Distributed Portfolio and Investment Risk Analysis on Global Grids

Portfolio ManagementPortfolio Management: An OverviewDescribe the portfolio approach to investing1.The portfolio perspective refers to evaluating individual investments by theircontribution on the risk and return of an investor’s portfolio.投资组合视角指的是通过投资组合对风险和回报的贡献来评估个人投资。

2.把所有钱用于买一只股票并不是一种portfolio perspective，把钱分散在多只证券中才能降低风险，增加收益。

3.One measure of the benefits of diversification is the diversification ratio. It iscalculated as the ratio of the risk of an equally weighted portfolio of n securities to the risk of a single security selected at random from the n securities.衡量多样化的好处之一是多样化比率。

它计算的是n证券等加权组合的风险与随机从n证券中选择的单一证券的风险之比。

4.If the average standard deviation of returns for the n stocks is 25%, and thestandard deviation of returns for an equally weighted portfolio of the n stocks is 18%, the diversification ratio is 18/25=0.72.Describe types of investors and distinctive characteristics and needs of each1.Individual investor个人投资者就是个人为了满足生活目标而进行理财的投资者，是牺牲当前消费以期获得未来更高水平消费的个人。

FOREIGN WITHHOLDING TA XESForeign Withholding TaxesDIMENSIONAL FUND ADVISORSAPRIL 2012Investors do not earn gross returns. ey earn netreturns—and the di erence is re ected in costs. e mostcommon costs include management fees, fund operatingexpenses, commissions, bid/ask spreads, market impact,cash drag, and taxes. Some of these costs are more visiblethan others, but they all impact investment performance,and investors should be aware of them.Unfortunately, most investors tend to focus only on coststhat are both quanti able and transparent. For example,the management expense ratio (MER) receives a lot of attention because it is quanti ed, fully disclosed, and signi cant. Meanwhile, other material costs may easily go unnoticed. is brief discusses the various ways that one of the more obscure costs—foreign withholding taxes— can a ect investors.Dividends received from non-Canadian investmentsare usually subject to foreign withholding taxes because most countries require tax to be withheld from dividend payments to foreign investors and be remitted to the local government. Although withholding taxes vary by country, 15% is the most common rate. Taxable investors in Canada receive a credit1 for the amount of foreign taxes paid (up to 15% of dividends), so the tax paid to foreign governments is substantially, if not completely, o set by a reduction in Canadian taxes.2UNITEDSTATESUK/EUROPECANADAASIAPACIFICC anadian investorsshould consider thepotential impact of foreign withholding taxes onreturns. Their tax statusand investment structuremay result in additional unrecoverable withholdingtaxes paid on their foreign dividends.1. Provided the fund makes the appropriate designation.2. W ithholding tax rates vary by country (i.e., some are more and some are less than the typical 15% rate). A maximum foreign tax credit of 15% is applied at the portfolio level, not by country, but if the weighted average withholding tax rate exceeds 15%, then the full amount would not be recovered and some withholding tax would apply.On the other hand, investors do not receive a refundable credit for foreign taxes paid in their non-taxable accounts such as Registered Retirement Savings Plans (RRSPs).In certain situations, Canadians with non-taxable accounts are e ectively taxed on dividends earned from non-Canadian securities.FOREIGN SECURITIES HELD DIRECTLYFigures 1 and 2 illustrate how foreign dividends flow to investors in a Canadian-domiciled fund that directly holds non-Canadian dividend-paying securities (e.g., a Dimensional fund). As shown in both figures, withholding taxes are applied to dividends paid by non-Canadian stocks. The Canadian fund distributes the remaining dividends to investors. Taxable investors receive a foreign tax credit to offset some or all of the withheld taxes, as indicated by the green shading. Non-taxable investors do not, as indicated by thered shading.FIGURE 1:DIMENSIONAL FUND HELD IN A TAXABLE ACCOUNT FIGURE 2:DIMENSIONAL FUND HELD IN A NON-TAXABLE ACCOUNT FOREIGN SECURITIES HELD INDIRECTLYForeign withholding taxes become more onerous and opaque when a US entity stands between a Canadian investor and non-US securities. Two prominent examples are when Canadian investors buy US-listed international equity exchange-traded funds (ETFs) that hold non-US securities, and Canadian-listed international equity “wrap” ETFs. Both scenarios may result in taxes being withheld and remitted twice on the same dividend payment. More important, the additional level of withholding tax cannot be recovered. e following explores these examples in more detail.Dimensional FundForeignDividendsDistributedas ForeignDividends* Not offset by tax credit.Canadian InvestorUnderlyUS-LForeignDividends* Not offset by ta** All or part offseForeignDividendsCanad Dimensional FundForeignDividendsDistributedas ForeignDividendsReceivesForeignCanadian InvestorUS-LisForeignDividends* Not offset by tax cr** All or part offset byForeignDividendsCanadiaUS-LISTED INTERNATIONAL EQUITY ETFsFigures 3 and 4 illustrate how dividends from non-US securities flow to Canadian investors in US ETFs. The first level of tax is withheld on dividend payments from the underlying portfolio to the US ETF. The ETF then distributes these dividends, net of expenses and the first level of withholding taxes, to Canadian investors. However, as shown in Figure 3, taxes are further withheld in taxable accounts and remitted to the US on the flowthrough of these dividends because the ETF is a US security paying a dividend to a foreign investor—in this case, a Canadian individual. In these accounts, the second level of withholding tax on the dividend payment from the US ETF can be substantially, if not completely, recovered by a foreign tax credit, but the first level of withholding tax on the dividends from the underlying portfolio to the US ETF cannot.FIGURE 3:US-LISTED INTERNATIONAL EQUITY ETF HELD IN A TAXABLE ACCOUNT Figure 4 illustrates the dividend flow for a non-taxable account. There is no tax withheld on dividends froma US security in non-taxable accounts, such as RRSPs. However, investors with non-taxable accounts do pay withholding taxes at the first level, when dividends flow from the underlying portfolio to the US-listed ETF. FIGURE 4:US-LISTED INTERNATIONAL EQUITY ETF HELD IN ANON-TAXABLE ACCOUNTCANADIAN-LISTED INTERNATIONAL EQUITY “WRAP” ETFs Figures 5 and 6 illustrate how dividends from non-US securities flow to investors in Canadian “wrap” ETFs. A “wrap” ETF is one that invests solely in a US ETF that holds an underlying portfolio of non-US securities. This structure is commonly used, among others, by international equity ETFs that hedge the Canadian dollar. The first level of tax is withheld on dividend payments from the underlying portfolio of foreign securities to the US ETF. The US ETF then distributes these dividends, net of expenses and the first level of withholding taxes, toUnderUSForeignDividenDistribuas ForeiDividen* Not offset by** All or part offCan“ForeignDividenCana* Not offset by tax credit.ForeignDividendsCanadian InvestorUnderlying PortfolioUS-Listed ETFForeignDividendsUnderlying PUS-ListedForeignDividendsDistributedas ForeignDividends* Not offset by tax credit.** All or part offset by foreigCanadian-L“Wrap”EForeignDividendsCanadian Inthe Canadian ETF. However, taxes are further withheld and remitted to the US on the flowthrough of these dividends because the ETF is a US security paying a dividend to a foreign investor—the Canadian-listed ETF. In taxable accounts (Figure 5), the second level of withholding tax on the dividend payment from the US ETF to the Canadian ETF can be substantially, if not completely, recovered by a foreign tax credit, but the first level of withholding tax on the dividends from the underlying portfolio to the US ETF cannot. However, as shown in Figure 6, there is (once again) no refundable credit for foreign taxes paid in non-taxable accounts. So, Canadians with non-taxable accounts are effectively taxed twice when investing in Canadian “wrap” ETFs that hold US ETFs invested in non-US securities. FIGURE 5:CANADIAN-LISTED INTERNATIONAL EQUITY “WRAP” ETF HELD IN TAXABLE ACCOUNT FIGURE 6:CANADIAN-LISTED INTERNATIONAL EQUITY “WRAP” ETFDistributedas ForeignDividends* Not offset by tax credit.Canadian-Listed“Wrap”ETFDividendsCanadian Investor Underlying PortfolioUS-Listed ETFForeignDividendsCanadian-ListedForeignDividendsThis presentation is distributed by Dimensional Fund Advisors Canada ULC for educational purposes only and should not be construed as investment advice or an offer of any security for sale.SUMMARYTable 1 summarizes how withholding taxes apply to international equity investments in both taxable and non-taxable accounts for the three structures discussed. In certain situations, Canadian investors in US ETFs or Canadian “wrap” ETFs holding non-US securities pay an extra level of withholding tax on dividends relative to investing in products that hold these types of securities directly (e.g., a Dimensional fund).IMPACT OF DOUBLE TAXATIONTable 2 contains estimates, for illustrative purposes, of the additional tax drag if the extra level of withholding tax applies. The actual amount incurred will depend on the dividend yield of the underlying portfolio and the withholding tax rate being charged.This extra level of withholding tax is often overlooked because it is buried in the financial statements of the US ETF, or underlying US ETF, rather than beingquantified and clearly conveyed like the cost components included in the MER. However, investors should take into account, among other cost considerations, the tax drag of alternative structures rather than simply comparing MERs.TABLE 2:ESTIMATED TAX DRAG OF ADDITIONAL LEVEL OFWITHHOLDING TAX** Does not account for the deduction of US ETF expenses against dividend income from the underlying portfolio.TABLE 1:SUMMARY OF WITHHOLDING TAX LEVELS* Withholding tax rates vary by country (i.e., some are more and some are less than the typical 15% rate). A maximum foreign tax credit of 15% is applied at the portfolio level, not by country, but if the weighted average withholding tax rate exceeds 15%, then the full amount would not be recovered and some withholding tax would apply.。

