the domestic currency?The answer is that all governments should issue debt that is indexed to their domestic price level. This is due to the fact that if debts are not indexed to the domestic price level, governments have the incentive to print money to repay those debts, thereby increasing domestic inflation which negatively impacts all of society.16. Describe your country’s system for financing higher education. Wh at are the roles played by households, voluntary non-profit organizations, businesses and government?SAMPLE ANSWER:In the United States, the vast majority of higher education is paid for by individuals through savings. These sums can be supplemented in whole or in part by government-guaranteed loans and through student loans and scholarships provided by universities themselves as well as by private foundations such as those provided by the Fulbright scholarship.17. Describe your country’s system for fin ancing residential housing. What are the roles played by households, businesses and government?SAMPLE ANSWER:In the United States individuals and individual borrowings from savings and loans, commercial banks and mortgage lending companies finance the vast majority of residential housing through individual equity savings. The government guarantees a certain amount of low income mortgages and local governments finance some low-income housing. Businesses play a role through the lending business as well as through the financial markets which provide liquidity for portfolios of certain standardized mortgages.18. Describe your country’s system for financing new enterprises. What are the roles played by households, businesses and government?SAMPLE ANSWER:In the United States, the vast majority of new enterprises is financed through individual savings and through initial public offerings made to the general public. These sources of financing are augmented by established firms which spend research and development (R&D) dollars developing new products and businesses and by venture capital institutions which also provide start-up financing.19. Describe your country’s system for financing medical research. What are the roles played by voluntary non-profit organizations, businesses and government?SAMPLE ANSWER:In the United States, medical research is financed both by non-profit organizations (such as universities and medical facilities as well as organizations such as the American Heart Association) as well as by businesses such as Merck, Johnson & Johnson and Genentech. The government is involved in research grants, primarily to universities.20. Assume there are only two stocks traded in the stock market, and you are trying to construct an index to show what has happened to stock prices. Let us say that in the base year the prices were $20 per share for stock 1 with 100 million shares outstanding and $10 for stock 2 with 50 million shares outstanding. A year later, the prices are $30 per share for stock 1 and $2 per share for stock 2. Using the two different methods explained in the chapter, compute stock indexes showing what has happened to the overall stock market. Which of the two methods do you prefer and why? (See appendix that follows.)DJI-Type Index = Average of Current Prices/Average of Base Prices * 100 = 106.67S&P-Type Index = (Weight of Stock 1 * Current Price of Stock 1 / Base Price of Stock 1 + Weight of Stock 2 * Current Price of Stock 2/Base Price of Stock 2) * 100 = 124The S&P-Type Index accurately reflects what has happened to the total market value of all stocks.。

版权所有，翻版必究第五章习题答案1. 已知某10年期零息票债券兑现值为1000，试对收益率为10%和9%分别计算当前价格。

并说明如果收益率下调10%，债券价格上涨的百分比。

解：(1)记P为买价，则有价值方程:P1(1 + 10%)10 = 1000P2(1 + 9%)10 = 1000解得:P1 = 385.54元P2 = 422.41元(2)收益率下降后P1(1 + 10% ×90%)10 = 1000P2(1 + 9% ×90%)10 = 1000解得:P1 = 422.41元，上涨百分比:9.56%; P2 = 458.93元，上涨百分比:8.65%。

2. 已知26周的短期国债的发行价格为9600元，到期兑现10,000元。

1〕按短期国债计算天数的典型方法计算贴现率；2〕假定投资期恰为半年，计算年收益率。

解：(1)由短期国债的定价公式10000(1 −Y dt360) = 9600解得：Y d = 7.91%(2)由定义设年换算收益率为i，则：9600(1 + i)12 = 10000解得：i = 8.51%3. 短期国债的贴现率均为8%，计算52 周国债与13 周短期国债的年利率之比。

52周实际天数已经超过360，如何处理；年利率之比是指等价年利率之比还是贴现率的比。

4. 某10年期面值为100元的债券半年名息率10%，到期兑现105元，如果收益率为半年换算8%，计算债券的买价。

北京大学数学科学学院金融数学系第1 页版权所有，翻版必究解：由基本公式：P = Fra n p i + Cv n = 100 ×5% ×13.5903 + 105 ×1.04¡20 = 115.875. 由债券价格计算公式，给出以下导数的计算公式，并解释其含义。

1) ∂P∂i , ∂P∂n和∂P∂g2) ∂n∂P和∂n∂P解：(1.1)由基本公式对i求导：∂P∂i= Fr(Da)n p i −nP(n + 1, i) < 0解释：债券的买价随着年限的增加而递减。