quantitative method 复习冲刺-数量
【1】微臣教育GRE数学---B班(强化练习班)__第一、二章
GRE 数学强化班 微臣教育 Weichen Education帅 旗 2014.08.19微臣教育Weichen Education •笔记本一个（用于记录讲课内容的重点）•草稿纸若干张（用于上课的习题练习）•手机里面最简单的计算器温馨提示，请同学们自备以下物品：第一章、GRE Quantitative概述微臣教育第一章、 GRE Quantitative 复习的四大注意方面•全面复习考点•熟悉数学词汇以及英语中数学语言的表达习惯•熟悉做题的技巧•总结归纳易错点微臣教育第二章 GRE Quantitative题型及答题策略在新GRE的数学考试里面，会遇到四种题型。
分别是：数量比较题 Quantitative Comparison Questions单项选择题 Multiple-choice questions—Select One Answer Choice不定项选择题 Multiple-choice questions—Select One or More Answer Choices 数字填空题 Numeric Entry Questions下面我们将简要的把各个题型的要求与解题技巧给大家呈现出来：微臣教育第二章 GRE Quantitative题型及答题策略微臣教育第一节 Quantitative Comparison questions（数量比较题）一、题目的要求Questions of this type ask you to compare two quantities—Quantity A andQuantity B, and then determine which of the following statements describes thecomparison.A.Quantity A is greater.B.Quantity B is greater.C.The two quantities are equal.D.The relationship cannot be determined from the information given.微臣教育第一节 Quantitative Comparison questions（数量比较题）微臣教育微臣教育4836n r s t +=+1. n, r, s, and t are positive integers .A. 2n+rB. 2s+t第一节 Quantitative Comparison questions （数量比较题）第二章 GRE Quantitative题型及答题策略第一节 Quantitative Comparison questions（数量比较题）A: The area of the triangular region B: The area of the circular region 微臣教育微臣教育3. Fields X and Y are to be enclosed with fencing that costs $ 24 per meter.X Y 470m 180m290m470m 470m470m B. The cost of the fencingneeded to enclose Y . A. The cost of the fencing needed to enclose X ;第二章 GRE Quantitative 题型及答题策略第一节 Quantitative Comparison questions （数量比较题）第二节 Multiple-choice Questions—Select One Answer Choice（单项选择题）一、题目的要求These questions are multiple-choice questions that ask you to select only one answerchoice from a list of five choices.微臣教育第二节 Multiple-choice Questions—Select One Answer Choice（单项选择题）1. If the sum of two positive integers is 24 and the difference of their squares is48, what is the product of the two integers?A 108B 119C 128D 135E 143微臣教育第二节 Multiple-choice Questions—Select One Answer Choice（单项选择题）2. If p and r are prime numbers, which of the following must also be prime?A prB p+rC pr+1D p2+r2E None of the above微臣教育第二节 Multiple-choice Questions—Select One Answer Choice（单项选择题）3. A certain theater has 100 balcony seats. For every $2 increase in the price of a balcony seatabove $10, 5 fewer seats will be sold. If all the balcony seats are sold when the price of each seat is $10, which of the following could be the price of a balcony seat if the revenue from the sale of balcony seats is $1,360?A $12B $14C $16D $17E $18微臣教育第三节 Multiple-choice Questions—Select One or More Answers（不定项选择题）一、题目的要求These questions are multiple-choice questions that ask you to select one or more answer choicesfrom a list of choices. A question may or may not specify the number of choices to select. Thesequestions are marked with square boxes beside the answer choices, not circles or ovals.Select one or more answer choices according to the specific question directions. If the question doesnot specify how many answer choices to select, select all that apply.•The correct answer may be just one of the choices or as many as all of the choices, depending on the question.•No credit is given unless you select all of the correct choices and no others.If the question specifies how many answer choices to select, select exactly that number of choices. 微臣教育第三节 Multiple-choice Questions—Select One or More Answers（不定项选择题）微臣教育第三节 Multiple-choice Questions—Select One or More Answers（不定项选择题）1. If n is any prime number greater than 2, which of the following CANNOT be a primenumber?A.n-4B.n-3C.n-1D.n+2E.n+5微臣教育第三节 Multiple-choice Questions—Select One or More Answers（不定项选择题）三、例题分析2. Each employee of a certain company is in either Department X or Department Y, and there aremore than twice as many employees in Department X as in Department Y. The average (arithmetic mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all of the employees of the company? Indicate all such amounts.A $26,000B $28,000C $29,000D $30,000E $31,000F $32,000G $34,000微臣教育第四节 Numeric Entry questions（数字填空题）一、题目的要求Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxes—one for the numerator and one for the denominator.To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.•First, click on the answer box—a cursor will appear in the box—and then type the number.•To erase a number, use the Backspace key.•For a negative sign, type a hyphen. For a decimal point, type a period.•To remove a negative sign, type the hyphen again and it will disappear; the number remains.•The Transfer Display button on the calculator will transfer the answer to the answer box.•Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.•Enter the exact answer unless the question asks you to round your answer.To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.•For a negative sign, type a hyphen; to remove it, type the hyphen again.•A decimal point cannot be used in a fraction.•The Transfer Display button on the calculator cannot be used for a fraction.•Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes. 微臣教育第四节 Numeric Entry questions（数字填空题）微臣教育第四节 Numeric Entry questions（数字填空题）1. An experiment has three possible outcomes ,I,J, and K. The probabilities of theoutcomes are 0.25,0.35,and 0.40, respectively. If the experiment is to be performed twice and successive outcomes are independent, what is the probability that K will not be an outcome either time?微臣教育第四节 Numeric Entry questions（数字填空题）2. The figure above shows a normal distribution with mean m and standard deviation d, including approximate percent of the distribution corresponding to the six regions shown.Suppose the heights of a population of 3,000 adult penguins are approximately normally distributed with a mean of 65 centimeters and a standard deviation of 5 centimeters.(a)Approximately how many of the adult penguins are between 65 centimeters and 75 centimeters tall?(b)If an adult penguin is chosen at random from the population, approximately what is the probabilitythat the penguin’s height will be less than 60 centimeters? Give your answer to the nearest 0.05.微臣教育第五节 GRE数学考试中常见的错误一、英文理解能力不够好二、考点遗忘的太多三、读题信息的遗漏四、忽略数量、单位的转换五、不知如何检查（A. 数据格式 B. 常识或者具体题目中估计 C. 计算失误）微臣教育第五节 GRE数学考试中常见的错误六、数学术语的不熟悉Addition (A+B) the sum of A and B the total of A and BSubtraction(A-B)微臣教育第五节 GRE数学考试中常见的错误六、数学术语的不熟悉Multiplication(A×B)A multiplied by BDivision (A÷B)A divided byB the quotient of A and BFactors and Divisors(A×B=C) C is divisible by A and by BC is a multiple of A and of B微臣教育。
有关高考英语冲刺复习的知识点
有关高考英语冲刺复习的知识点时光飞逝，暑假过去了，新学期开始了，不管情愿与否，无论准备与否，我们已走进高三，走近我们的梦!接下来是小编为大家整理的有关高考英语冲刺复习的知识点，希望大家喜欢!1 区别一、在句中作用不同限制性定语从句对被修饰的先行词有限定制约作用，使该词的含义更具体，更明确。
