Master in Money and Finance Goethe University Frankfurt Week 8: From factor models to asset pricing Fall 2012/2013 Please note the disclaimer on the last page
Announcements Solution to exercise 1 of problem set 1 will be on the website today. Rest of the problem set should be self-explanatory after reading the papers. Second problem set will be out next week. 2
Today Structure of this course General Stocks Bonds Evidence Observed returns and risk premia Stylized facts on the yield curve Foundations Forecastability & predictability Financial crisis etc. Valuation techniques Yields, duration, estimation of the yield curve Theory Asset pricing (factor model, APT) CAPM APT Expectations hypothesis Factor model Applications Technical analysis and trading rules Value factor Momentum factor Portfolio optimization and risk models Forecasting the yield curve Finance industry Overview asset management industry Guest lecture on industry factors Investment process 3
Topics of class 8 1. Stock selection in practice 2. Consumption-based asset pricing General asset pricing formula Equity premium puzzle 4
Portfolio construction: Inputs we have seen so far Workhorse model: Mean variance optimization Expected returns: success of certain characteristics like size and book-to-market see e.g. Fama and French (1993) Risk model (covariance matrix): procedure for choosing the risk model discussed e.g. by Chan, Karceski and Lakonishok (1999) Issue in portfolio optimization: The estimation error, which is implied by the plugin approach, might lead to inefficient portfolio choices and suboptimal Sharpe ratios, see. e.g. Jobson and Korkie (1980) Next: We present an entire investment process for constructing equity portfolios ( stock selection process ). Today s main reference is: Grinold, R.C., and R.N. Kahn (2000, 2 nd ed.) Active Portfolio Management: A Quantitative Approach Approach for Producing Superior Returns and Controlling Risk, NY: McGrawHill. 5
Roadmap for today We consider a typical institutional framework for an active equity manager. We see that he is asked to manage his portfolio close to some benchmark. Therefore we have to develop a new framework: Portfolio holdings, risk and return are now measured relative to the benchmark. The information ratio is used as a performance measure. It mirrors the Sharpe ratio that is common in the context of total returns. The information ratio is shown to have two sources: Manager s skill of forecasting future returns Number of independent forecasts The authors propose a refinement process for the single-stock forecasts, in which raw forecasts are controlled for volatility, skill and expectations. This process is sufficiently general to implement buy and sell recommendations and other simple ways of expressing views on single stocks. The modified forecasts can be used in a mean variance optimization. 6
Typical institutional framework Plan sponsors (e.g. pension fund or endowment funds) often determine the strategic asset allocation (long-term allocation of asset classes) either in-house or by using an external consultant. Given the strategic allocation a search process (so-called beauty contest) is often initiated to find the potential managers for a segment of the asset allocation, e.g. European stocks. Depending on the assets for the segment, several managers are often hired for the same segment for the sake of diversification. Often the manager is expected to be a specialist in his field. His job is to follow some benchmark index (e.g. MSCI Europe) and, ideally, to outperform it constantly after fees by moderately deviating from the benchmark (active management). Managers are often given constraints that ensure that they do not deviate too much from the benchmark. In other words, the sponsor is often significantly more tolerant to the risk associated with the asset class that with the risk, that the manager deviates too much from its benchmark. We introduce today the so-called tracking error which is a measure for the manager s deviation from the benchmark. In the institutional framework, managers are often given upper bound for the tracking error. 7
CAPM: Consensus Expected Returns cont. Notation used by Grinold and Kahn: CAPM relationship: New aspect: Use of the CAPM to separate the market risk and the residual risk Reason: The portfolio P of the manager is expected to closely track the market M (more concretely the mandated market segment). Therefore he should have a beta close to one. Factor model: residual return θ P be uncorrelated with the market return (ω: residual risk) portfolio variance: 8
CAPM: Consensus Expected Returns cont. Another aspect of the CAPM: The CAPM is a model of expected excess returns. Given a stock s beta we can back out the expected return expected by the market along the security market line. In other words, The CAPM provides us with a consensus expected returns for each asset. The consensus view leads to the market portfolio. Active management is about forecasting. Only expected returns different from the consensus view will lead to portfolios different from the market portfolio. One valid investment strategy (not the only one) is to concentrate on forecasting residual returns and keeping a portfolio beta close to one. Hereby the manager does not need to time the market. 9
Risk Grinold and Kahn favor the standard deviation (SD) as the key measure of risk. Reasons: close links to mean variance optimization, relatively stable measure, forecasting methods available, under normality alternative risk measure can be calculated using SD Risk and returns vary over the investment horizon. Any statistic that builds on risk and return should, for the sake of comparability, refer to the same horizon. Choosing a horizon of one year is quite common. Rule of thumb for annualizing risk of monthly returns: Underlying assumption: autocorrelation close to zero (it is only an approximation) Due to the institutional setup the manager is interested in active risk, i.e. the risk associated with the deviations of his portfolio (P) from the benchmark (B). The active risk is commonly measured by the tracking error which is defined as The tracking error is a key statistic for active managers in the aforementioned framework. Plan sponsors often impose constraints on the allowed tracking error. 10
