CFA一级笔记-第二部分 数量分析方法
2012CFA一级总结(数量统计部分)
数量统计部分设置计算器 [2nd] P/Y 1 [ENTER] [2nd][quit]一、几个利率的概念1. EAR—effective annual rate.EAR=(1+periodic rate)^m-1,periodic rate=年利率/m, m=每年compound的次数当m->无穷时,EAR=e^rate-1, rate[2nd][e^x]-1Ordinary annuities(现金流在期末)与annuities due(现金流在期初),需将计算器调整成BGN模式,[2nd][BGN][2nd][set]永久年金perpetuity， PV=PMT/IYUneven 现金流的PV和FV计算NPV[CF][2nd][CLR Work]Initial cash outlay [Enter][↓]period 1 cash flow [+-][Enter][↓][↓]period 2 cash flow [+-][Enter]……[↓][↓]period n cash flow [+-][Enter][NPV]discount rate [Enter][↓][CPT]FV要分别计算再累加2.BDY--Bank discount yield银行贴现率（T-bill以此报价）BDY =D/F*360/t其中D是折现额（face value-购买价格），F是面值！转换也是采用360，不是复利，该指标缺陷很大3.持有期收益Holding period returnHolding period yield-HPY=(P1-P0+Div)/P0顾名思义4.Money-weighted return and time-weighted return计算Money-weighted return资金加权,统计每期cash flow,计算使PV inflow=PV outflow的IRR计算器IRR[CF][2nd][CLR Work]Initial cash outlay [Enter][↓]period 1 cash flow [+-][Enter][↓][↓]period 2 cash flow [+-][Enter]……[↓][↓]period n cash flow [+-][Enter][IRR][CPT]time-weighted return时间加权:先计算每期的HPR,在进行算几何平均数。
CFA一级金融数量分析的三个难点剖析
全球最大的CFA(特许金融分析师)培训中心总部地址：上海市虹口区花园路171号A3幢高顿教育电话：400-600-8011网址： 微信公众号：gaoduncfa 1 CFA 一级金融数量分析的三个难点剖析 根据中心极限定理(Central Limit Theorem)，如果乒乓球的随机数容量很大，即不符合正态分布， 其样本均值将服从m 为总体均值，s 为总体标准差除以n 平方根的正态分布。
在估量数据的标准误差时，考生常常疑惑何时应当用标准差s(standard deviation)度量、何时又应该用s/√n (standard deviation divided by square root of n)度量。
考生须牢记，在计算样本均值的置信区间时，就要用s/√n 来度量误差。
举例来说，考虑100个标上了正态随机数的乒乓球， 这串随机数的均值(mean)是0，标准差(standard deviation)是10。
根据置信区间的计算，将有95%的随机数落在(-1.95*10，1.95*10)区间内。
现在考虑9个样本球，并假定这9个乒乓球的随机数均值为0，样本均值标准差为10/√9 = 10/3 = 3.33. 那么这9个样本球的均值有95%的概率落在(-1.96*3.33,1.96*3.33)区间内。
样本的规模越大，样本均值就越接近真实均值。
现在若考虑100个样本球随机数，均值标准差为10/√100 =10/10 = 1，则这100个随机数的均值95%的概率落在(-1.96,1.96)。
计算货币的时间价值，考生遇到的难点往往是计算在n 期时间后开始的(永续)年金的折现值。
需要注意的是，考生若将计算器设置在END 模式，计算出的现值即折现到第一个支付日的前一日。
各位考生，2015年CFA 备考已经开始，为了方便各位考生能更加系统地掌握考试大纲的重点知识，帮助大家充分备考，体验实战，网校开通了全免费的高顿题库（包括精题真题和全真模考系统），题库里附有详细的答案解析，学员可以通过多种题型加强练习，通过针对性地训练与模考，对学习过程进行全面总结。
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一级考试《数量分析》考试大纲
CFA一级考试《数量分析》考试大纲CFA一级考试大纲科目《数量分析》中的考纲变化：数量科目权重未变8-12%，但被移动到第一个部分，替代道德的位置。
CFA一级考试总体来说，本科目是新考纲中变动较大的科目，主要体现在协会更加注意学问的有用性，与当前金融行业的进展特别契合。
新增两个章节分别为：Organizing,Visualizing,And Describing Data以及Introduction To Linear Regression,同时删除Statistical Concepts and Market Returns，其余每一个reading 都有少量的增减。
变化分析：第一个新增的"Organizing,Visualizing,and Describing Data'其实是原本"Statistical Concepts and Market Returns'的进阶版本。
在新增的这个章节中，除了保留对数据描述性的分析介绍之外，更加注意对数据进行系统性的分类、整理和可视化分析，介绍了在实务中常用的一些数据分析手段和方法论，比如热力图、树状图、云图、气泡线图等。
另一个新增的章节"Introduction to Linear Regression'原本是二级数量中的一个章节，这个章节主要讲的是线性回归的基本规律和原理，是为二级数量中多元回归打基础的一个章节。
另外，线性回归是金融数据分析中常用的一个量化分析模型，更加偏重应用，突出了协会对量化建模的重视。
详细变动内容：STUDY SESSION 1-QUANTITATIVE METHODS（1）Reading 1.THE TIME VALUE OF MONEY（原Reading 6-R6）：（1）由solve time value of money problems for different frequencies ofcompounding变为calculate the solution for time value of moneyproblems with different frequencies of compounding，考纲描述变化。
CFA一级数量分析最新考点解析20200127
最新CFA一级数量分析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。
中博-诚通小编就为大家分享一下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)CFA原版书将数量分为两个session去阐述:第一个session主要围绕着计量的基本概念去展开;第二个session 则着眼于实际运用。
Reading 5和Reading 6:铺垫类的内容，为学习其他科目做一个铺垫，同时也是金融人的common sense的一部分。
Reading 7、Reading 8和Reading 9：描述性统计的内容。
Reading 10和Reading 11是推断性统计的内容。
Reading 12其实就是一个理论具体分析运用的过程，为分析师们提供了一些可实操重要指标。
CFA协会一篇针对投资风格的文章深入浅出地讲解了资产组合中如何通过收益率风险的分布情况对冲TailRisk(尾部风险)，作者调用了1981-2010这20年G7(加拿大、法国、德国、意大利、日本、英国和美国)国家主板市场股票收益率情况。
标准差、夏普比率大家都不陌生，当然真正的投资组合管理不会只让你看考虑收益率统计值和风险调整收益这么简单，实际中我们更需要发现问题，并解决(InvestmentStrategy).我们在量化中学的偏度(Skewness)这个概念很少有人用到，实际上这个概念在解读历史收益率分布时非常有用，因为它决定了我们投资组合的尾部风险(TailRisk)，更决定了我们下一步hedge的策略。
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。
CFA考试：投资分析的数量方法(投资工具)
第一章：货币的时间价值Chapter ⒈The Time Value of Money§⒈解释利息率是对投资者的不同风险予以回报的实际无风险利率和风险溢价的总和利息率和折现率（Interest Rates and Discount Rates）货币时间价值概念的基础：收益率（rates of return）、利息率（interest rate）、要求的收益率（required rates of return）、折现率（discount rates）、机会成本（opportunity costs）、通货膨胀（inflation）和风险（risk）。
