Asset Pricing

1. Assumptions
   1. All investors are small relative to market, price-takers
   2. One period model
   3. All investments are equally available, model does not include alternative investments
   4. No taxes
   5. No transaction costs
   6. Borrowing at risk free rate
   7. Homogenous assumptions - everyone has same mean-variance optimisation problem
2. CAPM in practice: How to measure
   1. Forecasting Beta
      1. Use historic data and run regression
         1. Based on EMH: equity market determines required returns
      2. Look up on Bloomberg
         1. 2 year - weekly or 5 year - monthly
      3. Evidence suggest that betas regress to average of 1. Blume, 1973
   2. Forecasting risk free rate
      1. Yield of long-term treasury bonds
      2. For shorter investments, shorter maturity bonds might be more appropriate
      3. No credit but default risk on t-bill
   3. Forecasting market premium
      1. S&P500 adequate market: Sufficient analyst covering
3. Evidence on CAPM: Testable Hypotheses
   1. Only risk measured by beta affects returns
      1. Historically true
      2. Anomalies
         1. Firm Size: Small firms generate higher returns. Banz, 1981
         2. Market-to-book ratio: Small generates higher returns. Statman, 1980; Rosenberg et al, 1985
         3. P/E ratio: Small generates higher returns. Basu, 1997
         4. Dividend yield: high generates higher returns.
         5. Momentum: Buy stock with high past-6 months returns. XXX
         6. Calendar effects
         7. Daniel & Titman, 1997: Portfolios of mispriced stocks
         8. Either: Investors systematically ignoring profitable opportunities OR: Positive-alpha strategies contain risks not caputred by CAPM
   2. Expected returns increase linearly with beta
      1. Empirical SML is flatter than CAPM predicts. Black et al 1972
      2. More recent estimations suggest it is flatter still. XXX
      3. Securities should have zero alpha. But large variance make it impossible to say alpha categorically zero
   3. Market portfolio is mean variance efficient
   4. CAPM Supporters say tests are bad, rather than theory
      1. Real Betas are not observed: based on historical values and delevered by today's leverage ratio
      2. Expected returns are not observed: actual returns need not equal expected returns for irrational investors or concern about certain event
      3. The market proxy is not correct: investors hold assets other than the ones used for CAPM, e.g. houses
      4. Data snooping: given enough characteristics, we can find some that by chance are related to estimation error of regression
      5. Anomalies can be explained by disaster. Gabaix, 2002
      6. CAPM has never actually been tested
         1. Fama & French, 2004. Use specific proxy for market portfolio, efficient from set of portfolios
         2. Roll, 1970. True market portfolio cannot be measured, all hypotheses around CAPM arise from mean variance efficiency
            1. Says one lacking assumption is that market portfolio must be identifiable
4. Consumption-based CAPM
   1. ICAPM: Takes into account consumption over multiple periods. Merton, 1973
   2. Rather than using mean-variance preferences, this links asset returns to consumption
      1. Investors as consumers that optimise their portfolio against a consumption tracking portfolio
         1. Risk of securities is measured with regard to covariance with aggregate consumption
            1. E(Rm) - Rf = A Cov(Rm,Rc): equity risk premium is driven by A (risk aversion) and covariance of market relative to consumption-tracking portfolio (Cov(Rm,Rc) can be assumed as Var(Rc) as return on market assumed equivalent to consumption tracking portfolio
      2. Investors value the flow of consumption associated with wealth
   3. Equity Risk Premium Puzzle. Mehra and Prescott, 1985
      1. Look at covariance of market with actual consumption
      2. Disparity between returns on bonds and stock is so great that it implies an implausibly high level of investor risk aversion (approx 6% too high 1889-1978). XXX
         1. Maybe covariance not correctly measured
         2. Maybe actual risk aversion is higher
         3. Maybe risk premium incorrect as period too short, sample size too small
         4. Hypothesis: Actual returns higher than expected returns in second half of 20th century. Fama & French
            1. Goetzmann & Ibbotson, 2005: Prior to 1792 only 3.66%
            2. Suggestion that US is outlier, however ERP high everywhere. Mehra & Prescott, 2008
         5. Survivorship bias in stock markets
         6. Behavioural Finance: Irrational investor behaviour, less averse, keen to maintain attained level of consumption
            1. Investors engage in narrow framing, seeing individual investment for inherent risk rather than market
      3. Maybe measuring consumption wrong, e.g. extrapolation from GDP survey and smoothing
         1. Savov, 2011. Garbage production as measure of consumption, manages to reduce relative risk significantly, yet small ERP remains
5. First model to specify what drives returns: BETA
6. Single-factor model

Nota:

• Intuition: If you have n factors, n securities can replicate any factor profile. Price of n+1st security must be determined by previous n securities. In practice # of factors << # of securities

1. E(Ri) = Rf + Summation(BETAijLAMBDAj) LAMBDA is risk premium on each factor, BETA it's sensitivity
2. No reliance on mean-variance analysis
3. Crucial Assumptions
   1. Arbitrage impossible in market equilibrium
   2. Securities' returns functions of (macro) factors
   3. Large number of traded assets
4. Limitations
   1. Lack of theoretical foundations for choice of factors
   2. Have all factors been considered?
5. Self-Financing Portfolio: Going long on some, short on others, weight sums to zero
   1. Small-Minus-Big: Firm size
   2. High-Minus-Low: Book-to-market ratio
   3. One-Year Momentum: PR1YR
   4. E.g. Fama-French-Carhart model: Mkt, SMB, HML, PR1YR
      1. Extensively used in event studies
      2. Also used in risk measurement of actively managed mutual funds
   5. Fama-French: Market Return, SMB, HML
      1. Jagannathan & Wang, 2006: 4th quarter consumption, FF SMB and HML linked to consumption beta, e.g. smaller firms associated with higher betas/consumption risk

1. Consumption is far less volatile than wealth
2. Barro, 2006: Both ERP and EVP solved by disasters
   1. Defined as falls in GDP per capita > 15%, almost all OECD countries experienced one in 20th century
   2. Gabaix, 2012: Suggest that disaster can also solve Daniel & Titman, 1997, anomalies
3. Shiller, 2003: Forward feedback model. Speculative prives rise, encouraging a fad until bubble
   1. Bubbles tend to form if people are conditioned to expect bubbles
   2. Kahneman & Tversky, 1974: Judgements are made closest to previous pattern without attention to whether new events will follow past pattern
   3. Smart investors amy reinforce feedback loop to make money. De Long et al, 1990