Investment Brief Note

经12-计18 张诗颖 2021011056

1 金融市场 Common Sense

• 金融机构：Investment Banking, Commercial Banking, Pension Fund, Mutual Fund, Hedge Fund, Private Equity（股权 / 证券）, Venture Capital

• 2008金融危机：Mortgage-Backed Securities, Collateralized Debt Obligations, Credit Default Swap

• Asset Classes：Fixed Income (Money / Capital Market) + Equity + Derivatives

  • Treasury bills, Certificates of Deposits (London InterBank Offered Rate), Commercial Paper, Bankers' Acceptances, Eurodollars, Repo and Reverse Repo, Federal Funds, Exchange-Traded Funds (Asset Management + Authorized Participants)
  • (Inflation-Protected) Treasury Notes and Bonds (par=1000, semi-annual compounding), Federal Agency Debt, International Bonds, Municipal Bonds, Corporate Bonds, Mortgage-Backed Securities
  • Equity, Options (Call/Put), Futures

• Market：Primary Market + Secondary Market

  • Private Placement, Initial Public Offering (underwriters + Securities and Exchange Commission + China Securities Regulatory Commission / 中国证监会, prospectus + road shows + bookbuilding), Seasoned Equity Offering

  • Limit Order Book: Limit-Buy, Limit-Sell, Stop-Buy, Stop-Loss Electronic Communication Networks, NASDAQ, New York Stock Exchange (+ Arca), Sarbanes-Oxley Act Broker / Dealer - Specialist (market maker)

  • Margin $=\frac{Equity}{Value}$$= \frac{Equity}{Value}$, Margin Account, Maintenance Margin, Margin Call

    $$\begin{array}{rl}Assets& |Liabilities\mathrm{\&}Equity\\ -------------& --------------\\ Shares\mathrm{\$}4,000\downarrow & |\mathbf{Margin}\mathbf{Loan}\mathrm{\$}2,000\\ \mathbf{Margin}\mathbf{Deposit}\mathrm{\$}0& |AccountEquity\mathrm{\$}2,000\downarrow \end{array}$$
    $$\begin{array}{rl}Assets& |Liabilities\mathrm{\&}Equity\\ -------------& --------------\\ Sale\mathrm{\$}3,000& |ShortPosition\mathrm{\$}3,000\uparrow \\ \mathbf{Margin}\mathbf{Deposit}\mathrm{\$}1,500& |AccountEquity\mathrm{\$}1,500\downarrow \end{array}$$

2 Capital Allocation

It is about speculation, not gamble...

1. Fisher Effect: $i=r+E(\pi )$$i = r + E(\pi)$，actually: $1+r=\frac{1+i}{1+\pi }$$1 + r = \frac{1 + i}{1 + \pi}$

2. $ExpectedReturnE(r)=\sum _i{p}_i{r}_i$$Expected\ Return\ E(r)\ =\ \sum_ip_ir_i$ , or: $\widehat{E(r)}=\frac{1}{n}\sum _{i=1}^n{r}_i$$\hat{E(r)}\ =\ \frac{1}{n}\sum_{i=1}^nr_i$

   $Variance{\sigma }^2=\sum _i{p}_i[r_i-E(r){]}^2$$Variance\ \sigma^2\ =\ \sum_ip_i[r_i-E(r)]^2$, or: ${\hat{\sigma }}^2=\frac{1}{n-1}\sum _{i=1}^n(r_i-\bar{r})$$\hat\sigma^2\ =\ \frac{1}{n-1}\sum_{i=1}^n(r_i-\overline{r})$ $EstimatedSkewness=[\widehat{(\frac{r-\bar{r}}{\hat{\sigma }})}{]}^3$$Estimated\ Skewness\ =\ [\hat{(\frac{r - \overline{r}}{\hat\sigma})}]^3$，正/左偏 = 高估risk，负/右偏 = 低估risk $EstimatedKurtosis=[\widehat{(\frac{r-\bar{r}}{\hat{\sigma }})}{]}^4-3$$Estimated\ Kurtosis\ =\ [\hat{(\frac{r - \overline{r}}{\hat\sigma})}]^4 - 3$，正峰度 = 低估risk

