Introduction to Asset Pricing: Overview, Motivation, Structure Lecture Notes Part H Zimmermann 1a Prof. Dr. Heinz Zimmermann Universität Basel WWZ Advanced Asset Pricing Spring 2016
2 Asset Pricing: Valuation of financial or nonfinancial assets under condition of risk, mostly under (strong or mild) equilibrium conditions: o General equilibrium (utility based) versus arbitrage models o omplete or incomplete market models (non traded or non spanned risk factors) o Representative agent or heterogeneous agents models What kind of assets? Asset pricing requires assumptions about risk (i.e. the underlying return distributions or risk factors) and/or preferences of the individuals typically, there is a tradeoff between the two. Well known models: o Bachelier Martingale model o Gordon constant growth model o apital Asset Pricing Model (Treynor, Sharpe, Lintner, Mossin) o State preference model (Arrow, Debreu, Beja, Rubinstein) o onsumption APM (Lucas, Breeden, ampbell) o Black Scholes option pricing model o Arbitrage pricing theory, exact (Ross) and approximate (hamberlain and Rothschild) o Fama French model o Fundamental Theorem of Asset Pricing (Harrison, Kreps, Dybvig, Ross, Delbeaen u.a.) o Hansen Richard model o Why are not all of these models so well known? Which can be found in textbooks? Empirical tests o Plot of beta versus realized returns o Returns of risk / or characteristics sorted portfolios o Performance tests of managed portfolios or funds o Test of mv efficiency of a given portfolio o Time variation and predictability of returns o Profitability of active strategies based on pricing characteristics/ anomalies
3 But is it so simple? o Relation between expectations and observations o Impact of portfolio formation (why portfolios at all?) o Estimation risk in parameters, idiosyncratic noise o Market proxies, benchmarks o Time variation (structural, statistical) o Out of sample validity o Selection biases o The relevant null hypothesis (there is a pricing relationship) a problem if the model is misspecified Asset pricing tests are always joint tests: the model and the formation of expectations Puzzles o Size, value, momentum o Temporal anomalies o Excess volatility (Shiller) o Equity premium (still a puzzle?) o The poor performance of consumption based models o Rebalancing premium Asset Pricing = a history of non testabilities o The Roll critique about the APM o The testability debate about the APT (Shanken, Roll and Ross etc.) o The Hansen critique about the testability of unconditional models Thus, the issues in asset pricing are: o How are the models related? o How can they be stated (or represented) in an empirically meaningful way? o What is economics, what statistics in asset pricing tests? Fischer Black (1986), JF In the end, a theory is accepted not because it is confirmed by conventional empirical tests, but because researchers persuade one another that the theory is correct and relevant. This is a challenging statement I wonder what you think at the end of this lecture.
4 The roadmap of the lecture o Microeconomic foundations of risk premiums: where do rp come from? Intertemporal portfolio selection and asset pricing o The standard approach to characterize asset pricing models (APM and extensions) Factor pricing models ( beta models) o A unifying representation of risk premiums, for theory and empirical work, and the role of non traded (non spanned) risk factors Stochastic discount factor models (SDF) o Model diagnostics or investment opportunities? Pricing errors o How conditioning information affects asset prices and performance measures onditioning information
5 Appendix: Empirical asset pricing examples 1 Standard approach: TSR with beta sorted portfolios Black (1993), JPM, survey paper but what are the cross sectional implications? And if the factors are not spanned, i.e. not traded portfolios? The Fama MacBeth two pass test methodology. The tests are typically performed with beta or characteristics sorted portfolios.
6 2 The Fama French hallenge: Risk (betas) versus characteristics Fama and French (1992), JF From Alex hinco s Blog
Goyal (2012), FMPM, survey paper 7
8 3 Porolio sorts and the market price of risk Daniel and Titman (2012), FR
9 4 Porolio sorts and the pricing of asset characteriscs unrelated to risk Excerpt from: Factor models, diversification, and factor pricing by arbitrage (APT) 4.0 (Spring 2012, Heinz Zimmermann) Ferson, Sarkissian and Simin (1999) provide a particularly interesting example how betas with respect to characteristics sorted spread portfolios seem to be related to the crosssection of expected returns (i.e., how characteristic sorted spread portfolio returns seem to be priced risk factors), when the characteristics are completely unrelated to systematic risk. 1 In order to separate L asset characteristics and K systematic risks within the same statistical model, an augmented factor model is used. In contrast to our earlier return model, the expected part is now the sum of two components: a constant E plus a sum of purely characteristic based, time varying (predictable) components r E Z e E t1 t t1 with t Z t : (56a) r ( N 1) vector of excess returns with unconditional expected excess returns t1 E and covariance matrix V ( N L) matrix of observed asset characteristics, assumed to be constant for simplicity, and normalized to have a cross sectional mean of zero e ( N 1) vector of unexpected returns t1 The characteristic based component is part of the conditional expectation E Er t t1 order to allow for time variation in expected returns, a ( L 1) vector Z of time varying, predictable coefficients 2 is used; it is not a systematic risk factor! Thus, the characteristics contribute to changing conditional return expectations, but do not add to the riskiness of the assets. Risk is determined by the unexpected part of the returns, e t 1.. In 1 This section borrows from the appendix of Ferson et al. (1999); the assumptions of the model are the same, but the derivation differs in some points. 2 A time-varying variable is predictable, if it changes randomly over time, but the realization is known at the beginning of each time interval.