货币的时间价值，反映了时间、现金流量和利息率三者之间的关系。
投资者偏好现在消费。
利息率是投资者推迟现在消费的回报。
在确定世界，利息率被认为是无风险（risk-free）利率。
一般是国家的短期债券，如美国的国库券（Treasury-bills, T-bills）。
在不确定的世界，有两个因素影响利息率：①通货膨胀。
贷款者承担通货溢价（inflation premium）和推迟消费的机会成本。
因此，货币的名义成本（nominal cost of money），由实际利率（real rate）和通货溢价组成。
②风险。
贷款者还承担了不履行风险（default risk）。
因此，利息率包括：名义的无风险利率和不履行风险溢价。
利息率的意义：①收益要求率。
即促使投资者放弃现在消费所要求的收益。
②折现率（利息率和折现率可以交互使用）。
③机会成本。
即投资者按某一选择行为而放弃其他选择所失去的价值。
影响利息率最重要的因素是：资金的供求关系。
§⒉计算整笔现金的终值（FV）和现值（PV）单一现金流量的终值（The Future Value of a Single Cash Flow）整笔现金流（或lump-sum investment）的终值计算公式（N的初值为0）：interest），即利息率乘原始本金。
CFA一级笔记-第二部分数量分析方法
CFA一级考试知识点第二部分数量分析方法名义利率等于实际利率加上预期通货膨胀率，而不是当期的实际通货膨胀率。
Holding period return,HPR持有期收益率Bank discount yield,BDY银行贴现利率，本金为F，价格为P，公式：F-PF * 360T Money market yield,MMY货币市场收益率F-PP * 360TEffective yield,EAY有效年利率（1+HPY）^365/t-1Money-weighted rate of return,MWR货币加权收益率（内部收益率）Time-weighted rate of return,TWR时间加权收益率（几个收益期间的几何平均）Bond equivalent yield,BEY债券等价收益率（irr的年化）货币加权受现金流入流出影响，因此时间加权更加广泛四种度量衡：名义尺度nominal scale（分类不排序）、排序尺度ordinal scale（排序进行比较，不能够加减，有优先次级，不成比例）、区间尺度interval scale（温度、评分，零不具备数学意义）、比例尺度ratio scale（常用最高级、身高、收入、资产收益率）算术平均arithmetic mean：相加后除以数据几何平均geometric mean：可以排除算术平均的极端值，相乘后开次方nX1*X2*X3。
计算多期平均价收益率n1+r11+r2….n-1调和平均harmonic mean：用于计算定投平均成本N/(i=1N1/X)，3期1元定投价格X1、X2、X3，总共3元买入了i=131/X份股票，调和平均成本即为3/i=131/X调和平均≤几何平均≤算术平均，等号成立只有X1=X2=X3时加权平均weighted mean：加入资产比重计算分位数L =(N+1)Y/100，N是样本数、Y是分位数位置数：四分位、五分位。
CFA考试：投资分析的数量方法
CFA考试：投资分析的数量方法Chapter ⒈ The Time Value of Money§⒈说明利息率是对投资者的不同风险予以回报的实际无风险利率和风险溢价的总和利息率和折现率〔Interest Rates and Discount Rates〕货币时刻价值概念的基础：收益率〔rates of return〕、利息率〔interest rate〕、要求的收益率〔required rates of return〕、折现率〔discount rates〕、机会成本〔opportunity costs〕、通货膨胀〔inflation〕和风险〔risk〕。
货币的时刻价值，反映了时刻、现金流量和利息率三者之间的关系。
投资者偏好现在消费。
利息率是投资者推迟现在消费的回报。
在确定世界，利息率被认为是无风险〔risk-free〕利率。
一样是国家的短期债券，如美国的国库券〔Treasury-bills, T-bills〕。
在不确定的世界，有两个因素阻碍利息率：①通货膨胀。
贷款者承担通货溢价〔inflation premium〕和推迟消费的机会成本。
因此，货币的名义成本〔nominal cost of money〕，由实际利率〔real rate〕和通货溢价组成。
②风险。
贷款者还承担了不履行风险〔default risk〕。
因此，利息率包括：名义的无风险利率和不履行风险溢价。
利息率的意义：①收益要求率。
即促使投资者舍弃现在消费所要求的收益。
②折现率〔利息率和折现率能够交互使用〕。
③机会成本。
即投资者按某一选择行为而舍弃其他选择所失去的价值。
阻碍利息率最重要的因素是：资金的供求关系。
§⒉运算整笔现金的终值〔FV〕和现值〔PV〕单一现金流量的终值〔The Future Value of a Single Cash Flow〕整笔现金流〔或lump-sum investment〕的终值运算公式〔N的初值为0〕：差不多概念：①简单利息〔simple interest〕，即利息率乘原始本金。
泽稷教育 CFA一级金融数量分析笔记
CFA一级金融数量分析笔记（上）Overview：CFA原版书将数量分为两个session去阐述。
第一个session主要围绕着计量的基本概念去展开第二个session则着眼于实际运用。
其实我们也可以自己就每个Reading的具体内容去做一个分类Reading 5和Reading 6是铺垫类的内容，为学习其他科目做一个铺垫，同时也是金融人的common sense的一部分。
Reading 7，Reading 8和Reading 9是描述性统计的内容。
Reading 10和Reading 11是推断性统计的内容。
Reading 12其实就是一个理论具体分析运用的过程，为分析师们提供了一些可实操重要指标。
小编在梳理数理笔记同时分上中下进行的，本次主要为大家奉上Reading 5 & Reading 6的内容；数量（中）为大家奉上描述性统计的内容；数量下奉上推断性统计和Reading 12的内容希望能给大家带来帮助~不管你是喜欢小编的内容，还是不喜欢有改进建议的，又或者是有想要小编推的内容，都可以留言哦~小编会根据大家的留言做适时调整的。
言归正传，正文开始！Reading 5 The time value of the money笔记一：同一个时间点的现金流才可直接相加；不同的时间点的钱如若要相加，就需要引入利率的概念将其等同起来笔记二：利率可作为要求的回报率（Required rate of return），折现率（Discounted rate）和现在消费的机会成本（Opportunity cost）笔记三：利率的组成部分（不考虑风险的情况）名义的无风险利率（nominal risk-free rate）= 实际的无风险利率（real risk-free rate）+ 预期的通货膨胀率（expected inflation rate）其中：实际的无风险利率是在假定没有预期通货膨胀的情况下，单期贷款的理论利率。
2019年CFA一级notes整理数量分析答案
数量分析答案Module Quiz 6.11.C Interest rates can be interpreted as required rates of return,discount rates,or opportunity costs of current consumption.A risk premium can be,but is not always,a component of an interest rate.(LOS 6.a, 6.b)2.A Real interest rates are those that have been adjusted for inflation.(LOS 6.b)3.C EAR=[(1+(0.18/12)]12−1=19.56%(LOS6.c)4.C N=30×12=360;I/Y=9/12=0.75;PV=–150,000(1−0.2)=–120,000;FV=0;CPT→PMT=$965.55(LOS6.d)5.C N=1×365=365;I/Y=12/365=0.0328767;PMT= 0;PV=–5,000;CPT→FV=$5,637.37(LOS 6.d)Module Quiz 6.21.C N=15;I/Y=9;PV=–1,000;PMT=0;CPT→FV= $3,642.48(LOS 6.e)2.A N=7;I/Y=8;FV=–10,000;PMT=0;CPT→PV= $5,834.90(LOS 6.e)3.A N=20;I/Y=10;PMT=–50,000;FV=0;CPT→PV= $425,678.19(LOS6.e)4.C This is an annuity due.Switch to BGN mode:N=15; PMT=–8,000;I/Y=11;FV=0;CPT→PV=63,854.92. Switch back to END mode.