3. Value at Risk：最差的情况损失多少，$VaR(1\mathrm{\%},normal)=Mean-2.33SD$$VaR(1\%, normal)\ =\ Mean-2.33SD$

   Expected Shortfall / CVaR：最差的情况平均损失多少

   

4. Normal Distribution: $E(GeometricAvergage)=E(ArithmeticAverage)-\frac{1}{2}{\sigma }^2$$E(Geometric\ Avergage)\ =\ E(Arithmetic\ Average)\ -\ \frac{1}{2}\sigma^2$

• Effective Annualized Rates of Return, Annual Percentage Return Holding Period Return (= Capital Gain + Dividend Yield)

• Sharp Ratio $=\frac{RiskPremium}{SDofexcessreturn}$$=\ \frac{Risk\ Premium}{SD\ of\ excess\ return}$, where $RiskPremium=E(r)-r_f$$Risk\ Premium\ =\ E(r)-r_f$

  Sortino Ratio $=\frac{RiskPremium}{LPSD}$$=\ \frac{Risk\ Premium}{LPSD}$, Lower Partial Standard Deviation: count only negative deviations

  Utility Function $=E(r)-\frac{1}{2}A{\sigma }^2$$=\ E(r) - \frac{1}{2}A\sigma^2$, (Mean-Variance Utility), is also denoted as Certainty Equivalent Rate, $r$$r$ and $\sigma$$\sigma$ in decimal

2.1 Markowitz

1. One Risky + Risk-Free

• Capital Allocation Line (Expected Return - SD) + Utility Function => Best Allocation

  $$\begin{array}{c}\{\begin{array}{rl}E({\mathbf{r}}_{\mathbf{p}})& =r_f+y[E(r_1)-r_f]\\ {\sigma }_{\mathbf{p}}& =y{\sigma }_1\end{array}\\ \to E({\mathbf{r}}_{\mathbf{p}})=\frac{E(r_1)-r_f}{{\sigma }_1}·{\sigma }_{\mathbf{p}}+r_f(CAL)\end{array}$$

  

  $$\begin{array}{c}Ma{x}_{\mathbf{y}}U=E({\mathbf{r}}_{\mathbf{p}})-\frac{1}{2}A{{\sigma }_{\mathbf{p}}}^2\\ \to {\mathbf{y}}^{\mathbf{\ast }}=\frac{E(r_1)-r_f}{A{\sigma }_1^2}(bestallocation)\end{array}$$

• Capital Market Line: CAL for market index + Risk-Free asset (short-term T-bill / money market fund)

2. Two Risky + Risk-Free

• Two Risky Assets

  Efficient Frontier => Optimal Risky Portfolio => Capital Allocation

  $$\begin{array}{c}\{\begin{array}{rl}E({\mathbf{r}}_{\mathbf{risky}})& =w_{1}E(r_1)+w_{2}E(r_2)\\ {\sigma }_{\mathbf{risky}}^{\mathbf{2}}& =w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2·Cov(r_1,r_2),\\ & ,whereCov(r_1,r_2)={\rho }_{12}{\sigma }_1{\sigma }_2\end{array}\\ (EfficientFrontier)\\ perfecthedge:{\rho }_{12}=-1,{\mathbf{w}}_{\mathbf{1}}=\frac{{\sigma }_2}{{\sigma }_1+{\sigma }_2},{\mathbf{w}}_{\mathbf{2}}=\frac{{\sigma }_1}{{\sigma }_1+{\sigma }_2}\to {\sigma }_{\mathbf{risky}}=0\end{array}$$

  

  $$\begin{array}{c}\mathbf{(}\mathbf{FOC})Ma{x}_{{\mathbf{w}}_{\mathbf{i}}}{S}_{risky}=\frac{E(r_{risky})-r_f}{{\sigma }_{risky}}s.t.w_1+w_2=1\\ \begin{array}{rl}solution:w_1& =\frac{E(R_1){\sigma }_2^2-E(R_2)Cov(R_1,R_2)}{E(R_1){\sigma }_2^2+E(R_2){\sigma }_1^2-[E(R_1)+E(R_2)]Cov(R_1,R_2)}\\ w_2& =1-w_1\end{array}\\ \to E({\mathbf{r}}_{\mathbf{p}})=\frac{E(r_{risky}^{\ast })-r_f}{{\sigma }_{risky}^{\ast }}·{\sigma }_{\mathbf{p}}+r_f(CAL)\end{array}$$

