Multi-factor Risk-return Relationships

This paper derives three multi-factor risk-return relationships each of which employs macro-economic variables in presenting the underlying factors that influence security returns. The first relationship holds if the underlying portfolio lies on the expected return-standard deviation efficient frontier, the second is valid when the underlying portfolio lies inside the efficient frontier and the third characterises security markets in which no arbitrage opportunities are present. An attempt is also made to appraise critically previous multi-factor risk-return relationships which rely on an expected return-standard deviation approach.     

Linear beta pricing with inefficient benchmarks in a given factor structure*

ABSTRACT

We show the equivalence between the zero-beta version of a multi-factor arbitrage pricing model and a linear pricing model utilizing undiversified inefficient benchmarks in a given factor structure. The resulting linear model is a two-beta model, with one beta related to the inefficient benchmark and another adjusting for its inefficiency. This linear model shows that there are only two distinctive and computable sources of risk, affecting security expected returns, despite the existence of several risk factors. In a short empirical example we demonstrate that the model can be employed to provide guidance and allow researchers to test for the validity of their selection of the underlying risk factors driving variations in security returns.     

On the Cross-sectional Relation between Expected Returns and Betas

There is an exact linear relation between expected returns and true “betas” when the market portfolio is on the ex ante mean-variance efficient frontier, but empirical research has found little relation between sample mean returns and estimated betas. A possible explanation is that market portfolio proxies are mean-variance inefficient. We categorize proxies that produce particular relations between expected returns and true betas. For the special case of a zero relation, a market portfolio proxy must lie inside the efficient frontier, but it may be close to the frontier.     

Systematic Risk and Empirical Research

We show here that risky asset returns generating processes stated in terms of factors which include both accounting and non-accounting based measures of risk (e.g. book to market ratios) imply, under fairly standard regularity conditions, that the Sharpe-Lintner-Black asset pricing model beta is a 'sufficient' statistic in the sense that it captures all important attributes of the returns generating process in a single number. We then derive the parametric relationship between betas based on inefficient index portfolios and betas based on the market or tangency portfolio. We demonstrate that the relationship between risky asset expected returns and betas computed on the basis of inefficient index portfolios is both consistent with the predictions of the Capital Asset Pricing Model and the multi-factor asset pricing models of Fama and French (1992, 1993, 1995 and 1996). The 'trick' is to realise that inefficient index portfolios are composed of the market portfolio and a collection of inefficient but self financing 'kernel' or 'arbitrage' portfolios. It then follows that there is a perfect linear cross sectional relationship between risky asset expected returns, betas based on inefficient index portfolios and the arbitrage portfolios. Hence, if we happen to stumble across variables that span the same subspace as the vectors representing the arbitrage portfolios, it is easy to create the illusion that risky asset expected returns depend on variables other than 'beta'.     

Constructing Asset Pricing Models With Specific Factor Loadings

We demonstrate how one can build pricing formulae in which factors other than beta may be viewed as determinants of asset returns. This is important conceptually as it demonstrates how the additional factors can compensate for a market portfolio proxy that is mis‐specified, and also shows how such a pricing model can be specified ex ante. The procedure is implemented by first selecting an ‘orthogonal’ portfolio which falls on the mean‐variance efficient frontier computed from the empirical average returns, variances and covariances on the equity securities of a large sample of firms. One then determines the inefficient index portfolio which leads to a vector of betas that when multiplied by the average return on the orthogonal portfolio, and which when subtracted from the vector of average returns for the firms comprising the sample, yields an error vector that is equal to the vector of numerical values for the variables that are to form the basis of the asset pricing formula. There will then be a perfect linear relationship between the vector of average returns for the firms comprising the sample, the vector of betas based on the inefficient index portfolio and such other factors that are deemed to be important in the asset pricing process. We illustrate computational procedures using a numerical example based on the quality of information contained in published corporate financial statements.     

Are the Latent Variables in Time-Varying Expected Returns Compensation for Consumption Risk?

Multibeta asset pricing models are examined using proxies for economic state variables in a framework which exploits time-varying expected returns to estimate conditional betas. Examples include multiple consumption-beta models and models where asset returns proxy for the state variables. When the state variables are not specified, the tests indicate two or three time-varying expected risk premiums in the sample of quarterly asset returns. Conditional betas relative to consumption generate less striking evidence against the model than betas relative to asset returns, but both the consumption and the market variables fail to proxy for the state variables.     