限制性定语从句不能被省略，否则句意就不完整。
而非限制性定语从句与先行词的关系不十分密切，只是对其作一些附加说明，不起限定制约作用。
如果将非限制性定语从句省去，主句的意义仍然完整。
二、外在表现形式不同限制性定语从句因与先行词的关系密切，所以不可以用逗号将其与主句隔开;而非限制性定语从句与先行词的关系不十分密切，所以可用逗号将其与主句隔开。
三、先行词内容有所不同大多数限制性和非限制性定语从句的先行词往往为某一个词或短语，而特殊情况下非限制性定语从句的先行词也可为整个主句，此时非限制性定语从句常由 which 引导。
四、关系词的使用情况有所不同(一)that 不可用于引导非限制性定语从句所有关系代词和关系副词均可引导限制性定语从句，大多数关系代词和关系副词可引导非限制性定语从句，但 that 不可。
(二)关系代词替代情况不同关系代词 whom 在限制性定语从句中作宾语时可用who 代替whom，但 whom 在非限制性定语从句中作宾语时不可用who 来代替。
同位语从句同位语从句就是在复合句中作名词的同位语的名词性从句。
1. 同位语从句的功能同位语从句对于名词进一步解释，说明名词的具体内容，一般由 that 引导，例如：1) The king’s decision that the prisoner would be set free surprised all the people.2) The order that all the soldiers should stay still is given by the general.2. 同位语在句子中的位置同位语从句有时可以不紧跟在它所说明的名词后面，而是被别的词隔开。
GRE(QUANTITATIVE)模拟试卷21(题后含答案及解析)
GRE（QUANTITATIVE）模拟试卷21(题后含答案及解析)题型有：1. 2. 3.1．n=l ×19? Column A Column BThe number of distinct10positive factors of nA．if the quantity in Column A is greaterB．if the quantity in Column B is greaterC．if the two quantity are equalD．if the relationship cannot be determined from the information given正确答案：B解析：本题的正确答案为(B)。
该题也就是让考生求n的正因子(positive factor)的个数。
因子求解法则是：因子数=(x+1)×(y+1)×(z+1)；根据此法则可得n的因子个数为(1+1)×(3+1)=82．Richard’s salary, which is greater than $10,000, is 75 percent of Sandra’s salary. Ted’s salary is 80 percent of Richard’s salary.Column A Column BSandra’s salaryTed’s salaryA．if the quantity in Column A is greaterB．if the quantity in Column B is greaterC．if the two quantity are equalD．if the relationship cannot be determined from the information given正确答案：A解析：Richard的薪水是Sandra薪水的75％，且多于$10,000，Ted的薪水是Richard薪水的80％。
Quantitative Data Analysis- Questionnaire Design
Professional and Academic Skills(PAS)-2Week-3Quantitative Data Analysis- Questionnaire Design andLevel MeasurementDate: 10/02/2020v Collecting Quantitative Data using Questionnaire v Questionnaire Designv Pilot Questionnairev Level of Measurement: Categorical Variable and Continuous Variablev Identify the main issues that you need to consider when preparing quantitative data for analysisCollecting Primary Data using QuestionnaireØQuestionnaire:“is a method/technique of data collection in which each person is asked to respond to same set of questions in a predetermined order”(De Vaus, 2014)Ø “It also widely used as an instrument” (Ekinci, 2015)ØIt provides efficient way of collecting responses from a large sample prior to quantitative analysisTypes of QuestionnaireSource: Saunders (2016)Advantages and disadvantages of questionnaireAdvantages◦Large sample size – can be collected relatively quickly◦Cost effective- large sample of the population can be contacted at relative cost◦Analysis easier and visualisation – closed questions are easier to analyse◦Anonymity- allows respondents to maintain their anonymity◦If completed anonymously and in private then personal questions may receive more valid answersDisadvantages◦Difficult for respondents to provide deep reasons behind consumers behaviour◦Unanswered questions- some questions will be ignored and left unanswered◦Dishonest answer- they may not be 100% truthful with their answers◦Respondents under no/little pressure to complete and return◦No flexibility re questions posed and no supplementary options available-Things to Consider to Produce a Good QuestionnaireüTo ensure that it will collect precise data that you require to answer your research question(s) and your research objectivesüYou are unlikely to have more than one opportunity to collect the dataüDesign of your questionnaire will affect the response rate and the reliability of the data you collect (clear, specific and unambiguous)üEasy questions at the start and open questions at endüFar more closed than open questionsü Types of questions you need to ask to collect your dataüNumber of questions you need to ask to collect your dataüVisual presentation of the questionnaire- Colourful, artistic appearance of the questionnaireValidity and ReliabilityValidity: refers to the ability of your questionnaire to measure what you intend to measure also know as measurement validity.Reliability- refers to the consistency of the research method used, For example if the same method was used again would it lead to the same data collected.For a questionnaire to be valid it must be reliableValidity and reliability of the data you collect and the response rate you achieve depend largely on the:- Design of your questions- Structure of your questionnaire- and the rigor of your pilot testingFactors that can impact ReliabilityØParticipants errorØParticipants biasØResearcher errorØResearcher biasHow many questions should a questionnaire have?◦For your MA dissertation time taken to complete the questionnaire should not exceed 10 -12 minutes.◦Effective design of the questions should allow you to include all necessary questions for your final project and should consist maximum 15-20 questions.How many need to be collected ?◦Around a minimum of 100 for your MA dissertation when conducting a mono method.Constructing your Questions- Closed and Open questionsOpen questions- refer to as open ended questions, allow respondents to give answers in their own wayFink (2013) Closed questions- refer to closed-ended questions or forced-choice questions, provide a number of alternative answers from which the respondent is instructed to choose. It is easier,quicker and require minimal writingThere are 6 types of closed questions:List question: where the respondent is offered a list of items, any of which may be selectedCategory question: where only one response can be selected from a given set of categoriesRanking question: where the respondent is asked to place something in orderRating question: in which a rating device is used to record responsesQuantity question: to which the response is a number giving the amountMatrix question: where responses to two or more questions can be collected using the same gridSource: Saunders, 2016Rating /Likert Scale questionsCategory questionQuantity questionSemantic differential Rating question- Series of bi-polar adjectives between which underlined spaces areprovided for the recipient to record their views/feelings.Example of Rank questionExample of Matrix QuestionConstructing the QuestionnaireØExplaining the purpose of the questionnairev Covering letter or welcome screen:self-completed questionnaires should be accompanied by a covering letter, email, text or SMS message which explains the purpose of of the surveyv Introducing the questionnaire- explain clearly and concisely why you want the respondents to complete the surveyExample of