  

• N Risky Assets: Markowitz Portfolio Optimization

  $$\begin{array}{c}\{\begin{array}{rl}E({\mathbf{r}}_{\mathbf{risky}})& =\sum _{i=1}^n{w}_{i}E(r_i)\\ {\sigma }_{\mathbf{risky}}^{\mathbf{2}}& =\sum _{i=1}^n\sum _{j=1}^n{w}_i{w}_{j}Cov(r_i,r_j)\\ & ={\mathbf{w}}^T\mathrm{\Sigma }\mathbf{w},\mathrm{\Sigma }=CovarianceMatrix\end{array}\\ \begin{array}{rl}& \mathbf{(}\mathbf{FOC})mi{n}_{\mathbf{w}}{\sigma }_{risky}^{2}s.t.E(r_{risky})=\sum _{i=1}^N{w}_i{r}_i=R\to EfficientFrontier\\ & \mathbf{(}\mathbf{FOC})Ma{x}_{\mathbf{w}}{S}_{risky}=\frac{E(r_{risky})-r_f}{{\sigma }_{risky}}s.t.\sum w_i=1\to CAL\\ \\ & AnotherApporach:\end{array}\\ \begin{array}{rl}& Define:\mathbf{R},\mathrm{\Sigma },\mathbf{w},U=r_p-\frac{1}{2}A{\sigma }_p^2\\ & Then:R_p={\mathbf{w}}^T\mathbf{R},{\sigma }_{\mathbf{p}}^{\mathbf{2}}={\mathbf{w}}^T\mathrm{\Sigma }\mathbf{w}\\ & \mathbf{(}\mathbf{FOC})mi{n}_{\mathbf{w}}U=R_p-\frac{1}{2}A{\sigma }_p^2\\ & \to \mathbf{w}=\frac{1}{A}{\mathrm{\Sigma }}^{-1}R(bestallocation)\end{array}\end{array}$$

• Detailed Analysis

  Define:

  $$\begin{array}{rl}AverageVariance\bar{\sigma }& =\frac{1}{n}\sum _{i=1}^n{\sigma }_i^2\\ AverageCovarianceCov& =\frac{1}{n(n-1)}\sum _{j=1,j\ne i}^n\sum _{i=1}^{n}Cov(r_i,r_j)\end{array}$$

  Then, we would have:

  $$\begin{array}{c}{\sigma }_p^2=\frac{1}{n}{\bar{\sigma }}^2+\frac{n-1}{n}Cov\\ (firmspecificrisk)(systematicrisk)\end{array}$$

  Risk pooling (long-term investment): increase overall risk, increase Sharpe ratio Risk sharing: limit pool size, reduce overall risk, increase Sharpe ratio

2.2 Index Model

1. Single Index Model

• Single Factor Model

  $$\begin{array}{c}{r}_i=E(r_i)+{\beta }_{i}m+e_i\\ whereE(m)=0,E(e_i)=0\\ \begin{array}{rl}& Characteristics:\\ & 1.{\sigma }_i^2={\beta }_i^2{\sigma }_m^2+{\sigma }^2(e_i)\\ & 2.Cov(r_i,r_j)=Cov({\beta }_{i}m+e_i,{\beta }_{j}m+e_j)={\beta }_i{\beta }_j{\sigma }_m^2\\ & 3.Corr(r_i,r_j)=\frac{{\beta }_i{\beta }_j{\sigma }_m^2}{{\sigma }_i{\sigma }_j}=Corr(r_i,r_m)\times Corr(r_j,r_m)\\ & 4.Cov(r_i,r_m)={\beta }_i{\sigma }_m^2,Corr(r_i,r_m)={\beta }_i\frac{{\sigma }_m}{{\sigma }_i}\end{array}\end{array}$$

