Quantitative Portfolio Theory & Performance Analysis

550.447 Quantitative Portfolio Theory & Performance Analysis Week of April 15, 013 & Arbitrage-Free Pricing Theory (APT) Assignment For April 15 (This Week) Read: A&L, Chapter 5 & 6 Read: E&G Chapters 8 & 16 For April (ext Week) Problems 8.5; 16.3, 16.4 Last Day of Class: Wednesday, May 1 st Final: Monday, May 13 th ; 9:00am oon In the classroom: Whitehead 03 1.1 1. Where we Are CAPM Performance Measures Relative Performance (to market, peers, risk adjusted, ) Enabling Rational Decisions (Sharpe-Dowd) and Arbitrage Free Pricing (APT) One last item about market timers and performance measurement 1.3 CAPM & Performance Timing Analysis What s That? Beta-Duration adjustment Up to now, CAPM measures have assumed constant risk and they measure risk-adjusted returns When risk modulation is used as a management strategy (market timing), we need to extend the CAPMbased measure to account for changing PF exposure to market risk We use Jensen s alpha and assess whether the result is through skill or luck 1.4 1

CAPM & Performance CAPM & Performance Treynor & Mazury Method (of Jensen s alpha) Uses a quadratic version of CAPM Market timers will lower their exposure to market risk in times of declining markets & increase their exposure when markets rise When markets decline the PF will loose less than it might otherwise When markets rise the PF will gain more than otherwise Therefore, the PF s return in excess of the risk-free rate might better be represented by a curve rather Treynor & Mazury Method (of Jensen s alpha) Consider the model characterized by the quadratic R R R R R R Pt Ft P P Mt Ft P Mt Ft Pt Where we have the PF return vector for the period studied Market return vector for the same period (measured with the same frequency as the PF) Risk-free rate over the same period The coefficients are estimated through regression and if delta is positive and significantly different from zero than a straight line 1.5 the manager has made good calls/decisions 1.6 CAPM & Performance CAPM & Performance Explicit timing models onparametric (option-based) model Assume the PF is modeled as a split between the risk-free rate and the risky asset (more generally, could be stocks and bonds) The investor modifies the split over time according to his anticipation of their relative performance; if perfect Only hold risk if it will out perform the risk-free asset Hold cash otherwise Therefore, the PF can be modeled as an investment in cash and a call on the better of the two assets Explicit timing models onparametric (option-based) model (continued) Consider the two conditional probabilities P 1 : the probability of an accurate forecast, given risk wins P : the probability of accuracy, given risk-free asset wins We have that f = P 1 + P -1and the manager has market timing ability if f > 0 (and is perfect if f = 1 ) The fraction f can be estimated using the following formula If the forecasts are not perfect, the manager will only hold a fraction, f, of options between -1 and 1 (1 is perfection) 1.7 1.8

CAPM & Performance Explicit timing models onparametric (option-based) model (continued) It 1 0 1yt t where, It 1 1 if manager forcasts that risk (stocks, etc) will perform better than risk-free rate; zero otherwise y 1 if risk actually did outperform the risk-free asset; zero otherwise t Where the coefficients are estimated through regression, and The coefficient α 0 is an estimate for 1 P 1 The coefficient α 1 is an estimate for P 1 + P -1 We then test the hypothesis α 1 > 0 1.9 CAPM & Performance Explicit timing models Parametric Model Like the nonparametric case, but formulation is based on CAPM The following model takes the manager s risk objectives into account depending on whether risk outperforms RPt RFt P 1P RMt RFt PDt RMt RFt t where, D 0, if R R 0 t Mt Ft Dt 1, if RMt RFt 0 The coefficients are estimated through regression and β allows us to evaluate the manager s ability If positive and significantly different from zero. Manager has good timing ability 1.10 CAPM & Performance Limitations (criticism?) of the CAPM Difficult to test as the true market PF is elusive One can only establish surrogates which may not be efficient Similarly, performance measures derived from CAPM suffer from the same limitation As the choice of surrogate can change conclusions about relative performance So Treynor and Jensen have been held in suspicion as betas may not be accurate ( true ) Others have developed methods for correcting measurement shortcomings Still others have moved away from the CAPM 1.11 Where we Are Single Index Model Week 5; introduced to get away from explicitly finding covariances for finding the optimal PF With the single index model we found a fast algorithm Used beta and idiosyncratic risks of the PF-member choices Other use for model is for predicting returns Generalizing the approach to multi-factors Move away from predictive ability of the market alone Identify other sources to explain and model returns Rationalize predictive ability (in the absence of a bid) through arbitrage reasoning of one price APT 1.1 3