Introduction for Interviewer-completed questionnaire….Lavrakas, 2016Closing the QuestionnaireØ at the end of the questionnaire you will need to explain clearly what you want your respondents to do with their completed questionnaire.Example:Saunders, 2016Pilot Testing and Assessing ValidityThe purpose of the pilot testing is to refine questionnaire so that the respondents will have no problems in answering the questions and there will be no problem in recording the dataØIt helps you to obtain some assessment of the questions, validity and likely reliability of the dataØTo ensure that the data collected will enable your investigative questions to be answeredØfor student questionnaire minimum number of a pilot is 10 (Flink, 2013)To ensure questionnaire’s face validity, as part of your Pilot you should try to get additional information about (Bell and Waters, 2014)-üHow long the question took to completeüthe clarity of the instructionüWhich, if any, questions were ambiguousüWhich, if any, question the respondents felt uneasyüwhether in the layout was clear and attractiveüany other commentsInternet QuestionnaireFor both Web and Mobile questionnaire:ØIt is important to have clear timetable that identifies the tasks need to be doneØA good response will depend on the recipient being motivated to answer the questionnaire and to send it backØEmail or SMS message and visual appearance will help to ensure a high level of responseØQuestionnaire design must be clear across all display mediaData collection- what to measureIf you intend to undertake quantitative analysis then you should consider:v The number of cases of data (sample size)v Type or types of data (scale of measurement)v Data layout and format required by the analysis softwarev Impact of data coding on subsequent analysis (for different types ofdata)v Process and checking the data error (checking your data for out of range values)When you collect data you need to decide on two thingsv What to measurev How to measureWhat to Measurev Usually we collect several measures on each person or thing of interestv Each thing we collect data about is called an observationv Each observation can be a person or an organisation, a product or a period in time Factual or demographic variables- age, gender, education, occupation, incomeAttitudes or opinion variables- record how respondents feel about something, or what they think is true or falsev Behaviour or event variables- contain data about what people did or what happened in past, happening or will happen in the future.Data collection- what to measure continue …..VariablesObservations Age SexIncome Brand preference 20Female 10,000Zara 30Female 35,000Cartier25Male 25,000 SuperdryLevel of MeasurementLevel of measurement: is the relationship between what is being measured and the numbers that represent what is being measured Variables can take many forms and levels. Quantitative data can be divided into two distinct groups:v Categorical variable◦Descriptive or Nominal data◦Ranked datav Numerical variable◦Continuous data◦Discrete DataCategorical DataCategorical data- refer to data whose values can not be measured numerically but can be classified into sets (categories) according to the characteristics that identify or describe the variable or placed in rank order (Brown and Saunders, 2008)i.e. your species (human, domestic cat, fruit bat), race, sex, age group, and educational levelv Descriptive/Nominal data- these data simply count the number of occurrences in each category of a variable. (a car manufacturer may categorize the types of cars as hatchback, Saloon and estate)Binary/dichotomous variable: a sub-type of nominal scale with only two categories.It names two distinct types of things such as: Male or Female, Dead or Alive; Yes or NoCategorical Data continuous…….v Ranked/ordinal variable- precise form of categorical data. When categories are ordered, the variable is known as an ordinal variable.v Ordinal variables will tell us the things that happenedv The order in which things occurredv But this data will not tell us differences between the values/points on a scaleExample 1- beauty pageant winners- first, second and thirdExample-2 : satisfaction levelUnsatisfied Very unsatisfied Very Satisfied Satisfied Somewhatsatisfied123 45Examples of Nominal and ordinal DataNumerical Variablev Numerical variables are those whose values are measured or counted numerically as quantities . (Brown and Saunders, 2008).v These data are more precise than categorical data as you can assign each data valuea position on a numerical scaleTypes of Numerical variable:v Continuous variable-a variable that has a changing value and it can take on infinitely many, uncountable number. i.e. time, a person’s weight, Age, number of customers etcv Interval data: for data to be interval, we must be certain that data on scalerepresent equal differences in the property being measured. (i.e. measure averageday time temperature during summer in London- 60-70 degrees Fahrenheit, 80-90….)Numerical Data continues ……v Ratio data : a ratio variable has all the properties of an interval variable, and also has an absolute value of 0.v Discrete variable- can take only certain values. Usually the whole number on the scale. i.e.v Example:rating your confidence level on 5 point scale.Number of Customers-12Number of shops- 10Number of friends- 8Stress Level0 1 2 3 4 5 6 7 8 9 10Seminar ActivityActivity-1:Demonstrate your choice of data collection method(s) that you plan to adopt in your study.Activity-2-In small groups, discuss about different types of questionnaire and which one you are likely to use.- What are the main attributes of questionnaires discussed on this PPT?-What scale of measurement questions you are likely to use in you questionnaire, provide at least three examples of Rating scales questions.。
CFA考试必考点精讲-数量分析方法：抽样与估计考点解析
CFA考试必考点精讲-数量分析方法：抽样与估计考点解析对于很多想参加CFA考试的同学来说，对于CFA的考试内容还不是很了解。
我就为大家分享一下CFA考试的考试科目：1、道德与职业行为标准(Ethics and Professional Standards)2、定量分析(Quantitative)3、经济学(Economics)4、财务报表分析(Financial Statement Analysis)5、公司理财(Corporate Finance)6、权益投资(Equity Investments)7、固定收益投资(Fixed Income)8、衍生工具(Derivatives)9、其他类投资（Alternative Investments）10、投资组合管理(Portfolio Management)推断统计是研究如何利用样本数据来推断总体特征的统计方法。
可以分为三个步骤：✔抽样（sampling）：采取一定的方法从总体中抽取一部分个体，组成样本（我们可以从总体中获得无数个样本）；✔估计（estimation）：根据样本信息估计总体特征；✔假设检验（hypothesis test）：利用样本信息判断在一定的置信水平下对总体的假设是否成立。
详述：抽样当总体中个体数量非常大时，我们不可能对个体进行一一观测。
例如，想要了解中国人的平均身高，我们就不可能对13亿人口都进行问卷调查，这时就需要进行抽样。
简单随机抽样假设一个总体内包含N个个体，如果通过逐个抽取的方法从中抽取n个个体组成一个样本，并且每次抽取时每个个体被抽到的概率相等，那么这样的抽样方法叫做简单随机抽样。
简单随机抽样的特征：✔每个个体被抽中的概率相等，个体之间是完全独立的。
✔总体个数N是有限的。
