First 5-8 Part one Portfolio management

这是一篇关于First 5-8 Part one Portfolio management的思维导图，讲述的是Modern portfolio theory and capital asset pricing model。

Portfolio management

• Modern portfolio theory and capital asset pricing model
  • Modern portfolio theory
    • Assumption of mpt
      • Capital market perfect
        • No taxes or transaction cost
        • Costless access to all available information
        • Perfect competition
      • Return are normally distributed
    • Measurements of return and risk单一证券
      • Return: average return (算术平均数）R=(R1+R2+...+Rn)/n
      • Risk:
    • Utility theory
      • Behavior of investors under uncertainty
        • Risk averse(风险厌恶）:最小风险相同收益，最大收益相同风险，默认为理性投资者
        • Risk neutral（风险中性）
        • Risk seeking（风险偏好）
      • utility function (效用函数）:对不同投资组合进行排序 U=E(R)-1/2A∂2
        • A＞0厌恶
        • A=0 中性
        • A＜0 喜好
      • Indifferent curve (无差别曲线）E (R)=U+1/2∂2 ，所有人都为风险厌恶 越往左走越喜爱风险
        • upward sloping and convex (get steeper) 边际效用递减（斜率变小），每增加一单位承担风险， 要求回报会更高需求侧，每条线上效用一致
        • for various types of investors
    • Efficient frontier
      • Return and risk of portfolio with two risky assets
        • Two risky asset: Rp=w1r1+w2r2加权平均
      • Risk of portfolio with risky asset:
      • 计算协方差
      • The risk of portfolio with two risky assets decreases as the assets within the portfolio are becoming less correlated.
        • 分散化投资总体风险变小
        • 可以抵消特定的风险敞口
        • 观察投资组合，不看个体的表现，不看个体风险，看总体风险
    • Optimal portfolio selection（最优投资组合） indifferent curve 与efficient frontier相切m点最优投资组合 indifferent curve much deeper slope往上倾斜越陡峭在上面和前面更为厌恶风险，且有效前沿相切更低
    • 有效组合:给定收益（expect return）风险（standard deviation）最小，给定风险收益最大
  • Capital asset pricing model
    • Capital market theory
      • A combination of risky and riskfree asset
    • Capital asset pricing model (CAPM) 计算出的收益率为承担那么多的风险所应该要求的回报为require return
      • Broke down the risk of an individual asset into specific risk and systematic risk,investors receive no compensation for taking on the specific
    • Capital allocation line (CAL) betaβ=1 有一个一致有效前沿的假设 有效前沿combine the minimum variance portfolio and the market portfolio
      • Selection among cal
        • Optimal CAL是CAL与有效前沿的切线
    • Optimal portfolio along CAL
      • 由风险厌恶因子（A)去看图
    • capital market line（CML）市场组合特殊的CAL线斜率最大，且与有效前沿相切于M点 同一条CML永远有效，CML相当于一个有效前沿 投资forecast for asset return相同 收入分布为无偏has no skewness 适用于完全分散化组合 well diversified 因为横坐标为总风险
      • market portfolio : 市场组合包括所有的风险资产risky asset 对应不同的权重 weighted
      • 公式:
    • 系统性风险与非系统性风险:非系统性风险可以被投资分散化而分散掉，系统风险无法被充分分散化点风险（天灾）承担非系统性风险不可被补偿，但可以被分散 Systematic: investor would be rewarded for bearing systematic risk Unsystematic risk: investor would not be rewarded for bearing unsystematic risk
      • β有正相关和负相关的 衡量非系统性风险
    • assumption of capm
      • 投资者做决定是看expected value 还有 standard deviation
      • 投资者持有相同时间的资产holding same time horizon mean not normal distribution
      • 资产可以被无限分化，allocation can be made in an investment of any partial amount
      • 无交易成本，无税收，无交易摩擦no transaction cost