• Single-Index Model

  $$\begin{array}{rl}E(R_i)=& {\alpha }_i+{\beta }_{i}E(R_M)\\ (totalRiskPremium)(no& nmarket)(market)\end{array}$$

  Regression: $R_i={\alpha }_i+{\beta }_i{R}_M+e_i$$R_i = \alpha_i + \beta_i R_M + e_i$

  Security Characteristic Line: $R_i=f(R_M)$$R_i = f(R_M)$ Ajusted Beta $=\frac{2}{3}\hat{\beta }+\frac{1}{3}\times 1$$=\ \frac{2}{3}\hat\beta + \frac{1}{3} \cross 1$

2. Treynor Black Process

an active portfolio (A) comprised of the n analyzed securities + the market index portfolio / passive portfolio (M), the (n+1)-th assets

$$\begin{gathered} \begin{array}{c}\{\begin{array}{rl}\mathbf{E}\mathbf{(}{\mathbf{R}}_{\mathbf{p}})=& {\alpha }_p+{\beta }_p·E(R_M)\\ =& (\sum _{i=1}^{n+1}{w}_i{\alpha }_i)+(\sum _{i=1}^{n+1}{w}_i{\beta }_i)·E(R_M)\\ {\sigma }_{\mathbf{p}}^{\mathbf{2}}=& {\beta }_p^2{\sigma }_M^2+{\sigma }^2(e_p)\\ =& (\sum _{i=1}^{n+1}{w}_i{\beta }_i{)}^2·{\sigma }_M^2+\sum _{i=1}^{n+1}(w_i^2{\sigma }^2(e_i))\\ & \to trackingerror\text{积}\text{极}\text{组}\text{合}\text{的}\text{个}\text{股}\text{风}\text{险}\end{array} \\ ,ma{x}_{\mathbf{w}}{S}_p=\frac{E(R_p)}{{\sigma }_P}\end{array} \end{gathered}$$

• TB procedure

  • initial position of each security in the active portfolio: $w_i^0=\frac{{\alpha }_i}{{\sigma }^2(e_i)}$$w_i^0 = \frac{\alpha_i}{\sigma^2(e_i)}$

  • scale to summation 1: $w_i=\frac{w_i^0}{\sum _{i=1}^n{w}_i^0}$$w_i = \frac{w_i^0}{\sum_{i=1}^nw_i^0}$

  • alpha of the active portfolio: ${\alpha }_A=\sum _{i=1}^n{w}_i{\alpha }_i$$\alpha_A = \sum_{i=1}^nw_i\alpha_i$

    residual variance of the active portfolio: ${\sigma }^2(e_A)=\sum _{i=1}^n{w}_i^2{\sigma }^2(e_i)$$\sigma^2(e_A) = \sum_{i=1}^nw_i^2\sigma^2(e_i)$

  • initial position of the active portfolio: $w_A^0=\frac{{\alpha }_A/{\sigma }^2(e_A)}{E(R_M)/{\sigma }_M^2}$$w_A^0 = \frac{\alpha_A/\sigma^2(e_A) }{E(R_M)/\sigma_M^2}$

    beta of the active portfolio: ${\beta }_A=\sum _{i=1}^n{w}_i{\beta }_i$$\beta_A = \sum_{i=1}^nw_i\beta_i$

  • adjust the initial posistion in the active portfolio: $w_A^{\ast }=\frac{w_A^0}{1+(1-{\beta }_A)w_A^0}$$w_A^* = \frac{w_A^0}{1+(1-\beta_A)w_A^0}$

    the optimal risky portfolio: $w_M^{\ast }=1-w_A^{\ast }$$w_M^* = 1 - w_A^*$, $w_i^{\ast }=w_A^{\ast }{w}_i$$w_i^* = w_A^*w_i$

  risk premium of the optimal risky portfolio: $E(R_p)=(w_M^{\ast }+w_A^{\ast }{\beta }_A)E(R_M)+w_A^{\ast }{\alpha }_A$$E(R_p)=(w_M^*+w_A^*\beta_A)E(R_M)+w_A^*\alpha_A$

  variance of the optimal risky portfolio: ${\sigma }_P^2=(w_M^{\ast }+w_A^{\ast }{\beta }_A{)}^2{\sigma }_M^2+[w_A^{\ast }\sigma (e_A){]}^2$$\sigma_P^2=(w_M^*+w_A^*\beta_A)^2\sigma_M^2+[w_A^*\sigma(e_A)]^2$