✔样本数n小于等于样本总体的个数N。
✔样本是逐个抽取的。
✔简单随机抽样是一种不放回的抽样。
分层随机抽样分层随机抽样首先将个体按一定的标准分为几类（或几层）；然后根据各类型个体数与总个体数的比例，确定每个类型应该抽取多少个个体；最后，根据随机抽样原则，从各类型中抽取相应数量的个体，组成最终的样本。
CFA资料-一级易考点-数量 quantitative method
知识点4-Quantitative Method:CV /Sharpe Ratio1.CV (Coefficient of Variation)✓s CV=X,表示单位回报对应的风险，越小越好。
2.Sharpe Ratio：✓P F PR - R Sharpe ratio = σ，单位风险对应的超额回报，越大越好。
知识点5-Quantitative Method:Covariance /Correlation1.Covariance✓衡量两个变量的线性关系。
✓取值范围：(-∞,+∞)，正值代表正相关，负值代表负相关。
2.Correlation✓公式i j i j i,j i,j i j i jCov(R ,R )Cov(R ,R )ρ=, r =σσs s 。
✓取值范围：(-1,+1)，+1/-1代表完全正/负相关，0代表没有线性关系。
知识点6-Quantitative Method:Normal Distribution1.正态分布的性质：✓X~N(μ,σ²)，由均值μ和方差σ²两个参数完全描述。
✓skewness=0，kurtosis=3.✓正态分布变量的线性组合还是服从正态分布。
2.正态分布的置信区间：✓68%confidence interval is [μ-σ,μ+σ].✓90%confidence interval is [μ-1.65σ,μ+1.65σ].✓95%confidence interval is [μ-1.96σ,μ+1.96σ].✓99%confidence interval is [μ-2.58σ,μ+2.58σ].3.正态分布的应用：✓Shortfall risk：回报低于shortfall level（R L ，最低可接受回报）的概率。
✓Safety-first Ratio：♦()p L pR R E SFRatio = σ-，表示预期回报与最低可接受回报（R L ）之间有几个标准差的安全边际，Safety-first Ratio 越大越好。
1.Quantitative Methods
Quantitative Methods1.An investment is expected to produce the cash flows of $500, $200, and $800 atthe end of the next three years. If the required rate of return is 12%, the present value of this investment is closest to:A.$835.B.$1,175.C.$1,235.Solution: BUsing your cash flow keys, CF0 = 0; CF1= 500; CF2 = 200; CF3 = 800; I/Y = 12; NPV = $1,175.29.Or you can add up the present values of each single cash flow.PV1: N = 1; FV = ‐500; I/Y = 12; CPT→PV= 446.43PV2 :N= 2; FV = ‐200; I/Y = 12; CPT→PV= 159.44PV3: N =3; FV = ‐800; I/Y = 12; CPT→PV= 569.42Hence, 446.43 + 159.44 + 569.42 = $1,175.29.2.An investor is looking at a $150,000 home. If 20% must be put down and thebalance is financed at 9% over the next 30 years, what is the monthly mortgage payment?A.$799.33.B.$895.21.C.$965.55.Solution: CN = 30* 12 = 360; I/Y = 9 / 12 = 0.75; PV =‐1 50,000*(1‐ 0.2) = ‐120,000; FV = 0; CPT → PMT = $965.553.If $10,000 is borrowed at 10% interest to be paid back over ten years, how muchof the second year’s payment is interest (assume annual loan payments)?A.$937.26.B.$954.25.C.$1,037.26.Solution: ATo get the annual payment, enter PV = ‐10,000; FV = 0; I /Y = 10; N = 10; CPT→ PMT = 1,627.45. The first year’s interest is $1,000 = 10,000 * 0.10, so the principal balance going into year 2 is 10,000‐(1627.45‐1000) = $9,372.55. Year 2 interest= $9,372.55*0.10 = $937.26.4.What is the effective annual rate for a credit card that charges 18% compoundedmonthly?A.15.38%.B.18.81%.C.19.56%.Solution: CEAR = [(1 + (0.18 / 12))12 – 1 = 19.56%5.What is the effective annual yield for a T‐bill that is selling for $99,000, with aface value of $100,000 and 95 days remaining until maturity?A. 3.79%.B. 3.94%.C. 4.50%.Solution: B(1 + 0.0101)365/95 ‐ 1= 3.94%6.What is the money market yield for a T‐bill that is selling for $99,000, with a facevalue of $100,000 and 95 days remaining until maturity?B. 3.83%.C. 3.90%.Solution: B(360 *0.0379) / [360 ‐(95* 0.0379)] =3.83%Or (1,000 / 99,000)(360 / 95)=3.83%7.A 175‐day T‐bill has an effective annual yield of 3.80%. Its money‐market yield isclosest to:A. 1.80%.B. 3.65%.C. 3.71%.Solution: CSince the effective yield is 3.8%, we know (1000/price)365/175 = 1.038And price = 1000/1.038175/365 = $982.28 per$1000 face.The money market yield is 360 * HPY = (360 / 175) * (1000 / 982.28 ‐ 1) = 360/175 * 0.01804 = 3.711%.Alternatively, we can get the HPY from the FAY of 3.8% as 1.038175/365 ‐1 = 1,804%Use the following data to answer Questions 8 through 9.XYZ Corp. Annual Stock Prices2003 2004 2005 2006 2007 2008 22% 5% ‐7% 11% 2% 11%8.What is the mean absolute deviation for XYZ stock returns?A. 5.20%.B.7.33%.C.29.0%.The mean absolute deviation is found by taking the mean of the absolute values of the deviations from the mean.(|22 ‐ 7.3| + |5 – 7.3| + |‐7 – 7.3| + |11 – 7.3| + |2 – 7.3| + |11 – 7.3|) / 6 = 79.Assuming that the distribution of XYZ stock returns is a sample, what is thesample standard deviation?A.9.8%.B.72.4%.C.96.3%.Solution: AThe sample standard deviation, s, is the square root of the sample variance.s = {[(22 ‐ 7.3)2 + (5 ‐ 7.3)2 ÷ (7 ‐ 7.3)2 ÷ (11 ‐ 7.3)2 + (2 ‐ 7.3)2 + (11 ‐ 7.3)2]/ (6 ‐ 1) }0.5 = (96.3)0.5 = 9.8%10.Which of the following is most accurate regarding a distribution of returns thathas a mean greater than its median?A.It is positively skewed.B.It is a symmetric distribution.C.It has positive excess kurtosis.Solution: AA distribution with a mean greater than its median is positively skewed, or skewed to the right. The skew “pulls” the mean.Note: Kurtosis deals with the overall shape of a distribution and not its skewness.Use the following conditional probabilities to answer Questions 11 through 14.State of the Economy Probability of theEconomic StateStockPerformanceConditional Probability ofStock PerformanceGood 0.30 Good 0.60 Neutral 0.30Poor 0.30Neutral 0.50 Good 0.30 Neutral 0.40 Poor 0.30Poor 0.20 Good 0.10 Neutral 0.60 Poor 0.3011.What is the conditional probability of having good stock performance in a pooreconomic environment?A.0.02.B.0.10.C.0.30.Solution: BGo to the poor economic state and read off the probability of good performance [i.e., P(good performance| poor economy) = 0.10].12.What is the total probability of having a good performance in the stock?A.0.35.B.0.65.C. 1.00.Solution: A(0.3)(0.6) + (0.5)(0.3) + (0.2)(0.l) = 0.35. This is the sum of all the joint probabilities for good performance over all states [i.e., > ∑P(economic state) P(good performance | economic state)].13.What is the joint probability of having a good economy and a neutral stock performance?A.0.09.B.0.20.C.0.03P(good economy and neutral performance) = P(good economy)P(neutral performance| good economy) = (0.3)(0.3) = 0.09.14.Given that the stock had good performance, the probability the state of theeconomy was good is closest to:A.0.35.B.0.46.C.0.51.Solution: CThis is an application of Bayes’ formula. P(good economy | good performance) = P(good stock performance |good economy) × P(good economy) / P(good stock performance).=()()()()()()()()0.60.30.30.6+0.50.3+0.20.1=35.018.0=0.514315.Ten percent of all college graduates hired stay with the same company for morethan five years. In a random sample of six recently hired college graduates, the probability that exactly two will stay with the same company for more than five years is closest to:A.0.098.B.0.114.C.0.185.Solution: ASuccess = staying for five years. [6! / 2!