Suppose returns, R i can be modeled as R R e i i i m i Where R m, is the return on the market index, and Where R m, e i are r.v. with std dev σ m, σ ei respectively e i is zero mean, and e i is uncorrelated w/r m : cov( ei, Rm) Eei Rm R m 0 Estimates of i, i, and ei from time series regression Finally, we assume, e i, e j are uncorrelated: E ee i j 0 1.13 Then we have Mean Return Ri i irm Variance of Return i i m ei Covariance of returns ij i i m If the single index model holds, then for a PF The expected return is R XRX X R i1 i1 i1 Variance P Xi i XiX jij P i i i i i i m i1 i1 j1 ji Xi i m XiXi jm Xi ei i1 i1 j1 i1 ji 1.14 So for the PF return we can write RP Xii XiiRm P PRm Giving a definition for PF alpha and beta P X ii P X ii i1 i1 As for PF risk X X X X XX i i j m Xi ei XiiXiim Xi ei i1 j1 i1 i1 i1 i1 X 1.15 We will return to this shortly but first 1.16 Or i1 i1 P i i m i i j m i ei i1 i1 j1 i1 ji P P m i ei i1 If we assume a large, equally weighted PF then 1 1 P P m ei i 1 Which is 1/ times the average residual risk in the PF And as the PF becomes large, the importance of the average residual risk diminishes to bcome insignificant 1/ Thus X P P m P m m i i i1 So the measure of the contribution of a security to the risk of a large PF is its beta often used a measure of a securities risk 4

The last thing for the single index model is parameterization for each security, i The model Ri i irm ei The alpha and beta are determined by regression on the returns series Rit Rit Rmt Rmt Beta im t1 i m Rmt Rmt t1 Alpha R R i it i mt 1.17 The last thing for the single index model is parameterization for each security, i The model Ri i irm ei Alpha and beta from regression on the returns series Model Statistics Estimate Error 1 ei Rit i irmt t1 Coefficient of Determination the square of the correlation im i m m im i im i m i a measure of how much of the variation in a single stock is due to a variation in the market Standard Error in beta for security i : 1.18 i ei m How explanatory is this parameterization? How much association in beta one period to the next? Beta s on large PF contain a great deal of information about future betas not so good for individual securities Can the predictive ability be improved? Both Blume & Vasicek (Baysian) propose techniques that lead to better betas than not adjusting Is beta the right criteria? Better to consider ability to give better correlations! This is what matters to PF optimization ij i j m ij Study of Correlation matrix forecast i j i j Use the historical correlation matrix itself Matrix from historical beta forecast Matrix from Blume-adjusted betas using prior periods Matrix from Vasicek-adjusted Bayesian technique Historical betas better predictors on PF than on stocks 1.19 1.0 5

CAPM vs. the Sharpe Single Factor Model Theoretical (the Market) vs. Empirical In moving to multi-factor models we find a similar, more powerful contrast though the increased sophistication of multiple factors makes the move challenging Arbitrage Theory in Pricing, vs. Adding Explanatory Factors (conjectured) and Regressing to get the Best Model Let s take a high level look Can the predictive ability be improved? Historical correlation matrix was worst by far All single index models did better surprising when it might be though that these loose information in achieving simplification! Bayesian Technique is best if results are forced to have stationary average correlation to the period where the model was fitted More can be said about other factors that influence beta and how fundamentals can provide improvements to the forecast in multifactor models 1.1 1. Rather than just a market variable w/return R m Assume other factors, e.g. Interest rates Inflation Industry/Sector Capitalization Book-equity/Market-equity Then we can postulate the relationship for the return, R i, and risk, σ i, on stock i 1.3 In terms of the factors (or indexes) as R a b I b I b I c i i i1 1 i ik K i The indexes, I j, are assumed to be uncorrelated Factors can always be converted into uncorrelated ones EI w/variance i IiI j I j 0 Ij In addition we assume: Residual of each stock has mean 0 and variance ci Indices have covariance w/residuals Eci I j I j 0 And each other E cc i j 0 1.4 6

