投资学笔记

基本目标

• Asset pricing: equity, bonds, derivatives

• Asset allocation

• Consumption timing

• Financial engineering

• AN INVESTMENT IS the current commitment of money or other resources in the expectation of reaping future benefits

• These investment decisions are made in a competitive environment where

  • higher returns usually can be obtained only at the price of greater risk
  • it is rare to find assets that are so mispriced as to be obvious bargains.

Overview

• Financial assets
  • Claims on real assets
  • Stocks, bonds, mutual funds, bank deposits（银行存款）, investment accounts（投资账户）, and cash
• real assets generate net income to the economy
• financial assets simply define the allocation of income or wealth among investors
  • 金融资产只是在投资者之间进行分配
• Financial assets are aggregated out and the total balance sheet for a country contains only real assets
• 金融资产流动性更高
• 金融资产更难定价

金融市场和经济

• informational role
  • 资金流向前景最好的公司
  • 与股价相关
• 投资时间
  • 通过securities将消费推迟到下个周期
• 分散风险
  • select securities consistent with their tastes for risk
  • the firms that need to raise capital as security can be sold for the best possible price
• Separation of Ownership and Management

Mechanisms to mitigate agency problems:

• Tie managers’ income to the success of the firm （stock options）
• Monitoring from the board of directors
• Monitoring from the large outside investors and security analysts
• Takeover threat

Investment Process

composition

• Portfolios
• Asset classes
  • stocks, bonds, real estate, commodities,…
• Asset Allocation
• The security selection
  • Investment choice of which particular securities to hold within each asset class
  • Security analysis: the valuation of particular securities

process

• “Top-down” approach
  • 先从资源配置出发，再具体到证券或股票
  • First step: Asset allocation
  • Second step: security selection
• “Bottom-up” approach
  • Investment based solely on security selection
  • Disadvantage: may result in unintended heavy weight of a portfolio in only one or another sector of the economy
  • Advantage: focus the portfolio on the assets that seem to offer the most attractive investment opportunities

Competitive Market

no “free lunches”

• $\color{red}{\text{Risk-Return Trade-Off}}$

  • If you want higher expected returns, you will have to pay a price by accepting higher risk

• Diversification

  • When we mix assets into diversified portfolios, we need to consider the interplay among assets and the effect of diversification on the risk of the entire portfolio\

• 有效市场假说

  • In fully efficient markets when prices quickly adjust to all relevant information, there should be neither underpriced nor overpriced securities

• Passive Management（被动投资）

  • 接受市场大盘
  • Holding a highly diversified portfolio（分散投资）
  • No attempt to find undervalued securities
  • No attempt to time the market

• Active Management（主动投资）

  • Finding mispriced securities（选股）
  • Timing the market（择时）

Players

• Demanders of capital – Firms（企业部门）
  • Net borrowers of fund: raise capital to pay for investments in plant and equipment
  • income generated by those real assets provides the returns to investors who purchase the securities issued by the firm.
• Suppliers of capital – Households（住户部门）
  • Net suppliers of capital: purchase the securities issued by firms
• Governments （政府）
  • Can be both borrowers or lenders

Financial Intermediaries:

• to bring the suppliers of capital (investors) together with the demanders of capital
• Pool and invest funds the primary social function is to channel household savings to the business sector

Players

• Commercial Banks
• Investment Companies
  • Mutual funds (共同基金）
    • pool and manage the money of many investors， who are assigned a prorated（按比例分配） share of the total funds according to the size of their investment
    • economies of scale (由规模经济决定）
    • large-scale trading and portfolio management
    • for small investors
  • Trust
    • design portfolios specifically for large investors with particular goals.
  • Investment Bank
    • Advice on how to price new securities (stocks ,bonds)
    • Sell newly issued securities to public in the primary market
    • Investors trade previously issued securities among themselves in the secondary markets
  • venture capital (VC, 风险投资）
    • Investments in Start-up companies: invest in them in return for an ownership stake in the firm
    • Source: venture capital funds, wealthy individuals known as angel investors, and institutions such as pension funds
    • Venture capital investors commonly take an active role in the management of a start-up firm
  • Private equity(PE）
    • also invest in distressed firms, or merger & acquisition for restructuring firms
• Insurance companies
• Credit unions

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Asset Classes and Financial Instruments

Money Market （货币市场）

• Money market assets
  • short-term debt, liquid, low risk
• Classification by market participants：
  • Interbank market （银行间市场, wholesale）
    • Participated by banks, non-bank financial institutions and large corporations
• Money market mutual funds （货币基金）
  • allow individuals to access the money market
  • Higher interest rate than bank deposit
  • Risk-free rate for individual

Money Market securities

• Treasury Bills: 政府发的短期债券
• Certificates of deposit（CD）
  • Time deposit with a bank （定期存款）
  • CDs issued in denominations greater than $100,000 are usually negotiable （ can be sold to another investor）
  • Short-term CDs （maturity lower than 3month） are highly marketable
• Eurodollars（欧洲美元）
  • Dollar-denominated time deposits in banks outside the U.S.
• Commercial paper（商业票据）
  • unsecured debt of a company
  • Short term （1-3 months）
  • Considered to be safe because a firm’s financial condition is predictable in short term
• Bankers’ Acceptances（银行承兑汇票）
  • An order to pay a bank on a future date by a bank’s customer
  • For international trade financing （国际贸易工具）
• Repurchase agreement or Repo（回购）
  • 见 #固定收益证券分析
  • Issuer of repo sells government securities to an investor, with an agreement at a slightly higher price
  • The increase in the price is the interest
  • Usually overnight
  • Short-term loans in interbank market backed by government securities
• Reverse repo (逆回购）
  • The buyer of repo, investor
• Interbank loans（银行间同业拆借）
  • Fed funds（联邦基金）: Very short-term loans between banks in US
  • The LIBOR Market（伦敦银行同业拆放利率市场）
    • LIBOR rate
      • LIBOR stands for London Interbank Offer Rate
      • a rate charged in an interbank lending market outside of the U.S. （largely centered in London）
      • typically quoted for 3-month loans
      • Except for Treasury bills, money market securities are not free of default risk
      • TED stands for Treasury–LIBOR spread
        • credit spread for banks