(6 — 2)! ] (0.10)2(0.90)6—2 =15(0.01)(0.656)=0.0984.16.For the standard normal distribution, P(0 < Z < 1.96) is:A.0.4713.C.0.4750.Solution: CFrom the table F(1.96) = 0.9750; thus, the answer is 0.9750 — 0.5 = 0.4750. Knowing that 95% lie between —1.96 and + 1.96, and that 0 is the midpoint, we can saythat 95%2=47.5% lie between 0 and + 1.96.17.Given a threshold level of return of 4%, use Roy’s safety‐first criterion to choose the optimal portfolio. Portfolio:Portfolio Portfolio A Portfolio B Portfolio CE(R) 5% 11% 18%σp 8% 21% 40%A. A.B. B.C. C.Solution: CSFR = (18—4) / 40 is the largest value.ing hypothesized parameter values and a random number generator to studythe behavior of certain asset returns is part of:A.historical analysis.B.Monte Carlo simulation.C.standardizing a random variable.Solution: BMonte Carlo simulation involves modeling asset prices or returns by generating random values for the risk factors that affect the price of a security.19.Approximately 95% of all observations for a normally distributed randomvariable fall in the interval:B.μ±2σ.C.μ± 3σ.Solution: Bμ ± 2σ. Approximately 95% of the outcomes for a normally distributed random variable are within two standard deviations of the mean.20.Best Computers, Inc., sells computers and computer parts by mail. A sample of25 recent orders showed the mean time taken to ship out these orders was 70hours with a sample standard deviation of 14 hours. Assuming the population is normally distributed, the 99% confidence interval for the population mean is:A.70 ± 2.80 hours.B.70 ± 6.98 hours.C.70 ± 7.83 hours.Solution: CSince the population variance is unknown and n < 30, the confidence interval is determined as X ± tα/2 (s/n) . Look up tα/2and df = n‐1 to get critical t‐value. t01 and df = 24 is 2.797. So, the confidence interval is 70 ± 2.797(14 / 5) = 70 ±2.797 (14/5) = 70±7.83.21.What is the most appropriate test statistic for constructing confidence intervalsfor the population mean when the population is normally distributed, but the variance is unknown?A.The z‐statistic at α with n degrees of freedom.B.The t‐statistic at α/2 with n degrees of freedom.C.The t‐statistic at α/2 with n ‐ 1 degrees of freedom.Solution: CUse the t‐statistic at α/2 and n ‐ 1 degrees of freedom when the population variance is unknown. While the z‐statistic is acceptable when the sample size is large, sample size is not given here, and the t‐statistic is always appropriate under these conditions.22. Which of the following statements about hypothesis testing is least accurate ? A. The power of test = 1 ‐ P(Type II error).B. If the computed z ‐statistic = —2 and the critical z ‐value = —1.96, the nullhypothesis is rejected.C. The calculated z ‐statistic for a test of a sample mean when the populationvariance is known is: z =02X-μ Solution: Cz =X-μ (σ2 is the variance) 23. Using a 5% level of significance and a hypothesis test structure of H 0: 21σ ≤24versus H a : 21σ> 24, the null hypothesis:A. cannot be rejected.B. should be rejected.C. cannot be tested using this sample information provided. Solution: AThe chi ‐square test is used to test hypotheses concerning a single population variance. Since this is a one ‐tailed test, the decision rule is to reject H 0 if X 2 > the critical chi ‐square value at c = 0.05 with df = 24.()()()22n-120n-1s 2425X ==σ24= 25.0 . Theright ‐tail critical chi ‐square value is 36.4 15. Since x 2 = 25≤36.415, H 0 cannot be rejected.24. A momentum indicator based on the ratio of price increases to price decreasesover the last 14 days is most likely a: A. stochastic oscillator.B. Relative Strength Index.C. moving average convergence/divergence oscillator. Solution: BThe RSI is calculated from the ratio of total price increases to total price decreases over a chosen number of days, then scaled to fluctuate between 0 and 100 using the formula RSI = 100 ‐ [100 1 (1 + ratio of increases to decreases)]. Stochastic oscillators are based on the highest and lowest prices over a chosen number of days. MACD oscillators are calculated based on exponentially smoothed moving averages. 25. The standard error of the sampling distribution of the sample mean for a samplesize of n drawn from a population with a mean of µ and a standard deviation of σ is:A. sample standard deviation divided by the sample size.B. sample standard deviation divided by the square root of the sample size.C. population standard deviation divided by the square root of the sample size. Solution: CThe formula for the standard error when the population standard deviation is known is σ=nσ26. For a skewed distribution, what is the minimum percentage of the observationsthat will lie between ±2.5 standard deviations of the mean based on Chebyshev’s Inequality? A. 56%. B. 75%. C. 84%. Solution: CApplying Chebyshev’s inequality, 1 ‐[1/(2.5)2] = 0.84, or 84%. 27. If a stock’s relative strength ratio increases, the stock is:A.increasing in price.B.outperforming its benchmark.C.increasing on high volume or decreasing on low volume.Solution: BIf the relative strength ratio (stock price / benchmark value) increases, the stock is outperforming the benchmark stock or index against which it is being measured. This does not imply that the stock is increasing in price; if the stock price decreases but the benchmark decreases by a larger percentage, the ratio will increase. Volume is not an input into a relative strength.28.A company is considering entering into a joint venture that will require aninvestment of $10 million. The project is expected to generate cash flows of $4 million, $3 million, and $4 million in each of the next three years, respectively.Assuming a discount rate of 10%, what is the project’s NPV?A.‐$879,000.B.‐$309,000.C.‐$243,000.Solution: ANPV = 4 / 1.10 + 3 / 1.102 + 4 / 1.103 ‐ $10 = ‐$0.879038 million, or‐$879,038.Calculator approach: CF0= ‐10; CF1 = 4; CF2 = 3; CF3 = 4; I = 10→NPV=‐$0879038 (million).29.At a charity ball, 800 names are put into a hat. Four of the names are identical.On a random draw, what is the probability that one of these four names will be drawn?A.0.004.B.0.005.C.0.010.Solution: BP(name I or name 2 or name 3 or name 4) = 1/800 + 1/800 + 1/800 + 1/800 = 4/800= 0.005。
CFA一级典型例题 Quantitative Method 定量方法