Changes in Expected Security Returns,Risk, and the Level of Interest Rates

Regressions of security returns on treasury bill rates provide insight about the behavior of risk in rational asset pricing models. The information in one-month bill rates implies time variation in the conditional covariances of portfolios of stocks and fixed-income securities with benchmark pricing variables, over extended samples and within five-year subperiods. There is evidence of changes in conditional “betas” associated with interest rates. Consumption and stock market data are examined as proxies for marginal utility, in a general framework for asset pricing with time-varying conditional covariances.     

Linear programing models for portfolio optimization using a benchmark

We consider the problem of constructing a perturbed portfolio by utilizing a benchmark portfolio. We propose two computationally efficient portfolio optimization models, the mean-absolute deviation risk and the Dantzig-type, which can be solved using linear programing. These portfolio models push the existing benchmark toward the efficient frontier through sparse and stable asset selection. We implement these models on two benchmarks, a market index and the equally-weighted portfolio. We carry out an extensive out-of-sample analysis with 11 empirical datasets and simulated data. The proposed portfolios outperform the benchmark portfolio in various performance measures, including the mean return and Sharpe ratio.     

Sources of risk and expected returns in global equity markets

This paper empirically examines multifactor asset pricing models for the returns and expected returns on eighteen national equity markets. The factors are chosen to measure global economic risks. Although previous studies do not reject the unconditional mean variance efficiency of a world market portfolio, our evidence indicates that the tests are low in power, and the world market betas do not provide a good explanation of cross-sectional differences in average returns. Multiple beta models provide an improved explanation of the equity returns.     

Heterogeneous Expectations,Short Sales Regulation,and the Risk-Return Relationship

This paper examines, in a Canadian context, the effect of short sales regulation on the risk-return relationship. It shows that, theoretically, the opportunity cost induced by short sales regulation is positively related to the dispersion of agents' beliefs and negatively related to the security's liquidity level. The model is tested over the sixty-month period from January 1985 through December 1989. All the 13,079 observations are pooled into a time series cross-sectional model. The results corroborate that a negative linear relationship links expected risky asset returns and the divergence of agents' beliefs. This negative relationship is consistent with the presence of opportunity costs resulting from short sales regulation when return beliefs are heterogeneous. However, the negative relationship between security returns and dispersion of beliefs is essentially confined to illiquid securities, that is, those monitored by a small number of analysts.     

Where do betas come from? Asset price dynamics and the sources of systematic risk

In this article we break asset's betas with common factors intocomponents attributable to news about future cash flows, realinterest rates, and excess returns. To achieve this decomposition,we use a vector autoregressive time-series model and an approximatelog-linear present value relation. The betas of industry andsize portfolios with the market are largely attributed to changingexpected returns. Betas with inflation and industrial productionreflect opposing cash flow and expected return effects. We alsoshow how asset pricing theory restricts the expected excessreturn components of betas.     

The Performance of Actively Managed International Mutual Funds

International mutual funds allow individual investors to diversify abroad at a reasonable cost. This paper tests whether international funds that actively engage in country and security selection outperform passive global benchmarks. We apply a mean-variance efficiency test that incorporates the practical prohibition against short sales of open-end mutual funds. Our tests reject the efficiency of the world equity market portfolio over the sample period, and our funds as a group outperform the inefficient world index. However, we find no evidence of security selectivity ability using a 12-country benchmark. We do find that active international funds provide global diversification benefits. Tests using the Positive Period Weight (PPW) measure of Grinblatt and Titman (1989), which is robust to nonlinearity in fund and benchmark returns, yield similar results.     

Monotonicity in asset returns: New tests with applications to the term structure,the CAPM,and portfolio sorts

Many theories in finance imply monotonic patterns in expected returns and other financial variables. The liquidity preference hypothesis predicts higher expected returns for bonds with longer times to maturity; the Capital Asset Pricing Model (CAPM) implies higher expected returns for stocks with higher betas; and standard asset pricing models imply that the pricing kernel is declining in market returns. The full set of implications of monotonicity is generally not exploited in empirical work, however. This paper proposes new and simple ways to test for monotonicity in financial variables and compares the proposed tests with extant alternatives such as t-tests, Bonferroni bounds, and multivariate inequality tests through empirical applications and simulations.     