Bond Market

$\color{red}{详见固定收益证券分析}$

• Treasury Notes and Bonds（中长期国债）
  • Inflation-Protected Treasury Bonds
    • TIPS
• Federal Agency Debt（联邦机构债券or政策性债）
  • 政策性金融机构债：国家开发银行（后改为开发性金融机构）、中国农业发展银行、进出口银行
• International Bonds
• Municipal Bonds（市政债券）
• Corporate Bonds（公司债券）
• Mortgages and Mortgage-Backed Securities（按揭与按揭支持证券）

Equity Securities

• Common stock（普通股or股票）: Ownership
  • Residual claim
  • Limited liability
• Preferred stock（优先股）: Perpetuity
  • Fixed dividends
  • Priority over common
  • Tax treatment
• American Depository Receipts（美国存托凭证）
  • Certificates traded in U.S. markets that represent ownership in shares of a foreign company（如百度、新 浪、搜狐等在美国上市的中国公司）
• CDR（Chinese Depository Receipt）

Stock Market Indexes

• Dow Jones
  • Includes 30 large blue-chip corporations
  • Computed since 1896
  • Price-weighted average
• Standard & Poor’s 500
  • Broadly based index of 500 firms
  • Market-value-weighted index
• Others
  • NYSE Composite
  • NASDAQ Composite
  • Wilshire 5000

Chinese Index

• 上证综指（全样本、市值加权）
• 深证综指（全样本、市值加权）
• 沪深300（流动性强、市值大、流通市值加权）
• 中证500（剔除沪深300、流动性和市值排序、流通市值加权）

Derivative Market

A derivative (证券衍生品or衍生品） is a security that gets its value from the values of another asset, such as commodity prices, bond and stock prices, or market index values.

• Options（期权）
  • Call（看涨期权）: Right to buy underlying asset at the strike or exercise price（执行价）
    • Value of calls decreases as strike price increases
  • Put（看跌期权）: Right to sell underlying asset at the strike or exercise price
    • Value of puts increase with strike price
  • Value of both calls and puts increases with time until expiration
  • 有价格
  • 权利而非义务
• Futures Contracts（期货合约）
  • An agreement made today regarding the delivery of an asset （or in some cases, its cash value） at a specified delivery or maturity date for an agreed- upon price, called the futures price, to be paid at contract maturity
  • Long position（买方）: Take delivery at maturity
  • Short position（卖方）: Make delivery at maturity
  • 必须执行交易

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Basics of Investing Mechanism

How Firms Issue Securities

• Primary Market
  • Market for newly-issued securities
  • Firms issue new securities through underwriter （investment banker） to public
• Secondary Market
  • Investors trade previously issued securities among themselves
• Companies
  • Privately Held Firms
    • Up to 499 shareholders
    • Raise funds through private placement
    • Lower liquidity of shares
    • Have fewer obligations to release financial statements and other information
  • Publicly Traded Companies
    • Raise capital from a wider range of investors through initial public offering, IPO
      • Seasoned equity offering: The sale of additional shares in firms that already are publicly traded
    • Public offerings are marketed by investment bankers or underwriters
    • Registration must be filed with the Security regulators （eg SEC, CSRC）
• 

中国股票发行制度及改革

• 审批制（examination and approval,1991-2000）
  • 采用行政和计划的办法分配股票发行的指标和额度
• 核准制（approval, 2001-2020）
  • 取消了指标和额度管理
  • 证券中介机构：发行是否符合市场条件
  • 证券监管机构：发行是否符合监管条件
• 注册制（registration, 2020-）
  • 市场化发行制度：发行公司股票的规模和定价由市场判断
  • 注册制注重信息披露：对申报文件的全面性、准确性、真实性和及时性作形式审查
  • 23年，全面注册制

IPO, initial public offering

• （under registration system）
  • File prospectus to security regulator
  • Road shows to publicize new offering
  • Determine demand for the new issue （Bookbuilding）
  • Degree of investor interest in the new offering provides valuable pricing information
• Underwriter bears price risk associated with placement of securities:
  • IPOs are commonly underpriced compared to the price they could be marketed （IPO discount）
  • Some IPOs are well overpriced others cannot even fully be sold

How securities are traded

• Types of Markets:
  • Direct search
    • Buyers and sellers seek each other
  • Brokered markets
    • Brokers search out buyers and sellers
  • Dealer markets
    • Dealers have inventories of assets from which they buy and sell
    • 注意broker和dealer的区别
    • Bid-ask spread is the profit for a dealer
  • Auction markets
    • Traders converge at one place to trade
• Types of Orders
  • Market orders
    • 直接交易，按照市场价
  • Price-contingent Order
    • Traders specify buying or selling price
    • 包括：Limit-Buy Order, Stop-Buy Order, Limit-Sell Order, Stop-Loss Order

Trading Mechanisms

• Dealer markets

• Electronic communication networks （ECNs）

  • True trading systems that can automatically execute orders

• Specialists markets

  • Maintain a “fair and orderly market”
  • Have been largely replaced by ECNs

• New trading strategies

  • Algorithmic Trading
    • The use of computer programs to make trading decisions
  • High-Frequency Trading
    • Special class of algorithmic with very short order execution time
  • Dark Pools
  • Trading venues that preserve anonymity, mainly relevant in block trading