10 The factor structure of systematic risk is then defined by the ( N N) matrices V ee ov e, e' ' (56b) where the betas are unexpected return sensitivities with respect to K risk factors, whereby it is again assumed that the covariance structure of the factors is the identity matrix, V. ff I K The key assumption is the characteristics are orthogonal to the systematic risk of the assets: ' 0 (57) LK Therefore, if standard asset pricing theory is maintained, there should be no risk premiums apart from those arising from systematic risk factors 3, in particularly not from asset characteristics unrelated to systematic risk as assumed by (57). But it can be shown that such premiums i.e. a pure anomaly can be easily established by characteristic based portfolio sorts: First, perform a sequence t of cross sectional regressions (SR) of excess returns on the L characteristics: r ˆ ˆ SRs t 1,..., T t1 t Z t uˆ t1 (58a) As shown in (54), the GLS factor estimates can be interpreted as excess returns unit characteristic portfolios : ˆ on L r p, t Zˆ t rˆ 1 1 1 ' V ' V r t1 r t1 p, t ' ee ee (58b) The T SRs generate L time series of length T of unit characteristics factor portfolio returns ˆ. Notice the following properties: the ( L 1) r p, t variances is vector of unconditional portfolio 3 In order to produce a pure characteristic-based anomaly, the model setup (56a) and (56b) is such that the factor-betas do not determine the cross-section of expected returns as they do in standard asset pricing models.
11 Var Zˆ t Var rˆ p, t Var ' r t Var Z t Var ' et1 and their ( L L) covariance matrix is ov rˆ p, t, rˆ p, t ' ' V ee Var ' Z I LL t ' e t1 (58c) (58d) The ( N L) covariance matrix is with the N asset returns is finally ov r t1, rˆ p, t ' ov Z t er 1, Z t ' et 1' ov Z, Z ' ove, e ' t t V ZZ r1 r1 V ee (58e) Second, the factor returns ˆ are used in a multivariate time series regression (TSR) to r p, t estimate the factor portfolio betas of the assets : r * * rˆ * TSR over t 1,... T t1 p p, t t1 p The OLS estimate of the N L beta matrix is * p (59a) ˆ* p ov r, rˆ ' ovrˆ, rˆ 1 t1 p, t p, t p, t ' Substituting the covariance matrices by (58d) and (58e) leads to ˆ* p V V ' V ZZ ee ee 1 V ZZ 1 ' V V ' V ee ee ee 1 (59b) V ZZ ' V ee 1 I LL
12 where the second expression (on the second line) follows directly from substituting the unitcharacteristic portfolios from (52), as in the derivation of (53c). Thus, if we assume 0 Z t Var so that V 0, we have the old result that the asset betas from using the ZZ unit characteristic portfolios are identical to the asset characteristics. But there is an additional term due to time varying expectations. However, the important insight is that both terms, and thus the estimated beta matrix, are proportional to the asset characteristics. In the third step, the estimated factor portfolios betas (we skip the ^ on the beta matrix to avoid confusion) are used in a cross sectional regression to estimate the market prices of (apparent) risks as well as potential mispricing: r ˆ * ˆ SRs t 1,..., T t1 * p F t uˆ t1 p The GLS cross sectional estimate is (60a) 1 1 ˆ F * 1 ' * * ' r p p p * f * ' r f rˆ* fp ˆ * (60b) and since the * matrix is proportional to the characteristics (59b), the factor fits Fˆ p can again be interpreted as excess returns on unit characteristic portfolios, and thereby confirming an apparent risk based factor pricing relationship E * * (61) p (in the absence of mispricing * 0 ). But this is, by construction, a pure anomaly because p it has been assumed that no risk factor is priced in the cross section of expected returns. This illustrates how characteristics based portfolio sorts make characteristics totally unrelated to risk appear as risk premiums.
13 List of abbreviations R A r A E E r V ov r, r' V rr Simple returns Excess returns, mostly over the riskfree rate Expected excess returns (vector) Variance covariance matrix of excess returns f F Unexpected factor shocks (vector) E F E f 0 Var f, f ' V ff, ' Expected value of factor shocks (vector) Variance covariance matrix of factor shocks (mostly assumed to be the identity matrix) Var Variance covariance matrix of residual returns from a factor model f f * r fp, E fp,... r, E f *, V,... * * f * f f fp, fp,...,, p p *, *, * p Unit beta factor portfolio Mimicking factor portfolio (or maximum correlation portfolio) Returns, expected returns etc. on unit beta portfolios Returns, expected returns and variancecovariance matrix on mimicking factor portfolios Mispricing of assets (zero under exact arbitrage pricing), or alpha (vector) Market price of risk of factors (vector) Betas of returns with respect to risk factors (matrix) Matrix of asset characteristics Parameter of the arbitrage pricing equation under the unit beta representation Parameters of the arbitrage pricing equation under the unit characteristic representation Parameters of the arbitrage pricing equation under the mimicking factor representation
Ferson / Sarkissian / Simin (1999), JFM 14
15 References Black, F. (1993): Beta and Return, Journal of Portfolio Management, 8 18. Daniel, K. and S. Titman (2012): Testing Factor Model Explanations of Market Anomalies, ritical Finance Review 1, 103 139. Fama E. and K. French (1992): The ross section of Expected Stock Returns, Journal of Finance 47, 427 486. Fama E. and K. French (2004): The apital Asset Pricing Model: Theory and Evidence, Journal of Economic Perspectives 18, 25 46. Ferson W., S. Sarkissian and T. Simin (1999): The Alpha Factor Asset Pricing Model: A Parable. Journal of Financial Markets 2, 49 68. Goyal A. (2012): Empirical cross sectional asset pricing: a survey, Financial Markets and Portfolio Management 26, 3 38