Portfolio Management: An OverviewOne measure of the benefits of diversification is the diversification ratio. It is calculated as the ratio of the risk of an equally weighted portfolio of n securities (measured by its standard deviation of returns) to the risk of a single security selected at random from the n securities.例子：If the average standard deviation of returns for the n stocks is 25%, and the standard deviation of returns for an equally weighted portfolio of the n stocks is 18%, the diversification ratio is 18 / 25 = 0.72.Foundations and endowments typically have long investment horizons, high risk tolerance, and, aside from their planned spending needs, little need for additional liquidity.Banks seek to keep risk low and need adequate liquidity to meet investor withdrawals as they occur.Insurance companies invest customer premiums with the objective of funding customer claims as they occur. Life insurance companies have a relatively long-term investment horizon, while property and casualty财产和意外保险(P&C) insurers have a shorter investment horizon because claims are expected to arise sooner than for life insurers.Sovereign wealth funds refer to pools of assets owned by a government.A defined contribution pension plan is a retirement plan in which the firm contributes a sum each period to the employee’s retirement account.In a defined benefit pension plan, the firm promises to make periodic payments to employees after retirement.There are three major steps in the portfolio management process:Step 1: The planning step begins with an analysis of the investor’s risk tolerance, return objectives, time horizon, tax exposure, liquidity needs, income needs, and any unique circumstances or investor preferences.This analysis results in an investment policy statement (IPS)that details the investor’s investment objectives and constraints.Step 2: The execution step involves an analysis of the risk and return characteristics of various asset classes to determine how funds will be allocated to the various asset types.in what is referred to as a top-down analysis, a portfolio manager will examine current economic conditions and forecasts of such macroeconomic variables as GDP growth, inflation, and interest rates, in order to identify the asset classes that are most attractive.Step 3: The feedback step is the final step. Over time, investor circumstances will change, risk and return characteristics of asset classes will change, and the actual weights of the assets in the portfolio will change with asset prices.Mutual funds are one form of pooled investments (i.e., a single portfolio that contains investment funds frommultiple investors). Each investor owns shares representing ownership of a portion of the overall portfolio. The total net value of the assets in the fund (pool) divided by the number of such shares issued is referred to as the net asset value (NA V) of each share.With an open-end fund, investors can buy newly issued shares at the NA V. Newly invested cash is invested by the mutual fund managers in additional portfolio securities. Investors can redeem their shares (sell them back to the fund) at NA V as well. All mutual funds charge a fee for the ongoing management of the portfolio assets, which is expressed as a percentage of the net asset value of the fund. No-load funds免佣基金do not charge additional fees for purchasing shares (up-front fees) or for redeeming shares (redemption fees). Load funds charge either up-front fees, redemption fees, or both.Closed-end funds are professionally managed pools of investor money that do not take new investments into the fund or redeem investor shares. The shares of a closed-end fund trade like equity shares (on exchanges or over-the-counter). As with open-end funds, the portfolio management firm charges ongoing management fees.T ypes of Mutual Funds：Money market funds invest in short-term debt securities and provide interest income with very low risk of changes in share value.Bond mutual funds invest in fixed-income securities. They are differentiated by bond maturities, credit ratings, issuers, and types.A great variety of stock mutual funds are available to investors. Index funds are passively managed; that is, the portfolio is constructed to match the performance of a particular index, such as the Standard & Poor’s 500 Index. Actively managed funds refer to funds where the management selects individual securities with the goal of producing returns greater than those of their benchmark indexes.Other Forms of Pooled Investments：Exchange-traded funds (ETFs) are similar to closed-end funds in that purchases and sales are made in the market rather than with the fund itself.【相同之处】【ETFs和close end fund不同之处】While closed-end funds are often actively managed, ETFs are most often invested to match a particular index (passively managed). With closed-end funds, the market price of shares can differ significantly from their NA V due to imbalances between investor supply and demand for shares at any point in time. Special redemption provisions for ETFs are designed to keep their market prices very close to their NA Vs.【ETFs和open end fund不同之处】ETFs can be sold short, purchased on margin, and traded at intraday盘中交易价prices, whereas open-end funds are typically sold and redeemed only daily, based on the share NA V calculated with closing asset prices.Investors in ETFs must pay brokerage commissions when they trade, and there is a spread between the bid price at which market makers will buy shares and the ask price at which market makers will sell shares.With most ETFs, investors receive any dividend income on portfolio stocks in cash, while open- end funds offer thealternative of reinvesting dividends in additional fund shares.One final difference is that ETFs may produce less capital gains liability compared to open- end index funds. This is because investor sales of ETF shares do not require the fund to sell any securities. If an open-end fund has significant redemptions that cause it to sell appreciated portfolio shares, shareholders incur a capital gains tax liability.A separately managed account is a portfolio that is owned by a single investor and managed according to that investor’s needs and preferences. No shares are issued, as the single investor owns the entire account.Portfolio Risk and Return: Part IHolding period return (HPR) is simply the percentage increase in the value of an investment over a given time period:The geometric mean return is a compound annual rate. When periodic rates of return vary from period to period, the geometric mean return < the arithmetic mean return:The money-weighted rate of return is the internal rate of return on a portfolio based on all of its cash inflows and outflows.Gross return refers to the total return on a security portfolio before deducting fees for the management and administration of the investment account. Net return refers to the return after these fees have been deducted.Note that commissions on trades and other costs that are necessary to generate the investment returns are deducted in both gross and net return measures.Pretax nominal return refers to the return prior to paying taxes.After-tax nominal return refers to the return after the tax liability is deducted.year when inflation is 2%. The investor’s approximate real return is simply 7 - 2 = 5%. The investor’s exact real return is slightly lower, 1.07 / 1.02 - 1 = 0.049 = 4.9%.A leveraged return refers to a return to an investor that is a multiple of the return on the underlying asset.The leveraged return is calculated as the gain or loss on the investment as a percentage of an investor’s cash investment. An investment in a derivative security, such as a futures contract, produces a leveraged return because the cash deposited is only a fraction一小部分of the value of the assets underlying the futures contract. Leveraged investments in real estate are very common: investors pay for only part of the cost of the property with their own cash, and the rest of the amount is paid for with borrowed money.small-capitalization stocks have had the greatest average returns and greatest risk over the period.Covariance measures the extent to which two variables move together over time. A positive covariance means that the variables (e.g., rates of return on two stocks) tend to move together. Negative covariance means that the two variables tend to move in opposite directions.Here we will focus on the calculation of the covariance between two assets’ returns using historical data.The covariance of the returns of two securities can be standardized by dividing by the product of the standard deviations of the two securities. This standardized measure of co-movement is called correlation and is computed as:A risk-averse investor is simply one that dislikes risk (i.e., prefers less risk to more risk). Given two investments that have equal expected returns, a risk-averse investor will choose the one with less risk (standard deviation).A risk-seeking (risk-loving) investor actually prefers more risk to less and, given equal expected returns, willchoose the more risky investment. A risk-neutral investor has no preference regarding risk and would be indifferent between two such investments.The variance of returns for a portfolio of two risky assets is calculated as follows:Note that portfol io risk falls as the correlation between the assets’ returns decreases. This is an important result of the analysis of portfolio risk: The lower the correlation of asset returns, the greater the risk reduction (diversification) benefit of combining assets in a portfolio. If asset returns were perfectly negatively correlated, portfolio risk could be eliminated altogether for a specific set of asset weights.For each level of expected portfolio return, we can vary the portfolio weights on the individual ass ets to determine the portfolio that has the least risk. These portfolios that have the lowest standard deviation of all portfolios with a given expected return are known as minimum-variance portfolios. T ogether they make up the minimum-variance frontier. On a risk versus return graph, the portfolio that is farthest to the left (has the least risk) is known as the global minimum-variance portfolio整体最小方差投资组合.Assuming that investors are risk averse, investors prefer the portfolio that has the greatest expected return when choosing among portfolios that have the same standard deviation of returns. Those portfolios that have the greatest expected return for each level of risk (standard deviation) make up the efficient frontier.An investor’s utility function效用函数represents the investor’s preferences in terms of risk and return (i.e., his degree of risk aversion).An indifference curve is a tool from economics that, in this application, plots combinations of risk (standard deviation) and expected return among which an investor is indifferent.a more risk-averse investor will have steeper indifference curves, reflecting a higher risk aversion coefficient. Combining a risky portfolio with a risk-free asset is the process that supports the two- fund separation theorem, which states that all investors’ optimum portfolios will be made up of some combination of an optimal portfolio of risky assets and the risk-free asset. The line representing these possible combinations of risk-free assets and theoptimal risky asset portfolio is referred to as the capital allocation line.Now that we have constructed a set of the possible efficient portfolios (the capital allocation line) Portfolio Risk and Return: Part IIThe line of possible portfolio risk and return combinations given the risk-free rate and the risk and return of a portfolio of risky assets is referred to as the capital allocation line (CAL).A simplifying assumption underlying modern portfolio theory (and the capital asset pricing model, which is introduced later in this topic review) is that investors have homogeneous expectationsDepending on their preferences for risk and return (their indifference curves), investors may choose different portfolio weights for the risk-free asset and the risky (tangency) portfolio. Every investor, however, will use the same risky portfolio. When this is the case, that portfolio must be the market portfolio of all risky assets because all investors that hold any risky assets hold the same portfolio of risky assets.只有与有效边界相切的那条才是CML。