• Bond Trading

  • Most bond trading takes place in the OTC market among bond dealers
  • NYSE Bonds is the largest centralized bond market of any U.S. exchange
  • Market for many bond issues is “thin” and is subject to liquidity risk

Trading Costs

• Brokerage Commission:
  • Fee paid to broker for making the transaction
• Explicit cost of trading
  • Spread: Difference between the bid and asked prices
• Implicit cost of trading
  • the price concession an investor may be forced to make for trading in quantities greater than those associated with the posted bid or ask price
  • 例如，买一50手，卖一30手，卖二20手，不能只买卖一的。

Brokerage account

• Cash account 现金账户
  • A brokerage account in which all transactions are made on a strictly cash basis.
• Margin account 信用账户（融资融券）
  • A brokerage account in which securities can be bought and sold on credit.

Margin（保证金）

• The portion of the value of an investment that is not borrowed in a margin purchase
  • Margin rate = Equity/Value of investment
• The portion that is borrowed incurs an interest
  • This interest is set by brokers
  • determined by the rate brokers pay to borrow money plus a service charge charge
• Initial Margin 初始保证金
  • the minimum margin requirement when you first purchase securities on margin
  • Currently 50% for stocks; you can borrow up to 50% of the stock value
• Maintenance Margin 维持保证金
  • the minimum margin that you must need to maintain in a margin account after your initial purchase.
• Margin Call
  • When the margin drops below the maintenance margin, the broker issues a margin call.
  • It requires the investor to bring the margin back up to an acceptable level （most of the time the initial margin level）
    • add new cash or securities, pay off the loan, or sell enough current securities
    • If investors do not act, the broker has the right to sell securities from the account.
• Margin is a Financial Leverage
  • the impact is to magnify both your gains and your losses

Short sale

• After short sales, the investor have a short position

• an investor who buys and owns shares of stock is said to be long in the stock or to have a long position.

• For short position, sell high buy low

• For long position, buy low sell high

• Risks

  • Long position in stocks:
    • Upside return (unlimited)
    • Downside losses (limited: -100%)
  • Short selling stocks:
    • Upside return (limited: +100%)
      • 股价最低到0，而且几乎不可能达到
    • Downside losses (unlimited)
      • 股价升值

Short sale mechanisms

• Proceeds from the short sale must be kept on account with the broker
• the short seller does not earn interest on the short proceeds and cannot use the proceeds for another transaction
• If any dividends are paid on the stock while you have a short position, you must pay them.
  • 分红必须返还
• A short position can be covered at any time before the securities are due to be returned.
• At any time, the security lender may call for the return of his shares
  • The security borrower (short seller) must buy shares on the market and return them to the lender (or he must borrow the shares from elsewhere).
  • 只要lender想要，必须返还

Short Sqeeze 挤兑

• A short squeeze
  • heavily shorted security moves sharply higher, forcing short sellers to close out their short positions and adding to the upward pressure on the stock.
  • 知道卖空的人多，故意抬高价格
  • Short sellers are being squeezed out of their short positions

Impact of short selling on financial market

• Pros
  • Enhance the liquidity and efficiency of the market.
  • Help security price discovery: exploits market mistakes about firms’ fundamentals. Curb “pump and dump”.
  • Uncover fraudulent accounting and other problems at companies
• Cons
  • Abusive short selling can be used for market manipulation:
    • aggressively driving down a stock price through heavy short selling, then buying the shares for a large profit
  • Lead to the undesired market volatility

Short selling for investors

• Pros
  • Some investors are better at identifying overpriced, bad companies than underpriced, good companies.
    • 相当于long position的反向
  • Many institutions won’t do short selling, leaving unexploited short selling opportunities from which you can benefit.
  • Hedging. A portfolio including both long and short positions in stocks which tend to move together will generally have lower volatility than one which has only long positions.
    • 对冲风险
• Cons
  • There’s unlimited downside potential and limited upside potential.
  • You’re fighting the trend of the market, which is, in the long run, up.
  • Money from a short sale is used as collateral for the owner of the borrowed shares. You aren’t earning interest on the proceeds from short sales.
  • You have to pay any dividends that are earned.
  • Another company could acquire the company you’re shorting, at a significant premium, thus driving up the share price.
  • Sometimes shares aren’t available to short.
  • The obligation to buy and return the shares whenever the lender call for the return of his shares

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Risk and Return

Interest rate

• Equilibrium real rates are determined by demand for funds supply and demand of funds
  • The supply of funds from Households
  • The demand for funds from Businesses
  • Government can provide either net supply of or demand for funds
  • 就是 #经济学原理 里面的供给需求线的交汇
• Equilibrium nominal rate
  • investors should be concerned with real returns
  • When expected inflation rate is higher, nominal interest rates should be higher
  • This higher nominal rate is necessary to maintain the expected real return offered by an investment
• Fisher’s hypothesis
  • Nominal rate should increase one-for-one with increases in the expected inflation rate
  • $$r_{nom}=r_{real}+E(i)$$
  • difficult to test the hypothesis because the equilibrium real rate also changes unpredictably over time

some returns

• APR: auunal percentage return
  • simple compounding
  • T < 1 year
  • $$APR=r_f(T)\times (1/T)$$
• EAR: effective annual return
  • T > 1 year
  • $$1+EAR=[1+r_f(T)]^{1/T}$$
• HPR: holding period return
  • $$HPR=\frac{P_1-P_0+D_1}{P_0}$$

excess return

• Excess return: the difference between the actual rate of return on a risky asset and the actual risk-free rate
• Risk premium is the expected value of the excess return, 是期望
  • Risk is measured by standard deviation
• Risk aversion
  • The degree to which investors are willing to commit funds to stocks

Time series analysis

• Estimated Expected Return through historical data

  • $$E(r)=\sum_{s=1}^Np(s)r(s)=\frac{1}{N}\sum_{s=1}^Nr(s)$$
  • 把每个数据点等价加权
  • If the time series of historical returns fairly represents the true underlying probability distribution, then the arithmetic average return from a historical period provides a forecast of the investment’s expected future HPR

• The arithmetic average provides an unbiased estimate of the expected future return.