(LOS 6.e)5.B The key to this problem is to recognize that it is a 4-year annuity due,so switch to BGN mode:N=4;PMT=–1,000;PV=0;I/Y=12;CPT→FV=5,352.84.Switch back to END mode.(LOS6.e)6.A9/0.11=$81.82(LOS 6.e)Module Quiz 6.31.B N=8;PV=–3;FV= 4.50;PMT=0;CPT→I/Y= 5.1989(LOS 6.e)2.C PV=–5,000;I/Y=12;FV=10,000;PMT=0;CPT→N = 6.12.Note to HP12C users:One known problem with the HP12C is that it does not have the capability to round.In this particular question,you will come up with7,although the correct answer is 6.1163.CFA Institute is aware of this problem,and hopefully you will not be faced with a situation on exam day where the incorrect solution from the HP is one of the answer choices.(LOS 6.e)3.B 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=569.42 Hence,446.43+159.44+569.42=$1,175.29.(LOS 6.e)4.B PV=–10,000;I/Y=9.5;N=5;FV=0;CPT→PMT= $2,604.36(LOS 6.e)5.B N=10;I/Y=11;FV=–60,000;PV=0;CPT→PMT= $3,588.08(LOS6.e)6.B Two steps:(1)Find the PV of the10-year annuity:N =10;I/Y=9;PMT=–4,000;FV=0;CPT→PV=25,670.63.This is the present value as of the end of Year4;(2)Discount PV of the annuity back four years:N=4;PMT=0;FV=–25,670.63;I/Y=9;CPT→PV=18,185.72.(LOS6.e) Module Quiz7.11.A NPV=4/ 1.10+3/(1.10)2+4/(1.10)3−$10 =–$0.879038million,or–$879,038Calculator approach:CF0=–10;CF1=4;CF2=3;CF3 =4;I=10;NPV=–$0.879038(million)(LOS7.a)2.A NPV=PV(cash inflows)−CF0=($1.8million/0.105)−$15million=$2,142,857.Accept the project.(LOS7.a)3.C As a perpetuity,the following relationship applies: $1.8million/IRR=$15million.Thus,IRR= 1.8/15= 12%.Since IRR>cost of capital(hurdle rate),accept the project.(LOS7.a)4.B If the NPV and IRR methods give conflicting decisions when selecting among mutually exclusive projects,always select the project with the greatest positiveNPV.(LOS7.b)5.C A project will have a negative NPV if its IRR is less than the firm’s cost of capital.(LOS7.b)6.C Both projects should be accepted since both projects have positive NPVs andwill thus increase shareholder wealth.(LOS7.b)7.C(LOS7.c)Module Quiz7.21.C The money-weighted rate of return is the IRR of aninvestment’s net cashflows.(LOS7.d)2.C On your financial calculator,solve for IRR: Calculating Money-Weighted Return With the TI Business Analyst II Plus®(LOS7.d)3.C HPR1=(65+2)/50−1=34%,HPR2=(140+4)/ 130−1=10.77%Time-weighted return=[(1.34)(1.1077)]0.5−1=21.83% (LOS7.d)4.B(1,000/100,000)×(360/95)= 3.79%(LOS7.e)5.A(100,000−99,000)/99,000= 1.01%(LOS7.e)6.B(1+0.0101)365/95−1= 3.94%(LOS7.e)7.B(360×0.0379)/[360–(95×0.0379)]= 3.83%,or (1,000/99,000)(360/95)= 3.83%(LOS7.e)8.C This is actually the definition of the holding period yield.The other answers aretrue statements regarding the bank discount yield.(LOS7.e)8.CModule Quiz8.11.B Poker hand rankings are an ordinal scale.We can saya pair outranks a highcard(therefore,it is not a nominal scale),but we cannot say a pair is some amountgreater than a high card(therefore,it is not an interval scale).(LOS8.a)2.C Intervals within a frequency distribution should always be nonoverlapping andclosed ended so that each data value can be placed into only one interval.Intervalshave no set width and should be set at a width so that data is adequatelysummarized without losing valuable characteristics.(LOS 8.b)3.B An interval is the set of return values that an observation falls within.Simplycount the return intervals on the table—there are five of them.(LOS8.c)4.C The sample size is the sum of all of the frequencies in the distribution,or3+7+3+2+1=16.(LOS8.c)5.C The relative frequency is found by dividing thefrequency of the interval by thetotal number of frequencies.(LOS8.c)6.A In a histogram,the intervals are on the horizontal axis and the frequency is onthe vertical axis.(LOS8.d)Module Quiz8.21.A[22%+5%+–7%+11%+2%+11%]/6=7.3%(LOS8.e)2.B To find the median,rank the returns in order and take the middle value:–7%,2%,5%,11%,11%,22%.In this case,because there is an even number of observations,the median is the average of the two middle values,or(5%+11%)/2=8.0%.