      • 无买卖限制 short sale is allowed unlimitedly
      • 所有参与者可以借入与借出在无风险利率borrow and lend
      • Investor have homogeneous expectations or belief
      • 所有个人投资者的投资决定都不影响市场价格所有人都是散户
      • 所有的资产包括人力成本都可以被交易
      • Market portfolio should be the one with highest sharp rate of an portfolio
      • Investors have access to the same information
      • Return are normally distributed and investors' expectations for risk and return are identical
    • components of capm E (Rm)-Rf price per unit of beta risk capm模型中投资者关注utility效用 不关注尾部情况 偏度和尾度皆关注尾部情况 betaβ＞1说明，其期望收益大于市场组合
      • 变形:
        • security market line（SML）证券市场线
          • comparison of CML和SML
  • Performance measures
    • Sharpe performance index (spi)承担一单位总风险的超额回报,两种资产计算相同，说明风险报酬比相同，资产无区别，总风险包括系统性和非系统性风险，E (Ri)和information ratio中的E (Ri)是相等的 use total volatility 夏普比率未考虑well disversified portfolio
      • mean annualized rate of return (平均收益比）大于market portfolio 的收益，所以对非系统性风险进行一些补偿，理论上对充分分散组合是不补偿的
      • 知道market portfolio 的 sharp ratio求组合的夏普比率
    • Treynor performance index (TPI) 适合well diveraified（对非系统风险充分分化） using beta as denominator 一般认为指数为充分分散化的组合 performance of index fun
      • according capm TPI与alpha measure相比，＞即正alpha
    • Jensen's performance index (JPI)=实际收益率➖理论收益 beta必须为同一个值
    • Sortino ratio: 关注 downside risk 亏损风险，应用于上下波动的情况，适合不对称回报的资产
    • Tracking error跟踪误差loss relative to benchmark benchamark基准主动管理，absolute risk（绝对风险，不对比基准benchmark），relative risk（相对风险，对比benchmark），必然大于0，无负数 TE 相当于 residual risk
    • Information ratio调过风险管理，越大越好，主动投资beating a benchmark
• The arbitrage pricing theory and multifacto models of risk and return
  • arbitrage pricing theory
    • Comparison
      • Capm only one systematic risk factor, 过多的假设对于投资者行为
      • APT 有多个systematic risk factors，只假设there are no arbitrage opportunity无套利机会
    • assumption假设
      • return can be described by a factor model can be explained by systemic factors（specific risk非系统性风险）apt为只有系统性风险的模型
      • Sufficient securities diversify unsystematic risk
      • No arbitrage opportunity
      • Suggest that multiple factors(多个因素） markets expected rksk premium can help explain the expected rate of return on a risky asset
      • 不假设投资者是理性的投资者，不一定会选择有效的组合，认为投资者可以是风险偏好也可以是厌恶
    • APT公式:关注纯因素模型
      • 非系统性风险可以被分散， 在APT模型中只有系统性风险可以带来回报
      • 一家定律（the law of one price: 商品有相同的市场价格），一价定律不成立则有套利空间
      • pure factor portfolio 用于对冲
    • Fama French three factor model (三因素模型）
  • Multifactor model计算真实的回报率
    • manage exposure to economy wide risk factors(systemic risk系统性风险） 真实值与预测值的差异
      • single factor model
    • hedging exposure to multiple factors纯因素组合用于对冲
    • underbaseline expectation 即为多因素模型，industrial production可视为GDP
• SMB 小盘股减去大盘股 HML 高减低
• Assumption:1. the market portfolio should have the highest sharpe ratio and including all inevitable 2.expected excess return for the market are known to all investors
• E (risky portfolio)= return×probability E(Ram)=return×probability
• 投资者通过APT模型可以create a zero beta portfolio with zero net investment (可以通过long和short来使得beta为0，对冲风险，是一个手段方法），在现实中，有套利机会则投资者会把套利机会消除，即去做套利使得套利机会去除