• Geometric (Time-Weighted) Average

  • $$g=TV^{1/n}-1, \text{where } TV_n=(1+r_1)(1+r_2)…(1+r_n)$$
  • $TV_n$ is the Terminal value of the investment

• 几何平均总小于算术平均

  • For normal distribution: $E(\text{Geometric average})=E(\text{Arithmetric average})-\sigma^2/2$

• Using historical data with n observations, we could estimate variance is

  • $$\hat{\sigma}^2=\frac{1}{n}\sum_{s=1}^n[r(s)-\bar{r}]^2$$

• historical performance measure: geometric

• Forecast: arithmetic

Sharpe ratio

investments typically entail accepting some risk in return for the more excess return

• If we are willing to face risk, then we can expect to earn a risk premium
• the trade-off between reward and risk

$$\color{red}{\text{Sharpe ratio}=\frac{\text{Risk premium}}{\text{SD of excess return}}}$$

• 通常来说，high risk high return

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Risk Aversion and Capital Allocation to Risky Assets

speculation vs. gamble

• speculation （投机）
  • Taking considerable risk for a commensurate gain
    • Reject a fair game
    • Only accept risk for risk premium
    • Speculators are Rational
• gamble （赌博）
  • Bet on an uncertain outcome for enjoyment
    • Accept a fair game or a negative value game
    • Do not requires risk premium for taking risks
    • Gamblers are irrational, subject to behavioral bias

risk aversion and utility

• Risk-averse Investors are speculators in capital markets
  • Investors invest in portfolios of assets
  • reject investment portfolios that are fair games or worse
• Risk aversion measures investor’s preference for risk
  • Different investors are different in risk aversion
• assess the risk aversion level of an investor by his utility score
• Risk averse investors are willing to consider:
  • Risk-free assets
  • Speculative investments with positive risk premiums
• Risk tolerance: inverse of risk aversion

Mean variance utility

• A = coefficient of risk aversion
• $$U=E(r)-\frac{A}{2}\sigma^2$$
• Risk averse investor: A>0
• Risk neutral investor: A=0
• Risk lover investor （gambler） A<0

Certainty equivalent rate, CER

• 确定等价收益
  • the rate that a risk-free investment an investor requires to provide the same utility as the risky investment
• An investment can be desirable only if its certainty equivalent return exceeds that of the risk-free
  • The difference is the risk premium

Indifference curve

• An investor’s own indifference curves do not intersect
• Indifference curves of two investors might intersect
• Indifference curves have positive slopes
• In a set of indifference curves, the highest offers the greatest utility

Capital Allocation

• The choice among broad asset classes : stocks, bonds, cash
• The most fundamental decision in investments
• The simplest way to control risk is to manipulate the fraction of the portfolio invested in risk-free assets versus in the risky assets

Capital Allocation Line （CAL）

• Investment opportunity set formed with a risky asset and a risk-free asset
• 只有一个risky asset和risk-free asset之间
• 
• invertors 在这条线与Utility的切线点可以达到效用最大化
• optimal position in risky:
  • $$\color{red}{y^*=\frac{E(r_p)-r_f}{A\sigma_p^2}}$$

Passive strategies

• The passive strategy avoids any direct or indirect security analysis
• Supply and demand forces may make such a strategy a reasonable choice for many investors
• A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks such as the S&P 500

Capital Market Line, CML

• a capital allocation line formed investment in two passive portfolios
  • Virtually risk-free short-term T-bills (or a money market fund)
  • Fund of common stocks that mimics a broad market index
• 是由risk-free asset与market index连线

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MPT and CAPM

Diversification

portfolio可以分散公司的自有风险，不能分散公司的系统性风险

Portfolio of two risky assets

• 假设是debt和equity
• Portfolio return $$E(r)=w_DE(r_D)+w_EE(r_E)$$
• Portfolio Variance $$\sigma_p^2=w_D^2\sigma_D^2+w_E^2\sigma_E^2+2w_Dw_ECov(r_D,r_E)$$
• $$Cov(r_D,r_E)=\rho_{DE}\sigma_D\sigma_E$$
• 当相关系数为1时，没有任何风险被分散
• 当相关系数为-1时，可能构造出一个风险为0的组合，完全对冲风险

Global Minimum Variance Portfolio （MVP）

• The minimum variance portfolio is the portfolio composed of the risky assets that has the smallest standard deviation; the portfolio with least risk
  • 当相关系数>0时，不可能减小风险
  • 当相关系数=0时，总风险可能比两个分别都小
  • 当相关系数<0时，可能构造出risk-free

portfolio return and variance

• 这条线是Opportunity Set of risky assets

Determination of the optimal overall portfolio

• Optimal portfolio selection for N risky assets and one risk free asset:
  • Efficient frontier: identify the risk–return combinations available from the set of risky assets
    • 找到所有risky asset组成的组合的可能性边界，不考虑risk-free asset
  • Security selection: Determination of the optimal risky portfolio
    • 找Sharpe ratio最大的点，也就是切点
  • Capital allocation: Allocation of the complete portfolio to risk-free versus the risky portfolio depends on personal preference
    • 在这条切线上找与Utility相切点

