Abstract:
This thesis consists of five theoretical essays, each focusing on an aspect of mean variance based capital asset pricing theory.
Essay 1 : Beta Loss, Beta Quotient, and Bias in Estimation. This essay extends the work by Camp and Eubank t171 which introduced the Beta Quotient. The essay proposes the additional concept of Beta Loss as an indicator of the economic loss resulting from a choice of an inefficient portfolio. The essay also shows that estimators of the Beta Quotient and Beta Loss, based on sample standard deviations of portfolio and market rates of return, are biased but that the bias would, in most practical applications, be expected to be very small.
Essay 2 : Risk Decomposition and Estimation of Undiversifiable Risk. This essay shows that the decomposition of variance into undiversifiable and diversifiable risks, typically associated. only with the basic form of the CAPM due to Sharpe 1821, Lintner [55] and Mossin [66], (S-L-M), can also be applied to the extended CAPM due to Black [4]. In each case the decomposition can be derived directly from the asset pricing model. Market models have relevance only to empirical estimation. The essay also shows that if the proportion of undiversifiable risk is estimated from the one factor market model when the two factor model is appropriate, then the estimate will be negatively biased. Between asset pricing domains the bias increases with increasing size of the ratio of the variances of the global minimum variance portfolio and the market portfolio. This ratio is the asymptotic limiting value of the bias within a domain and the bias is largest for portfolios with an absolute small beta.
Essay 3 : A New Statement of the Extended Capital Asset Pricing Model. This essay proposes that the extended CAPM be restated to the form
µj = µo + (µm - µo) θj
where
(i) µi is the expected return on portfolio i and the subscripts o and m refer, respectively, to the global minimum variance portfolio, and the market portfolio;
(ii) θj = (βj -βo)/)(1 -βo)
where
βi = σim/σ2m
In this form, the risk margin is stated relative to an efficient portfolio (just as in the S-L-M CAPM) and thereby emphasises the fundamental concept of this theory of asset pricing, viz., that investors hold portfolios that are mean variance efficient. It is shown that the proposed statement leads to a generalised form of the risk decomposition associated with the S-L-M model, and also facilitates development of a Capital Market Line for the extended model.
Essay 4 : Risk Decomposition: Variance or Standard Deviation; A Re-examination and Extension. This essay extends the work by Ben-Horim and Levy [2] which argued for a risk decomposition based on σ rather than σ2. Their analysis showed that decomposition of σ2 is wrong when β<0 and that, in general, this procedure produces incorrect estimates of undiversifiable risk. the essay shows that those conclusions also apply to the extended CAPM if risk is defined, as (σ2-σ2o)½.
Essay 5 : Potential performance of a Set of N Risky Assets in the Absence of Riskless Borrowing and Lending. This essay extends the Jobson and Korkie [43], [45] potential performance concept for a set of N risky assets, to the case where there are no opportunities for riskless borrowing and lending. Using the statement derived in Essay 3, the essay develops a generalised potential performance measure. The measure remains a function only of the vector of mean returns and the variance covariance matrix, is an index of diversifiable risk and provides a basis for preference orderings among portfolios.