FHS Finance: 3rd Week

Plan For Today

• Empirically Motivated Asset Pricing Models
  • Other sources of return predictability
  • What if investors care about risk beyond portfolio variance
• General models of risk-return trade-off
  • Arbitrage Pricing Theory & Factor Models
  • factor models easily generalize and accommodate anomalies
  • trade CAPM economic assumptions for assumptions about risk
  • Link CAPM, Factor Models & Stochastic Discount Factor (SDF) approaches

Empirically Motivated Asset Pricing Models

• CAPM: if the market portfolio is efficient, securities should not have alphas significantly different from 0
  • For individual stocks, standard errors of the alpha estimates are large, so it is hard to conclude that alphas are statistically different from 0
  • But for portfolios sorted on several characteristics it is possible to show significant alphas (possible to find stocks that in the past have not plotted on the SML)
  • i.e., such characteristics can be used to pick portfolios that produce high average returns

Empirically Motivated Asset Pricing Models

• One set of asset pricing models that has emerged in recent years are empirical asset pricing models
• Idea: capture the cross-section of expected returns without specifying the forces that drive asset prices
  • models that fit the data on returns; arguably useful in selecting portfolios

Other Sources Of Return Predictability

1. The Size Effect

• Stocks with lower market capitalizations have higher average returns (Banz 1981)
• Small stocks have higher betas, but returns too large to be explained by CAPM
• Still true but weaker today

Other Sources Of Return Predictability

1. The Size Effect

• Turns out that the return difference is concentrated in January
• One explanation: tax-loss selling
  • taxable investors realize capital losses to get deduction in current tax year
  • affects small stocks more: more volatile (can have larger losses) and less liquid (selling drives prices down)
• January effect weakened in recent years

Other Sources Of Return Predictability

2. The Book to Market, or Value Effect

• Value measured by book-market ratio: (Book Value of Equity) / (Market Value of Equity)
• Alternatives: earnings-price or dividend-price ratios
• Stocks with high book-market ratio ("value stocks") delivered higher returns than stocks with low ratios ("growth stocks")

Other Sources Of Return Predictability

2. The Book to Market, or Value Effect

• Value stocks have lower betas than growth stocks
• extra puzzling!

Other Sources Of Return Predictability

2. The Book to Market, or Value Effect

• Possible explanation: higher returns for value stocks reflect a risk premium
  • BUT value outperforms growth in most years, except in boom periods (dot-com bubble & 2014-2020)
  • BUT value less risky than growth on usual measures (volatility, market beta)
  • BUT growth stocks often earn lower returns than safe Treasuries

Other Sources Of Return Predictability

3. The Momentum Effect

• Momentum Strategy is to buy stocks that have had past high returns (and shorting stocks that have had past low returns)
• When the market portfolio is efficient, past returns should not predict future risk adjusted returns
• Instead, best performing stocks over the past six months have positive alphas over the next year
• Conversely, the worse performing stocks have negative alpha over the next year
  • Effects are large: top 10% outperforms the bottom 10% by around 100 basis points per month for the subsequent six months
• Violates the weak form of EMH!

Other Sources Of Return Predictability

3. The Momentum Effect

Sharpe Ratio of 12 Month Trend Strategy

• Holds for many different assets (Moskowitz et al 2012)

Reactions to Anomalies

Explanation	Details	Checks
It's all data-snooping	Try many factors on a finite sample, you may find spurious predictability (does not hold out of sample)	Predict out-of-sample, using mechanism of first principles. MacLean et al (2015): out of 200+ factors, 26% drop post sample, 32% post public
It will be arbitraged away	Reduction in average anomaly return after sample period or publication of source
It's all omitted risk factors	Factors to be captured in a multifactor return model (How do we know we capture risk?)	Measure risk, risk aversion
It's mispricing	Anomalies reflect market mispricing (market inefficiency) driven by investor psychology	Psych. Mechanisms; Why don't rational arbitrageurs eliminate this effect?