Markowitz Portfolio Optimization Model

• Method 1

  • $$\sigma_p^2=\sum_{i=1}^N\sum_{j=1}^Nw_iw_jCov(r_i,r_j)$$
  • $$min_{w_i}\sigma_p^2 \quad\text{, so that } \quad E(r_p)=\sum_{i=1}^Nw_ir_i=R$$

• Method 2

  • $$max_{w_i}S_p=\frac{E(r_p)-r_f}{\sigma_p} \quad \text{subject to } \quad \sum w_i=1$$

• CAL: introduce the risk-free asset

Separation Principle

• Everyone invests in P, regardless of their degree of risk aversion
  • More risk averse investors put more in the risk-free asset

Assumptions

• Investor behavior
  • Investors are rational, mean-variance optimizers.
  • Their investment horizon is a single period
  • Investors use identical input list for efficient frontier (homogeneous expectation)
• Market structure
  • All assets are publicly held and trade on public exchanges
  • Investors can borrow or lend at a common risk-free rate, and they can take short positions on traded securities
  • No taxes
  • No transaction costs

Market portfolio

Equilibrium Conditions

• Each investor’s risky portfolio holding is

  • $$y=\frac{E(r_m)-r_f}{A\sigma_m^2}$$

• We aggregate all investors together, In equilibrium, all borrowing and lending cancelled

  • Assume average investor’s risk averse $\bar{A}$
  • y = 1
  • thus, $E(R_m)=\bar{A}\sigma_m^2$

• portfolio risk

  • $$\sigma_p^2=\sum_{i=1}^n\sum_{j=1}^nw_iw_jCov(r_i,r_j)$$

• market portfolio risk:

  • $$\sigma_p^2=\sum_{i=1}^Nw_iCov(r_i,\sum_{j=1}^Nw_jr_j)$$
  • $$\sigma_m^2=\sum_{i=1}^Nw_iCov(r_i,r_m)$$
  • 每只stock对整体风险的贡献是这只股票与市场的Cov，不是自己的Variance

• $$\frac{E(r_m)-r_f}{\sigma_m^2}=\frac{E(r_i)-r_f}{Cov(r_i,r_m)}$$

• $$\text{Since }\beta_i=\frac{Cov(r_i,r_m)}{\sigma_m^2}\text{ , thus }\quad \color{red}{E(r_i)-r_f=\beta_i(E(r_m)-r_f)}$$

• this is CAPM.

Security Market Line, SML

CAPM Application and Limitation

• CAPM holds for the overall portfolio
• Application: Provide a benchmark model for risk premium
  • Application in corporate finance
  • Cost of capital
  • IPO pricing
• Limitation: CAPM only accounts for market risk
  • Systematic risk other than market risk such as Non-tradable risk, Liquidity risk, inflation, interest rate …

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Index Model

• Problems with Markovitz model

  • The covariance matrix is too large to be estimated accurately
    • For example, the covariance of n=50 stocks consists of $(n^2 –n)/2=1325$ estimations
    • Inconsistent estimation of correlation matrix

• Factor Model

  • The actual rate of return on any security
    • $r_i=E(r_i)+\text{unanticipated surprise}$

Single factor model

• Single factor model: decompose risk （the unanticipated surprise） into
  • A systematic risk factor （denoted as m）
  • Firm specific risk（denoted as $e_i$）
  • Covariance between stocks are due to their covariance with systematic risk factor m
• $$\color{red}{r_i=E(r_i)+\beta_im+e_i}$$
  • $$E(m)=0,E(e_i)=0$$
  • m 和 $e_i$ 是独立的
  • $E(r_i)$ 无方差
  • $\beta_i$, the sensitivity for firm i
  • the total variance of $r_i$ is
    • $$\sigma_i^2=\beta_i^2\sigma_m^2+\sigma^2(e_i)$$
  • Cov between two firm’s return is
    • $$Cov(r_i,r_j)=Cov(\beta_im+e_i,\beta_jm+e_j)=\beta_i\beta_j\sigma_m^2$$
  • 当n很大时，firm-specific risk被消去了，只剩系统风险

single index model

• When single factor is a market index （or generally well-diversified portfolio）

• Security characteristic line

  • $$R_i(t)=\alpha_i+\beta_iR_m(t)+e_i$$
  • 

• Expected Return–Beta Relationship

  • $$E(R_i)=\alpha_i+\beta_iE(R_m)$$
  • $$\color{red}{R-square=\frac{\beta_i^2\sigma_m^2}{\beta_i^2\sigma_m^2+\sigma^2(e_i)}}$$

• Single index model: a framework that separates macroeconomic and security analysis

  • Macroeconomic analysis : estimate the risk premium and risk of the market index.
  • Statistical analysis : estimate the beta coefficients of all securities and their residual variances, $\sigma^2(e_i)$.
  • Asset pricing model: The portfolio manager uses the estimates for the market-index risk premium and the beta coefficient of a security to establish the expected return of that security
  • Security analysis: expected return forecasts （alphas） are derived from various security-valuation models

Alpha

• Alpha（α） is a nonmarket premium （abnormal return）
  • Depends on factor model
  • α may be large if you think a security is underpriced and therefore offers an attractive expected return
  • CAPM （or other competitive equilibrium model） predicts α will be zero
  • Security analysis: each security analyst comes up with his or her own estimates of alpha
  • A portfolio manager who has no special information about a security nor insight unavailable to the general public will take the security’s alpha value as zero
• alpha is the key variable that tells us whether a security is a good or a bad buy
  • A positive-alpha security is a bargain and therefore should be over-weighted