(LOS8.e)3.C The mode is the value that appears most often,or11%. (LOS8.e)4.C The range is calculated by taking the highest value minus the lowest value.22%−(–7%)=29.0%(LOS8.g)5.B The mean absolute deviation is found by taking the mean of the absolute valuesof the deviations from the mean.(|22−7.3|+|5−7.3|+|–7−7.3|+|11−7.3|+|2−7.3|+|11−7.3|)/6=7.33%(LOS8.g)6.C The population variance,σ2,is found by taking the mean of all squared deviations from the mean.σ2=[(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=80.2%2(LOS8.g)7.B The population standard deviation,σ,is found by taking the square root of the population variance.σ==(80.2%2)0.5=8.96%(LOS8.g)8.C The sample variance,s2,uses n–1in the denominator.s2=[(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)=96.3%2(LOS8.g)9.A The sample standard deviation,s,is the square root of the sample variance.σ==(96.3)0.5=9.8%(LOS8.g)10.A(15%+19%–8%+14%)/4=10%(LOS8.e)11.A(1.15×1.19×0.92×1.14)0.25−1=9.45%(LOS8.e) PROFESSOR’S NOTEThis question could have been answered very quickly since the geometric mean must be less than the arithmetic mean computed in the preceding problem.12.B(LOS8.e)13.C With eight observations,the location of the65th percentile is:(8+1)×65/100= 5.85observationsThe fifth observation is7and the sixth observation is9, so the value at 5.85observations is7+0.85(9–7)=8.7. (LOS8.f)Module Quiz8.31.C Applying Chebyshev’s inequality,1−[1/(2.5)2]=0.84,or84%.(LOS8.h)2.C The portfolio’s Sharpe ratio is(6.0%–1.0%)/ 2.5% = 2.0.(LOS8.i)3.B A distribution that has a greater percentage of small deviations from the meanand a greater percentage of extremely large deviationsfrom the mean will beleptokurtic and will exhibit excess kurtosis(positive). The distribution will bemore peaked and have fatter tails than a normal distribution.(LOS8.j)4.A A distribution with a mean greater than its median is positively skewed,orskewed to the right.The skew“pulls”the mean.Note: Kurtosis deals with theoverall shape of a distribution and not its skewness.(LOS 8.k)5.C Normal distributions have kurtosis of three and excess kurtosis of zero.(LOS8.l)6.B The arithmetic mean is appropriate for forecasting single period returns.Thegeometric mean is appropriate for forecasting compound returns over more than a single period.(LOS8.m)Module Quiz9.11.C An event is said to be exhaustive if it includes all possible outcomes.(LOS9.a)2.C Probabilities may range from0(meaning no chance ofoccurrence)through1(which means a sure thing).(LOS9.b) 4.C By the multiplication rule of probability,the joint probability of two events,P(AB),is the product of a conditional probability,P(A|B),and an unconditional probability,P(B).(LOS9.d,9.e)5.C There is no intersection of events when events are mutually exclusive.P(A|B)=P(A)×P(B)is only true for independent events.Note that since A and B are mutually exclusive(cannot both happen),P(A|B)and P(AB) must both be equalto zero.(LOS9.f)6.B P(name1or name2or name3or name4)=1/800+ 1/800+1/800+1/800=4/800=0.005.(LOS9.f)7.B One or the other may occur,but not both.(LOS9.f) Module Quiz9.21.C Two events are said to be independent if the occurrence of one event does not affect the probability of the occurrence of the other event.(LOS9.g)2.B The three outcomes given for economic growth are mutually exclusive and exhaustive.The probability that economic growth is positive but less than3%is100%–25%–25%=50%.Using the total probability rule,theprobability that the share price increases is(80%)(25%)+ (40%)(50%)+(10%)(25%)=42.5%.(LOS9.h,9.i)3.A Covariance may have any value.Correlation is bounded by–1and 1.(LOS9.k)Module Quiz9.31.B E(X|Y=1)=(0.2)(0)+(0.4)(5)+(0.4)(10)=6E(X|Y=2)=(0.1)(0)+(0.8)(5)+(0.1)(10)=5E(X)=(0.3)(6)+(0.7)(5)= 5.30(LOS9.l)2.B Expected value=(1/4)(1+2+3+4)= 2.5Variance=(1/4)[(1−2.5)2+(2−2.5)2+(3−2.5)2+(4−2.5)2]=1.25Note that since each observation is equally likely,each has25%(1/4)chance of occurrence.