证券投资英语怎么说证券投资是狭义的投资，是指企业或个人购买有价证券，借以获得收益的行为。

那么你知道证券投资用英语怎么说吗?下面跟着店铺一起来学习一下吧。

证券投资的英语说法1：portfolio investment证券投资的英语说法2：investment securities证券投资相关英语表达：证券投资委员会 Securities Investments Commission证券投资顾问 securities investment adviser国际证券投资 International equity investment组合证券投资理论 portfolio theory证券投资信托基金 securities investment trust fund证券投资经理 Securities Investment Manager ;证券投资的英语例句：1. In addition, we hold two point two billion dollars'worth of investment securities.另外我行持有价值22亿美元的证券投资.2. In this paper, the portfolio investment model with probability criterion is investigated.提出一种基于概率准则的新型组合证券投资模型.3. Foreign capitals include foreign direct investment, international loan and international portfolio investment.外资包括外商直接投资、国际信贷和国际证券投资.4. Portfolio Investment: It covers all international transactions in assets and liabilities.证券投资.它包括所有资产和负债的国际交易.5. Benchmark return is an important concept in theinvestment decision process.基准收益率是证券投资决策过程中需要考虑的一个重要概念.6. In this paper the return and risk of portfolio are analyzed.文章首先分析了组合证券投资的收益率和风险.7. Chinese entrepreneurs to enhance income securities investment funds issue on the 24 th.华商收益增强证券投资基金24日发行.8. Focus on portfolio investment contract translation, translation technology contracts, intellectual property law.专注证券投资合同翻译, 技术合同翻译, 知识产权法.9. On the further regulate the securities investment fund business valuation notice.关于进一步规范证券投资基金估值业务的公告.10. Many countries restrict ability of foreigners to invest in such securities.许多因国家限制外国人对这类证券投资的比例.11. Galaxy Yintai fiscal dividend securities investment funds to update placement prospectuses.银河银泰理财分红证券投资基金更新招募说明书.12. Professional securities investment consulting companies have coMMitted low - level error when the.专业证券投资咨询公司也有犯低级错误的时候.13. Newton had suffered a disappointing setback in investment securities.牛顿曾经在证券投资中惨遭挫败.14. Therefore, risk analysis and control of securities investment have great practical significance.因此, 进行证券投资风险分析和控制,有着重要的现实意义.15. If you are serious about portfolio investment, we must use wisdom.如果您对证券投资是认真的, 就必须使用大智慧.。

英语介绍大数定律The Law of Large NumbersThe law of large numbers is a fundamental concept in probability theory and statistics that describes the tendency of an average of a large number of independent and identically distributed random variables to converge to a specific value as the number of variables increases. This principle is essential in understanding and predicting the behavior of various phenomena, from financial markets to weather patterns, and is widely used in various fields, including science, engineering, and decision-making.The law of large numbers can be understood as follows: as the number of independent observations or trials of a random experiment increases, the average or sample mean of the observations will converge to the expected value or population mean of the underlying probability distribution. This means that as the sample size grows larger, the observed average will become increasingly close to the true average or expected value of the random variable.There are two main types of the law of large numbers: the weak lawof large numbers and the strong law of large numbers. The weak law of large numbers states that the sample mean will converge to the expected value in probability, which means that the probability that the absolute difference between the sample mean and the expected value is greater than a given positive number approaches zero as the sample size increases. The strong law of large numbers, on the other hand, states that the sample mean will converge to the expected value with probability one, which means that the sample mean will almost surely (with probability one) converge to the expected value as the sample size approaches infinity.The law of large numbers has several important implications and applications. First, it provides a theoretical foundation for the use of sample statistics, such as the sample mean and sample variance, to estimate population parameters. This is crucial in statistical inference, where researchers often use sample data to draw conclusions about the underlying population.Second, the law of large numbers is the basis for the central limit theorem, which states that the distribution of the sample mean of a large number of independent and identically distributed random variables will converge to a normal distribution, regardless of the underlying distribution of the individual random variables. This result is essential in statistical analysis and hypothesis testing, as it allows researchers to use the normal distribution to make inferences aboutpopulation parameters based on sample data.Third, the law of large numbers is widely used in decision-making and risk management. For example, in the insurance industry, the law of large numbers is used to predict the likelihood of claims and set appropriate premiums for various insurance policies. Similarly, in finance, the law of large numbers is used to manage portfolio risk and make investment decisions.Furthermore, the law of large numbers has important implications in the field of machine learning and artificial intelligence. In these domains, the law of large numbers is used to justify the use of large datasets and the application of algorithms that rely on the convergence of sample statistics to population parameters.It is important to note that the law of large numbers assumes that the random variables being observed are independent and identically distributed. Violations of these assumptions can lead to biased or misleading results, and it is crucial to carefully consider the underlying assumptions of the law of large numbers in any practical application.In conclusion, the law of large numbers is a fundamental concept in probability theory and statistics that has far-reaching implications in various fields. It provides a theoretical foundation for understandingand predicting the behavior of complex systems and is essential in decision-making, risk management, and the development of advanced technologies. As the amount of data and computational power continues to grow, the importance of the law of large numbers is likely to increase, making it an increasingly crucial tool in the modern world.。