Implications Of Non-Zero Alphas

• If small stock or high book-market portfolios do have positive alphas, one can draw one of two conclusions:
  1. Investors are systematically ignoring profitable opportunities

     If the CAPM correctly computes risk premiums, but investors are ignoring opportunities to earn extra returns without bearing any extra risk, it is because they are unaware of them or the costs to implement the strategies are larger than the value generated by undertaking them

  2. The positive-alpha trading strategies contain risk that investors are unwilling to bear and that the CAPM does not capture

Factor Models

• Consider a "single-index" decomposition:

$$ E(r_{it})-r_{f}=\alpha_{i}+\beta_{im}(E(r_{mt})-r_{f})+\epsilon_{it} $$

• Can always decompose returns as above, for adequate choice of $\epsilon_{it}$
• BUT to be informative about the cross-section of stock returns, the decomposition requires more structure
  • structure = restrictions on residual term $\epsilon_{it}$
• CAPM obtains in equilibrium where $\alpha_{i}=E[\epsilon_{it}]=0$

Factor Models

Key assumption:

Errors are mean zero, uncorrelated across stocks and with factors

$E(\epsilon_{it})=0$

$Cov(\epsilon_{it},\epsilon_{jt})=0,$

$Cov(\epsilon_{it},r_{mt})=0$

• After controlling for the factor, residual risks $\epsilon_{it}$ are uncorrelated across stocks
• hence return on each stock i is "driven by" exposure to the factor
• Essentially assume that systematic risk is driven by factor
• $\epsilon_{it}$ is stock's specific, idiosyncratic risk
• Unlike CAPM, we do not derive this from an equilibrium model

Covariance Across Stocks

$$ Cov(r_{it},r_{jt}) $$ $$ =Cov((\alpha_{i}+\beta_{im}(r_{mt}-r_{f})+\epsilon_{it}),(\alpha_{j}+\beta_{jm}(r_{mt}-r_{f})+\epsilon_{jt})) $$

• Constant $\alpha_j, r_f$, hence no covariation
• $Cov(\epsilon_{it},\epsilon_{jt})=Cov(\epsilon_{it},r_{mt})=0$ by assumption

$$ Cov(r_{it}, r_{jt}) = Cov(\beta_{im}r_{mt}, \beta_{jm}r_{mt}) = \beta_{im}\beta_{jm}\sigma_{m}^{2} $$

Factor Models

• Given that researchers have identified a number of factors that influence the return on an asset, factor models arguably help us to see how asset prices are correlated with each other
  • not only because of co-movement with the market but also due to these other factors

Arbitrage Pricing Theory

• Proposed by Ross (1976)
• Given the process that generates security returns, asset prices are derived from the process of arbitrage
• APT is based on the law of one price: two items that are the same cannot sell at different prices
• Equilibrium in asset prices exists when it is impossible for arbitrage to exist
• No arbitrage - cornerstone of capital market theory; violation would indicate grossest form of market irrationality

Arbitrage Pricing Theory

Assumptions

• Homogenous expectations
• There exists a process generating security returns

APT does not need

• Deep theory about investor behaviour
• Mean-Variance Analysis; pricing can be affected by influences beyond mean and variance

APT predicts a SML linking expected returns to risk, but the path it takes to the SML is quite different to CAPM

Arbitrage Pricing Theory

Consider portfolio of N assets indexed by i

$$ r_{pt}-r_{f}=\alpha_{p}+\beta_{pm}(r_{mt}-r_{f})+\epsilon_{pt} $$

where $w_i$ are weights of individual assets in portfolio, and

$$ \alpha_{p}=\sum_{i=1}^{N}w_{i}\alpha_{i} $$ $$ \beta_{pm}=\sum_{i=1}^{N}w_{i}\beta_{im} $$ $$ \epsilon_{pt}=\sum_{i=1}^{N}w_{i}\epsilon_{it} $$

Arbitrage Pricing Theory

Variance of residual $\epsilon_{pt}$

$$ Var(\epsilon_{pt})=\sum_{i=1}^{N}w_{i}^{2}Var(\epsilon_{it}) $$

• Shrinks fast with N provided no single weight $w_i$ is large diversification
• Simplest case: portfolio with N equally weighted assets, $w_i = \frac{1}{N}$
• All stocks have same idiosyncratic variance $Var(\epsilon_{it}) = \sigma^2$

$$ Var(\epsilon_{pt}) = \frac{\sigma^2}{N} $$

Arbitrage Pricing Theory

Well-diversified portfolio

• has enough stocks, with a small enough weight in each one
• residual risk has negligible variance $Var(\epsilon_{pt})$, meaning $\epsilon_{pt}$ can be neglected

Portfolio returns are thus

$$ r_{pt}-r_{f}=\alpha_{p}+\beta_{pm}(r_{mt}-r_{f}) $$

• BUT then we must have $\alpha_p = 0$, otherwise there is a pure arbitrage opportunity