(LOS9.l)3.A(LOS9.l)4.C This is an application of Bayes’formula.As the tree diagram below shows,the updated probability that earnings per share are greater than$2is(LOS9.n)5.B(LOS9.o)6.B Since the order of the top three finishers matters, we need to use the permutation formula.(LOS9.o)Module Quiz10.11.B Time is usually a continuous random variable;the others are discrete.(LOS10.a)2.B For a continuous distribution p(x)=0for all X; only ranges of value of X have positive probabilities. (LOS10.a)4.B From the table.(LOS10.d)5.C(0.04+0.11+0.18+0.24+0.14+0.17)=0.88(LOS 10.c)6.B(0.14+0.17+0.09+0.03)=0.43(LOS10.d)7.C(0.18+0.24+0.14+0.17)=0.73(LOS10.d)8.A0+1(0.11)+2(0.18)+3(0.24)+4(0.14)+5(0.17)+ 6(0.09)+7(0.03)= 3.35(LOS10.d)9.A There may be any number of independent trials,each with only two possible outcomes.(LOS10.e)10.B With only two possible outcomes,there must be some positive probability for each.If this were not the case, the variable in question would not be a random variable, and a probability distribution would be meaningless.It does not matter ifone of the possible outcomes happens to be zero.(LOS10.e)13.C Success=passing the exam.Then,E(success)=np= 15×0.4= 6.(LOS10.f)14.C F(x)is the cumulative probability,P(x<20)here. Because all the observations in this distribution are between4and10,the probability of an outcome less than 20is100%.(LOS10.h)Module Quiz10.21.A Normal distributions are symmetrical(i.e.,have zero skewness)and their kurtosis is equal to three.(LOS10.i)2.B To describe a multivariate normal distribution,we must consider the correlations among the variables as well as the means and variances of the variables.(LOS10.j)3.C z=(100−175)/25=–3,F(–3)=1−0.9987= 0.0013.(LOS10.k)4.C1−F(2),where F(2)equals0.9772.Hence,1−0.9772= 0.0228.(LOS10.k)5.C1−F(–1)=F(1)=0.8413.There is an84.13% probability that a randomly chosen income is not more than one standard deviation below the mean.(LOS10.k)6.B This is true by the formula for z.(LOS10.l)7.C By the symmetry of the z-distribution and F(0)=0.5. Half the distribution lies on each side of the mean.(LOS10.l)Module Quiz10.31.C SFR=(18−4)/40=0.35is the largest value.(LOS 10.m)2.A SFR=(5−0)/8=0.625is the largest value.(LOS 10.m)3.A Using the tables,the cdf for–1.3is9.68%,which is the probability of returns less than2%.(LOS10.m)4.B A lognormally distributed variable is never negative. (LOS10.n)5.B ln(23/20)=0.1398(LOS10.o)6.B ln(2)=0.6931(LOS10.o)7.B Monte Carlo simulation involves modeling asset prices or returns by generating random values for the risk factors that affect the price of a security.(LOS10.p,9.q)Module Quiz11.11.B In a simple random sample,each element of the population has an equal probability of being selected. Choice C allows for an equal chance,but only if there are 100elements in the population from which the random sample is drawn.(LOS11.a)2.C Suppose you have a population of10,000employees.If you take100samples of50employees each,the distribution of the100sample means is the sampling distribution.(LOS11.a)3.C An example might be the difference between a particular sample mean and theaverage value of the overall population.(LOS11.b)4.B The sampling error is the difference between the population parameter and thesample statistic.(LOS11.b)5.B Cross-sectional data is a set of data that are all collected as of the same point intime.(LOS11.c,11.d)6.C Sample sizes of30or greater are typically considered large.(LOS11.e)11.A Efficiency,consistency,and unbiasedness are desirable properties of an estimator.(LOS11.g)Module Quiz11.21.A As the degrees of freedom get larger,the t-distribution approaches the normal distribution.As the degrees of freedom fall,the peak of the t-distribution flattensand its tails get fatter(more probability in the tails—that’s why,all else the same,the critical t increases as the df decreases).