Quantitative Method1. A 24 year old is using the following information to plan her retirement:Current age 24Expected retirement age 68Life expectancy 93Current annual expenditures $30,000Expected inflation rate ofCurrent expenditures until retirement 3%Expected return on investment 8%She assumes her consumption expenditures will increase with the rate of inflation, 3 percent, until she retires. Upon retiring she will have end-of-year expenditures equal to her consumption expenditure at age 68. The minimum amount that she must accumulate by age 68 in orderto fund her retirement is closest to:A. $928,000.B. $1,176,000.C. $1,552,000.Answer: BHer consumption spending (currently $30,000 annually) increases with the rate of inflation (3%) over the next 44 years until she retires. Her annual consumption spending at the time she retires will be $110,143.57 (PV = 30,000, %I = 3, N = 44, solve for FV). To support that level of spending for 25 years of retirement assuming an 8% return on her retirement account, she must accumulate $1,175,756 by her retirement date (PMT = 110,143.57, N = 25, %I = 8, solve for PV).2.A graphic depiction of a continuous distribution that shows the left tail to be longer than the right tail is best described as having:A. lepto-kurtosis.B. positive skewness.C. negative skewness.Answer: CA negatively skewed distribution appears as if the left tail has been pulled away from the mean.3. The manager of a pension fund determines that during the past five years 85 percent of the stocks in the portfolio have paid a dividend and 40 percent of the stocks have announced a stock split. If 95 percent of the stocks have paid a dividend and/or announced a stock split, the joint probability of a stock paying a dividend and announcing a stock split is closest to:A. 30%.B. 45%.C. 55%.-1-Answer: AThe probability that at least one of two events will occur is the sum of the probabilities of the separate events less the joint probability of the two events. P(A or B) = P(A) + P(B) – P(AB) 95% = 85% + 40% – P(AB); therefore P(AB) = 30%4. A portfolio manager gathers the following information about three possible asset allocations: Allocation Expected annual return Standard deviation of returnI13% 6%II 26% 14%III 32% 20%The manager’s client has stated that her minimum acceptable return is 8 percent. Based on Roy’s safety-first criterion, the most appropriate allocation is:A. IB. IIC. IIIAnswer: BRoy’s safety-first ratio = [E(R P) –R L] / σP with the optimal portfolio having the highest ratio. The safety-first ratios for the three allocations are:Allocation Safety-first ratioI 0.83II 1.29III 1.205. A sample of 100 observations drawn from a normally distributed population has a sample mean of 12 and a sample standard deviation of 4. Using the extract from the z-distribution given below, find the 95% confidence interval for the population mean. The 95% confidence interval is closest to:Cumulative Probabilities for a Standard Normal DistributionP(Z ≤ x) = N(x) for x ≥ 0 or P(Z ≤ z) = N(z) for z ≥ 0x or z 0 0.01 0.02 0.03 0.04 .05 .06 .07 .08 .091.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817A. 7.840 to 27.683B. 11.216 to 12.784C. 11.340 to 12.660Answer = BThe 95% confidence interval uses z0.025 as the reliability factor. The cumulative probability value closest to 0.975 provides the appropriate value of z0.025 which is z0.025 = 1.96. The confidence interval-2--3-is formed as:α2s X z ?± . In this problem, 12 ± 1.96 × (4 / √100) = 12 ± 1.96 × 0.4. Thus the6. Using the sample results given below, drawn as 25 paired observations from their underlyingdistributions, test if the mean returns of the two portfolios differ from each other at the 1% level of statistical significance. Assume the underlying distributions of returns for each portfolio are normal and that their population variances are not known.Portfolio 1 Portfolio 2 DifferenceMean Return 17.00 21.25 4.25Standard Deviation 15.50 15.75 6.25t-statistic for 24 df and at the 1% level of statistical significance = 2.807Based on the paired comparisons test of the two portfolios, the most appropriate conclusion is:A. reject the hypothesis that the mean difference equals zero as the computed test statistic exceeds2.807.B. accept the hypothesis that the mean difference equals zero as the computed test statistic exceeds2.807.C. accept the hypothesis that the mean difference equals zero as the computed test statistic is less than 2.807.Answer = A The test statistic is:()0d /d d S μ- , where d is the mean difference,μ0d is the hypothesizeddifference in the means, S d is the sample standard deviation of differences, and n is the sample size. In this case, the test statistic equals: (4.25-0)/(6.25/√25)= 3.40. As 3.40 > 2.807, we reject the null hypothesis that the mean difference is zero.7. The joint probability of returns, for securities A and B, are as follows:Joint Probability Function of Security A and Security B Returns(Entries are joint probabilities)Return on security B=30% Return on security B=20%Return on security A= 25% 0.60 0Return on security A= 20% 0 0.40The covariance of the returns between securities A and B is closest to:A. 3.B. 12.C. 24.Answer: BExpected return on security A = 0.6 × 25% + 0.4 × 20% = 15% + 8% = 23%Expected return on security B = 0.6 × 30% + 0.4 × 20% = 18% + 8% = 26%Cov (R A,R B) = 0.6 [(25 − 23)(30 −26)] +0.4 [(20 − 23)(20 − 26)]= 0.6(2 × 4) + 0.4(−3 × −6) = 0.6(8) +0.4(18) = 4.8 + 7.2 = 12.8. A two-tailed t-test of the null hypothesis that the population mean differs from zero has a p-value of 0.0275. Using a significance level of 5%, the most appropriate conclusion is:A. reject the null hypothesis.B. accept the null hypothesis.C. the chosen significance level is too high.Answer = AThe p-value is the smallest level of significance at which the null hypothesis can be rejected. In this case, the given p-value is less than the given level of significance and we reject the null hypothesis.9. If the distribution of the population from which the samples are drawn is positively skewed, and given that the sample size is large, the sampling distribution of the sample means is most likely:A. approximately normally distributed.B. to have a variance equal to that of the entire population.C. to have a mean smaller than the mean of the entire population.Answer: AThe central limit theorem establishes that the sampling distribution of sample means will be approximately normal, will have a mean equal to the population mean, and will have a variance equal to the population variance divided by the sample size.10. Which of the following statements about a normal distribution is least accurate? A normal distribution:A. has an excess kurtosis of 3.B. is completely described by two parameters.C. can be the linear combination of two or more normal random variables.Answer: AA normal distribution has a kurtosis of 3. Its excess kurtosis (kurtosis – 3.0) equals zero.-4-。
CFA一级培训项目 quantitative methods 金融计算器使用 数量 1页版
me Value of Money
> About perpetuity • Definition: A perpetuity is a set of level never-ending sequential cash flows, with the first cash flow occurring one period from now.