CHAPTER 5: RISK, RETURN, AND THE HISTORICALRECORDPROBLEM SETS1.The Fisher equation predicts that the nominal rate will equal the equilibriumreal rate plus the expected inflation rate. Hence, if the inflation rate increasesfrom 3% to 5% while there is no change in the real rate, then the nominal ratewill increase by 2%. On the other hand, it is possible that an increase in theexpected inflation rate would be accompanied by a change in the real rate ofinterest. While it is conceivable that the nominal interest rate could remainconstant as the inflation rate increased, implying that the real rate decreasedas inflation increased, this is not a likely scenario.2.If we assume that the distribution of returns remains reasonably stable overthe entire history, then a longer sample period (i.e., a larger sample) increasesthe precision of the estimate of the expected rate of return; this is aconsequence of the fact that the standard error decreases as the sample sizeincreases. However, if we assume that the mean of the distribution of returnsis changing over time but we are not in a position to determine the nature ofthis change, then the expected return must be estimated from a more recentpart of the historical period. In this scenario, we must determine how far back,historically, to go in selecting the relevant sample. Here, it is likely to bedisadvantageous to use the entire data set back to 1880.3.The true statements are (c) and (e). The explanations follow.Statement (c): Let = the annual standard deviation of the riskyσinvestments and = the standard deviation of the first investment alternative1σover the two-year period. Then:σσ⨯=21Therefore, the annualized standard deviation for the first investment alternative is equal to:σσσ<=221Statement (e): The first investment alternative is more attractive to investors with lower degrees of risk aversion. The first alternative (entailing a sequence of two identically distributed and uncorrelated risky investments) is riskierthan the second alternative (the risky investment followed by a risk-freeinvestment). Therefore, the first alternative is more attractive to investorswith lower degrees of risk aversion. Notice, however, that if you mistakenly believed that time diversification can reduce the total risk of a sequence ofrisky investments, you would have been tempted to conclude that the firstalternative is less risky and therefore more attractive to more risk-averseinvestors. This is clearly not the case; the two-year standard deviation of the first alternative is greater than the two-year standard deviation of the second alternative.4.For the money market fund, your holding-period return for the next yeardepends on the level of 30-day interest rates each month when the fund rolls over maturing securities. The one-year savings deposit offers a 7.5% holding period return for the year. If you forecast that the rate on money marketinstruments will increase significantly above the current 6% yield, then themoney market fund might result in a higher HPR than the savings deposit.The 20-year Treasury bond offers a yield to maturity of 9% per year, which is 150 basis points higher than the rate on the one-year savings deposit;however, you could earn a one-year HPR much less than 7.5% on the bond if long-term interest rates increase during the year. If Treasury bond yields rise above 9%, then the price of the bond will fall, and the resulting capital losswill wipe out some or all of the 9% return you would have earned if bondyields had remained unchanged over the course of the year.5. a.If businesses reduce their capital spending, then they are likely todecrease their demand for funds. This will shift the demand curve inFigure 5.1 to the left and reduce the equilibrium real rate of interest.b.Increased household saving will shift the supply of funds curve to theright and cause real interest rates to fall.c.Open market purchases of U.S. Treasury securities by the FederalReserve Board are equivalent to an increase in the supply of funds (ashift of the supply curve to the right). The FED buys treasuries withcash from its own account or it issues certificates which trade likecash. As a result, there is an increase in the money supply, and theequilibrium real rate of interest will fall.6. a.The “Inflation-Plus” CD is the safer investment because it guarantees thepurchasing power of the investment. Using the approximation that the realrate equals the nominal rate minus the inflation rate, the CD provides a realrate of 1.5% regardless of the inflation rate.b.The expected return depends on the expected rate of inflation over the nextyear. If the expected rate of inflation is less than 3.5% then the conventionalCD offers a higher real return than the inflation-plus CD; if the expected rateof inflation is greater than 3.5%, then the opposite is true.c.If you expect the rate of inflation to be 3% over the next year, then theconventional CD offers you an expected real rate of return of 2%, which is0.5% higher than the real rate on the inflation-protected CD. But unless youknow that inflation will be 3% with certainty, the conventional CD is alsoriskier. The question of which is the better investment then depends on yourattitude towards risk versus return. You might choose to diversify and investpart of your funds in each.d.No. We cannot assume that the entire difference between the risk-freenominal rate (on conventional CDs) of 5% and the real risk-free rate (oninflation-protected CDs) of 1.5% is the expected rate of inflation. Part of thedifference is probably a risk premium associated with the uncertaintysurrounding the real rate of return on the conventional CDs. This impliesthat the expected rate of inflation is less than 3.5% per year.7.E(r) = [0.35 × 44.5%] + [0.30 × 14.0%] + [0.35 × (–16.5%)] = 14%σ2 = [0.35 × (44.5 – 14)2] + [0.30 × (14 – 14)2] + [0.35 × (–16.5 – 14)2] = 651.175σ = 25.52%The mean is unchanged, but the standard deviation has increased, as theprobabilities of the high and low returns have increased.8.Probability distribution of price and one-year holding period return for a 30-year U.S. Treasury bond (which will have 29 years to maturity at year-end):Economy Probability YTM Price CapitalGainCouponInterest HPRBoom0.2011.0%$ 74.05-$25.95$8.00-17.95% Normal growth0.508.0100.00 0.008.008.00 Recession0.307.0112.2812.288.0020.289.E(q) = (0 × 0.25) + (1 × 0.25) + (2 × 0.50) = 1.25σq = [0.25 × (0 – 1.25)2 + 0.25 × (1 – 1.25)2 + 0.50 × (2 – 1.25)2]1/2 = 0.8292 10.(a) With probability 0.9544, the value of a normally distributedvariable will fall within 2 standard deviations of the mean; that is,between –40% and 80%. Simply add and subtract 2 standarddeviations to and from the mean.11.From Table 5.4, the average risk premium for the period 7/1926-9/2012 was:12.34% per year.Adding 12.34% to the 3% risk-free interest rate, the expected annual HPR for the Big/Value portfolio is: 3.00% + 12.34% = 15.34%.12.(01/1928-06/1970)Small BigLow2High Low2HighAverage 1.03% 1.21% 1.46%0.78%0.88% 1.18%SD8.55%8.47%10.35% 5.89% 6.91%9.11%Skew 1.6704 1.6673 2.30640.0067 1.6251 1.6348Kurtosis13.150513.528417.2137 6.256416.230513.6729(07/1970-12/2012)Small BigLow2High Low2HighAverage0.91% 1.33% 1.46%0.93% 1.02% 1.13%SD7.00% 5.49% 5.66% 4.81% 4.50% 4.78%Skew-0.3278-0.5135-0.4323-0.3136-0.3508-0.4954Kurtosis 1.7962 3.1917 3.8320 1.8516 2.0756 2.8629No. The distributions from (01/1928–06/1970) and (07/1970–12/2012) periods have distinct characteristics due to systematic shocks to the economy and subsequent government intervention. While the returns from the two periods do not differ greatly, their respective distributions tell a different story. The standard deviation for all six portfolios is larger in the first period. Skew is also positive, but negative in the second, showing a greater likelihood of higher-than-normal returns in the right tail. Kurtosis is also markedly larger in the first period.13.a%88.5,0588.070.170.080.01111or i i rn i rn rr =-=+-=-++=b.rr ≈ rn - i = 80% - 70% = 10%Clearly, the approximation gives a real HPR that is too high.14.From Table 5.2, the average real rate on T-bills has been 0.52%.a.T-bills: 0.52% real rate + 3% inflation = 3.52%b.Expected return on Big/Value:3.52% T-bill rate + 12.34% historical risk premium = 15.86%c.The risk premium on stocks remains unchanged. A premium, thedifference between two rates, is a real value, unaffected by inflation.15.Real interest rates are expected to rise. The investment activity will shiftthe demand for funds curve (in Figure 5.1) to the right. Therefore theequilibrium real interest rate will increase.16. a.Probability distribution of the HPR on the stock market and put:STOCKPUT State of the Economy Probability Ending Price +Dividend HPREnding Value HPR Excellent 0.25$ 131.0031.00%$ 0.00-100%Good 0.45 114.0014.00$ 0.00-100Poor 0.25 93.25−6.75$ 20.2568.75Crash0.05 48.00-52.00$ 64.00433.33Remember that the cost of the index fund is $100 per share, and the costof the put option is $12.b.The cost of one share of the index fund plus a put option is $112. Theprobability distribution of the HPR on the portfolio is:State of the Economy Probability Ending Price +Put +Dividend HPRExcellent 0.25$ 131.0017.0%= (131 - 112)/112Good 0.45 114.00 1.8= (114 - 112)/112Poor 0.25 113.50 1.3= (113.50 - 112)/112Crash0.05 112.000.0= (112 - 112)/112c.Buying the put option guarantees the investor a minimum HPR of 0.0%regardless of what happens to the stock's price. Thus, it offers insuranceagainst a price decline.17.The probability distribution of the dollar return on CD plus call option is:State of theEconomyProbability Ending Value of CD Ending Value of Call Combined Value Excellent0.25$ 114.00$16.50$130.50Good0.45 114.00 0.00114.00Poor0.25 114.00 0.00114.00Crash 0.05 114.00 0.00114.0018.a.Total return of the bond is (100/84.49)-1 = 0.1836. With t = 10, the annual rate on the real bond is (1 + EAR) = = 1.69%.1.18361/10 b.With a per quarter yield of 2%, the annual yield is = 1.0824, or8.24%. The equivalent continuously compounding (cc) rate is ln(1+.0824) =.0792, or 7.92%. The risk-free rate is 3.55% with a cc rate of ln(1+.0355) =.0349, or 3.49%. The cc risk premium will equal .0792 - .0349 = .0443, or4.433%.c.The appropriate formula is , σ2(effective )= e2 × m (cc ) ×[e σ2(cc )‒1]where . Using solver or goal seek, setting thetarget cell to the known effective cc rate by changing the unknown variance(cc) rate, the equivalent standard deviation (cc) is 18.03% (excel mayyield slightly different solutions).d.The expected value of the excess return will grow by 120 months (12months over a 10-year horizon). Therefore the excess return will be 120 × 4.433% = 531.9%. The expected SD grows by the square root of timeresulting in 18.03% × = 197.5%. The resulting Sharpe ratio is120531.9/197.5 = 2.6929. Normsdist (-2.6929) = .0035, or a .35% probabilityof shortfall over a 10-year horizon.CFA PROBLEMS1.The expected dollar return on the investment in equities is $18,000 (0.6 × $50,000 + 0.4× −$30,000) compared to the $5,000 expected return for T-bills. Therefore, theexpected risk premium is $13,000.2.E(r) = [0.2 × (−25%)] + [0.3 × 10%] + [0.5 × 24%] =10%3.E(r X) = [0.2 × (−20%)] + [0.5 × 18%] + [0.3 × 50%] =20%E(r Y) = [0.2 × (−15%)] + [0.5 × 20%] + [0.3 × 10%] =10%4.σX2 = [0.2 × (– 20 – 20)2] + [0.5 × (18 – 20)2] + [0.3 × (50 – 20)2] = 592σX = 24.33%σY2 = [0.2 × (– 15 – 10)2] + [0.5 × (20 – 10)2] + [0.3 × (10 – 10)2] = 175σY = 13.23%5.E(r) = (0.9 × 20%) + (0.1 × 10%) =19% $1,900 in returns6.The probability that the economy will be neutral is 0.50, or 50%. Given aneutral economy, the stock will experience poor performance 30% of thetime. The probability of both poor stock performance and a neutral economy is therefore:0.30 × 0.50 = 0.15 = 15%7.E(r) = (0.1 × 15%) + (0.6 × 13%) + (0.3 × 7%) = 11.4%。