So we can write expected Returns as Ri ai bi 1I1biI bikik The Variance as i bi1i1bii bikik ci And the Covariance as ij bb i1 j1i 1bb i ii bikbjkik Which allows an easy representation for any portfolio 1.5 Often multi-factor models are implemented as industry models The first factor is the market index The remaining factors are industry factors uncorrelated to the market and each other How well does that work when the parameters are estimated from historical data? These models fall between the 1-factor model and the full correlation matrix More factors better reproduce the historical matrix 1.6 However, this does not imply that future correlation matrices are forecast more accurately More important is how methods effect the return or profit in using one technique over the other Techniques Principal Component Analysis (PCA) Extracts from past values of the variance-covariance matrix the best reproducing factors; usually in order of significance Let Standard Industry Classifications define the indexes Pseudo-Industries are defined from stocks with highly correlated returns 1.7 More important is how methods effect the return or profit in using one technique over the other Results PCA Empirical studies of stocks indicate that this multi-factor technique is inferior to a single index model! Researchers studying Standard Industry Classification for multifactor modes concluded single index models more desirable Pseudo-industries models performed no better than standard classifications 1.8 7

As more factors seem to add more noise than real information, smoothing/averaging techniques have been tested Averaging over all pair-wise correlations as a forecast for each performed pair-wise correlation (overall mean model) performed best! Outperformed the single-index model, the multi-index models, and the historical correlation matrix itself not unlike a similar result found in the single factor world Alternatively, pair-wise averaging within industries was no better 1.9 There is hope as fundamental multi-index models seem to work well! Fama-French found that market capitalization and the ratio of book equity to market value of equity adds significant insight Factors market return in excess of T-bills, and portfolios that mimic the impact of the factors Size variable free of book to market effect Book to market free of size variable A number of studies have pursued this line successfully 1.30 There is hope as fundamental multi-index models seem to work well! Chen, Roll, and Ross worked with a different set of indices Default Risk credit spreads Curve shape term structure Unexpected deflation Unexpected change in growth rate in real final sales The market return (S&P500) uncorrelated with the above Good indication of superior performance 1.31 We put this into perspective more formally looking to moving away from CAPM Theoretical rationale for the models Choosing factors and parameterization Application Risk analysis PF evaluation Arbitrage-Free Pricing Theory (APT) Less restrictive assumptions than CAPM A familiar approach 1.3 8

APT Moving away from CAPM CAPM is based on very strong theoretical assumptions not always respected by markets In an effort to develop a more general model with simplifying assumptions, a family of models has emerged called multi-factor models An alternative to CAPM, but strongly influenced by it e.g. linearity, but making no assumptions about the behavior of investors Allow asset returns to be described by factors other than the market index Arbitrage Models The Arbitrage-Free Pricing Theory (APT) Less restrictive assumptions than CAPM CAPM assumes returns are normally distributed, APT makes no assumption about return distributions, per se o assumptions about investor utility functions only that investors are risk averse These assumptions allow empirical validation Two paradigms: arbitrage & empirical observations1.33 1.34 APT APT Arbitrage Models APT Less restrictive assumptions than CAPM APT explains returns through common factors, but instead of the well defined single factor in CAPM, the model employs K factors and is therefore more general The problem is to determine K and the nature of the factors 1.35 The Intuitive Idea Most Important the law of one price The model of returns R a b I b I b I e i i i1 1 i ik K i Where ai the expected level of return for stock i if all indices are 0 I j the value of index j that impacts the return on stock i bij the sensitivity of return on stock i to index j ei unexplained random residual w/mean 0 & variance ei With Eei I j I j 0 and E ee i j 0 1.36 9

APT APT The Intuitive Idea Even though we start with the model conceptualized as a mere extension of the single factor model motivated under the CAPM; The key to APT is in demonstrating how (under what conditions) one can go from a multi-index model to a description of equilibrium; A demonstration quite a bit different from the unpopular assumptions/requirements under CAPM and its relationship to equilibrium 1.37 A simple demonstration and example Consider the -factor model R a b I b I e i i i1 1 i i With E ee i j 0 In a well diversified PF, residual risk is diversified away and only systematic risk matters; Systematic Risk in the PF are from b i1 and b i. With a concern only for risk and return, for any PF p an investor need focus only on three attributesrp, bp, and bp R X R and b X b and b X b with X 1 p i i p1 i i1 p i i i i1 i1 i1 i1 1 1.38 APT APT A simple demonstration and example Consider next the three diversified PFs From this we can find the modelri 7.75 5bi13.75bi And the expected risk and return measures of any PF made up of these is R X R and b X b and b X b with X 1 p i i p1 i i1 p i i i i1 i1 i1 i1 Where = 3 1.39 A simple demonstration and example Any PF made up of these 3 lie in the plane described by the span of A, B, and C Consider a PF E with return 15% and risk factors bi 1 0.6 and bi 0.6 From our spanning set we can find a PF D where 1 1 1 bp 1 1.0 0.5 0.30.6 and 3 3 3 1 1 1 bp 0.6 1.0 0.0.6 3 3 3 The risk from PF D equals PF E but the expected 1 1 1 return on PF D is RD 15 14 1013 3 3 3 1.40 10