• PV = present value
• PMT = amount of each periodic payment
• FV= future value
10-73
行业•创新•增值
♦ Time Value of Money
> An example of ordinary annuities （后付年金）: > Example: Tom makes a saving plan with $1000 annual deposit at the end
15-73
行业•创新•增值
定量研究方法(Quantitative Research Method)
什么是定量研究？定量研究一般是为了对特定研究对象的总体得出统计结果而进行的。
定性研究具有探索性、诊断性和预测性等特点，它并不追求精确的结论，而只是了解问题之所在，摸清情况，得出感性认识。
定性研究的主要方法包括：与几个人面谈的小组访问，要求详细回答的深度访问，以及各种投影技术等。
在定量研究中，信息都是用某种数字来表示的。
在对这些数字进行处理、分析时，首先要明确这些信息资料是依据何种尺度进行测定、加工的，史蒂文斯（S. S. Stevens）将尺度分为四种类型，即名义尺度、顺序尺度、间距尺度和比例尺度。
[编辑]定量研究的四种测定尺度及特征名义尺度所使用的数值，用于表现它是否属于同一个人或物。
顺序尺度所使用的数值的大小，是与研究对象的特定顺序相对应的。
例如，给社会阶层中的上上层、中上层、中层、中下层、下下层等分别标为“5、4、3、2、1”或者“3、2.5、2、1.5、1”就属于这一类。
只是其中表示上上层的5与表示中上层的4的差距，和表示中上层的4与表示中层的3的差距，并不一定是相等的。
5、4、3 等是任意加上去的符号，如果记为 100、50、10 也无妨。
间距尺度所使用的数值，不仅表示测定对象所具有的量的多少，还表示它们大小的程度即间隔的大小。
不过，这种尺度中的原点可以是任意设定的，但并不意味着该事物的量为“无”。
例如，O°C 为绝对温度273°K，华氏32°F。
名义尺度和顺序尺度的数值不能进行加减乘除，但间距尺度的数值是可以进行加减运算的。
然而，由于原点是任意设定的，所以不能进行乘除运算。
例如，5℃和 10℃之间的差，可以说与15℃和20℃之间的差是相同的，都是5°C。
但不能说 20℃就是比5℃高4倍的温度。
比例尺度的意义是绝对的，即它有着含义为“无”量的原点0。
长度、重量、时间等都是比例尺度测定的范围。
比例尺度测定值的差和比都是可以比较的。
例如：5分钟与10 分钟之间的差和10分钟与15分钟之间的差都是5 分钟，10 分钟是2分钟的5倍。
Exercise lecture 1
N [( µ small − µl arg e , (σ
2 small
/ nsmall ) + (σ
2 l arg e
/ nl arg e )]
Empirical problem: Class size and educational output
2 • If σ small and σ l2arge are known, then this approximate normal distribution can be used to compute p-values for the test of the null hypothesis that
14
Compute the difference-of-means t-statistic:
Size small large
t=
Y 657.4
650.0
Ys − Yl
2 ss
SY 19.4 17.9
= 657.4 − 650.0
19.42 238
n 238 182
7.4 = 4.05 = 1.83
13
2. Hypothesis testing
Difference-in-means test: compute the t-statistic,
t=
Ys − Yl
2 ss
ns
+
sl2 nl
Ys − Yl = SE (Ys − Yl )
(remember this?)
where SE(Ys – Yl ) is the “standard error” of Ys – Yl , the subscripts s and l refer to “small” and “large” STR districts, and 1 ns ss2 = (Yi − Ys ) 2 (etc.) ∑ ns − 1 i =1
方法-数量科目学习计划表
视频：Time-Series Analysis 题目：原版书课后题
1.巩固回忆知识点 （配合科目重难点 资料),也可自己总结重难点 2.做数量历年Mock题，记忆考点，找 题感
☆☆ ☆☆☆
做第三模块押题
6月16日
押题密卷解析-M3
第三模块总复习
M3(QM+Port+FI+D er）
做2018年第三模块对应 1-2套题目，巩固相应 模块知识点（留1-2套
☆☆☆
掌握Multiple Regression and Issues in Regression Analysis
视频：Multiple Regression and Issues in Regression Analysis 题目：原版书课后题
☆☆☆
掌握Time-Series Analysis
1.知识巩固 2.刷题找题感
考前模拟），记录问题
考前专场答疑-M3
6月17日 6月18日-6月19日
6月19日
☆☆☆
考前模拟
完整的做1-2套2018年 剩下的mock题
6月20日-6月22日
☆☆☆
科目
Quantitative Methods
占比 5%-10%
Case数 1
学习天数 7
天数
Day 1-Day 2 Day
数量科目学习计划表
任务
对应资料
考试重要程度
掌握Correlation and Regression
视频：Correlation and Regression 题目：原版书课后题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
Quantitative MethodWeight : 12%ContentØQuantitative Method: Basic ConceptsüThe Time Value of Money (☆☆)üDiscounted Cash Flow Applications (☆☆)üStatistical Concepts and Market Returns (☆☆☆)üProbability Concepts (☆☆)Content (Cont.)ØQuantitative Method: ApplicationüCommon Probability Distributions (☆☆)üSampling and Estimation (☆☆☆)üHypothesis Testing (☆☆☆)üTechnical Analysis (☆)只要 Good Good Study,分数一定 Day Day Up!加油，必胜！The Time Value of MoneyThe Time Value of Money ContentInterest rate TVM problemInterpretation Components Conversion PV/FVNumber of periods (n) Size of payment (PMT) Discount rate(I/Y)Interest rate conversionØSimple interest vs. Compounding interestØStated rate (r s ) vs. Effective rate (EAR)üFor continuous compounding:Interest Rate()m m s r EAR = 1+Periodic interest rate -1=(1+)-1ms r EAR=e -1If the stated annual interest rate is 9% and the frequency of compounding is daily, the effective annual rate (EAR) is closest to:A. 9.42%.B. 9.86%.C. 9.00%.3659%EAR=(1+)-1=9.42%365Answer: AA financial contract offers to pay €1,200 per month for five years with the first payment made immediately. Assuming an annual discount rate of 6.5%, compounded monthly the present value of the contract is closest to:A. €61,663.B. €63,731.C. €61,330.Answer: AUsing a financial calculator: N = 60; the discount rate, I/Y = (6.5%/12) = 0.54166667; PMT = €1,200; Future value = €0; Mode = BGN; Calculate present value (PV): PV = -€61,662.62.Pay attention to the sign and direction of cash flows.Thanks!Discounted Cash Flow ApplicationsDiscounted Cash Flow ApplicationsContentProject returnMoney market yield NPV & IRRBank discount yield Bond equivalent yieldEffective annual yield Money market yieldPortfolio returnHolding period returnTime weighted return vs.Money weighted returnTime-weighted return (TWR)ØThe compound return that $1 initially invested in the portfolio over a stated measurement period.[]1N12n TWR = (1+HPR )(1+HPR )(1+HPR )-1⨯⨯⨯ Money-weighted return (MWR)ØSimilar to IRR.()1N 0N CF CF CF + +...+ = 01+MWR 1+MWRTWR vs. MWRØ如果组合money的数量和return随时间同向变化，MWR大于TWR；如果反向变化，TWR大于MWR。
ØTWR不受cashflow影响，MWR受cashflow影响。
üTWR适用于组合经理不能控制cashflow的情形，MWR适用于组合经理能控制cashflow的情形。
Ø计算TWR的subperiod的时间长度可以不相同，MWR 一定要相同。
Money Market YieldsMoney market yieldsØBank discount yield (BDY)BDY = (Discount/Face Value) × (360/Days to maturity)ØMoney market yieldMMY = (Discount/Price) × (360/Days to maturity)ØBond equivalent yieldBEY = (Discount/Price) × (365/Days to maturity)ØEffective annual yieldEAY = (1 + Discount/Price)365/Days -1BDY < MMY < BEY < EAYAn analyst gathers the following information about a common stock investment:Date Amount (€) Stock purchase (1 share) 15 January 2006 86.00 Stock purchase (1 share) 15 January 2007 94.00 Stock sale (2 shares) 15 January 2008 212.00 The stock does not pay a dividend. The money-weighted rate of return on the investment is closest to:A. 11.02%.B. 11.60%.C. 11.89%.Answer: BThe money-weighted rate of return is the IRR based on the cash flows related to the investment. In this case, a cash outflow of €86 occurs at t=0, another outflow of €94 occurs at t=1, and an inflow of €212 occurs at t=2. Using a financial calculator, the IRR of these cash flows is 11.60%.The dollar discount on a US Treasury bill with 91 days until maturity is $2,100. The face value of the bill is $100,000. The bank discount yield of the bill is closest to:A. 8.58%.B. 8.40%.C. 8.31%Answer: CBDY = (2,100/100,000)×(360/91) = 0.083077 ~ 8.31%.Thanks!Statistical Concepts and MarketReturnsStatistical Concepts and Market ReturnsContentData distributionCentral tendencySkewnessKurtosis DispersionBasic concepts Measurement scales Presentation of dataBasic ConceptsMeasurement scalesØNominal scale: categorizing the data.