武汉微博教育整理分享—计算机英语第三版课后答案（五）练习答案e-BusinessSection A I．Fill in the blanks with the information given in the text:1．commercial; network2．catalogs; engines3．currencies; checks4．relationship-based; purses5．Signatures6．cash; token7．Public8．SymmetricII．Translate the following terms or phrases from English into Chinese andvice versa:1．user authentication 用户认证2．electronic purse 电子钱包3．information filter 信息过滤4．data integrity 数据完整性5．smart card 智能卡6．HTML 超文本标记语言7．symmetric key cryptosystem 对称密钥密码系统8．message authentication code 信息鉴定码9．unauthorized access control 未授权访问控制10．electronic catalog 电子目录11．electronic money (或cash) 电子货币12．search engine 搜索引擎13．digital signature 数字签名14．user interface 用户界面15．EFT (Electronic Funds Transfer) 电子资金转帐16．public key cryptosystem 公钥密码系统17．PDA (personal digital assistant) 个人数字助理18．hypertext link 超文本链接19．3D image 三维图像20．credit card 信用卡III．Fill in each of the blanks with one of the words given in the following list, making changes if necessary:The term electronic commerce encompasses the entire process of buying, selling, and advertising products and services using electronic communications media. But when people talk about (e-commerce) today, they're usually referring to the emerging marketplace on the Internet. (Retailers) of all kinds are now scrambling to establish themselves and reach (customers) in this new marketplace.The major obstacle to e-commerce is the problem of how to pay for (products) and services online. How can electronic (payments) be made convenient, reliable, and (secure) for consumers and retailers alike? As with every other Internet (technology), a global standard for online (transactions) must emerge beforee-commerce becomes a widely accepted (practice).The drive to establish a standard for (electronic) money is well underway. Software developers, banks, and (credit) card companies are all pushing transaction systems to (online) merchants, each betting that their system will become the (standard) way to pay for things online. As a Web (storefront) owner, you must investigate all the (options) and offer as many choices to your customers as they demand, while keeping (costs) and complexities from getting out of hand.IV．Translate the following passage from English into Chinese.A computer virus is a program tha t “infects” computer files (usually other executable programs) by inserting in those files copies of itself. This is usually done in such a manner that the copies will be executed when the file is loaded into memory, allowing them to infect still other files, and so on. Viruses often have damaging side effects, sometimes intentionally, sometimes not.A virus that propagates (传播) itself across computer networks is sometimes referred to as a “Worm”, especially if it is composed of many separate segments distr ibuted across the network. A “Trojan Horse”, though technically not a virus, is a program disguised as something useful, which when run does something equally devious (阴险的) to the computer system while appearing to do something else.计算机病毒是通过在计算机文件（通常是其他可执行程序）中插入自身的拷贝来“传染”这些文件的程序。