The Industry Version of the Index Model

• Adjusted beta
  • $\text{Adjusted beta}=\frac{2}{3}\text{estimated beta}+\frac{1}{3}$
  • Why adjusted beta
    • empirically, beta coefficients seem to move toward 1 over time
    • statistically, the average beta over all securities is 1. Before estimating the beta of a security, our best forecast would be that it is 1.
• Predicting beta
  • One simple approach would be to collect data on beta in different periods and then estimate a regression equation
    • $\text{current beta}=a+b\times (\text{past beta})$
  • Different predicting variables
    • $\text{current beta}=a+b_1\times (\text{past beta})+b_2\times (\text{firm size})+b_3\times (\text{debt ratio})$

Portfolio construction and the single index model

• A simple way to achieve diversification: include the S&P 500 index portfolio as one of the assets of the portfolio
• Passive portfolio: If no security, portfolio manager(PM) would hold the S&P 500 as a passive portfolio
  • It gives broad market exposure without the need for expensive security analysis
• Active portfolio: With security analysis, the PM may be able to devise an active portfolio that can be mixed with the index to provide a better risk–return trade-off
• Under single index model, the optimal portfolio can be derived explicitly, consisting of
  • an active portfolio（A） comprised of the n analyzed securities
  • the market index portfolio, the （n + 1）th asset （passive portfolio ,M）

构造方法

1. 计算各个股票的权重为 $w_i^0=\alpha_i/\sigma^2(e_i)$
2. 归一化 $w_i=w_i^0/(\sum w_i^0)$
3. 计算active portfolio的α $\alpha_A=\sum w_i\alpha_i$
4. 计算active portfolio的残差 $\sigma^2(e_A)=\sum w_i^2\sigma^2(e_i)$
5. 计算active portfolio的初始权重 $w_A^0=(\alpha_A/\sigma^2(e^A))/(E(R_M)/\sigma^2M)$
6. 计算active portfolio的beta $\beta_A=\sum w_i\beta_i$
7. 调整初始权重 $w_A^*=w_A^0/(1+(1-\beta_A)w_A^0)$
8. 组合的权重为 $w_M^*=1-w_A^\quad w_i^=w_A^*w_i$

• 相关数据：

  • $$E(R_p)=(w_M^*+w_A^\beta_A)E(R_M)+w_A^\alpha_A$$
  • $$\sigma^2_P=(w_M^*+w_A^\beta_A)^2\sigma_M^2+(w_A^\sigma(e_A))^2$$
  • $$\color{red}{S_p^2=S_M^2+(\frac{\alpha_A}{\sigma(e_A)})^2}$$
  • 其中 $(\frac{\alpha_A}{\sigma(e_A)})^2$ 是 information rato
    • measures the extra return we can obtain from security analysis compared to the firm-specific risk
    • the information ratio if optimized, is always increasing with increasing number of active security
    • if a security’s alpha is negative, the security will assume a short position in the optimal risky portfolio

• The index portfolio is an efficient portfolio only if all alpha values are zero from security analysis

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CAPM, Multifactor model and APT

Single factor model

• $$R_i=E(R_i)+\beta_iF+e_i$$
  • $R_i$ is excess return on security
  • $F$ is Surprise in macro-economic factor
    • F could be positive or negative but has expected value of zero
  • $\beta_i$ Factor sensitivity or factor loading or factor beta
  • $e_i$ Firm specific events （zero expected value）

index model and CAPM

• CAPM indicates $\alpha_i$ is 0
• $E(R_i)=\beta_iE(R_m)$
• assets should be on the security market line （SML）

CAPM

• The assumptions of the CAPM
  • Single period investment （vs intertemporal risk premium fluctuation）
  • investors face the same opportunity set
  • No frictions
  • All assets are traded: no untradable asset risk
  • All investors are rational mean-variance optimizers with homogeneous expectations
• Systematic Risk beyond CAPM
  • Inflation risk
    • Uncertainty due to inflation rate
  • Interest rate risk (Real)
  • Volatility risk
  • Liquidity risk
    • Price risk due to transaction costs/trading friction
  • Untradable asset risk
    • labor income risk(unemployment risk)
    • entrepreneurial risk
    • housing price risk
  • For long horizon investors, concerns about future risk premium changes over economic cycle
    • Macro economic factors

Multifactor model

• Possible common macro-economic factors

  • Business cycle （GDP Growth）
  • Interest Rates
  • Inflation Rates
  • Liquidity

• Possible factors due to extra risk premium

  • size
  • Value
  • Many firm characteristics seem to predict future returns （active current research area）

• Example

  • $$R_i=E(R_i)+\beta_{iGDP}GDP+\beta_{iIR}IR+e_i$$
  • $$E(R_i)=\beta_{iGDP}RP_{GDP}+\beta_{iIR}RP_{IR}$$
    • RP stands for risk premium

• multifactor model

  • Multiple risk exposures, each risk factor has its own risk premium
  • Estimate a beta or factor loading for each factor using multiple regression
    • $$R_it=\sum_{j=1}^k\beta_{ij}F_j+\epsilon_{it}$$

• APT

  • Multifactor version of security market line
  • $$E(R_i)=\sum_{j=1}^k\beta_{ij}E(R_j^F)$$

well-diversified portfolio

• if a portfolio is well diversified, its firm-specific or nonfactor risk becomes negligible
• 

APT（Arbitrage Pricing Theory）

• The Law of One Price :

  • if two assets are equivalent in all respects, they should have the same market price

• Risk free arbitrage （a very strong assumption）:

  • any investor will want to take an infinite position in it

• no-arbitrage condition:

  • rules out the existence of arbitrage opportunities

• In deriving APT, we assume risk-free arbitrage

• Arbitrage pricing theory for any well-diversified portfolio:

• $$\color{red}{E(r_i)=r_f+\beta_{i1}[E(r_1)-r_f]+\beta_{i2}[E(r_2)-r_f]}$$

• Can be extended to multifactor model

• if this equation is to be satisfied by all well-diversified portfolios, it must be satisfied by almost all individual securities