  Sharp Ratio on the optimal risk portfolio: $S_p^2=S_M^2+[\frac{{\alpha }_A}{\sigma (e_A)}{]}^2(={IR}^2=\sum _{i=1}^n[\frac{{\alpha }_i}{\sigma (e_i)}{]}^2)$$S_p^2 = S_M^2 + [\frac{\alpha_A}{\sigma(e_A)}]^2\ (\ ={IR}^2=\sum_{i=1}^n[\frac{\alpha_i}{\sigma(e_i)}]^2\ )$

• hedged security return $=R_i-{\beta }_i{R}_M={\alpha }_i+e_i$$=\ R_i - \beta_iR_M\ =\ \alpha_i+e_i$

• comparison: Index Model vs. Full-Covariance Matrix

  

2.3 CAPM

Market Portfolio (M^*): 包括投资者所有可交易的证券或资产

$$E(R_M^{\ast })=E(r_M^{\ast })-r_f=\bar{A}{\sigma }_{M^{\ast }}^2$$

Correspondingly, we generate the concept Capital Market Line

• Expected Return - Beta relationship: $E(R_i)=E(r_i)-r_f={\beta }_i(E(r_m)-r_f)={\beta }_{i}E(R_m)$$E(R_i) = E(r_i) - r_f = \beta_i(E(r_m)-r_f)=\beta_iE(R_m)$

  $$\begin{array}{rl}& marketreturn:\\ & E(R_M)=\sum _{i=1}^N{w}_{i}E(R_i)\\ & marketportfoliorisk:\\ & {\sigma }_{\mathbf{M}}^{\mathbf{2}}={\mathbf{w}}^T\mathrm{\Sigma }\mathbf{w}=\sum _{i=1}^N\sum _{j=1}^N{w}_i{w}_{j}Cov(r_i,r_j)\\ & =\sum _{i=1}^N{w}_{i}Cov(r_i,\sum _{j=1}^N{w}_j{r}_j)=\sum _{i=1}^N{w}_i·Cov(r_i,r_M)\to riskcontributio{n}_i\\ & \mathbf{(}\mathbf{Equilibrium}\mathbf{Condition}):E(R_M)=\bar{A}{\sigma }_M^2\\ & \to E(R_i)=\bar{A}Cov(R_i,R_M)=\frac{E(R_M)}{{\sigma }_M^2}Cov(R_i,R_M)\\ & Define:{\beta }_i\equiv \frac{Cov(R_i,R_M)}{{\sigma }_M^2}\\ & Therefore:E(R_i)={\beta }_{i}E(R_M)\end{array}$$

• Security Market Line: $E(r)=f(\beta )$$E(r) = f(\beta)$

  

2.4 APT and Factor Model

APT: arbitrage if ${\alpha }_i>0$$\alpha_i \gt 0$

$$\begin{gathered} \begin{array}{c}portfolioP:R_i={\alpha }_p+{\beta }_p{R}_M,e_{p}diversifiedaway\\ \{\begin{array}{r}{w}_p=\frac{1}{1-{\beta }_p}\\ w_M=\frac{-{\beta }_p}{1-{\beta }_p}\end{array}\to {\beta }_z=w_p{\beta }_p+w_M{\beta }_M=0 \\ \\ \to {\alpha }_z=w_p{\alpha }_p=\frac{1}{1-{\beta }_p}{\alpha }_p\end{array} \end{gathered}$$

Suppose $k=2$$k=2$, arbitrage pricing portfolio Q (portfolio A: $E(R_A)={\beta }_{A1}{R}_1^F+{\beta }_{A2}{R}_2^F$$E(R_A) = \beta_{A1}R_1^F + \beta_{A2}R_2^F$, if $E(R_A)\ne price(A)$$E(R_A) \ne price(A)$, then we can arbitrage by construting Q)