Arbitrage Pricing Theory

Pure arbitrage opportunity

• Suppose $\alpha_p \neq 0$
• short $\beta_{pm}$ units of the market and long one unit of the portfolio
  • fund both positions with riskless borrowing and lending
• this gives a riskless exact return of $\alpha_p$
• Idea: the exact same portfolio returns can be reproduced with $\alpha_p = 0$

Individual assets:

• If $\alpha_p \sim 0$ for all well-diversified portfolios, then “almost all” individual assets have $\alpha_i$ very close to 0, a few assets can be mispriced and have nonzero $\alpha_i$
• If sufficiently many assets are mispriced, they can be grouped into a well diversified portfolio and create an arbitrage opportunity

APT and CAPM

	APT	CAPM
Result	$r_{pt}-r_{f}=\beta_{pm}(r_{mt}-r_{f})$	$E(r_{i})=r_{f}+\beta_{i}(E(r_{m})-r_{f})$
Key Assumptions	Systematic risk captured by factors, with residual risks uncorrelated across stocks. Absence of arbitrage.	Equilibrium of demand and supply of assets, given mean-variance prefs and rational expectations.
Pros	Generalises to multiple factors, matches data better. Robust to CAPM assumptions: non-financial assets, homogeneity of expectations, zero mispricing, etc.	Equilibrium of demand and supply of assets, clarifies the risk-return trade off with diversification, explains why only systematic risk matters.
Cons	It is not an equilibrium model, rather a description of data. Why is systematic risk captured by factors? What are those factors?	Does not match the evidence well, implementation issues.

Multifactor Models

• Generalize to a Multifactor Model
  • K portfolios capturing the common influence of K sources of risk

$$ r_{it}-r_{f}=\alpha_{i}+\sum_{k=1}^{K}\beta_{ik}(r_{kt}-r_{f})+\epsilon_{it} $$

Key assumption:

• Residuals are uncorrelated across stocks
• Prediction is (again) that $\alpha_i = 0$ for almost all stocks
• This is restrictive if $K \ll N$

Multifactor Models

In practice, no single factor accounts for all the correlations among individual stocks:

What might be some factors that drive correlations among stocks?

• There are common industry effects
• Cyclical companies are sensitive to industrial production
• Leveraged companies are sensitive to interest rates
• Value stocks seem to move together

Multifactor Models

How do we pick the factors for a multifactor model?

• First factor is (almost always) the return on a broad market index
• Use variance-covariance matrix of asset returns to extract most important common factors
  • however, the number of such factors increases with the number of stocks
  • recall that the theory is useful only if $K \ll N$
• Use our economic intuition to identify these factors
  • factors investors ought to care about or seem to care about

Gap in our analysis: need theory to explain which factors should arise

Multifactor Models

Factors we ought to care about	Factors investors seem to care about
macroeconomic factors inflation, GDP growth trade shocks factors related to changing investment opportunities riskless interest rate credit spreads	Portfolios with common characteristics known to be linked to average returns (size, value, etc.) Fama-French 3-factor model: market, small minus big firms (SMB), high minus low book-market firms (HML) 4th factor: momentum (MOM)

Four Factor Returns

Commonly Selected Factors

Currently the most popular choice for the multifactor models use the excess return of the market, SMB, HML, and PR1YR portfolios

Fama-French-Carhart (FFC) Factor Specification

$$ E[R_{s}]=r_{f}+\beta_{s}^{MKT}(E[R_{MKT}]-r_{f})+\beta_{s}^{SMB}E[R_{SMB}] + \beta_{s}^{HML}E[R_{HML}]+\beta_{s}^{PR1YR}E[R_{PR1YR}] $$

Fama-French (FF) Factor Specification

$$ E[R_{s}]=r_{f}+\beta_{s}^{MKT}(E[R_{MKT}]-r_{f})+\beta_{s}^{SMB}E[R_{SMB}] + \beta_{s}^{HML}E[R_{HML}] $$

Commonly Selected Factors

• FF and FFC are widely used, but dispute about whether it is a significant improvement over the CAPM
• Claim to "explain" 90% of diversified portfolios returns, compared to 70% with the CAPM
• 2015: FF extended the model with 2 factors based on profitability and investment. For US data, adding these made the HML factor redundant
• FF and FFC used extensively in event studies in academic papers, although CAPM also used
• But this is an empirical model - no underlying theory - and no guarantee that the future will be like the past
• You can find more details on Ken French's website: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