The Security Market Line of the APT

• Security market line for well-diversified portfolios
  • $$E(R_P)=\beta_PE(R_F)$$

APT vs. CAPM

APT	CAPM
Using any systematic risk factor	Model is based on an inherently unobservable “market” portfolio
Does not assume investors as mean-variance investors	rest on mean-variance efficiency. The actions of many small investors restore CAPM equilibrium
Apply to well-diversified portfolio	Apply to all assets

Fama-French Three-Factor Model

• Size anomaly
  • the market value of outstanding equity
  • firms with small size outperforms large ones after accounting for CAPM market risk
• Value anomaly
  • book value per share divided by stock price
  • Firms with high book to market ratios outperforms lower ones
• Factors constructed based on the size and value anomaly
  • SMB = Small Minus Big （firm size）
  • HML = High Minus Low （book-to-market ratio）
• $$R_{it}=\alpha_i+\beta_{iSMB}SMB_t+\beta_{iHML}HML_t+\beta_{iM}R_{Mt}+e_{it}$$
• While SMB and HML are not themselves obvious candidates for relevant risk factors
  • may proxy for hard-to-measure more fundamental variables
  • firms with high book-to-market ratios are more likely to be in financial distress
  • small stocks may be more sensitive to changes in business conditions

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Efficient Market Hypothesis

• 1953年 Maurice Kendall 发现股价的波动时随机游走的。该怎么解释？

EMH

• Efficient Market
  • Market prices reflect all relevant information at a given time
• New information is unpredictable
  • if it could be predicted, then the prediction would be part of today’s information.
• Stock prices that change in response to new （unpredictable） information also must move unpredictably.
  • Stock price should follow a random walk if market is efficient
  • Efficient market ≠ perfect foresight market（无未知信息）

implication1

• EMH implies that there are no undervalued or overvalued financial assets
  • A forecast about favorable future performance leads to favorable current performance, as market participants rush to trade on new information.
  • Result: Prices change until expected returns are exactly commensurate with risk.
  • the price is “right”

reasons

• 为什么市场是有效的
  • driving force：competition and profit-motive
  • 当基数很大时很小的 $\alpha$ 也能产生很高的收益
  • 导致大基金公司倾向于寻找relevant information
• 为什么百分百有效不可能
  • 只有当寻找信息的投入likely to 产生超额收益时才会愿意去投入时间和精力
  • 在市场均衡时，有效的信息收集活动应当被奖励
  • the degree of efficiency differs across various markets, depending on the information environment

Information Set for Market efficiency

• Strong-form efficiency: all information of any kind, public or private, are available
• Semistrong form efficiency: all publicly available information
• Weak form efficiency: past price and volume

EMH Implications 2

Technical analysis

• search for recurrent and predictable patterns in stock price
• Weak form efficiency implies technical analysis is useless
• Rationale
  • a sluggish response of stock prices to fundamental supply and demand factors
  • if the stock price responds slowly enough, the analyst will be able to identify a trend that can be exploited

Fundamental analysis

• using economic and accounting information to predict stock prices
  • Study past earnings and balance sheets information
  • detailed economic analysis （including the quality of the firm’s management, the firm’s standing within its industry, the prospects for the industry, etc）
• Semi-strong form EMH implies Fundamental Analysis is useless

Information analysis

• whether the information you obtained is already priced in the market?
• Whether the information source is trust worthy?
• How the information is processed?

inspiration

• try to find firms better than everyone’s estimates
• try to find poorly runned firms that are not that bad

Active and passive portfolios

• Active Management
  • An expensive strategy
  • Suitable only for very large portfolios
• Passive Management: No attempt to outsmart the market
  • Accept EMH
  • Index Funds and ETFs
  • Very low costs
• 据统计，基金经理平均水平没有跑过大盘

Event study

• If security prices reflect all currently available information, then price changes must reflect new information
• 因此，measure the importance of an event by examining price changes during the period in which the event occurs
• Event study（短期事件分析）
  • Method
    • a technique to assess the impact of a particular event on a firm’s stock price
    • If market is efficient, the event should not impact price before or after the announcement of the event
    • Choose a time window （usually 10 days） around the event date to account for the information leakage and possible sluggish reactions
  • Key issue: Isolating the part of a stock price movement that is attributable to a specific event
    • Use of a “market model” to approximate for all other information impact on the stock return
    • 也就是单因子模型寻找 $e_t$
  • cumulative abnormal return （CAR）, the sum of all abnormal returns over the time period
  • application
    • the Regulators use event studies to measure illicit gains captured by traders who may have violated insider trading or other securities laws
    • Event studies are also used in fraud cases, where the courts must assess damages caused by a fraudulent activity.

Test EMH

statistical issues

• Magnitude Issue
  • Only managers of large portfolios can earn enough trading profits to make the exploitation of minor mispricing worth the effort.
• Selection Bias Issue
  • Only unsuccessful investment schemes are made public
  • good schemes remain private.
• Lucky Event Issue

金融市场异常

• Anomaly is defined to be a deviation from a the CAPM （or other benchmark model）
• Return patterns that are unexplained by the CAPM are considered to be anomalous.
• Usually a significantly positive intercept （正的 $\alpha$）

Weak-form test （弱式检验）

• Returns over the Short Horizon
  • Momentum
    • Good or bad recent performance continues over short to intermediate time horizons
• Returns over Long Horizons
  • negative long-term serial correlation in the stock return performance
• Behavioral explanation
  • Episodes of overshooting followed by correction
  • Overreaction leads to positive serial correlation （momentum） over short time horizons
  • Subsequent correction of the overreaction leads to negative serial correlation over longer horizons
  • stock price fluctuates around its fair value