中级会计教材英文原版以下为您生成 20 个关于中级会计教材相关的内容示例：---1. **Accrual basis accounting**- 英文释义：An accounting method where revenues and expenses are recorded when they are earned or incurred, regardless of when the cash is received or paid.- 短语：accrual basis of accounting（权责发生制会计）- 单词：accrual（应计；应计项目）- 用法：“The company uses accrual basis accounting to provide a more accurate picture of its financial performance.”（该公司采用权责发生制会计来更准确地反映其财务业绩。

）- 双语例句：Accrual basis accounting is more complex but provides a better reflection of economic reality.（权责发生制会计更复杂，但能更好地反映经济现实。

）2. **Amortization**- 英文释义：The gradual reduction of the value of an intangible asset or a loan over a period of time.- 短语：amortization expense（摊销费用）- 单词：amortize（摊销；分期偿还）- 用法：“The amortization of the patent is calculated over its useful life.”（该专利的摊销是在其使用寿命内计算的。

CHAPTER 3Valuing BondsAnswers to Problem Sets1. a. Does not change. The coupon rate is set at time of issuance.b. Price falls. Market yields and prices are inversely related.c. Yield rises. Market yields and prices are inversely related.Est. Time: 01-052. a. If the coupon rate is higher than the yield, then investors must beexpecting a decline in the capital value of the bond over its remaining life.Thus, the bond’s price must be greater than its face value.b. Conversely, if the yield is greater than the coupon, the price will be belowface value and it will rise over the remaining life of the bond.Est. Time: 01-053. The yield over six months is 2.7/2 = 1.35%.The six-month coupon payment is $6.25/2 = $3.125.There are 18 years between today (2012) and 2030; since coupon payments are listed every six months, there will be 36 payment periods.Therefore, PV = $3.125 / 1.0135 + $3.125 / (1.0135)2 + . . . $103.125 / (1.0135)36 = $150.35.Est. Time: 01-054. Yields to maturity are about 4.3% for the 2% coupon, 4.2% for the 4% coupon,and 3.9% for the 8% coupon. The 8% bond had the shortest duration (7.65years), the 2% bond the longest (9.07 years).The 4% bond had a duration of 8.42 years.Est. Time: 01-055. a. Fall. Example: Assume a one-year, 10% bond. If the interest rate is 10%,the bond is worth $110/1.1 = $100. If the interest rate rises to 15%, the bond isworth $110/1.15 = $95.65.b. Less (e.g., see 5a —if the bond yield is 15% but the coupon rate is lower at 10%, the price of the bond is less than $100).c.Less (e.g., with r = 5%, one-year 10% bond is worth $110/1.05 = $104.76).d. Higher (e.g., if r = 10%, one-year 10% bond is worth $110/1.1 = $100, while one-year 8% bond is worth $108/1.1 = $98.18).e.No. Low-coupon bonds have longer durations (unless there is only one period to maturity) and are therefore more volatile (e.g., if r falls from 10% to 5%, the value of a two-year 10% bond rises from $100 to $109.3 (a rise of 9.3%). The value of a two-year 5% bond rises from $91.3 to $100 (a rise of 9.5%).Est. Time: 01-056. a. Spot interest rates. Yield to maturity is a complicated average of theseparate spot rates of interest.b. Bond prices. The bond price i s determined by the bond’s cash flows andthe spot rates of interest. Once you know the bond price and the bond’s cash flows, it is possible to calculate the yield to maturity.Est. Time: 01-057. a. 4%; each bond will have the same yield to maturity.b.PV = $80/(1.04) + $1,080/(1.04)2 = $1,075.44.Est. Time: 01-058. a. PV ()221110515r r +++=b.PV ()2110515y y +++=c. Less (it is between the one-year and two-year spot rates). Est. Time: 01-059. a. The two-year spot rate is r 2 = (100/99.523).5 – 1 = 0.24%.The three-year spot rate is r 3 = (100/98.937).33 – 1 = 0.36%. The four-year spot rate is r 4 = (100/97.904).25 – 1 = 0.53%. The five-year spot rate is r 5 = (100/96.034).2 – 1 = 0.81%.b. Upward-sloping.c. Higher (the yield on the bond is a complicated average of the separatespot rates).Est. Time: 01-0510. a. Price today is $108.425; price after one year is $106.930.b. Return = (8 + 106.930)/108.425 - 1 = .06, or 6%.c. If a bond’s yie ld to maturity is unchanged, the return to the bondholder isequal to the yield.Est. Time: 01-0511. a. False. Duration depends on the coupon as well as the maturity.b. False. Given the yield to maturity, volatility is proportional to duration.c. True. A lower coupon rate means longer duration and therefore highervolatility.d. False. A higher interest rate reduces the relative present value of (distant)principal repayments.Est. Time: 01-0512.Est. Time: 06-1013. 7.01%; the extra return that you earn for investing for two years rather than oneyear is 1.062/1.05 – 1 = .0701.Est. Time: 01-0514. a. Real rate = 1.10/1.05 – 1 = .0476, or 4.76%.b. The real rate does not change. The nominal rate increases to 1.0476 ×1.07 – 1 = .1209, or 12.09%.Est. Time: 01-0515. With annual coupon payments:=+⎥⎦⎤⎢⎣⎡⨯-⨯=1010(1.06)100(1.06)0.0610.0615PV €92.64 Est. Time: 01-0516. a. $10,231.64(1.026)10,000(1.026)0.02610.0261275PV 2020=+⎥⎦⎤⎢⎣⎡⨯-⨯=b.Interest Rate PV of Interest PV ofFace Value PV of Bond1.0% $5,221.54 $9,050.63 $14,272.17 2.0% 4,962.53 8,195.44 13,157.973.0% 4,721.38 7,424.70 12,146.08 4.0% 4,496.64 6,729.71 11,226.365.0% 4,287.02 6,102.71 10,389.736.0% 4,091.31 5,536.76 9,628.067.0% 3,908.41 5,025.66 8,934.07 8.0% 3,737.34 4,563.87 8,301.219.0% 3,577.18 4,146.43 7,723.61 10.0% 3,427.11 3,768.89 7,196.00 11.0% 3,286.36 3,427.29 6,713.64 12.0% 3,154.23 3,118.05 6,272.28 13.0% 3,030.09 2,837.97 5,868.06 14.0% 2,913.35 2,584.19 5,497.54 15.0% 2,803.49 2,354.13 5,157.62Est. Time: 06-1017.Purchase price for a six-year government bond with 5% annual coupon:1,108.34(1.03)1,000(1.03)0.0310.03150PV 66$=+⎥⎦⎤⎢⎣⎡⨯-⨯= The price one year later is equal to the present value of the remaining five years of the bond:1,091.59(1.03)1,000(1.03)0.0310.03150PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯= Rate of return = [$50 + ($1,091.59 – $1,108.34)]/$1,108.34 = 3.00% Price one year later (yield = 2%):1,141.40(1.02)1,000(1.02)0.0210.02150PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯=Rate of return = [$50 + ($1,141.40 – $1,108.34)]/$1,108.34 = 7.49%.Est. Time: 06-1018. The key here is to find a combination of these two bonds (i.e., a portfolio ofbonds) that has a cash flow only at t = 6. Then, knowing the price of the portfolio and the cash flow at t = 6, we can calculate the six-year spot rate. We begin byspecifying the cash flows of each bond and using these and their yields tocalculate their current prices:Investment Yield C1. . . C5C6Price6% bond 12% 60 . . . 60 1,060 $753.3210% bond 8% 100 . . . 