Adding value by active management Define: active portfolio weights (holdings) as active variance as in beta notation alpha is the expected residual return We can decompose the (total) expected return on asset n as Hence The risk-free rate i F can be interpreted as the time premium. The risk premium is given by β n μ B. Δf B measures the difference between the expected return on the benchmark in the near future and the long-run expected return. It is a way to model benchmark timing (not further discussed). Finally, α refers to the expected return arising from stock selection. 11
Adding value by active management Forecast on the expected excess return on asset n: (f B is the forecast of the expected excess return of the benchmark) The expected utility of portfolio P is given by (mean variance framework) where λ T measures the aversion to total risk. Since plan sponsors exhibit different preferences towards different sources of risk, we can split risk into its components (each source is associated with its own risk aversion parameter) The last two components measure the manager s ability to add value by taking active positions. Hence the manager seeks to maximize the value added (VA) which can be defined by 12
Information Ratio The portfolio excess return can be used in the regression The residual return is given by The alpha is the forecast of the residual return By construction, the alpha of the benchmark portfolio is zero. The information ratio of any portfolio P is defined as the annualized residual return divided by the annualized residual risk (assuming ω>0) Given a set of alphas, the information ratio of IR is the largest attainable value of IR(p) under the assumption that the investors behaves optimally. Properties: IR can be negative, IR of the benchmark is zero, the IR is universal in the respect that it is independent of the investor s/manager s risk aversion /aggressiveness. Please note the similarity to the Sharpe ratio SR. SR relates absolute risk to absolute returns and IR relates residual risk to residual return. 13
Information Ratio Interpretation: Ex post the information ratio is a performance measure of active managers. Ex ante the information ratio says how good you think you are. We will see that it can be used to state our confidence in the forecasts. This can be done because evidence shows that similar values for the IR occurs in mutual fund data. Grinold and Kahn give the following rules of thumb: IR=0.5 is good, IR=0.75 is very good, IR=1 is exceptional When managers are evaluated by the rank/quantile in a peer group then a pattern like on the RHS often comes up In the investment industry, a good manager is considered to belong consistently to the top quartile. 14
Information Ratio cont. 1. IR can be re-interpreted as a budget constraint for the manager, stating the maximum attainable alpha: 2. Recall: objective to maximize the value added from residual returns (simplified version) We can construct a indifference curve for a constant value of VA. 1 and 2 can be depicted in a figure. P* is the optimal choice for our portfolio. 15
Fundamental Law of Active Management Define the strategy s breath (BR) and the information coefficient (IC) The sources of the IR are described by the fundamental law of active management: (for a derivation/proof see the Grinold and Kahn book) Maximizing the value added determines also the optimal level of residual risk Hence 16
Fundamental Law of Active Management cont. Example: Suppose we want to achieve a IR of 0.5 (good) A stock selector follows 100 stocks and makes 4 bets for each stock per year. He needs an IC of 0.025 to achieve the desired level of IR. The fundamental law shows that a moderate level of skill (measured by the IC) is sufficient to generate good outperformance. The fundamental law describes the gains from expanding the number of stocks from 100 to 200 (assuming independent bets). Rule of thumb for ICs: good (0.05), great (0.1), world-class (0.15) Issues: Bets are assumed to be independent. If we have two bets that are correlated with a coefficient of γ, then we need to replace IC by IC(com) Multivariate case: ϕ = vola*ic(adjusted)*score IC(adjusted)=IC*ρ -1 (g) ρ -1 (g): inverse of correlation matrix Strongest issue: Investors are assumed to behave optimal in the mean variance sense without restriction. Restrictions seriously disturb the trade-off. 17
Forecasting the Alphas On this slide we briefly sketch the theoretical background for the process on the next slide Basic forecasting formula: transforming raw forecasts into refined forecasts Univariate case: ϕ = cov(r,g) * (g-e(g))/var(g) by definition: corr(x,y)*sd(x)*sd(y)=cov(x,y) Naive (consensus) forecast is given e.g. by the CAPM ϕ = corr(r,g)*sd(r)*sd(g) * (g- E(g))/Var(g) ϕ = corr(r,g)*sd(r)*(g-e(g))/sd(g) Refined forecast: ϕ = SD(r)*corr(r,g)*Z(g) with Z(g)=(g-E(g))/SD(g) Alpha is given by ϕ = vola*ic*score 18
Forecasting the Alphas cont. Refinement process: 1. Compute score (Z-score) for the forecasts g(t): 2. Scale the score by applying the forecasting rule of thumb Example: regression model Least-squares estimates Forecasts g(t) are transformed to scores according to step 1. Refined forecasts of step 2 are computed via The refinement process converts raw forecasts into refined forecasts controlling for three factors: expectations, skill and volatility. 19