Semistrong form test

• Size effect: small-firm effect
  • higher average returns on small stocks than large stocks. Beta cannot explain the difference.
• Value effect: book-to-market ratios
  • higher average returns on value stocks than growth stocks. Beta cannot explain the difference.
  • Value firms: Firms with high B/M, E/P, D/P, B/P, or CF/P. The notion of value is that physical assets can be purchased at low market prices.
  • Growth firms: Firms with low above ratios. The notion is that high price relative to fundamentals reflects uncapitalized growth opportunities, or real options
• Post-Earnings Announcement Price Drift
  • earnings surprise
    • Difference between actual earning and market expectations of earning
    • Market expectations of earnings: measured by averaging the published earnings forecasts of analysts or by applying trend analysis to past earnings（一致预期）
  • EMH suggets response to announcement should be quick
    • For earnings announcement, positive (negative) surprise should lead to immediate positive (negative) stock returns
    • Many studies find market price responses to earnings announcements are slow

Strong form test

• Inside Information
• 不会预期市场强有效，因为
  • trading on inside information is illegal
  • Security regulators （SEC or CSRC） require all insiders to register their trading activity

Explanations of anomalies

• rational explanations
  • Fama French: explained by risk premiums
  • Measuring abnormal performance: rely on a specific asset pricing model
  • joint hypothesis problem: is it alpha or beta
  • distinguish luck from skill
• Behavioral explanations
  • evidence of inefficient markets
  • irrational investors

Mutual fund performance

• The conventional performance benchmark today is a four-factor model, which employs:
  • the three Fama-French factors
    • the return on the market index, and returns to portfolios based on size and book-to-market ratio
    • SMB，HML，$R_m-R_f$
• plus a momentum factor （a portfolio constructed based on prior-year stock return）

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Bonds

#见投资学与固定收益证券分析

一些要点：

• The bond price curve is convex
• The longer the maturity, the more sensitive the bond’s price to changes in market interest rates
• Current Yield
  • Bond’s annual coupon payment divided by the bond price
• For premium bonds （bond price above par value）
  • Coupon rate > Current yield > YTM
• Altman’s z-score model for bankruptcy prediction
  • $$Z=3.1\frac{EBIT}{\text{Total Assets}}+1.0\frac{Sales}{Assets}+.42\frac{\text{Shareholders Equity}}{Total Liability}+ .85\frac{\text{Retained Earnings}}{\text{Total Assets}}+.72\frac{\text{Working Capital}}{\text{Total Assets}}$$
  • Z < 1.23 , 有风险
  • 1.23 < Z < 2.9 , 灰色区域
  • Z > 2.9 , 安全
• Default yield spread
  • The difference between the promised YTM and the expected YTM

yield curve

• Investors require a liquidity premium （or risk premium） to hold a longer-term bond
  • $f_n$ generally exceeds $E(r_n)$
  • The excess of $f_n$ over $E(r_n)$ is the liquidity premium （or term risk premium, or simply risk premium）
  • The yield curve has an upward bias built into the long-term rates because of the risk premium
• An upward sloping curve could indicate:
  • Rates are expected to rise （Expectation hypothesis）
  • and/or
  • Investors require large term risk premiums to hold long term bonds （liquidity preference/Risk premium theory）
• Inverted yield curve indicate that interest rates are expected to fall and signal a recession

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Equity Valuation Models

感觉和 #公司金融 讲的又很像，简单写一下对照着看看吧。

Methods of Valuation

• Market value of equity: stock price too volatile
• Book values of equity: common shareholders equity
  • based on historical cost
  • It is possible, but uncommon, for market value to be less than book value.
  • “Floor” or minimum value of the liquidation value per share
  • Useful in bankruptcy
• Replacement value:
  • Liquidation value, value through liquidation of firm’s assets
  • Usually a measure for merger and acquisition
  • Tobin’s q is the ratio of market price to replacement cost
• Intrinsic value: the present value of all cash payments to the investor in the stock, discounted at the appropriate risk-adjusted interest rate
  • Dividend Discount Models （DDM）
    • The growth rate in dividends （g） can be estimated in a number of ways
      • Using the company’s historical average growth rate
      • Using an industry median or average growth rate
      • Using the sustainable growth rate.
        • if a firm can generate return on investment （ROE） greater than its cost of capital, it will retain part or all earnings to sustain its growth
        • $$g=ROE\times b$$
  • Free Cash Flow Models （useful for firms without stable dividend payments）

Price-Earning ratios and growth

• The value of the firm can be decomposed into two parts:
  • the value of the assets already in place, the no-growth value of the firm
  • Plus the NPV of its future investments, present value of growth opportunities
  • $$P_0=\frac{E_1}{k}+PVGO$$
  • $E_1$, earnings per share; k, required rate of return; PVGO, 未来预期增长机会现值
• The ratio of $PVGO/(E_1/k)$ is the ratio of firm value due to growth opportunities to value due to assets already in place
• Price-Earning ratio:
  • $$\color{red}{\frac{P_0}{E_1}=\frac{1}{k}(1+\frac{PVGO}{E/k})}$$
  • P/E rises dramatically with PVGO
  • High P/E indicates that the firm has ample growth opportunities.
  • 根据DDM, $P_0=\frac{D_1}{r-g}$, 则
    • $$\frac{P_0}{E_1}=\frac{D_1/EPS}{k-g}=\frac{1-b}{k-ROE\times b}$$
  • 当ROE、b增大时P-E增大
  • 当风险变大时，k增加，P-E减小

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记

这门投资学的课程内容含量确实不算高，金融课之间的许多知识点也是相通的，可以互相参考借鉴着学习并复习。