100 1,100 $1,092.46 From the cash flows in years 1 through 5, we can see that buying two 6% bonds produces the same annual payments as buying 1.2 of the 10% bonds. To seethe value of a cash flow only in year 6, consider the portfolio of two 6% bondsminus 1.2 10% bonds. This portfolio costs:($753.32 × 2) – (1.2 ⨯ $1,092.46) = $195.68The cash flow for this portfolio is equal to zero for years 1 through 5 and, for year 6, is equal to:(1,060 × 2) – (1.2 ⨯ 1,100) = $800Thus:$195.68 ⨯ (1 + r6)6 = $800r6 = 0.2645 = 26.45%Est. Time: 06-1019. Downward sloping. This is because high-coupon bonds provide a greaterproportion of their cash flows in the early years. In essence, a high-coupon bond is a ―shorter‖ bond than a low-coupon bond of the same maturity.Est. Time: 01-0520. a.Year Discount Factor Forward Rate1 1/1.05 = 0.9522 1/(1.054)2 = 0.900 (1.0542 /1.05) – 1 = 0.0580 = 5.80%3 1/(1.057)3 = 0.847 (1.0573 /1.0542 ) – 1 = 0.0630 = 6.30%4 1/(1.059)4 = 0.795 (1.0594 /1.0573 ) – 1 = 0.0650 = 6.50%5 1/(1.060)5 = 0.747 (1.0605 /1.0594 ) – 1 = 0.0640 = 6.40%b. i. 5%, two-year bond:$992.79(1.054)10501.0550PV 2=+=ii. 5%, five-year bond:$959.34(1.060)1,050(1.059)50(1.057)50(1.054)501.0550PV 5432=++++=iii. 10%, five-year bond:$1,171.43(1.060)1,100(1.059)100(1.057)100(1.054)1001.05100PV 5432=++++=c. First, we calculate the yield for each of the two bonds. For the 5% bond, this means solving for r in the following equation:5432r)(11,050r)(150)(150)(150150$959.34+++++++++=r r r r = 0.05964 = 5.964%For the 10% bond:5432r)(11,100r)(1100r)(1100r)(1100r 1100$1,171.43+++++++++=r = 0.05937 = 5.937%The yield depends upon both the coupon payment and the spot rate at the time of the coupon payment. The 10% bond has a slightly greater proportion of its total payments coming earlier, when interest rates are low, than does the 5% bond. Thus, the yield of the 10% bond is slightly lower.d. The yield to maturity on a five-year zero-coupon bond is the five-year spot rate, here 6.00%.e. First, we find the price of the five-year annuity, assuming that the annual payment is $1:Now we find the yield to maturity for this annuity:r = 0.05745 = 5.745%$4.2417(1.060)1.059)(11.057)(11.054)(111.051P V 5432=++++=5432r)(11r)(11r)(11r)(11r 11$4.2417+++++++++=f. The yield on the five-year note lies between the yield on a five-year zero-coupon bond and the yield on a five-year annuity because the cash flowsof the Treasury bond lie between the cash flows of these other twofinancial instruments during a period of rising interest rates. That is, theannuity has fixed, equal payments; the zero-coupon bond has onepayment at the end; and the bond’s payments are a combination of these. Est. Time: 06-1021. To calculate the duration, consider the following table similar to Table 3.4:The duration is the sum of the year × fraction of total value column, or 6.395 years.The modified duration, or volatility, is 6.395/(1 + .04) = 6.15.The price of the 3% coupon bond at 3.5%, and 4.5% equals $969.43 and $911.64, respectively. This price difference ($57.82) is 5.93% of the original price, which is very close to the modified duration.Est. Time: 06-1022.a. If the bond coupon payment changes from 9% as listed in Table 3.4 to 8%,then the following calculation for duration can be made:A decrease in the coupon payment will increase the duration of the bond, as theduration at an 8% coupon payment is 5.778 years.The volatility for the bond in Table 3.4 with an 8% coupon payment is:5.778/(1.04) = 5.556.The bond therefore becomes less volatile if the couponpayment decreases.b. For a 9% bond whose yield increases from 4% to 6%, the duration can becalculated as follows:There is an inverse relationship between the yield to maturity and the duration.When the yield goes up from 4% to 6%, the duration decreases slightly. Thevolatility can be calculated as follows: 5.595/(1.06) = 5.278. This shows that the volatility decreases as well when the yield increases.Est. Time: 06-1023. The duration of a perpetual bond is: [(1 + yield)/yield].The duration of a perpetual bond with a yield of 5% is:D5 = 1.05/0.05 = 21 yearsThe duration of a perpetual bond yielding 10% is:D10 = 1.10/0.10 = 11 yearsBecause the duration of a zero-coupon bond is equal to its maturity, the 15-year zero-coupon bond has a duration of 15 years.Thus, comparing the 5% perpetual bond and the zero-coupon bond, the 5%perpetual bond has the longer duration. Comparing the 10% perpetual bond and the 15-year zero, the zero has a longer duration.Est. Time: 06-1024. Answers will differ. Generally, we would expect yield changes to have thegreatest impact on long-maturity and low-coupon bonds.Est. Time: 06-1025. The new calculations are shown in the table below:26. We will borrow $1,000 at a five-year loan rate of 2.5% and buy a four-year strippaying 4%. We may not know what interest rates we will earn on the last year(4 5), but our $1,000 will come due, and we put it under our mattress earning0% if necessary to pay off the loan.Let’s turn to present value calculations: As shown above, the cost of the strip is$854.80. We will receive proceeds from the 2.5% loan = $1,000/(1.025)5 =$883.90. Pocket the difference of $29.10, smile, and repeat.The minimum sensible value would set the discount factors used in year 5 equal to that of year 4, which would assume a 0% interest rate from year 4 to 5. Wecan solve for the interest rate where 1/(1 + r)5 = 0.8548, which is roughly 3.19%. Est. Time: 06-1027.a. If the expectations theory of term structure is right, then we can determinethe expected future one-year spot rate (at t = 3) as follows: investing $100in a three-year instrument at 4.2% gives us $100(1 + .042)3 = 113.136.Investing $100 in a four-year instrument at 4.0% gives us $100 × (1+.04)4= 116.986. This reveals a one-year spot rate from year 3 to 4 of ($116.98– 113.136)/113.136 = 3.4%.b. If investing in long-term bonds carries additional risks, then the riskequivalent of a one-year spot rate in year 3 would be even less (reflectingthe fact that some risk premium must be built into this 3.4% spot rate).Est. Time: 06-1028.a. Your nominal return will be 1.082 -1 = 16.64% over the two years. Yourreal return is (1.08/1.03) × (1.08/1.05) - 1 = 7.85%.b. With the TIPS, the real return will remain at 8% per year, or 16.64% overtwo years. The nominal return on the TIPS will equal (1.08 × 1.03) × (1.08× 1.05) – 1 = 26.15%.Est. Time: 01-0529. The bond price at a 5.3% yield is:1,201.81(1.053)1,000(1.053)0.05310.0531100PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯= If the yield decreases to 5.9%, the price would rise to:1,173.18(1.059)1,000(1.059)0.05910.0591100PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯=30. Answers will vary by the interest rates chosen.a. Suppose the YTM on a four-year 3% coupon bond is 2%. The bond is selling for:08.038,1$3=+⎥⎦⎤⎢⎣⎡⨯-⨯=44(1.02)1,000(1.02)0.0210.0210PVIf the YTM stays the same, one year later the bond will sell for:84.028,1$3=+⎥⎦⎤⎢⎣⎡⨯-⨯=33(1.02)1,000(1.02)0.0210.0210PVThe return over the year is [$30 + (1,028.84 - 1,038.08)]/$1,038.08= 0.02, or 2%.b. Suppose the YTM