APT APT A simple demonstration and example Equivalently, we could have used our factor model RD 7.75 5bi 13.75bi 7.75 50.6 3.750.6 13 By the law of one price (no arbitrage) two PF with the same risk cannot have different expected return, arbitrageurs will sell the rich PF, D, and buy the cheap PF, E, until the zero investment riskless return vanishes Indeed, sell $100 of D, use the $100 to buy E Collect the $ at the end of period 1.41 In general, the return in the expected return, (b i1, b i )-space (risk-factor), can be represented as Ri 0 1bi 1bi In the example: 0 7.75; 1 5; and ; 0 3.75 This is the general equilibrium model under APT for a -factor (index) model The λ j have the interpretation as follows: When both b i1 and b i are zero, the return of this zero risk PF is λ 0 ; and if there is a risk-free asset with return, R F, then λ 0 = R F If either b i1 or b i are zero, with the other equal to 1, then λ j will capture the return of a unit change in risk in b ij 1.4 APT APT In general, the return in the expected return, b i1, b i, space, can be represented as Ri 0 1bi 1bi The λ j have the interpretation as follows: In general then, for j = 1,, with Ri 0 1bi 1bi and 0 RF R R R j j F j F if either b i1 or b i are zero, with the other equal to 1 Where the return (or risk price), λ j, associated with bearing (a unit of) the risk represented by index I j is quantified through the quantity of this risk, the b ij, for asset i For the assets, j, of our spanning set, we have the Price of Risk, λ j, and Quantity of Risk, b ij, which is familiar with previous no-arbitrage arguments/developments 1.43 APT A Generalization Again using Ri ai bi 1I1biIei, take the expectation and subtract so Ri Ri bi 1I1I1 I1bi I I ei A sufficient condition for APT to hold is that there be enough securities in the market st. a PF with the following characteristics can be formed Xb 0 and Xb 0 and X 0 and Xe 0 i i1 i i i i i i1 i1 i1 i1 Residual risk is approximately zero 1.44 11

APT APT APT A Generalization A PF with the following characteristics 1 Xb 0 and Xb 0 and X 0 and Xe 0 i i i i i i i i1 i1 i1 i1 The PF has no risk and there is no investment Therefore it should produce no return XR i i 0 The 1 st i1 3 equations says that the vector of security proportions is orthogonal to the b s and a vector of 1 s As we have shown, if this is so, this implies that the vector of security proportions is orthogonal to the vector of expected returns 1.45 APT A Generalization A PF with the following characteristics 1 Xb 0 and Xb 0 and X 0 and Xe 0 i i i i i i i i1 i1 i1 i1 If a vector is orthogonal to -1 vectors => orthogonal to the th vector, then the th vector can be expressed as a linear combination of the -1 vectors In particular, the vector of returns can be expressed as a linear combinations of a vector of 1 s, and b s or Ri 0 1bi 1bi And this must hold for all portfolios 1.46 APT APT APT A Generalization A PF with the following characteristics The λ can be evaluated as before by forming 3 PF w/ 1. b 0 and b 0 p1 p. b 1 and b 0 p1 p 3. b 0 and b 1 p1 p And we find R R b R R I b R R i F i1 1 F 1 i F R b b i 0 1 i1 i Or generally R R b R R I b R R I b R R i F i1 1 F 1 i F ij J F 1.47 APT A Generalization A PF with the following characteristics R b b i 0 1 i1 i Defining 0 RF and j Rj RF lets us write this as Ri RF bi 1R1RF I1bi R RF I bij RJ RF 0 1bi 1bi JbiJ The strength of APT approach is it s basis on the no arbitrage condition It holds for any subset of securities o need to identify all risky assets or market PF to test Reasonable over a class of assets such as all stocks 1.48 or all S&P index stocks (or YSE, etc.) 1