ØOrdinal scale: sorting the data into categories that are ordered.ØInterval scale: ranking the data with equal differences between scale values.üAdding and subtracting are meaningfully.ØRatio scale: interval scales with true zero point as the origin.üAdding, subtracting, multiplying and dividing aremeaningfully.Central Tendency Central tendencyMedian Arithmetic mean: 预测单期回报Harmonic mean: 计算多次等额购买同一资产的平均成本Weighted mean: 计算投资组合回报Geometric mean: 计算多期平均回报Mean ModeHarmonic Mean ≤Geometric Mean ≤Arithmetic MeanDispersionRangeVariance & standard deviation Mean absolute deviation (MAD)Absolute dispersionRelative dispersion Quantiles Quartile Quintile Decile PercentileChebyshev’s Inequality Sharpe ratio Coefficient of variation (CV)QuantilesØFormula for location of data in ascending order:Where: n = the number of data.y = the y th percentile.y y L =(n+1)100Coefficient of variation (CV)Ø , a measure of risk per unit of mean return, thus the lower is better.Sharpe ratio Ø , a measure of excess return per unit of risk, thus the higher is better.s CV=X Relative Dispersion P F P R - R Sharpe ratio = σAdvantage : no units of measurement, so permits direct comparisons of dispersions across different data sets.DispersionChebyshev’s inequalityØFor any distribution with finite variance, the minimum percentage of observations that lie within k standard deviations of the mean would be 1-1/k2, given k>1.P(μ-kσ < x < μ+kσ) ≥ 1-1/k2SkewnessSkewness (S k)ØS k = 0 → symmetrical distribution.üMean = median = modeØS k > 0 → positively (right) skewed distribution.üMean > median > modeüFatter/longer right tailØS k < 0 → negatively (left) skewed distribution.üMean < median < modeüFatter/longer left tailKurtosisKurtosisØLeptokurtic: kurtosis > 3, excess kurtosis > 0.ü More peaked, fatter tailed than normal distribution.ØMesokurtic: kurtosis = 3, excess kurtosis = 0.ü Identical to normal distribution.ØPlatykurtic: kurtosis < 3, excess kurtosis < 0.ü Less peaked, thinner tailed than normal distribution.The following information is available on three portfolios: Portfolio Mean Return (%) Std. Dev. (%)D 10 20E 18 15F 6 3The risk-free rate is 4%. The portfolio that has the best risk-adjusted performance as measured by the Sharpe ratio is:A. Portfolio F.B. Portfolio E.C. PortfolioD.Answer: BThe Sharpe ratio is defined as: S p = (R p – R F)/s p .In this case, S D = (10 – 4)/20 = 0.30;S E = (18 – 4)/15 = 0.9333;S F = (6 – 4)/3 = 0.6667.The portfolio with the best risk-adjusted performance as measured by the Sharpe ratio is the one with the highest Sharpe ratio: Portfolio E.Thanks!Probability ConceptsProbability ConceptsMultiplication ruleTotal probability ruleAddition ruleAlgorithms of probability Probability statistics Basic concepts Counting problemCovariance & correlation Bayes’ formula Contenti j i,j i jCov(R ,R )ρ=σσBasic ConceptsBasic conceptsØRandom variable & outcomes & eventØRelationship among eventsüMutually exclusive events & exhaustive eventsüIndependent events & dependent eventsØTypes of probabilityüEmpirical probability vs. priori probability vs. subjective probabilityüOdds for vs. odds againstüUnconditional probability vs. conditional probabilityMultiplication ruleØP(AB)=P(A︱B)P(B)üFor independent events: P(A︱B)=P(A) → P(AB)=P(A)P(B) Addition ruleØP(A + B)= P(A) + P(B) - P(AB)üFor mutually exclusive events:P(A + B)= P(A) + P(B)Total probability ruleØP(A) = P(A|S1)P(S1) + P(A|S2)P(S2) + …… + P(A|S n)P(S n) Bayes’ formulaØP(B A)P(A B)=P(A)P(B)An analyst has established the following prior probabilities regarding a company’s next quarter’s earnings per share (EPS). Prior probabilitiesEPS below consensus 40%EPS equal or exceed consensus 60%Several days before releasing its earnings statement, the company announces a cut in its dividend. Given this information, the analyst revises his opinion regarding the likelihood that the company will have EPS below the consensus estimate.He estimates the likelihoods the company will cut the dividend as reported below.P(Cut div│EPS below consensus): 70%;P(Cut div│EPS equal or exceed consensus): 20%.Using Bayes’ formula, the updated (posterior) probability that the company’s EPS are below the consensus is closest to:A. 24%.B. 70%.C. 85%.Practice 1Answer: BFirst, calculate the unconditional probability for a cut in dividends:P(Cut div) = 0.4×0.7 + 0.6×0.2 = 0.4.Then update the probability of EPS falling below the consensus: P(EPS below consensus│Cut div)= [P(Cut div│EPS below)÷P(Cut div)]×P(EPS below)= [0.7 ÷ 0.4]×0.4 = 70%.Thanks!Common Probability DistributionsCommon Probability DistributionsDiscrete uniform distributionDiscrete distributionsContinuous distributions Basic concepts Simulation Binomial distribution Continuous uniform distribution Normal distribution t-distributionContentCommon Probability DistributionsBasic conceptsØDiscrete distribution vs. Continuous distributionØProbability function vs. Probability density function ØCumulative probability functionBinomial distributionØThe number of successes in n Bernoulli trials, assuming that the probability of success (p) is constant for all trials.üE(X) = np.üVar(X) = np(1-p).x x n-xx n-xn n!P(x) = C p (1-p)= p (1-p)(n-x)!x!Normal distributionØProperties of normal distribution:üCompletely described by mean and variance.üSkewness = 0 (symmetric about the mean).üKurtosis = 3.üLinear combination of normally distributed randomvariables is also normally distributed.üProbabilities decrease further from the mean, but the tails go on forever.68% confidence interval = μ +/- 1σ90% confidence interval = μ +/- 1.65σ95% confidence interval = μ +/- 1.96σ99% confidence interval = μ +/- 2.58σNormal distribution (Cont.)ØProbability for a given intervalStandard normal distribution (z-distribution)ØNormal distribution with μ = 0 and σ = 1.ØStandardization :Lognormal distributionØX is normally distributed, then e x is lognormal distributed.üBounded from below by zero.üPositively skewed.X-μz=σNormal distribution: applicationØShortfall risk: the risk that portfolio return will fall below the minimum acceptable level (R L, shortfall level).ØSafety-first ratio: Shortfall riskp LpE(R)-RSFRatio=σSFRatio × σMinimizing shortfall risk = Maximizing safety-first ratio。