SDF Approach

Recall the dividend discount factor:

$$ P_{i}=\frac{E[X_{i}]}{1+r} $$

where $X_i$ is the stochastic payoff tomorrow

In general, different payoffs $X_i$ are discounted at different rates

$$ P_{i}=E[MX_{i}] $$

where M is the stochastic discount factor

• discount more (less) states with low (high) marginal utility

Connection: SDF & CAPM

Divide by the price of the asset

$$ 1=E\left[\frac{MX_{i}}{P_{i}}\right]=E[M(1+r_{i})] $$

• Holds for all assets, including risk-free asset and market portfolio

$$ 1=E[M(1+r_{i})]=E[M(1+r_{f})]=E[M(1+r_{m})] $$

• $r_i, r_f, r_m$ are all related through the SDF

Result: if SDF is $M=a-br_{m}$ then CAPM beta equation holds

$$ E[r_{i}-r_{f}]=\beta_{i}E[r_{m}-r_{f}] $$

Connection: SDF & CAPM

$$ 1=E[M(1+r_{i})]=E[M(1+r_{f})] $$

• Subtracting the two sides, we have: {$cov(a,b)=E(ab)-E(a)E(b)$}

$$ 0=E[M(r_{i}-r_{f})]=E[M]E[r_{i}-r_{f}]+Cov(M,r_{i}) $$

This implies:

$$ E[r_{i}-r_{f}]=\frac{-Cov(M,r_{i})}{E[M]} $$

The return of an asset depends negatively on its covariance with SDF

Connection: SDF & CAPM

• Plug in the assumption $M=a-b~r_{m}$ to get

$$ E[r_{i}-r_{f}]=b*\frac{Cov(r_{m},r_{i})}{E[M]} $$

• Assumption on M ensures asset return depends on covariance with market portfolio

Rewrite: reward-risk ratio is the same for all assets

$$ \frac{E[r_{i}-r_{f}]}{Cov(r_{m},r_{i})}=\frac{b}{E[M]} $$

Connection: SDF & CAPM

Equate reward-risk ratio for asset i and the market portfolio m:

$$ \frac{E[r_{i}-r_{f}]}{Cov(r_{m},r_{i})}=\frac{E[r_{m}-r_{f}]}{Var(r_{m})} $$

Rewriting this gives the CAPM pricing relation:

$$ E[r_{i}-r_{f}]=\beta_{i}E[r_{m}-r_{f}] $$

SDF Interpretation

Key assumption:

• SDF is linear in the market return
• To get a positive excess return on the market, the relation between the SDF and the market return needs to be negative
• SDF is proportional to marginal utility

Third way to derive the CAPM beta pricing equation

• CAPM follows from the assumption that investors' marginal utility declines linearly when the market goes up investors care about market risk

SDF Interpretation

Key assumption for multifactor model:

• SDF is linear function of K portfolio returns
• Fama-French model: SDF linear in market return, SMB return, HML return
• But why should these returns correlate with investors' marginal utility?
  • SMB: may correlate with missing components of wealth
    • moves with the returns on private businesses, which are mostly small firms

{Next lecture: behavioural drivers of multifactor models}

Aggregate Stock Market

• Previous discussions: understanding the differences in average returns across stocks
• What about the aggregate returns?
  • returns on a broad market index e.g. S&P500
• puzzles emerge about the level and variance of returns
  1. Equity Premium Puzzle
  2. Excess Volatility Puzzle

1. Equity Premium Puzzle

• Equities are riskier than bonds:
  • Std deviation (S&P) = 20%
  • Std deviation(T-bills) = 4%
• Mehra & Prescott (1985)
• Equities should command higher returns
  • How much higher?
  • Depends on risk aversion

$$ \mathbb{E}[R_{stocks}]-A\sigma_{stocks}^{2} \quad \text{vs} \quad \mathbb{E}[R_{bonds}]-A\sigma_{bonds}^{2} $$

A is Arrow-Pratt coefficient of ARA

1. Equity Premium Puzzle

• Return earned by a risky security in excess of that earned by a relatively risk-free T-bill is an order of magnitude greater than can be explained by risk aversion!
• Furthermore, Consumption-CAPM suggests:

$$ E[r_{stocks}]=r_{bonds}-cov(r_{stocks},M) $$

where M is SDF (marginal utility of consumption)

$$ E[r_{stocks}] - r_{bonds} = -Cov(r_{stocks}, M) $$

• Equity premium is high when stock returns & marginal utility highly negatively correlated
  • e.g. high premium for stocks with low returns in periods of low consumption
  • BUT consumption varies very little!