on a four-year 3% coupon bond is 4%. The bond is sellingfor:70.963$3=+⎥⎦⎤⎢⎣⎡⨯-⨯=44(1.04)1,000(1.04)0.0410.0410PV If the YTM stays the same, one year later the bond will sell for:25.972$3=+⎥⎦⎤⎢⎣⎡⨯-⨯=33(1.04)1,000(1.04)0.0410.0410PVThe return over the year is [$30+(972.25-963.70)]/$963.70 = 0.04, or 4%.Est. Time: 06-1031. Spreadsheet problem; answers will vary.Est. Time: 06-1032. Arbitrage opportunities can be identified by finding situations where the impliedforward rates or spot rates are different.We begin with the shortest-term bond, Bond G, which has a two-year maturity.Since G is a zero-coupon bond, we determine the two-year spot rate directly by finding the yield for Bond G. The yield is 9.5%, so the implied two-year spot rate (r 2) is 9.5%. Using the same approach for Bond A, we find that the three-yearspot rate (r 3) is 10.0%.Next we use Bonds B and D to find the four-year spot rate. The followingposition in these bonds provides a cash payoff only in year four: a long position in two of Bond B and a short position in Bond D.Cash flows for this position are:[(–2 ⨯ $842.30) + $980.57] = –$704.03 today[(2 ⨯ $50) – $100] = $0 in years 1, 2 and 3[(2 ⨯ $1,050) – $1,100] = $1,000 in year 4 We determine the four-year spot rate from this position as follows:4)4r (1$1,000$704.03+= r 4 = 0.0917 = 9.17%Next, we use r 2, r 3, and r 4 with one of the four-year coupon bonds to determine r 1. For Bond C:978.74r 1120(1.0917)1,120(1.100)120(1.095)120r 1120$1,065.2814321++=++++= r 1 = 0.3867 = 38.67%Now, in order to determine whether arbitrage opportunities exist, we use thesespot rates to value the remaining two four-year bonds. This produces thefollowing results: for Bond B, the present value is $854.55, and for Bond D, thepresent value is $1,005.07. Since neither of these values equals the currentmarket price of the respective bonds, arbitrage opportunities exist. Similarly, the spot rates derived above produce the following values for the three-year bonds: $1,074.22 for Bond E and $912.77 for Bond F.Est. Time: 11-1533. We begin with the definition of duration as applied to a bond with yield r and anannual payment of C in perpetuity:We first simplify by dividing both the numerator and the denominator by C :The denominator is the present value of a perpetuity of $1 per year, which isequal to (1/r ). To simplify the numerator, we first denote the numerator S andthen divide S by (1 + r ):Note that this new quantity [S /(1 + r )] is equal to the square of denominator in the duration formula above, that is:Therefore:Thus, for a perpetual bond paying C dollars per year:++++++++++++++++++=t 32t 32r)(1C r)(1C r)(1C r 1C r)(1tC r)(13C r)(12C r 11C DUR ++++++++++++++++++=t32t 32r)(11r)(11r)(11r 11r)(1t r)(13r)(12r)(11DUR+++++++++=++1t 432r)(1t r)(13r)(12r)(11r)(1S 2t 32r)(11r)(11r)(11r 11r)(1S ⎪⎪⎭⎫ ⎝⎛+++++++++=+ 22r r 1S r 1r)(1S +=⇒⎪⎭⎫ ⎝⎛=+rr 1r)/(11r r 1DUR 2+=⨯+=Est. Time: 06-1034. We begin with the definition of duration as applied to a common stock with yield rand dividends that grow at a constant rate g in perpetuity:We first simplify by dividing each term by [C (1 + g )]:The denominator is the present value of a growing perpetuity of $1 per year,which is equal to [1/(r - g )]. To simplify the numerator, we first denote thenumerator S and then divide S by (1 + r ):Note that this new quantity [S/(1 + r )] is equal to the square of denominator in the duration formula above, that is:Therefore:Thus, for a perpetual bond paying C dollars per year:Est. Time: 11-15++++++++++++++++++++++++++=t t 3322t t 3322r)(1g)C(1r)(1g)C(1r)(1g)C(1r 1g)C(1r)(1g)tC(1r)(1g)3C(1r)(1g)2C(1r 1g)1C(1DUR ++++++++++++++++++++++++=--t1t 322t 1t 322r)(1g)(1r)(1g)(1r)(1g 1r 11r)(1g)t(1r)(1g)3(1r)(1g)2(1r 11DUR ++++++++++++=++-1t 2t 4232r)(1g)t(1r)(1g)3(1r)(1g)2(1r)(11 r)(1S 2t 1t 322r)(1g)(1r)(1g)(1r)(1g 1r 11r)(1S ⎪⎪⎭⎫ ⎝⎛++++++++++++=+- 22g)(r r 1S g r 1r)(1S -+=⇒⎪⎪⎭⎫ ⎝⎛-=+g r r 1g)](r /[11g)(r r 1DUR 2-+=-⨯-+=35. a. We make use of the one-year Treasury bill information in order to determine the one-year spot rate as follows:1r 1$100$93.46+= r 1 = 0.0700 = 7.00%The following position provides a cash payoff only in year two: a longposition in 25 two-year bonds and a short position in 1 one-year Treasurybill. Cash flows for this position are:[(–25 ⨯ $94.92) + (1 ⨯ $93.46)] = –$2,279.54 today[(25 ⨯ $4) – (1 ⨯ $100)] = $0 in year 1(25 ⨯ $104) = $2,600 in year 2 We determine the two-year spot rate from this position as follows:2)2r (1$2,600$2,279.54+= r 2 = 0.0680 = 6.80%The forward rate f 2 is computed as follows:f 2 = [(1.0680)2/1.0700] – 1 = 0.0660 = 6.60%The following position provides a cash payoff only in year 3:a long position in the three-year bond and a short position equal to (8/104)times a package consisting of a one-year Treasury bill and a two-yearbond.Cash flows for this position are:[(–1 ⨯ $103.64) + (8/104) ⨯ ($93.46 + $94.92)] = –$89.15 today[(1 ⨯ $8) – (8/104) ⨯ ($100 + $4)] = $0 in year 1[(1 ⨯ $8) – (8/104) ⨯ $104] = $0 in year 21 ⨯ $108 = $108 in year 3 We determine the three-year spot rate from this position as follows:3)3r (1$108$89.15+= r 3 = 0.0660 = 6.60%The forward rate f 3 is computed as follows:f 3 = [(1.0660)3/(1.0680)2] – 1 = 0.0620 = 6.20%b.We make use of the spot and forward rates to calculate the price of the 4% coupon bond:The actual price of the bond ($950) is significantly greater than the pricededuced using the spot and forward rates embedded in the prices of theother bonds ($931.01). Hence, a profit opportunity exists. In order to takeadvantage of this opportunity, one should sell the 4% coupon bond shortand purchase the 8% coupon bond.Est. Time: 11-1536. a. We can set up the following three equations using the prices of bonds A, B,and C:Using bond A: $1,076.19 = $80/(1+r 1) + $1,080/(1+r 2)2Using bond B: $1,084.58 = $80/(1+r 1) + $80/(1+r 2)2 + $1,080 / (1+r 3)3Using bond C: $1,076.20 = $80/(1+r 1) + $80/(1+r 2)2 + $80/(1+r 3)3 + $1,080/(1+r 4)4 We know r 4 = 6% so we can substitute that into the last equation. Now wehave three equations and three unknowns and can solve this with variablesubstitution or linear programming to get r 1 = 3%, r 2 = 4%; r 3 = 5%, r 4 = 6%.b.We will want to invest in the underpriced C and borrow money at thecurrent spot market rates to construct an offsetting position. For example,we might borrow $80 at the one-year rate of 3%, $80 at the two-year rateof 4%, $80 at the three-year rate of 5%, and $1,080 at the four-year rate of6%. Of course the PV amount we will receive on these loans is $1,076.20.Now we purchase the discounted bond C at $1,040 and use the proceedsof this bond to repay our loans as they come due. We can pocket thedifference of $36.20, smile, and repeat. Est. Time: 11-15$931.01(1.062)(1.066)(1.07)1040(1.066)(1.07)40(1.07)40P =++=。