(LOS11.i) 4.C Use the t-statistic atα/2and n–1degrees of freedom when the populationvariance is unknown.While the z-statistic is acceptable when the sample size islarge,sample size is not given here,and the t-statistic is always appropriate underthese conditions.(LOS11.j)5.A When the sample size is large,and the central limit theorem can be relied uponto assure a sampling distribution that is normal,either the t-statistic or the z-statistic is acceptable for constructing confidence intervals for the populationmean.The t-statistic,however,will provide a more conservative range(wider)at agiven level of significance.(LOS11.j)8.A The primary example of look-ahead bias is using year-end financial information in conjunction with market pricing data to compute ratios like the price/earnings(P/E).The E in the denominator is typically not available for30–60days after the end of the period.Hence,data that was available on the test date(P)is mixed with information that was not available(E).That is,the P is “ahead”of the E.(LOS11.k)9.C Mutual fund performance studies are most closely associated with survivorship bias because only the better-performing funds remain in the sample over time.(LOS11.k) Module Quiz12.11.C To test whether the population mean is greater than 20,the test would attempt to reject the null hypothesis that the mean is less than or equal to20.The null hypothesis must always include the“equal to”condition. (LOS12.a)2.C Rejecting the null when it is actually true is a TypeI error.A Type II error is failing to reject the null hypothesis when it is false.The significance level equals the probability of a Type I error.(LOS12.b,12.c)3.C A Type I error is rejecting the null hypothesis when it’s true.The probability of rejecting a false null is [1−Prob Type II]=[1−0.60]=40%,which is called thepower of the test.A and B are not necessarily true,since the null may be false and the probability of rejection unknown.(LOS12.c)4.A The power of a test is1−P(Type II error)=1−0.15 =0.85.(LOS12.d)Module Quiz12.21.B With a p-value of3%,the manager can reject the null hypothesis(that abnormal returns are less than or equal to zero)using a significance level of3%or higher. Although the test result is statistically significant at significance levels as small as3%,this does not necessarily imply that the result is economically meaningful.(LOS12.e,12.f)2.A With a small sample size,a t-test may be used if the population is approximately normally distributed.If the population has a nonnormal distribution,no test statistic is available unless the sample size is large.(LOS12.g)3.B This is a two-tailed test with14–1=13degrees of freedom.From the t-table, 2.160is the critical value to which the analyst should compare the calculated t-statistic.(LOS12.g)Module Quiz12.31.C When the variances are assumed to be unequal,we just calculate thedenominator(standard error)differently and use both sample variances to calculate the t-statistic.(LOS12.h) 2.A Since the observations are likely dependent(both related to market returns),a paired comparisons(mean differences)test is appropriate and is based on a t-statistic.(LOS12.h,12.i)3.C There is no consistent relationship between the mean and standard deviation of the chi-square distribution or F-distribution.(LOS12.j)4.B The F-test is the appropriate test.(LOS12.j)5.C A test of the population variance is a chi-square test.(LOS12.j)Module Quiz13.11.A Technical analysis assumes persistent trends and repeating patterns in market prices can be used to forecast price behavior.Technical analysts believe prices reflect supply and demand,but that buying and selling can be motivated by both rational and irrational causes.Volume,along with price,is important informationto a technical analyst.(LOS13.a)2.B Candlestick charts show the open,high,low,and close for each trading period.Line charts use only the closing price.Point-and-figure charts do not necessarily show distinct trading periods.