Forecasting the Alphas cont. The alpha refinement process is quite flexible with respect to the raw forecasts that can be processed. Example for raw forecasts: Buy and sell recommendations of equity analysts: We can assign a score of 1 to buy recommendations and a value of -1 to sell recommendations. Fractiles: We can use the Fama and French decile portfolios (sorted e.g. by book-to-market ratio) and assign raw score from 1 to 10. Rankings: This generalizes the case of fractiles. We can use a wide universe of stocks (say 1000) and rank them according to some characteristic from 1 to 1000. 20
Portfolio construction The refined alphas can be used in the mean variance optimizer. In this exercise, the stocks were assigned some random alpha (for the sake of simplicity). These were used in an unrestricted mean-variance optimizer, focusing on active return and risk, instead of residual return and risk. Then two restrictions were imposed in the optimization: non-negative holdings and maximum holdings of 5%. Not shown here: Using the alphas in a constrained optimization is equivalent to using modified alphas in an unconstrained optimization (shown in the last column). More simple portfolio techniques may be based on a so-called screen. For example: rank stocks by refined alpha, take the top 50 stocks and build an equally-weighted portfolio. 21
Review: Major insights Active management is forecasting. The consensus view leads to the benchmark. The Information Ratio (IR) measures the value-added of the manager. The Fundamental Law of Active Management describes the sources of IR: IR = IC Square root of Breath Information Coefficient (IC) is the correlation of forecast returns with their subsequent realizations. It is a measure of skill. Breath is the number of independent forecasts available per year. In the alpha refinement process, the alphas control for volatility, skill and expectations: Alpha = Volatility IC Score The Score is a normalized asset return forecast. 22
Topics of class 8 1. Stock selection in practice 2. Consumption-based asset pricing General asset pricing formula Equity premium puzzle 23
Consumption-based asset pricing models Reference: chapters 1 and 9 in Cochrane, John H. (2005, 2nd ed.) Asset Pricing, Princeton University Press, Princeton. CAPM: Assumptions: mean-variance utility, one-period optimization Resulting market risk is global but no direct links to macroeconomics: Consumption-based asset pricing models: Derive from the maximization problem of consumers The same optimization problem is used by modern macroeconomic models (real business cycle, New Keynesian) to determine the consumption-savings decision over time, hence the macro-finance often builds on this direct link. 24
Consumption-based asset pricing models cont. Utility function: (in terms of consumption over time) where c t is consumption in period t. Setup: Household can save via investing in an asset that is traded at price p t.and gives a payoff x t. ξ t denotes the amount of the asset that the investor decides to hold. Maximization problem: s.t. where β is the subjective discount factor of the household and e t the endowed level of consumption. Central asset pricing formula: (given by the first-order conditions) 25
Consumption-based asset pricing models cont. Single-period utility function Example: power utility Example: log utility (power utility with lambda approaching 1) Rewriting general asset pricing formula: where m denotes the so-called stochastic discount factor (SDF). Intuition of the expression discount factor : e.g. under certainty it can be shown that the reciprocal of the risk-free rate is used to discount the future payoff 26
Consumption-based asset pricing models cont. The asset pricing formula is general i.e. different specifications of the payoff and the price lead to pricing formulas of various asset classes There is also a nominal version of the SDF: (Π denotes the consumer price index) 27
Topics of class 8 1. Stock selection in practice 2. Consumption-based asset pricing General asset pricing formula Equity premium puzzle 28
Equity premium puzzle Reference: Mehra and Prescott (2003) The Equity Premium in Retrospect Consumption-based model: Single-period utility with constant relative risk aversion (CRRA) Resulting asset pricing formula 29
Equity premium puzzle cont. Applying asset pricing formula to equity : Applying asset pricing formula to a riskless one-period bond: Combining both expressions: Assuming iid lognormal returns and in equilibrium c=y: 30
Equity premium puzzle cont. Substituting the specified utility function After some further algebraic transformations we arrive at the expression μ x and σ x describe the moments of the payoff (i.e. dividends) Simplifying further the model delivers a description for the log equity premium...and the risk-free rate 31
Equity premium puzzle cont. Final model: Calibration: Statistics for US data covering 1889-1978 Empirical puzzle: Free parameters: alpha (risk aversion) and beta (discounting factor) No choice of alpha and beta fits the empirically observed facts for the risk-free rate and the equity premium jointly! Example: alpha=10, beta=0,99 risk-free rate=12,9%, equity premium=1,4% (vs. 6% in the data) Risk-free rate typically too high, equity premium too low 32
Resolving the equity premium puzzle Solutions in the academic literature include employing more complex utility functions (e.g. habit formation, Epstein-Zin recursive utility) changing the way how expectations are build relaxing the assumption of complete markets (idiosyncratic risks that cannot be insured/diversified away like income shocks) In the financial industry the use of so-called valuation models is a more popular way of modeling the equity premium. They typically build on present value relationships, which can be derived from the asset pricing formula. Variants of the class of models are also commonly known as dividend discount models (DDM). We proceed with valuation models 33
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