Optimal granularity for portfolio choice

Many optimization-based portfolio rules fail to beat the simple 1/N rule out-of-sample because of parameter uncertainty. In this paper we suggest a grouping strategy in which we first form groups of equally weighted stocks and then optimize over the resulting groups only. This strategy aims at balancing the trade-off between the benefits from optimization and the losses from estimation risk. We rely on Monte-Carlo simulations to illustrate the performance of the strategy, and we derive the optimal group size for a simplified setup. Furthermore, we show that estimation risk also has an impact via the criterion by which the assets are sorted into groups (like the expected excess returns or betas), but does not negate the grouping approach. We relate our work to linear asset pricing models, and we conduct out of sample back-tests in order to confirm the validity of our grouping strategy empirically.     

Options Trading and the CAPM

This article studies equilibrium asset pricing when agents facenonnegative wealth constraints. In the presence of these constraintsit is shown that options on the market portfolio are nonredundantsecurities and the economy's pricing kernel is a function ofboth the market portfolio and the nonredundant options. Thisimplies that the options should be useful for explaining riskyasset returns. To test the theory, a model is derived in whichthe expected excess return on any risky asset is linearly related(via a collection of betas) to the expected excess return onthe market portfolio and to the expected excess returns on thenonredundant options. The empirical results indicate that thereturns on traded index options are relevant for explainingthe returns on risky asset portfolios.     

On the power of cross-sectional and multivariate tests of the CAPM

This paper examines the power of the cross-sectional and multivariate tests of the CAPM under ideal conditions. When the CAPM is true the positively weighted market portfolio is MV-efficient and securities plot on the security market line. When the CAPM is false an alternative asset pricing model determines prices. An examination of the population intercepts, slopes and R² from cross-sectional regressions of expected returns on betas indicates that all three are unreliable indicators of whether the CAPM holds. Simulation analysis of the power of the cross-sectional tests expands on and reinforces the analysis based on the population values. The Gibbons et al. (1989) multivariate test fares much better.     

On the Feasibility of Insurers' Investment Policies

This article calls attention to a difficulty with insurers' investment policies that seems to have been overlooked so far. There is the distinct possibility that insurers cannot satisfy the demands of different stakeholders in terms of expected returns and volatility. While using the capital asset pricing model as the benchmark, this article distinguishes two groups of stakeholders that impose additional constraints. One is “income security” in the interest of current beneficiaries and older workers; the other is “predictability of contributions” in the interest of contributing younger workers and sponsoring employers. It defines the conditions for which the combination of these constraints results in a lack of feasibility of investment policy. Minimum deviation from the capital market line is proposed as the performance benchmark in these situations.     

On the Cross-Sectional Relation between Expected Returns, Betas, and Size

In this paper, I set up scenarios where the mean-variance capital asset pricing model is true and where it is false. Then I investigate whether the coefficients from regressions of population expected excess returns on population betas, and expected excess returns on betas and size, allow us to distinguish between the scenarios. I show that the coefficients from either ordinary least squares or generalized least squares regressions do not allow us to tell whether the model is true or false.     

Notes on dynamic factor pricing models

These notes discuss three aspects of dynamic factor pricing (i.e., APT) models. First, the diversifiable component of returns is unpredictable in a no-arbitrage world. Second, conditional factor loadings or betas have an unconditional factor structure when returns follow an unconditional factor structure, which provides a link between conditional and unconditional factor pricing models. Third, the estimation of dynamic factor pricing models is easily simplified in large cross sections when returns follow an unconditional factor structure. These results aid in the interpretation of existing applications and identify some of the issues in the formulation and estimation of dynamic factor pricing models.     

Cross-sectional determinants of New Zealand share market returns

We present evidence of the cross-sectional relation between security returns, beta, firm size and book-to-market ratio over the period 1971 to 1993 on the New Zealand sharemarket. Our results suggest that the NZSE-40 market index is not a mean-variance efficient market proxy—the betas calculated with respect to it being of little use for explaining expected returns cross-sectionally. Also, there is a significant positive relation between book-to-market ratio and average return.