For $1 investment, invest: ${\beta }_{A1}$$\beta_{A1}$ to factor portfolio 1 (with return $r_1$$r_1$) ${\beta }_{A2}$$\beta_{A2}$ to factor portfolio 2 (with return $r_2$$r_2$) $1-{\beta }_{A1}-{\beta }_{A1}$$1 - \beta_{A1} - \beta_{A1}$ to risk free asset (with return $r_f$$r_f$)

• Fama-French Three-Factor Model

  $$\begin{array}{c}{R}_{it}={\alpha }_i+{\beta }_{iM}{R}_{Mt}+{\beta }_{iSMB}SM{B}_t+{\beta }_{iHML}HM{L}_t+e_t\\ whereSMB\to firmsize,HML\to book\mathrm{\_}to\mathrm{\_}market\end{array}$$

3 Frontier Topics

3.1 EMH

Cummulative Abnormal Return

• Weak Form: market trading data, technical analysis Momentum: positive serial correlation and negative long-term serial correlation
• Semi-Strong Form: all publicly available information, fundamental analysis Dividend Ratio (D/P): mostly 1% ~ 10%, (+) Earning Yield (E/P): P/E mostly 12 ~ 25 Bond Spread（高低信用评级债的利差）: (+) with market risk premium Fama-French Three Factor Model
• Strong Form: including private details, insider trading

3.2 Behavioral Finance

• Cognitive Bias / Heuristics 认知偏差 / 启发式认知 Availability bias 可得性偏差：错误的概率预期（忽略样本偏差、先验为主） Representativeness 代表性偏差：强加因果/关联，回归效应 Overconfidence 过度自信：交易frequency越高，平均回报越低 Conservatism 保守主义：slow react to new information
• Behavioral Bias / Framing 行为偏差 / 框架效应 Convex utility function Prospect Theory (Loss Aversion) Frame effect (refrence point matters, regret avoidance, mental accounting )
• Limit to Arbitrage 有限套利 noise-trader risk, implementation cost, model risk, sentiment index

Other Brief Notes

1. Bond pricing: Invoice Price = Flat Price + Accrued Interest (in % of par)

2. Callable Bond: (an example) Face value (par value): $1,000 Coupon rate: 6% Coupon payment frequency: semi-annual Maturity: 10 years Call date: After 5 years Call price: 105% of face value

3. Determinants of bond saftey Coverage ratios: earnings to fixed cost Leverage ratios: debt-to-equity Liquidity ratios: current asset to current liability Profitability ratios Cash flow-to-debt ratio

4. Default Bond: recovery rate, default yield spread

5. Interest Rate Uncertainty and Forward Rates Forward Rate: future short rates implied from current yield curve Expectation Hypothesis: $f_2=E(r_2)$$f_2 = E(r_2)$ (expected future short rate) Liquidity premium: $f_n-E(r_n)>0$$f_n - E(r_n) > 0$, and shall increase over maturity length

6. Synthetic forward loan

   

7. Economic Indicators: leading / coincident / lagging indicators

   

   Industry Analysis: cyclical / defensive industries（防御性行业） Sensitivity to the Business Cycle: sensitivity to sales (necessities vs discretionary goods) operating leverage: the split between fixed and variable costs (higher leverage, more sensitive), Degree of Operating Leverage: $DOL=\frac{PercentagechangeinProfits}{PercentagechangeinSales}=1+\frac{Fixedcosts}{profits}$$DOL=\frac{Percentage\ change\ in\ Profits}{Percentage\ change\ in\ Sales} = 1+\frac{Fixed\ costs}{profits}$ financial leverage: the use of borrowing (higher interest rate, higher sensitivity) 财务杠杆的比例和CAPM中的β等比例放缩 Sector Rotation: 预计好的行业based on business cycle 的 investment strategy inflation rate $\uparrow$$\uparrow$ $\Rightarrow$$\Rightarrow$ P/E Ratio $\downarrow$$\downarrow$ (lowers P and boost nomial E)