1. Equity Premium Puzzle

US equities

Data Set	% real return market index (mean)	% real return on a riskless security (mean)	% equity premium (mean)
1802-1998 (Siegel)	7.0	2.9	4.1
1871-1999 (Shiller)	6.99	1.74	5.75
1889-2000 (M-P)	8.06	1.14	6.92
1926-2000	8.8	0.4	8.4

In terms of terminal value, if you invested $1 in stocks and bonds

• From 1802-1997, your real return would be $558,945 and $276
• From 1926-2000, your real return would be $266.47 and $1.71

1. Equity Premium Puzzle

Is the US an outlier?

• Survivorship bias (many stock markets have had major interruptions - either due to revolutions, wars or hyperinflation)
• and the time series are longest for the "best" countries

1. Equity Premium Puzzle

• Equity premiums very large in many places
• Theory with standard measures of risk aversion predicts an annual equity premium of 1.5% a year

Country	Period	Market index	Riskless	Equity premium
United States	1802-2004	8.38%	3.02%	5.36%
UK	1900-2005	7.4%	1.3%	6.1%
Japan	1900-2005	9.3%	-0.5%	9.8%
Germany	1900-2005	8.2%	-0.9%	9.1%
France	1900-2005	6.1%	-3.2%	9.3%
Sweden	1900-2005	10.1%	2.1%	8.0%
Australia	1900-2005	9.2%	0.7%	8.5%
India	1991-2004	12.6%	1.3%	11.3%

1. Equity Premium Puzzle

Explanations

Candidate explanation	Findings
Stock risk is actually higher	Rare disaster risk (Barro 2006): with 1.7% chance of 40% default we generate a premium of 5.4%
Risk aversion is actually higher	High risk aversion can produce high risk premium for equity BUT cannot simultaneously explain low returns on bonds
Loss aversion	If stocks evaluated separately from bonds (narrow framing) then higher volatility entails severe loss aversion for stocks

1. Equity Premium Puzzle

Explanations

Other considerations:

• Incorrect measurement of market equity risk premium
  • Period too short or sample size too small
  • Goetzmann and Ibbotson (2005) find equity risk premium between 1792 and 1926 was only 3.66% in the US
• We observe realized, not expected, returns
  • Fama and French 2002 suggest that the second half of the 20th century saw higher than expected returns

2. Excess Volatility Puzzle

If we buy and hold a stock forever, price should be

$$ P_{t}=\frac{D_{t+1}}{r-g} $$

How to test this?

• Take a stock index
• Compute present value of future realized dividends
• Compare index's market price to discounted cash flow

2. Excess Volatility Puzzle

S&P500 (left) and Dow Jones Industrial Average (right)

2. Excess Volatility Puzzle

Excess volatility still observed in recent times

2. Excess Volatility Puzzle

Empirical evidence	Standard Finance Account	Behavioural Finance Account
Price levels probably not correct, particularly when extreme. But prices do fluctuate around "fundamentals" suggesting some predictability.	Preferences may change over time: sometimes investors are more risk averse, sometimes less (theory does not explain when or why). Time-varying disaster risk (but cannot measure it).	Investors may have wrong expectations about firm performance: sometimes too optimistic, sometimes too pessimistic. Need to understand the psychology of belief formation.

2. Excess Volatility Puzzle

• Shiller 2003 puts forward feedback models: "A price to price feedback theory". Speculative prices rise and this success encourages a fad and further price increases, if the loop continues sufficiently, this leads to bubbles

"Many individuals grew suddenly rich. A golden bait hung temptingly out before the people, and one after another, they rushed to the tulip marts, like flies around a honey-pot.... At last, however, the more prudent began to see that this folly could not last forever. Rich people no longer bought the flowers to keep them in their gardens, but to sell them again at cent per cent profit. It was seen that somebody must lose fearfully in the end. As this conviction spread, prices fell, and never rose again." - Charles McKay "Extraordinary Popular Delusions and the Madness of Crowds" (describing the 17th Century 'tulipmania')

Summary

• CAPM does not take into account some sources of return predictability
  • APT and multifactor models do a better job
  • SDF provides these models with a utility framework
  • While it can match the data, this framework does not explain it
• Puzzles about aggregate stock market
  • Equity premium puzzle
  • Excess volatility puzzle