(LOS13.b)3.A The downtrend reached a support level where buying demand sustained the price.A resistance level is a price at which selling pressure emerges that stops an uptrend. The change in polarity principle holds that breached support levels become resistance and breached resistance levels become support.With no information given on the stock’s history,we cannot determine whether$30had once been a resistance level.(LOS13.c)4.A Bollinger bands are based on the standard deviation of prices over some number of the most recent periods.An RSI is based on the sums of positive and negative price changes during a period.An ROC oscillator is based on the difference between the most recent closing price and the closing price a given number of periods earlier.(LOS13.e)5.B The RSI is calculated from the ratio of total priceincreases to total price decreases over a chosen number of days,then scaled to fluctuate between0and100using the formula RSI=100−[100/(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.(LOS13.e)6.B“More bullish”means investors expect prices to increase in the near term.Increasing margin debt suggests investors are bullish and buying aggressively.Increases in put volume relative to call volume,or in the number of shares sold short,indicate bearish investor sentiment. (LOS13.e)7.C The Kondratieff wave is a54-year cycle.(LOS13.f)8.C The value 1.618is the ratio of large consecutive Fibonacci numbers.Technical analysts who employ Elliott wave theory frequently use Fibonacci ratios to setprice targets.(LOS13.g)9.B If the relative strength ratio(stock price/ benchmark value)increases,the stockis outperforming the benchmark stock or index against which it is being measured.This does not imply that thestock 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 ratio.(LOS13.e,13.h)。
CFA-LEVEL-I-数量总结PPT课件
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1. Over the past 240 months, an investor’s portfolio had a mean monthly return of 0.79%, with a standard deviation of monthly returns of 1.16%. According to Chebyshev’s inequality, the minimum number of the 240 monthly returns that fall into the range of −0.95% to 2.53% is closest to:
Holding period yield (HPY)=(ending value/begin value)-1
Bank discount yield (BDY)=(discount/face value)x(360/days to maturity) simple interest
Money market yield (MMY)=(discount/price)x(360/days to maturity) simple interest
A. 80. B. 107. C. 133. 2. You have developed a set of criteria for evaluating distressed credits. Companies that do not receive a passing score are classed as likely to go bankrupt within 12 months. You gathered the following information when validating the criteria: • Forty percent of the companies to which the test is administered will go bankrupt
CFA考试：投资分析的数量方法(投资工具)
第五章：正态概率分布Chapter ⒌Common Probability Distributions本章的内容，是四种概率分布及它们的应用，即：①the uniform；②the binomial；③the normal；④the lognormal。
本章的其他数量工具：①Hypothesis testing；②regression analysis；③time-series analysis。
§⒈定义和解释概率分布（Probability Distributions）概率分布（Probability Distributions），即将随机变量可能结果的概率予以特定。
每个随机变量都有描述它的概率分布，概率分布的方式有两种：①概率函数（probability functions）。
②累积分布函数（cumulative distribution functions／distribution functions／cdf§⒉区别：连续的随机变量和不连续（discrete）的随机变量随机变量，是一个未来结果不确定的数。
随即变量有两种类型：不连续的随机变量（discrete random variable）、连续的随机变量（continuous random variable）。
变量的结果能予以历数（个数有限）的随机变量，为不连续的随机变量。
§⒊描述某特定变量可能结果的集合§⒋定义一个概率函数（Probability function）并说明它的关键特征概率函数的表示方法是：P（X ＝x），它表示随机变量的值为x的概率。
不连续随机变量的概率函数，可以缩写为p（x）；连续随机变量的概率函数用f（x）表示，称之为概率密度函数（Probability density functions／density／pdf）。
29概率函数有两个关键特征：⑴0≤p（x）≤1；⑵随机变量X所有值的概率的总和等于1。
CFA-LEVEL-I-数量总结
1. Over the past 240 months, an investor’s portfolio had a mean monthly return of 0.79%, with a standard deviation of monthly returns of 1.16%. According to Chebyshev’s inequality, the minimum number of the 240 monthly returns that fall into the range of −0.95% to 2.53% is closest to:
3. A portfolio manager annually outperforms her benchmark 60% of the time. Assuming independent annual trials, what is the probability that she will outperform her benchmark four or more times over the next five years?
