QM3_Statistics Concepts and Market Returns-LMLPHP

Basic Concepts

Terms

Descriptive Statistics
  • Describes the important aspects of large data sets.

    • 统计
    • 概率
    • 分布
Inferential statistics
  • Involves making forecasts, estimates, or judgments about a larger group from the smaller group.

    • 预测
    • 估计
    • 判断

QM3_Statistics Concepts and Market Returns-LMLPHP

Measurement scales

QM3_Statistics Concepts and Market Returns-LMLPHP

考点:
  • 给描述, 判断是哪种尺度
  • 给尺度, 判断孰强孰弱

Frequency distribution

Central Tendency (第一维度,中心趋势)

Mean

Calculation
  • Arithmetic mean (算术平均)

    • Population Mean

      • QM3_Statistics Concepts and Market Returns-LMLPHP
    • Sample Mean
      • QM3_Statistics Concepts and Market Returns-LMLPHP
  • Geometric mean (几何平均)
    • QM3_Statistics Concepts and Market Returns-LMLPHP
  • Harmonic mean (调和平均, I级考试不考)

    • QM3_Statistics Concepts and Market Returns-LMLPHP
  • Weighted mean (加权平均)

    • QM3_Statistics Concepts and Market Returns-LMLPHP
    • 样本均值中相当于权重都是1/n, 而weighted mean就是不等权重(w1,w2,...wn).
Properties (性质)
  • Arithmetic mean : 单期收益率的表现

    • focus on average single-period performance
    • sensitive to extreme values
  • Geometric mean: 多期收益率的表现

    • focus on multi-period performance
  • Weighted mean: 多用于计算期望值 (算期望就是算加权平均)
    • userd to calculate the portfolio return/expected value based on probabilities
  • Harmonic Mean <= Geometric Mean <= Arithmetic Mean
  • Median 中位数 与 Mode 众数

    • 例: 一组数, 1,1,2,4,8.
    • median: 一共有五个数, 中间的数是2, 所以median是2. 若这个数组是1,2,4,8. 中位数则是(2+4)/2 = 3.
    • mode : 1出现了两次, 所以众数是1.

Quantile (分位点) **

Definition
  • A value at or below which a stated fraction of the data lies.

    • Quantiles 四分位点
    • Quintiles 五分位点
    • Deciles 十分位点
    • Percentiles 百分位点
    • QM3_Statistics Concepts and Market Returns-LMLPHP
Calculation
  • Step 1: formula for location of data in ascending order (必须先把所有数据从小到大排列)
  • Step 2: 用公式计算
    • QM3_Statistics Concepts and Market Returns-LMLPHP
  • 例: for data with 17 observations, find out the location of 3rd quintile.

    • QM3_Statistics Concepts and Market Returns-LMLPHP
    • 注:      1. value 中10和11的顺序写错了, 数值应该是要按顺序排列的.
      • 2. 如果要计算3rd quintile这个位置上的值的话, 应该是(20+23)/2.    
考点
  • 描述

    • 例: 第一个四分位点 --> 有25%的数小于第一个四分位点(因数据是ascending order排列的,所以是小于).
  • 计算
    • Ly = (n+1)y/100 (算location)
    • 算value (算特定分位点的数值)

金融有风险, 风险有不确定性, 所以用离散程度来度量风险, 方差或者标准差就是用来度量离散程度的;

金融中的收益用均值 mean 来度量.

Risk <-- uncertainty <-- dispersion <-- variance, standard deviation

Dispersion (第二维度,离散程度,即偏离均值的程度)

Absolute dispersion (绝对离散程度)

Range (范围)
  • Maximum Value - Minimum Value
Mean Absolute Deviation (MAD, 均值绝对偏差)
  • MAD <= 西格玛
  • QM3_Statistics Concepts and Market Returns-LMLPHP
Variance (方差)
  • MAD是绝对值, 不好计算,所以平方之后就引入了方差.
  • Population 总体
    • QM3_Statistics Concepts and Market Returns-LMLPHP
  • Sample 样本

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Standard deviation (标准差, 把方差开根号)
  • Population 总体

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  • Sample 样本

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    • n-1 是为了满足无偏性或者自由度

QM3_Statistics Concepts and Market Returns-LMLPHP

Relative dispersion (相对离散程度) ***

Coefficient of variation (CV, 变异系数)
  • 每赚一块钱所承担的风险
  • Calculation
    • QM3_Statistics Concepts and Market Returns-LMLPHP
    • s: 样本标准差 (代表风险); x拔: 样本均值(代表收益)
  • Characteristics

    • CV has no units of measurement
    • a measure of risk per unit of mean return
    • the lower the better
Sharpe ratio (夏普比率)
  • 每承担单位风险所获得的超额收益率
  • Calculation
    • QM3_Statistics Concepts and Market Returns-LMLPHP
  • Characteristics

    • Sharpe ratio has no units of measurement
    • a measure of exccess return per unit of risk
    • the higher the better
考点
  • 计算

    • CV
    • Sharpe ratio
  • 描述

    • CV: 每赚一块钱所承担的风险
    • Sharpe ratio: 每承担单位风险所获得的超额收益
  • 性质
    • 变异系数CV越小越好
    • Sharpe ratio越大越好
Chebyshev's inequality (切比雪夫不等式)
  • 概念

    • For any distribution with finite variance, the minimum percentage of observations that lie within k standard deviation of the mean would be 1-1/k*k, given k>1.
    • 对任何一组观测值, 个休落在均值周围k个标准差之内的概率不小于1-1/k*k, 对任意k>1.
  • 例题
    • QM3_Statistics Concepts and Market Returns-LMLPHP
考点
  • 已知k, 需要计算概率1-1/k*k
  • 已知概率, 需要反算出k, 再算出区间
  • 已知区间, 需要计算k, 再算出概率

Skewness (第三维度,偏度) ***

QM3_Statistics Concepts and Market Returns-LMLPHP

肥尾: 取到极端值的概率较大

Kurtosis (第四维度,峰度) **

QM3_Statistics Concepts and Market Returns-LMLPHP

正态分布的峰度定义为3.

T-分布有特殊, 是低峰肥尾. ? 哪一章提到?

Z-分布?

04-02 11:10
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