survive 12 months, is 0.85:P(pass test | survivor) = 0.85. A. What is P(pass test | nonsurvivor)? B. Using Bayes’ formula, calculate the probability that a company is a survivor, given that
Portfolio 1 2 3
Expected return
22%
Standard deviation
40%
Roy’s Safety-First value 0.35 0.64
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几何平均geometric mean：可以排除算术平均的极端值，相乘后开次方 。计算多期平均价收益率
调和平均harmonic mean：用于计算定投平均成本N/ ，3期1元定投价格X1、X2、X3，总共3元买入了
份股票，调和平均成本即为3/
调和平均≤几何平均≤算术平均，等号成立只有X1=X2=X3时
天空多云。50%的雨天的早上是多云的，大约40%的日子早上是多云的。平均30天里一般只有3天会下雨，10%。
会下雨的可能性是 P(雨|云)。
P(雨|云) = P(雨)·P(云|雨) /P(云)
P(雨) 是今天下雨的概率 = 10%
P(云|雨) 是在下雨天早上有云的概率 = 50%
P(云) 早上多云的概率 = 40%
遍历事件：掷骰子，要么单数要么双数，单数和双数都为遍历事件，不可能还有其他的情形。遍历事件的概率和为1。
独立事件P（AￜB）=P（A）
事件概率的三类：经验概率empirical probability、先验概率priori probability、主观概率subjective probability，前两类又为客观概率。
对比两家公司的例题出现标准差、平均值时使用。
夏普比例 ，资产收益率-无风险收益率，除以总体标准差，用于衡量资产P在单位风险下的差额收益，比例越大越好。
偏度skewness：偏度大于0，数据出现右偏，也叫正偏。偏度小于0，数据出现左偏，也叫负偏
主要观察尾巴，而不是肩膀，右边尾巴长为右偏。左边尾巴长为左偏（与意识中概念不相同）。管理学中的长尾理论实际上是就是数据的左偏和右偏。
CFA一级考试知识点
第二部分 数量分析方法
名义利率等于实际利率加上预期通货膨胀率，而不是当期的实际通货膨胀率。
Holding period return,HPR持有期收益率
Bank discount yield,BDY银行贴现利率，本金为F，价格为P，公式： *
Money market yield,MMY货币市场收益率 *
正态分布μ表示均值σ表示标准差
横轴区间（μ-σ,μ+σ）内的面积为68%。
横轴区间（μ-1.65σ,μ+1.65σ）内的面积为90%。
横轴区间（μ-1.96σ,μ+1.96σ）内的面积为95%。
横轴区间（μ-2.58σ,μ+2.58σ）内的面积为99%。
正态分布有负数，设置对数正态分布来弥补，对数正态分布没有负数、概率密度函数是正偏，主要用来衡量资产价格。
加权平均weighted mean：加入资产比重计算
分位数L =(N+1)Y/100，N是样本数、Y是分位数位置数：四分位、五分位。
总体方差variance： 开根号后即为总体标准差standard deviation。
样本方差： 开根号后即为样本标准差。
以上公式用于衡量收入偏离均值的平均距离值。
变异系数coefficient of variation：CV=S/X平均值，样本标准差除以样本均值。用于对比不同资产（农业银行股份、茅台股份），一单位均值收益率承受的风险，系数越低越好。
无需记忆公式记住偏度是有关 的表达式，反映的是数据分布的对称性。相对于Xi而言右向偏离X平均值较多为右偏。
众数被定义为概率分布的最高点。
左偏时：均值＜中位数＜众数。
右偏时：众数＜中位数＜均值。
偏度是衡量尾巴长度是偏左还是偏右，峰度是进一步衡量尾部的厚度。
无需记忆公式记住偏度是有关 的表达式，衡量尾部厚度。峰度越高说明xi偏离x平均值的极端值越多，尾部越后。反之则尾部较薄，实际运用中通常用正态分布峰度为3作为基准，峰度高于3为高峰，为尖峰肥尾，峰度低于3为低峰，为矮峰瘦尾。
P(雨|云) =0.1×0.5/0.4=0.125，下雨的概率只有12.5%
300只股票，设置多个条件，符合标准的概率为条件相乘后率联动性，但是斜方差有缺陷，取值是负无穷到正无穷，没有考虑量纲。
随机变量的期望是以概率为权重，所有可能结果的加权平均，记为E（x）
贝叶斯定理bayes’formula是一种在已知其他概率的情况下求概率的方法：
公式ABABAB
P(A|B) 是在 B 发生的情况下 A 发生的概率；
P(A) 是 A 发生的概率；
P(B|A) 是在 A 发生的情况下 B 发生的概率；
P(B) 是 B 发生的概率。
货币加权受现金流入流出影响，因此时间加权更加广泛
四种度量衡：名义尺度nominal scale（分类不排序）、排序尺度ordinal scale（排序进行比较，不能够加减，有优先次级，不成比例）、区间尺度interval scale（温度、评分，零不具备数学意义）、比例尺度ratio scale（常用最高级、身高、收入、资产收益率）
而正态分布主要用来衡量收益率。
超峰度与峰度不同，超峰度是以超过3为基准，正态分布的超峰度为0。
Leptokurtic高峰
Mesokurtic 平峰
Platykurtic低峰
概率论基础
随机变量的部分组成的集合称为随机事件，简称为事件event
事件的三个分类：互斥事件mutually exclusive events、遍历事件exhaustive events、独立事件indepengdent events
Effective yield,EAY有效年利率（1+HPY）^365/t-1
Money-weighted rate of return,MWR货币加权收益率（内部收益率）
Time-weighted rate of return,TWR时间加权收益率（几个收益期间的几何平均）
Bond equivalent yield,BEY债券等价收益率（irr的年化）
经验概率通过历史数据和客观规律来估计。比如股票分红推算
先验概率通过逻辑分析而不是历史数据。比如抛硬币
主观概率：依据个人主观判断，在数据量小时发挥作用。
赔率odds：P(E)/(1-P(E))，一匹马获胜真实概率1/8，赔率应该是（1/8）/(1-1/8)=1/7，即1赔7。
条件概率conditional probability，在已知B发生的情况下，事件A发生的概率，记为P（AￜB）